The Finite Element Method for Solid and Structural Mechanics Sixth edition
Professor O.C. Zienkiewicz, CBE, FRS, FREng is Professor Emeritus at the Civil and Computational Engineering Centre, University of Wales Swansea and previously Director of the Institute for Numerical Methods in Engineering at the University of Wales Swansea, UK. He holds the UNESCO Chair of Numerical Methods in Engineering at the Technical University of Catalunya, Barcelona, Spain. He was the head of the Civil Engineering Department at the University of Wales Swansea between 1961 and 1989. He established that department as one of the primary centres of finite element research. In 1968 he became the Founder Editor of the International Journal for Numerical Methods in Engineering which still remains today the major journal in this field. The recipient of 27 honorary degrees and many medals, Professor Zienkiewicz is also a member of five academies - an honour he has received for his many contributions to the fundamental developments of the finite element method. In 1978, he became a Fellow of the Royal Society and the Royal Academy of Engineering. This was followed by his election as a foreign member to the US Academy of Engineering (1981), the Polish Academy of Science (1985), the Chinese Academy of Sciences (1998), and the National Academy of Science, Italy (Academia dei Lincei) (1999). He published the first edition of this book in 1967 and it remained the only book on the subject until 1971. Professor R.L. Taylor has more than 40 years' experience in the modelling and simulation of structures and solid continua including two years in industry. He is Professor in the Graduate School and the Emeritus T.Y. and Margaret Lin Professor of Engineering at the University of California at Berkeley. In 1991 he was elected to membership in the US National Academy of Engineering in recognition of his educational and research contributions to the field of computational mechanics. Professor Taylor is a Fellow of the US Association of Computational Mechanics - USACM (1996) and a Fellow of the International Association of Computational Mechanics- IACM (1998). He has received numerous awards including the Berkeley Citation, the highest honour awarded by the University of California at Berkeley, the USACM John von Neumann Medal, the IACM Gauss-Newton Congress Medal and a Dr.-Ingenieur ehrenhalber awarded by the Technical University of Hannover, Germany. Professor Taylor has written several computer programs for finite element analysis of structural and non-structural systems, one of which, FEAP, is used world-wide in education and research environments. A personal version, FEAPpv, available from the publisher's website, is incorporated into the book.
The Finite Element M ethod for Solid and Structural Mechanics Sixth edition O.C. Zienkiewicz, CBE, FRS
UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in Engineering, Barcelona Previously Director of the Institute for Numerical Methods in Engineering University of Wales Swansea
R.L. Taylor
Professor in the Graduate School Department of Civil and Environmental Engineering University of California at Berkeley Berkeley, California
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Butterworth-Heinemann is an imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First edition published in 1967 by McGraw-Hill Fifth edition published by Butterworth-Heinemann 2000 Reprinted 2002 Sixth edition 2005 Reprinted 2006 (twice) Copyright 9 2000, 2005, O.C. Zienkiewicz and R.L. Taylor. Published by Elsevier Ltd. All rights reserved The rights of O.C. Zienkiewicz and R.L. Taylor to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone: (+44) (0) 1865 843830; fax: (+44) (0) 1865 853333; email:
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British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN-13:978-0-7506-6321-2 ISBN-10:0-7506-6321-9 For information on all Butterworth-Heinemann publications visit our website at books.elsevier.com Printed and bound in Great Britain 06 07 08 09 10 10 9 8 7 6 5 4 3
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Dedication This book is dedicated to our wives Helen and Mary Lou and our families for their support and patience during the preparation of this book, and also to all of our students and colleagues who over the years have contributed to our knowledge of the finite element method. In particular we would like to mention Professor Eugenio Ofiate and his group at CIMNE for their help, encouragement and support during the preparation process.
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Contents
P refa c e
1.
2.
,
,
xiii
General problems in solid mechanics and non-linearity 1.1 Introduction 1.2 Small deformation solid mechanics problems 1.3 Variational forms for non-linear elasticity 1.4 Weak forms of governing equations 1.5 Concluding remarks References
4 12 14 15 15
Galerkin method of approximation- irreducible and mixed forms 2.1 Introduction 2.2 Finite element approximation- Galerkin method 2.3 Numerical integration - quadrature 2.4 Non-linear transient and steady-state problems 2.5 Boundary conditions: non-linear problems 2.6 Mixed or irreducible forms 2.7 Non-linear quasi-harmonic field problems 2.8 Typical examples of transient non-linear calculations Concluding remarks 2.9 References
17 17 17 22 24 28 33 37 38 43 44
Solution of non-linear algebraic equations 3.1 Introduction 3.2 Iterative techniques 3.3 General remarks - incremental and rate methods References
46 46 47 58 60
Inelastic and non-linear materials 4.1 Introduction 4.2 Viscoelasticity - history dependence of deformation 4.3 Classical time-independent plasticity theory 4.4 Computation of stress increments
62 62 63 72 80
1 1
viii
Contents
4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13
Isotropic plasticity models Generalized plasticity Some examples of plastic computation Basic formulation of creep problems Viscoplasticity- a generalization Some special problems of brittle materials Non-uniqueness and localization in elasto-plastic deformations Non-linear quasi-harmonic field problems Concluding remarks References
85 92 95 100 102 107 112 116 118 120
5.
Geometrically non-linear problems - finite deformation 5.1 Introduction 5.2 Governing equations Variational description for finite deformation 5.3 5.4 Two-dimensional forms A three-field, mixed finite deformation formulation 5.5 A mixed-enhanced finite deformation formulation 5.6 Forces dependent on deformation- pressure loads 5.7 Concluding remarks 5.8 References
127 127 128 135 143 145 150 154 155 156
6.
Material constitution for finite deformation 6.1 Introduction 6.2 Isotropic elasticity 6.3 Isotropic viscoelasticity 6.4 Plasticity models 6.5 Incremental formulations 6.6 Rate constitutive models 6.7 Numerical examples 6.8 Concluding remarks References
158 158 158 172 173 174 176 178 185 189
7.
Treatment of constraints - contact and fled interfaces 7.1 Introduction 7.2 Node-node contact: Hertzian contact 7.3 Tied interfaces 7.4 Node-surface contact 7.5 Surface-surface contact 7.6 Numerical examples 7.7 Concluding remarks References
191 191 193 197 200 218 219 224 224
Pseudo-rigid and rigid-flexible bodies 8.1 Introduction 8.2 Pseudo-rigid motions 8.3 Rigid motions
228 228 228 230
Contents
8.4 8.5 8.6 9.
Connecting a rigid body to a flexible body Multibody coupling by joints Numerical examples References
Discrete element methods 9.1 Introduction 9.2 Early DEM formulations Contact detection 9.3 9.4 Contact constraints and boundary conditions Block deformability 9.5 9.6 Time integration for discrete element methods 9.7 Associated discontinuous modelling methodologies 9.8 Unifying aspects of discrete element methods 9.9 Concluding remarks References
10. Structural mechanics problems in one dimension- rods 10.1 Introduction 10.2 Governing equations 10.3 Weak (Galerkin) forms for rods 10.4 Finite element solution: Euler-Bernoulli rods 10.5 Finite element solution: Timoshenko rods 10.6 Forms without rotation parameters 10.7 Moment resisting frames 10.8 Concluding remarks References 11. Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements 11.1 Introduction 11.2 The plate problem: thick and thin formulations 11.3 Rectangular element with comer nodes (12 degrees of freedom) 11.4 Quadrilateral and parallelogram elements 11.5 Triangular element with comer nodes (9 degrees of freedom) 11.6 Triangular element of the simplest form (6 degrees of freedom) 11.7 The patch test- an analytical requirement 11.8 Numerical examples 11.9 General remarks 11.10 Singular shape functions for the simple triangular element 11.11 An 18 degree-of-freedom triangular element with conforming shape functions 11.12 Compatible quadrilateral elements 11.13 Quasi-conforming elements 11.14 Hermitian rectangle shape function 11.15 The 21 and 18 degree-of-freedom triangle 11.16 Mixed formulations - general remarks
234 237 240 242 245 245 247 250 256 260 267 270 271 272 273 278 278 279 285 290 305 317 319 320 320 323 323 325 336 340 340 345 346 348 357 357 360 361 362 363 364 366
ix
x
Contents
11.17 11.18 11.19 11.20 11.21
Hybrid plate elements Discrete Kirchhoff constraints Rotation-free elements Inelastic material behaviour Concluding remarks- which elements? References
368 369 371 374 376 376
12. 'Thick' Reissner-Mindlin plates- irreducible and mixed formulations 12.1 Introduction 12.2 The irreducible formulation- reduced integration 12.3 Mixed formulation for thick plates 12.4 The patch test for plate bending elements 12.5 Elements with discrete collocation constraints 12.6 Elements with rotational bubble or enhanced modes 12.7 Linked interpolation- an improvement of accuracy 12.8 Discrete 'exact' thin plate limit 12.9 Performance of various 'thick' plate elements - limitations of thin plate theory 12.10 Inelastic material behaviour 12.11 Concluding remarks - adaptive refinement References
382 382 385 390 392 397 405 408 413
13. Shells 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8
426 426 428 429 431 435 440 440 441 450
as an assembly of fiat elements Introduction Stiffness of a plane element in local coordinates Transformation to global coordinates and assembly of elements Local direction cosines 'Drilling' rotational stiffness- 6 degree-of-freedom assembly Elements with mid-side slope connections only Choice of element Practical examples References
14. Curved rods and axisymmetric shells 14.1 Introduction 14.2 Straight element 14.3 Curved elements 14.4 Independent slope-displacement interpolation with penalty functions (thick or thin shell formulations) References 15. Shells as a special case of three-dimensional analysis - Reissner-Mindlin assumptions 15.1 Introduction 15.2 Shell element with displacement and rotation parameters 15.3 Special case of axisymmetric, curved, thick shells 15.4 Special case of thick plates
415 419 420 421
454 454 454 461 468 473
475 475 475 484 487
Contents
15.5 15.6 15.7 15.8
Convergence Inelastic behaviour Some shell examples Concluding remarks References
487 488 488 493 495
16. Semi-analytical finite element processes - use of orthogonal functions
and 'finite strip' methods 16.1 Introduction 16.2 Prismatic bar 16.3 Thin membrane box structures 16.4 Plates and boxes with flexure 16.5 Axisymmetric solids with non-symmetrical load 16.6 Axisymmetric shells with non-symmetrical load 16.7 Concluding remarks References
498 498 501 504 505 507 510 514 515
17. Non-linear structural problems- large displacement and instability 17.1 Introduction 17.2 Large displacement theory of beams 17.3 Elastic stability- energy interpretation 17.4 Large displacement theory of thick plates 17.5 Large displacement theory of thin plates 17.6 Solution of large deflection problems 17.7 Shells 17.8 Concluding remarks References
517 517 517 523 526 532 534 537 542 543
18. Multiscale modelling 18.1 Introduction 18.2 Asymptotic analysis 18.3 Statement of the problem and assumptions 18.4 Formalism of the homogenization procedure 18.5 Global solution 18.6 Local approximation of the stress vector 18.7 Finite element analysis applied to the local problem 18.8 The non-linear case and bridging over several scales 18.9 Asymptotic homogenization at three levels: micro, meso and macro 18.10 Recovery of the micro description of the variables of the problem 18.11 Material characteristics and homogenization results 18.12 Multilevel procedures which use homogenization as an ingredient 18.13 General first-order and second-order procedures 18.14 Discrete-to-continuum linkage 18.15 Local analysis of a unit cell 18.16 Homogenization procedure - definition of successive yield surfaces
547 547 549 550 552 553 554 555 560 561 562 565 567 570 572 578 578
xi
xii
Contents
18.17 Numerically developed global self-consistent elastic-plastic constitutive law 18.18 Global solution and stress-recovery procedure 18.19 Concluding remarks References
580 581 586 587
19. Computer procedures for finite element analysis 19.1 Introduction 19.2 Solution of non-linear problems 19.3 Eigensolutions 19.4 Restart option 19.5 Concluding remarks References
590 590 591 592 594 595 595
Appendix A Appendix B
597 604 609 619
Author index Subject index
Isoparametric finite element approximations Invariants of second-order tensors
Preface It is thirty-eight years since the The Finite Element Method in Structural and Continuum Mechanics was first published. This book,which was the first dealing with the finite element method, provided the basis from which many further developments occurred. The expanding research and field of application of finite elements led to the second edition in 1971, the third in 1977, the fourth as two volumes in 1989 and 1991 and the fifth as three volumes in 2000. The size of each of these editions expanded geometrically (from 272 pages in 1967 to the fifth edition of 1482 pages). This was necessary to do justice to a rapidly expanding field of professional application and research. Even so, much filtering of the contents was necessary to keep these editions within reasonable bounds. In the present edition we retain the three volume format of the fifth edition but have decided not to pursue having three contiguous volumes - rather we treat the whole work as an assembly of three separate works. Each one is capable of being used without the others and each one appeals perhaps to a different audience. Though naturally we recommend the use of the whole ensemble to people wishing to devote much of their time and study to the finite element method. The first volume is renamed The Finite Element Method: Its l~asis and Fundamentals. This volume covers the topic starting from a physical approach4o.solve problems in linear elasticity. The volume then presents a mathematical framework from which general problems may be formulated and solved using variational and Galerkin methods. The general topic of shape functions is also presented for situations in which the approximating functions are C Ocontinuous. The two- and three-dimensional problems of linear elasticity are then presented in a unified manner using higher order shape functions. This is followed by consideration of quasi-harmonic problems governed by Laplace and Poisson differential equations. The patch test is introduced and used as a means to guarantee convergence of the method. We also cover in some depth solution forms using mixed methods with special consideration given to problems in which incompressibility can occur. The solution of transient problems is presented using semi-discrete formulations and finite element in time concepts. The volume concludes with a presentation of coupled problems. In this volume we consider more advanced problems in solid and structural mechanics while in a third volume we consider applications in fluid dynamics. It is our intent that the present volume can be used by investigators familiar with the finite element method N
xiv Preface
at the level presented in the first volume or any other basic textbook on the subject. However, the volume has been prepared such that it can stand alone. The volume has been reorganized from the previous edition to cover consecutively two main subject areas. In the first part we consider non-linear problems in solid mechanics and in the second part linear and non-linear problems in structural mechanics. In Chapters 1 to 9 we consider non-linear problems in solid mechanics. In these chapters the special problems of solving non-linear equation systems are addressed. We begin by restricting our attention to non-linear behaviour of materials while retaining the assumptions on small strain. This serves as a bridge to more advanced studies later in which geometric effects form large displacements and deformations are presented. Indeed, non-linear applications are of great importance today and of practical interest in most areas of engineering and physics. By starting our study first using a small strain approach we believe the reader can more easily comprehend the various aspects which need to be understood to master the subject matter. We cover in some detail formulations of material models for viscoelasticity, plasticity and viscoplasticity which should serve as a basis for applications to other material models. In our study of finite deformation problems we present a series of approaches which may be used to solve problems including extensions for treatment of constraints such as near incompressibility, rigid and multi-body motions and discrete element forms. The chapter on discrete element methods was prepared by Professor Nenad Bi6ani6 of the University of Glasgow, UK. In the second part of the volume we consider problems in structural mechanics. In this class of applications the dimension of the problem is reduced using basic kinematic assumptions. We begin the presentation in a new chapter that considers rod problems where two of the dimensions of the structure are small compared to the third. This class of problems is a combination of beam bending, axial extension and torsion. Again we begin from a small strain assumption and introduce alternative forms of approximation for the Euler-Bernoulli and the Timoshenko theory. In the former theory it is necessary now to use C 1 interpolation (i.e. continuous displacement and slope) to model the bending behaviour, whereas in the latter theory use of C O interpolation is permitted when special means are included to avoid 'locking' in the transverse shear response. Based upon the study of rods we then present a detailed study of problems in which only one dimension is small compared to the other two. Building on the results from rods we present a coverage for thin plates (Kirchhoff theory), thick plates (Reissner-Mindlin theory) and their corresponding forms for shells. We then consider the problem of large strains and present forms for buckling and large displacements. The volume includes a new chapter on multi-scale effects. This is a recent area of much research and the chapter presents a summary of some notable recent results. We are indebted to Professor Bernardo Schrefler of the University of Padova, Italy, for preparing this timely contribution. The volume concludes with a short chapter on computational methods that describes a companion computer program that can be used to solve several of the problem classes described in this volume. We emphasize here the fact that all three of our volumes stress the importance of considering the finite element method as a unique and whole basis of approach and that it contains many of the other numerical analysis methods as special cases. Thus, imagination and knowledge should be combined by the readers in their endeavours.
Preface
The authors are particularly indebted to the International Centre of Numerical Methods in Engineering (CIMNE) in Barcelona who have allowed their pre- and postprocessing code (GiD) to be accessed from the web site. This allows such difficult tasks as mesh generation and graphic output to be dealt with efficiently. The authors are also grateful to Professor Eric Kasper for his careful scrutiny of the entire text. We also acknowledge the assistance of Matt Salveson who also helped in proofreading the text.
Resources to accompany this book
Complete source code and user manual for program FEAPpv may be obtained at no cost from the publisher's web page: http://books.elsevier.com/companions/or from the author's web page: http://www.ce.berkeley.edu/~rlt OCZ and RLT
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General problems problems in solid mechanics mechanics and non-linearity 1.1 Introduction Many introductory texts on the finite element method discuss the solution for linear I 3 problems of of elasticity and field equations. 1-3 - In practical applications the limitation of linear elasticity, or more generally of linear behaviour, often precludes precludes obtaining an accurate assessment of the solution because because of the presence of 'non-linear' effects and/or because the geometry has a 'thin' dimension in one or more directions. In this book we describe extensions to the formulations introduced to solve linear problems to permit solutions to both classes of of problems. Non-linear behaviour of solids takes two forms: material non-linearity and geometric non-linearity. The simplest form of of non-linear material behaviour is that of of elasticity for which the stress is not linearly proportional to the strain. More general situations are those in which the loading and unloading response of the material is different. Typical here is the case of classical elastic-plastic behaviour. When the deformation of a solid reaches a state for which the undeformed and deformed shapes are substantially different a state offinite deformation occurs. In this strain-displacement or equilibrium equations case it is no longer possible to write linear strain-displacement on the undeformed geometry. Even before finite deformation exists it is possible to observe buckling or load bifurcations in some solids and non-linear equilibrium effects need to be considered. The classical Euler column, where the equilibrium of this class equation for buckling includes the effect of axial loading, is an example of of problem. When deformation is large the boundary conditions can also become nonlinear. Examples are pressure loading that remains normal to the deformed body and also the case where the deformed boundary interacts with another body. This latter example defines a class known as contact problems and much research is currently performed in this area. An example of a class of problems involving non-linear effects behaviour and contact is the analysis of of a rolling in deformation measures, material behaviour tyre. A typical mesh for a tyre analysis is shown in Fig. 1.1. The cross-section shown character of a tread. is able to model the layering of rubber and cords and the overall character The full mesh is generated by sweeping the cross-section around the wheel axis with a variable spacing in the area which will be in contact. A formulation in which the mesh is fixed and the material rotates is commonly used to perform the analysis.4-7 analysis. 4-7
2 General problems in solid mechanics and non-linearity
(a) Tyre cross-section.
(b) Full mesh.
Fig. 1.1 Finite element mesh for tyre analysis.
Generally the accurate solution of solid problems which have one (or more) small dimension(s) compared to the others cannot be achieved efficiently using standard two- or three-dimensional finite element formulations. Traditionally separate theories of structural mechanics are introduced to solve this class of problems. A plate is a fiat structure with one thin (small) direction which is called the thickness. A shell is a curved structure in space with one such small thickness direction. Structures with two small dimensions are called beams, frames, or rods. A primary reason why use of standard two- or three-dimensional finite element formulations do not yield accurate solutions is the numerical ill-conditioning which results in their algebraic equations. In this book we combine the traditional approaches of structural mechanics with a much stronger link to the full three-dimensional theory of solids to obtain formulations which are easily solved using standard finite element approaches. This book considers both solid and structural mechanics problems and formulations which make practical finite element solutions feasible. We divide the volume into two main parts. In the first part we consider problems in which continuum theory of solids continues to be used, whereas in the second part we focus attention on theories of structural mechanics to describe the behaviour of rods, plates and shells. In the present chapter we review the general equations for analysis of solids in which deformations remain 'small' but material behaviour includes effects of a nonlinear kind. We present the theory in both an indicial (or tensorial) form as well as in the matrix form commonly used in finite element developments. We also reformulate the equations of solids in a variational (Galerkin) form. In Chapter 2 we present a general scheme based on the Galerkin method to construct a finite element approximate solution to problems based on variational forms. In this chapter we consider both irreducible
Introduction
and mixed forms of finite element approximation and indicate where the mixed forms have distinct advantages. Here we also show how the linear problems of solids for steady state and transient behaviour become non-linear when the material constitutive model is represented in a non-linear form. Some discussion on the solution of transient non-linear finite element forms is included. Since the form of the inertial effects is generally unaffected by non-linearity, in the remainder of this volume we shall primarily confine our remarks to terms arising from non-linear material behaviour and finite deformation effects. In Chapter 3 we describe various possible methods for solving non-linear algebraic equations. This is followed in Chapter 4 by consideration of material non-linear behaviour and completes the development of a general formulation from which a finite element computation can proceed. In Chapter 5 we present a summary for the study of finite deformation of solids. Basic relations for defining deformation are presented and used to write variational (Galerkin) forms related to the undeformed configuration of the body and also to the deformed configuration. It is shown that by relating the formulation to the deformed body a result is obtained which is nearly identical to that for the small deformation problem we considered in the small deformation theory treated in the early chapters of this volume. Essential differences arise only in the constitutive equations (stress-strain laws) and the addition of a new stiffness term commonly called the geometric or initial stress stiffness. For constitutive modelling we summarize in Chapter 6 alternative forms for elastic and inelastic materials. Contact problems are discussed in Chapter 7. Here we summarize methods commonly used to model the interaction of intermittent contact between surfaces of bodies. In Chapter 8 we show that analyses of rigid and so-called pseudo-rigid bodies 8 may be developed directly from the theory of deformable solids. This permits the inclusion in programs of options for multi-body dynamic simulations which combine deformable solids with objects modelled as rigid bodies. In Chapter 9 we discuss specialization of the finite deformation problem to address situations in which a large number of small bodies interact [multi-particle or granular bodies commonly referred to as discrete element methods (DEM) or discrete deformation analysis (DDA)]. In the second part of this book we study the behaviour of problems of structural mechanics. In Chapter 10 we present a summary of the behaviour of rods (beams) modelled by linear kinematic behaviour. We consider cases where deformation effects include axial, bending and transverse shearing strains (Timoshenko beam theory 9) as well as the classical theory where transverse effects are neglected (Euler-Bernoulli theory). We then describe the solution of plate problems, considering first the problem of thin plates (Chapter 11) in which only bending deformations are included and, second, the problem in which both bending and shearing deformations are present (Chapter 12). The problem of shell behaviour adds in-plane membrane deformations and curved surface modelling. Here we split the problem into three separate parts. The first combines simple flat elements which include bending and membrane behaviour to form a faceted approximation to the curved shell surface (Chapter 13). Next we involve the addition of shearing deformation and use of curved elements to solve axisymmetric shell problems (Chapter 14). We conclude the presentation of shells with a general form using curved isoparametric element shapes which include the effects of bending,
3
4 General problems in solid mechanics and non-linearity shearing, and membrane deformations (Chapter 15). Here a very close link with the full three-dimensional analysis will be readily recognized. In Chapter 16 we address a class of problems in which the solution in one coordinate direction is expressed as a series, for example a Fourier series. Here, for linear material behaviour, very efficient solutions can be achieved for many problems. Some extensions to non-linear behaviour are also presented. In Chapter 17 we specialize the finite deformation theory to that which results in large displacements but small strains. This class of problems permits use of all the constitutive equations discussed for small deformation problems and can address classical problems of instability. It also permits the construction of non-linear extensions to plate and shell problems discussed in Chapters 11-15 of this volume. We conclude the descriptions applied to solids in Chapter 18 with a presentation of multi-scale effects in solids. In the final chapter we summarize the capabilities of a companion computer program (called FEAPpv) that is available at the publisher's web site. This program may be used to address the class of non-linear solid and structural mechanics problems described in this volume.
1.2.1 Strong form of equations- indicial notation In this general section we shall describe how the various equations of solid mechanics* can become non-linear under certain circumstances. In particular this will occur for solid mechanics problems when non-linear stress-strain relationships are used. The chapter also presents the notation and the methodology which we shall adopt throughout this book. The reader will note how simply the transition between forms for linear and non-linear problems occurs. The field equations for solid mechanics are given by equilibrium behaviour (balance of momentum), strain-displacement relations, constitutive equations, boundary conditions, and initial conditions. 1~ In the treatment given here we will use two notational forms. The first is a cartesian tensor indicial form and the second is a matrix form (see reference 1 for additional details on both approaches). In general, we shall find that both are useful to describe particular parts of formulations. For example, when we describe large strain problems the development of the so-called 'geometric' or 'initial stress' stiffness is most easily described by using an indicial form. However, in much of the remainder, we shall find that it is convenient to use a matrix form. The requirements for transformations between the two will also be indicated. In the sequel, when we use indicial notation an index appearing once in any term is called a free index and a repeated index is called a dummy index. A dummy index may only appear twice in any term and implies summation over the range of the index. * More general theories for solid mechanics problems exist that involve higher order micro-polar or couple stress effects; however, we do not consider these in this volume.
Small deformation solid mechanics problems 5 Thus if two vectors ai and bi each have three terms the form ai bi implies aibi -- a l b l -Jr a2b2 -Jr-a3b3
Note that a dummy index may be replaced by any other index without changing the meaning, accordingly aibi =~ a j b j
Coordinates and displacements
For a fixed Cartesian coordinate system we denote coordinates as x, y, z or in index form as x l, x2, x3. Thus the vector of coordinates is given by X
: Xlel -4- x2e2 -+- x3e3 : xi ei
in which ei are unit base vectors of the Cartesian system and the summation convention described above is adopted. Similarly, the displacements will be denoted as u, v, w or ul, u2, u3 and the vector of displacements by u2e2 + u3e3 -- ui ei
u -- u~e~ +
Generally, we will denote all quantifies by their components and where possible the coordinates and displacements will be denoted as xi and ui, respectively, in which the range of the index i is 1,2, 3 for three-dimensional applications (or 1,2 for twodimensional problems).
Stra in-displacemen t relations
The strains may be expressed in Cartesian tensor form as (1.1) and are valid measures provided deformations are small. By a small deformation problem we mean that I~ij[ < < 1
and
Ioj2jl < < lIE/jI[
where [-I denotes absolute value and II 9 [I a suitable norm. In the above o.)ij denotes a small rotation given by ~vij -
l/Oui
OuJ /
~
Ox i
Ox j
(1.2)
and thus the displacement gradient may be expressed as Oui Oxj
: ~ij + (a')ij
(1.3)
6 General problems in solid mechanics and non-linearity
Equilibrium equations - balance of momentum
The equilibrium equations (balance of linear momentum) are given in index form as
ffji,j
Jr
bi
--
i, j = 1, 2, 3
p/~i,
(1.4)
where aij are components of (Cauchy) stress, p is mass density, and bi are body force components. In the above, and in the sequel, we use the convention that the partial derivatives are denoted by
Of f,i-- OXi
Of
and
3~ = -~-
for coordinates and time, respectively. Similarly, moment equilibrium (balance of angular momentum) yields symmetry of stress given in indicial form as (1.5)
(Tij - - a j i
Equations (1.4) and (1.5) hold at all points xi in the domain of the problem f2.
Boundary conditions
Stress boundary conditions are given by the traction condition ti - - f f j i n j
(1.6)
- - ti
for all points which lie on the part of the boundary denoted as Ft. A quantity with a 'bar' denotes a specified function. Similarly, displacement boundary conditions are given by Ui -- Ui
(1.7)
and apply for all points which lie on the part of the boundary denoted as F~. Many additional forms of boundary conditions exist in non-linear problems. Conditions where the boundary of one part interacts with another part, so-called contact conditions, will be taken up in Chapter 7. Similarly, it is necessary to describe how loading behaves when deformations become large. Follower pressure loads are one example of this class and we consider this further in Sec. 5.7.
Initial conditions
Finally, for transient problems in which the inertia term p/~i is important, initial conditions are required. These are given for an initial time denoted as 'zero' by U i ( X j , O) ~"
di(xj)
and
l ~ i ( X j , O) "~ ~3i(Xj)
in f2
(1.8)
It is also necessary in some problems to specify the state of stress at the initial time.
Constitutive relations
All of the above equations apply to any material provided the deformations remain small. The specific behaviour of a material is described by constitutive equations which relate the stresses to imposed strains and, often, other sources which cause deformation (e.g. temperature).
Small deformation solid mechanics problems 7 The simplest material model is that of linear elasticity where qUite generally _(0) ) - ~kl
O'ij - - C i j k l ( e k l
(1.9a)
in which Cijkl are e l a s t i c m o d u l i and _~0~ kl are strains arising from sources other than displacement. For example, in thermal problems strains result from change in temperature and these may be given by C(0) kl - - a u [ r - To] (1.9b) in which akl are coefficients of linear expansion and T is temperature with To a reference temperature for which thermal strains are zero. For linear i s o t r o p i c materials these relations simplify to O'ij = /~(~ij (Ekk - ekk _(0) ) + 2 #
and
E(klO)
- - t~ijO~ [ T -
(eij
_(0) ) -- e-.ij
To]
(1.10a) (1 10b)
w h e r e A and # are Lam6 elastic parameters and a is a scalar coefficient of linear expansion. 1~ In addition, ~Sij is the Kronecker delta function given by
1; ~ij - -
O;
for i = j for i # j
Many materials are not linear nor are they elastic. The construction of appropriate constitutive models to represent experimentally observed behaviour is extremely complex. In this book we will illustrate a few classical models of behaviour and indicate how they can be included in a general solution framework. Here we only wish to indicate how a non-linear material behaviour affects our formulation. To do this we consider non-linear elastic behaviour represented by a strain-energy density function W in which stress is computed as ll OW
O'iJ - -
tgeij
(1.11)
Materials based on this form are called h y p e r e l a s t i c . When the strain-energy is given by the quadratic form 1
/-,
W -- ~EijCijklEkl
,.(0)
-- Eijt...ijklakl
(1.12)
we obtain the linear elastic model given by Eq. (1.9a). More general forms are permitted, however, including those leading to non-linear elastic behaviour.
1.2.2 Matrix notation
Ixllxll
In this book we will often use a matrix form to write the equations. In this case we denote the coordinates as x=
y Z
=
x2 X3
(1.13)
8 General problems in solid mechanics and non-linearity
lullul
and displacements as U --"
1)
--
(1.14)
U2
w
u3
For two-dimensional forms we often ignore the third component. The transformation to matrix form for stresses is given in the order
~--[~ll
~22
~33
~12
~23
~311 ~
= [~xx ~yy
~zz
~xy
~yz
~zxl ~
C22
E33
"?'12 ')'23 ")/31]T
Eyy
Ezz
")/xy
(1.15)
and strains by ~=
JEll
--[Cxx
~yz
(1.16)
~zx] T
where symmetry of the tensors is assumed and 'engineering' shear strains are introduced as
")'ij = 2cij,
i ~= j
(1.17)
to make writing of subsequent matrix relations in a concise manner. The transformation to the six independent components of stress and strain is performed by using the index order given in Table 1.1. This ordering will apply to many subsequent developments also. The order is chosen to permit reduction to twodimensional applications by merely deleting the last two entries and treating the third entry as appropriate for plane or axisymmetric applications. The strain-displacement equations are expressed in matrix form as e = ,Su
(1.18)
with the three-dimensional strain operator given by -O S T =
0
-
0
o
~2
0
o 0 0 ~3
O ~
~1
0
0
O0
~3
0
~2
0 ~1-
Table 1.1 Index relation between tensor and matrix forms Form
Index value
Matrix Tensor (1, 2, 3)
1
2
3
4
5
6
11
22
33
12 21
23 32
31 13
Cartesian (x, y, z)
xx
yy
zz
xy yx
yz zy
zx xz
Cylindrical (r, z, 0)
rr
zz
O0
rz zr
zO Oz
Or rO
Small deformation solid mechanics problems 9 The same operator may be used to write the equilibrium equations (1.4) as S r o r + b = pii
(1.19)
The boundary conditions for displacement and traction are given by u=fi
OnFu
where Gr =
and
t-Grtr=t
[i n20 00 0
nl 0
n3
onFt
(1.20)
0 n3] 0
n3 n2
nl
in which n = (n~, n2, n3) are direction cosines of the normal to the boundary r'. We note further that the non-zero structure of S and G are the same. For transient problems, initial conditions are denoted by u(x, 0) - d(x)
and
ti(x, 0) - ~(x)
in f2
(1.21)
The constitutive equations for a linear elastic material are given in matrix form by or - O(e - e0)
(1.22)
where in Eq. (1.9a) the index pairs ij and kl for Cijkl are transformed to the 6 x 6 matrix D terms using Table 1.1. For a general hyperelastic material we use or =
OW
(1.23)
Oe
1.2.3 Two-dimensional problems There are several classes of two-dimensional problems which may be considered. The simplest are plane stress in which the plane of deformation (e.g. Xl - x2) is thin and stresses 0 3 3 - - 7-13 - - 7-23 - - 0 ; and plane strain in which the plane of deformation (e.g. X 1 - - X 2 ) is one for which e 3 3 " - - ")/13 - - ")/23 - - 0. Another class is called axisymmetric where the analysis domain is a three-dimensional body of revolution defined in cylindrical coordinates (r, 0, z) but deformations and stresses are two-dimensional functions of r, z only.
Plane stress and plane strain
For plane stress and plane strain problems which have mation, the displacements are assumed in the form U =
Ul(Xl,X2, t)} u2(Xl,X2, t)
X l --
X2
as the plane of defor-
(1.24)
10 General problems in solid mechanics and non-linearity and thus the strains may be defined by: 11 o
=
Cll C22 C33
=Su+e3
o
o
=
0
0
0
0
Ox 2
Ox l
"712
0
{ul)+ o U2
C33
(1.25)
0
Here the C33 is either zero (plane strain) or determined from the material constitution by assuming 0"33is zero (plane stress). The components of stress are taken in the matrix form ( 1.26) or.T = { 0"11 0"22 0"33 712 } where 0"33 is determined from material constitution (plane strain) or taken as zero (plane stress). We note that the local 'energy' term E = crr e
(1.27)
does not involve ~33 for either plane stress or plane strain. Indeed, it is not necessary to compute the 0"33 (or c33) until after a problem solution is obtained. The traction vector for plane problems is given by t=Grcr
where
Gr
~---
~.,[n, ]
[,,
0
l'/2
0 0
n21] nl j
(1.28)
and once again we note that ,5 and G have the same non-zero structure.
Axisymmetricproblems
Ixlllrl
In an axisymmetric problem we use the cylindrical coordinate system
X --
X2
x3
=
Z
(1.29)
0
This ordering permits the two-dimensional axisymmetric and plane problems to be written in a very similar manner. The body is three dimensional but defined by a surface of revolution such that properties and boundaries are independent of the 0 coordinate. For this case the displacement field may be taken as u--
u2(xl, x2, t) u3(xl, x2, t)
and, thus, also is taken as independent of 0.
-- ] Uz(r, z, t) kuo(r, z, t)
(1.3o)
Small deformation solid mechanics problems The strains for the axisymmetric case are given by "11 -0
o 0
~00
E'll E'22 C33
%z %o ")'Or
")q2 %3 ")'31
Err Ezz __
=Su=
o
&2
1
0
{Ul}
X1
a
o
OX2
OX1
0
0
0
0
U2 U3
(1.31)
0
(o 1) 0Xl
X1
The stresses are written in the same order as 03"--{fill
0"22 0"33 TI2
(1.32)
7"23 7"31}
Similar to the three-dimensional problem the traction is given by t =Grtr
where
Gr =
[o 00 n2 0
0 0
nl 0
0 n2
01
(1.33)
nl
where we note that n3 cannot exist for a complete body of revolution. Once again we note that S and G have the same non-zero structure. We note that the strain-displacement relations between the u 1, u2 and u3 components are uncoupled. If the material constitution is also uncoupled between the first four and the last two components of strain (i.e. the first four stresses are related only to the first four strains) we may separate the axisymmetric problem into two parts: (a) a part which depends only on the first four strains which are expressed in ul, u2; and (b) a problem which depends only on the last two shear strains and u3. The first problem is sometimes referred to as torsionless and the second as a torsion problem. However, when the constitution couples the effects, as in classical elastic-plastic solution of a bar which is stretched and twisted, it is necessary to consider the general case. The torsionless axisymmetric problem is given by -0 "Crr ~
"C11
E00
C22 E33
'ffr z
")/12
Ezz
=
=Su=
o
o
1 X1
0
Ul 0
0
0
.OqX2
OX1 -
U2
(1.34)
with stresses given by Eq. (1.26) and tractions by Eq. (1.28). Thus the only difference in these two classes of problems is the presence of the Ul/Xl for the third strain in
11
12 General problems in solid mechanics and non-linearity
the axisymmetric case (of course the two differ also in the domain description of the problem as we shall point out later). iii.......ii '~::'~::~'~~::~'~'............... ~'~::~::~~:~.!i...~............~.~.i...ii....~i.!.i.i.i.i.?~.i.~.!......ii ............................................. ::'::'::::::i~::...................... :': ~ !", i"~'~ii",iiiiiiTiiii'?:'::::"ii"............................................................. ~ ii............................... ': : ~~iiiii : : : iiiii.................................... ii..i.i.iii.. i.iii.ii~.. .i~:~:i.M.i.iT~.~.~.iii!i.i~ii~' ii.~.i~.~i',.~'i~i.,iiiii!iiii!iiiiiiiiiiiiiiii!iili i.i.i.i.i.i.i 'ii, iiii!i'i'i',~ii,i',ii'~i'iii',i',,ii',iiii~,',',',' ii",",i'~iii~ i'~~,'~',",',"~"~ ',}i"~i',",~~,,'"",,' !',' ~ii,"i,i',',i~,i",",i i',~,i"~ ~i~::~,::~i~:::i.i.~i~i~::.-~::~..::~i~-.---::!ii-~v-.::::::~:::::::--~,~..........~. ", !~iil :!i:~ '~......... ::....~i~i~ .:::........, ::~ii .... i~ '~::.... :N~ ~: ...... ::........~::~:':~,~!i:ii~i~i~i' ::~ ,i':!i ~ii. '~s !', iL,,!|~ ...~...~?~g~?~:~~:~
For an elastic material as specified by Eq. (1.23), the above equations may be given in a variational form when no inertial effects are included. The simplest form is the potential energyprinciple where
l"IpE= ~ W ( , . q u ) d ~ 2 - ~ u r b d ~ 2 - f r
urtdF
(1.35)
t
The first variation yields the governing equation of the functional ~rlee =
~(su)r O,Nu d~ -
~urb d~2 -
a s 16
~uridl -' = 0
(1.36)
t
After integration by parts and collecting terms we obtain ~l-Ipe=-- ~ ~ u r ( S r t r + b )
d~2 (1.37)
+fr
~ur ( G r a -
i) d[" = 0
t
where O'--
OW 08u
When W is given by the quadratic form (1.12) we recover the linear problem given by Eq. (1.22). In this case the form becomes the principle of minimum potential energy and the displacement field which renders W an absolute minimum is an exact solution to the problem, ll We note that the potential energy principle includes the strain-displacement equations and the elastic model expressed in terms of displacement-based strains. It also requires the displacement boundary condition to be stated in addition to the theorem. It is, however, the simplest variational form and only requires knowledge of the displacement field to be valid. This form is a basis for irreducible(or displacement) methods of approximate solution. A general variational theorem, which includes all the equations and boundary conditions, is given by the Hu-Washizu variational theorem. 17 This theorem is given by II HW(U ,
0")
f
[W(e) + err (Su - e)] dr2
,If2
(1.38)
_s t
u
in which t = Gr o'. The proof that the theorem contains all the governing equations is obtained by taking the variation of Eq. (1.38) with respect to u, s and o'. Accordingly,
Variational forms for non-linear elasticity taking the variation of (1.38) and performing an integration by parts on 3(,Su) we obtain
5Fl~w = f Ssr [ ~ - ~r] d~2 + f,o'r
[Su-s]
d~-fr
,t r (u-O)dF
(1.39)
u
-f'ur(sr~r+b)
d~+fr
'ur (t-i)
dF=0
t
and it is evident that the Hu-Washizu variational theorem yields all the equations for the non-linear elastostatic problem. We may also establish a direct link between the Hu-Washizu theorem and other variational principles. If we express the strains s in terms of the stresses using the Laurant transformation U (rr) + W (e) = rr Te (1.40) we recover the Hellinger-Reissner variational principle given by 18-2~
1-IMR(U, ~r) = ~_ [crrSu -- U(~r)] dr2
- f urbdf~-fruridF-frtr(u-fi,
dF
(1.41)
In the linear elastic case we have, ignoring initial strain and stress effects, 1
U (o') -- ~ oij Sijkl O'kl
(1.42)
where Sijkl are elastic compliances. While this form is also formally valid for general elastic problems. We shall find that in the non-linear case it is not possible to find unique relationsfor the constitutive behaviour in terms of stress forms. Thus, we shall often rely on use of the Hu-Washizu functional as the basis for a mixed formulation. We may also establish a direct link to the minimum potential energy form and the Hu-Washizu theorem. If we satisfy the displacement boundary condition (1.20) apriori the integral term over 1", is eliminated from Eq. (1.38). Generally, in our finite element approximations based on the Hu-Washizu theorem (or variants of the theorem) we shall satisfy the displacement boundary conditions explicitly and thus avoid approximating the Fu term. If we then satisfy the strain-displacement relations a priori then the Hu-Washizu theorem is identical with the potential energy principle. In constructing finite element approximations, the potential energy principle is a basis for developing displacement models (also referred to as irreduciblemodels1) whereas the Hu-Washizu form is a basis for developing mixedmodels.1As we will show in Chapter 2 mixed methods have distinct advantages in constructing robust finite element formulations. However, there are also advantages in having a finite element formulation where the global problem is expressed in a displacement form. Noting how the Hu-Washizu form reduces to the potential energy principle provides a link on treating the reductions to their approximate counterparts (see Sec. 2.6).
13
14 General problems in solid mechanics and non-linearity One advantage of a variational theorem is that symmetry conditions are automatically obtained; however, a distinct disadvantage is that only elastic behaviour and static forms may be considered. In the next section we consider an alternative approach of weak forms which is valid for both elastic or inelastic material forms and directly admits the inertial effects. We shall observe that for the elastostatic problem a weak form is equivalent to the variation of a theorem.
A variational (weak) form for any set of equations is a scalar relation and may be constructed by multiplying the equation set by an appropriate arbitrary function which has the same free indices as in the set of governing equations (which then becomes a dummy index and sums over its range), integrating over the domain of the problem and setting the result to zero. 1'17
1.4.1 Weak form for equilibrium equation For example, in indicial form the equilibrium equation (1.4) has the free index i, thus to construct a weak form we multiply by an arbitrary vector with index i and integrate the result over the domain f2. Virtual work is a weak form in which the arbitrary function is a virtual displacement ~ui, accordingly using this function we obtain the form
(~I'Ieq -- f ~ t~ui [pu i - o'ji,j -- b i ] d S2 :
0
Generally stress will depend on strains which are derivatives of displacements. Thus, the above form will require computation of second derivatives of displacement to form the integrands. The need to compute second derivatives may be reduced (i.e. 'weakened') by performing an integration by parts and upon noting the symmetry of the stress we obtain
'l-leq :f
t~uipuid~Wf t~Eij(uk)ffijd~-f
'uibid~2-fr'uitid~--O (1.43)
where virtual strains are related to virtual displacements as
t~'Cij(Uk) = l (t~Ui,j -~-(~Uj,i)
(1.44)
This may be further simplified by splitting the boundary into parts where traction is specified, Ft, and parts where displacements are specified, Fu. If we enforce pointwise all the displacement boundary conditions* and impose a constraint that ~u~ vanishes on Fu, we obtain the final result
t~I'Ieq--f t~uipi~lid~"~f-f t~Eij(Uk)O'ijd~'~--f (~uibid~'~-fF (~ui-tid~'~--- 0 t
(1.45) * Alternatively, we can combine this term with another from the integration by parts of the weak form of the strain--displacement equations.
References References 15 15
in matrix form as as or in
1n
r
r
r
T T 80 ( ~ I ' Ieq e q -= L 8uT ~uT piidQ piidQ + + L 8(SU)T ~(Su)r udQ o ' d Q -- L Bu ~uT bdQ b d Q -- f r Bu ~ u Ttdr tdF = = 0
in
in
ir,
t
(1.46)
The first term is the virtual work of internal inertial forces, the second the virtual work of the internal stresses and the last two the virtual work of body and traction forces, respectively. The above weak form provides the basis from which a finite element formulation of equilibrium may be deduced for general applications. It is necessary to add appropriate expressions for the strain-displacement and constitutive equations to complete a problem formulation. Weak forms for these may be written immediately from the of the Hu-Washizu principle given in Eq. (1.39). variation ofthe We note that the form adopted to define the matrices of stress and strain permits the internal work of stress and strain to be written as Eij O'ij - - "cTo " :
(1.47)
0 "T
Similarly, the internal virtual work per unit volume may be expressed by (~W = : &ij (~"Cij aij o'ij = : 8e (~~ TT u or" 8W
(1.48)
In Chapter Chapter 4 we will discuss this in more detail and show that constructing constructing constitutive equations of six components components of of stress and strain must be treated appropriately equations in terms of in reductions reductions from the original original nine tensor components. components.
i 1.5 i ii!iConcluding ii ijii remarks iii iii!! iiii !
i
i iiliii ii!iii!iii i iiiiiiii
In this chapter chapter we we have summarized summarized the basic basic steps needed needed to formulate a general general The formulation has has been been presented presented in in a strong strong small-strain solid solid mechanics mechanics problem. problem.' The small-strain terms of of partial partial differential differential equations equations and and in a wweak terms of of integral integral fform o r m in terms e a k fform o r m in terms expressions. expressions. We We have have also also indicated indicated how how the the general general problem problem can can become become non-linear. non-linear. In the the next next chapter chapter we we describe describe the the use use of of the the finite element element method method to to construct construct In approximate solutions solutions to to weak weak forms forms for non-linear non-linear transient transient solid solid mechanics mechanics problems. problems. approximate
References
iii !~ !iiiiiii!i!iiiiiii!i!!i!iiiii!iii!i!i!iiiiiiiiiiii!iii! i!i i i!i i
ili i ilii2iiili! ill ii ii ii !i
iiiiiiiiii iiiiiiii!i]!i!!!i i iii iii
i
ill ii i
i i i ii i!ii!!ii••i•ii••••iii••iii!••i•i••ii!•••ii•ii••!•ii!••i••••••ii••]•iiii!ii••ii
1. O.C. Zienkiewicz, R.L. R.L. Taylor Taylor and J.Z. Zhu. The The Finite Finite Element Element Method: Method: Its Its Basis Basis and and FundaFundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. mentals. Butterworth-Heinemann, 2. 2. T.J.R. T.I.R. Hughes. Hughes. The The Finite Finite Element Element Method: Method: Linear Linear Static Static and and Dynamic Dynamic Analysis. Analysis. Dover PubliPublications, New New York, York, 2000. 3. R.D. R.D. Cook, D.S. D.S. Malkus, MaIkus, M.E. M.E. Plesha Plesha and R.J. R.I. Witt. Witt. Concepts Concepts and and Applications Applications of of Finite Finite 3. Element John Wiley Wiley & Sons, New New York, York, 4th 4th edition, edition, 2001. 2001. Element Analysis. Analysis. John 4. 4. EF. de S. Lynch. Lynch. A A finite finite element element method method of of viscoelastic viscoelastic stress stress analysis analysis with with application to rolling rolling contact contact problems. problems. International International Journal Journal for for Numerical Numerical Methods Methods in in Engineering, Engineering, to 1:379-394, 1:379-394, 1969. 1969.
16 General problems in solid mechanics and non-linearity 5. J.T. Oden and T.L. Lin. On the general rolling contact problem for finite deformations of a viscoelastic cylinder. Computer Methods in Applied Mechanics and Engineering, 57:297-367, 1986. 6. P. le Tallec and C. Rahier. Numerical models of steady rolling for non-linear viscoelastic structures in finite deformation. International Journal for Numerical Methods in Engineering, 37:1159-1186, 1994. 7. S. Govindjee and P.A. Mahalic. Viscoelastic constitutive relations for the steady spinning of a cylinder. Technical Report UCB/SEMM Report 98/02, University of California at Berkeley, 1998. 8. H. Cohen and R.G. Muncaster. The Theory of Pseudo-rigid Bodies. Springer, New York, 1988. 9. S.P. Timoshenko and J.M. Gere. Theory of Elastic Stability. McGraw-Hill, New York, 1961. 10. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 11. I.S. Sokolnikoff. The Mathematical Theory of Elasticity. McGraw-Hill, New York, 2nd edition, 1956. 12. L.E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, NJ, 1969. 13. A.P. Boresi and K.P. Chong. Elasticity in Engineering Mechanics. Elsevier, New York, 1987. 14. P.C. Chou and N.J. Pagano. Elasticity: Tensor, Dyadic and Engineering Approaches. Dover Publications, Mineola, NY, 1992. Reprinted from 1967 Van Nostrand edition. 15. I.H. Shames and EA. Cozzarelli. Elastic and Inelastic Stress Analysis. Taylor & Francis, Washington, DC, 1997. (Revised printing.) 16. F.B. Hildebrand. Methods ofApplied Mathematics. Prentice-Hall (reprinted by Dover Publishers, 1992), 2nd edition, 1965. 17. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd edition, 1982. 18. E. Hellinger. Die aUgemeine Aussetze der Mechanik der Kontinua. In E Klein and C. Muller, editors, Encyclopedia der Mathematishen Wissnschafien, volume 4. Tebner, Leipzig, 1914. 19. E. Reissner. On a variational theorem in elasticity. Journal of Mathematics and Physics, 29(2): 90-95, 1950. 20. E. Reissner. A note on variational theorems in elasticity. International Journal of Solids and Structures, 1:93-95, 1965.
22
i~!iii~ii~i~ii!!!~!~i~i~ii!~!i~!~!ii!i!!!i!i~i~!~iii~iiii~i~i!i~i!i
method of approximation Galerkin method --irreducible irreducible and mixed mixed forms
iJ!ii~iii!iii!!i!iii~i~i!iiiiii~i!iiii!i~~iiiiiii~iiiiii!i!iii~iiiiiii~ii~ii~iii!~i~iiiii~i~iiiiiiiiiiiiiiii~iiiiiiiiii~iiiiiii!iiiiiiii~ii!ii~iiii~iii~i!iii!iiiiiiiii!~iiiii~!iiiiiiiii~i~i~iiiiii~iii~i~iii~iiiii~!iiii~i~i!i~iii~iiiii!~ii~i~iiii iiiii~i~d~i~~iii~iiiiiiii~2~iiii~ii~i~i~ii~i~ii!~ii~i~i~ii~i
2.1 Introduction
In the previous chapter we presented the basic equations for problems in non-linear solid mechanics in which strains remain small. We showed that the equations can be presented in a strong form as a set of partial differential equations or alternatively in terms of a variational principle or weak form expressed as an integral over the domain of interest. In the present chapter we use the weak form to construct approximate solutions based on the finite element method. This results in a Galerkin method for which general properties are well known. 1-4 1-4 Although it is assumed that the reader is familiar with finite element methods for small deformation linear problems, we present a full summary of the basic steps to construct a solution for the transient problem. We emphasize the differences between linear and non-linear effects as well as the numerical procedures used to establish the final discrete form of the equations which is the form used in computer analysis. We irreducible and mixed mixed forms of approximation. The mixed forms also consider both irreducible are introduced to overcome deficiencies arising in use of low order elements based on irreducible forms. In particular, in this chapter we consider a mixed form appropriate for use in problems in which near incompressible behaviour can occur. In the second part of this book, we consider forms for structural problems where so-called 'shear locking' can occur in bending of thin rods, plates and shells. chapter by applying the methods developed for the equations of We conclude this chapter solid mechanics to that for thermal analysis based on a non-linear form of the quasiharmonic equation.
i!ii!ii2.2 iiiiii~'ii'~'~i~iiii!i!iiiiiiiiiiiii' "'Finite ~~!~i '~'~"~i i~iiii~i~iii~ii~~' ~ii~element i~i~~:ii~i:~i!i!~ii!~ii!i!i~i!ii~iii~ii~iapproximation i~i~!iiiiii~ii~ii!i~ii~i!iiii~!~iiii~ii~i~~iiiii~!iii~~~i ii!i~i-i~i~iii~iiGalerkin .i!~.i.i~.i.i~ii~i~i~iii!~~ii~ii~iiii~ii:.:i.~i:.!i~ii!~!~i i~i!i~i~imethod ii~ii~ii~ii!~iii~i~~i~ii.~i:.i:.~iiii~i~ii~i!i!i~!ii~iii~ii~i~iii!i~i~iii~ii~ii~i~~ii~ii~iiiiiii~ii~i~ii iiii~i~iii~ii~!iii~i~i~i~i~i~iii~i!~i!i!i!i!iiii~i~i!~i~ii!iiii~i~iiii~!~i~i!~i~iiiii~iii~i~i!i!ii~iiiii~iiiiiiii!ii~i~iiiii~i!~i~iii~i~i~i~ii!i!ii~iiiiiiii~iii~i~ii~i!~i~!iii!~iiiii~iii~i~i~ii~ii!~i~iiii~i~ The finite element approximation to a problem starts by dividing the domain of interest, Q, f2, into a set of subdomains (called elements), Qe, fie, such that f2
= ~
~'2e
(2.la) (2.1a)
ee
Similarly the boundary is divided into subdomains as
L:r L:r L:rI~ue
rr~r= ,~ F - ~--~ Fee = -- ~-~ Ftete + -~- ~ e
et et
eu eu
ue
(2.tb) (2.1b)
18 Galerkinmethod of approximation -irreducible and mixed forms where Fte is a boundary segment on which tractions are specified and F/./e o n e where displacements are specified. We note that in general the domain of a finite element analysis is an approximation to the true domain which depends on the boundary shape of elements. The weak form for the governing equations is written for the problem domain (2 and also written as a sum over the element domains. Thus, the weak form given in Eq. (1.46) for the equilibrium equation becomes
r
~ (~'leq : ~-~ [ J~ 'uTpiidf2 + J~ e
e
et
= ~
6urbdf2] e
te
t~leIe -+- y ~ e
'(Su)To'df2-f~ e
t~leIt = 0
et
In the above r e a r e terms within the domain ~'~e of each element and ~l~It those which belong to traction boundary surfaces Fte. A Galerkin method of solution is obtained using approximations to the dependent variables and their virtual forms. 1'2 For an irreducible (or displacement) finite element form we only need approximations for u and ~u. In order for a variational theorem or a weak form to be split into the additive sum indicated in Eq. (2.2) the highest derivatives appearing in the functional must be at least piecewise continuous so that all the integrals exist and no contributions across interelement boundaries are present.* For a functional containing a variable with a highest derivative of order m + 1 the functions used to approximate the variable must have all derivatives up to order m continuous in the entire domain, f2 - such functions are called C m. For the weak forms considered for problems in solid mechanics we will encounter functionals which contain only first derivatives and thus will need only C O functions for the approximation. Indeed some functions in mixed forms will have no derivatives and these may be approximated by discontinuous functions in f2. Generally, one should respect the order of approximation (i.e. C m) where an exact solution can have discontinuous behaviour. For solid mechanics there are discontinuities in the displacement at material interfaces and at some singular load forms (e.g. point loads or line loads). Material interfaces are real; however, use of a point or line load is not and, when used, is an approximation to a physical action. Use of functions with added continuity over C m can be beneficial where solutions are smooth. Thus there are some forms of interpolation being introduced in recent literature that have increased smoothness. In this volume, however, we will generally present only those forms which provide C m continuity. In the first part of this volume concerning problems in solid mechanics approximations will be made with C O functions. In the second part concerning structural mechanics problems for rods, plates and shells we shall need C 1 functions for some formulations, whereas in others we can still use C Oforms. There are formulations which violate the continuity conditions leading to so-called incompatible approximation (e.g. see Wilson et al.6). Strang termed such approxima* It is possible to add interelement jump terms to a functional leading to a D i s c o n t i n u o u s see Cockburn e t al. 5 for more information.
Galerkinformulation
-
Finite element approximation-Galerkin method
tions a v a r i a t i o n a l c r i m e 7 but showed convergence could still be achieved provided certain requirements were met. Most incompatible formulations that perform well have subsequently been shown to be members of a valid mixed formulation (e.g. see Simo and Rifai8). In the forms given for problems in solid mechanics we will use interpolations which satisfy the C Orequirement; however, in the study of thin plates we will present some forms which violate the C 1 requirement. In addition to the continuity requirement it is necessary for C m functions to possess complete polynomials to order m + 1 to ensure that the derivatives up to order m + 1 can assume constant values. Both of the above requirements are covered in standard introductory texts on the finite element method (e.g. see reference 2 or 3). They remain equally valid for the study of non-linear problems - both for forms with material non-linearity as well as those with large deformations where kinematic conditions are non-linear. The patch test also remains valid in assessing the available continuity and derivatives present in any approximation (see reference 2 for a general discussion on the patch test for irreducible and mixed finite element formulations).
Displacementapproximation
A finite element approximation for displacements is given by u(x,
t) ~ fi -
~
Nb(X)fib(t) = N(x)fi(t)
(2.3)
b
where Nb are element shape functions, fib(t) are time dependent nodal displacements and the sum ranges over the number of nodes associated with an element. Alternatively, in i s o p a r a m e t r i c f o r m 2 the expressions are given by (as shown in Fig, 2.1 for a four-node two-dimensional quadrilateral) U(~, t) ~ ~t(~, t) -- ~
Nb(~)fib(t) --
N(~)fi(t);
b Nb(~')'~b -- N(~)'~
x(~) - ~
(2.4a)
b
~2
~2
1
I
I
r
( 1
._1~. f l
~,
(a) Element in ~ccoordinates.
1
:> Xl
r
(b) Element in x coordinates.
Fig. 2.1 Isoparametric map for 4-node two-dimensional quadrilateral.
19
20
Galerkin method of approximation -irreducible and mixed forms
where ~ represent nodal coordinate parameters and ( are the parametric coordinates for each element. An approximation for the virtual displacement is given by
611(~) ~ 61_1(~) -- Z
a
Na(~)lla -- N(~)u
(2.4b)
A summary of procedures used to construct shape functions for some isoparametric elements is included in Appendix A.
Derivatives
The weak forms presented in Chapter 1 all include first derivative of displacements. For the isoparametric approximation given in Eq. (2.4a) we need first derivatives of the shape functions with respect to xj. These are computed using the chain rule as:
ONa O~i or in matrix form
OXj ONa
=
ONa
(2.5)
O~i OXj
ON.
= J ~
(2.6a)
where
" 01~ ON a 0( =
ON N 01~ --~
;
ONa 0x =
ON,
Ox 1
~X1 ON ~ ON,
0~10~l Ox1 Ox2. 0~2 0~2 OXl Ox2
;
J=
~
-0~3
OX2
0~3
OX3 O~1 Ox3 0~2 Ox3
(2.6b)
0~3
in which J is the Jacobian transformation between x and ~. Using the above the shape function derivatives are given by
ONa = j-10Na
(2.6c)
In two-dimensional problems only the first two coordinates are involved, thus reducing the size of J to a 2 x 2 matrix. In the sequel we will often use the notation
ONa ONa Oxj = Na,xj and O~i = Na,~,
(2.7)
Strain-displacement equations
Using (1.18) the strain--displacement equations are given by -- S u ,~ ~--~(,-qNb)fib = ~ Bbfib -- Bfi b b
(2.8)
In a general three-dimensional problem the strain matrix at each node of an element is defined by [Nb,xl 0 0 Nb,x2 0 Nb,x3] B~-/ 0 Nb,x2 0 Nb,xl Nb,x3 0 J (2.9a) L0 0 Nb,x3 0 Na,x2 Nb,x,
Finite element approximation-Galerkin method
For the two-dimensional plane stress, plane strain and torsionless axisymmetric problem the strain matrix at a node is given by
B~-
[N~x,
0 CNb/Xl Nb,x2] Nb,x2 0 Nb,x,
(2.9b)
where c = 0 for plane stress and strain and c - 1 for the torsionless axisymmetric case. For the axisymmetric problem with torsion the strain matrix becomes
rNb,xl 0 Nb/Xl BTb-- [ 0 Nb,x: 0
0
0
Nb,x2 Nb,x,
0
0 0
0 0
]
]
(2.9c)
Nb,x2 (Nb,x, -- Nb/X~)
Weak form
Substituting the above forms for displacement and strains into the weak form of equilibrium given in Eq. (2.2) yields, for a single element,
(~I"Ieq- "e
(~1-1 T
[f~ e
N r pN dr2 fi +
f~ e
B r or dr2 -
f~ e
Nrb dr2 -
Ji te
N r t dF
(2.10)
]
Performing the sum over all elements and noting that the virtual parameters 6fi are arbitrary we obtain a semi-discrete problem given by the set of ordinary differential equations (2.11a) M a + P(or) = f where
M(e);
M = ~
e = y ~ e(e) and f = ~
e
e
f(e)
(2.1 l b)
e
with the element arrays specified by
M(e)=[NrpNdf2;
P(e)(or)-[Brordf2
J ~'~e
J ~"~e
and f ( e ) = [ N r b d f 2
+fNrtdF
J ~'~e
te
(2.11c) The term P is often referred to as the stress divergence or stressforce term. While the form for the arrays given above is valid for all problem classes the volume element differs and is given by: df2 dr2 dr2 dr2
= dXl dx2 dx3;
= dXl dx2;
-- h3 dXl dx2; = 2 7rxl dxl dxa;
General three-dimensional problems, Plane strain problems, Plane stress problems, Axisymmetric problems.
In the above we assume a unit thickness in the x3 direction for plane strain, h3 is the thickness of a plane stress slab; and the factor 2 7r in axisymmetric problems results from the integration of f dx3 = f d0 of the body of revolution.* In the sequel we will discuss the finite element form for solids in a general context using the coordinates, xi, displacements, ui, etc. Unless otherwise stated, we will also assume that the forms for B, f2e, dr2, etc. are always replaced by that appropriate for the problem class considered (i.e. plane stress, plane strain, axisymmetric, or general three dimensions).
* Someprograms,includingFEAPpvavailableat the publisher'sweb site, omitthe factor271"in axisymmetric forms.
21
22 22
Galerkin method method of of approximation approximation -irreducible - irreducible and and mixed mixed forms forms Galerkin
displacement method method 2.2.1 Irreducible displacement In In the the case case of of linear linear elasticity elasticity the the constitutive constitutive equations equations are are given given by by Eq. Eq. (1.22) (1.22) and and using Eq. Eq. (2.8) (2.8) an an irreducible irreducible displacement displacement method method results results22 with with using p(e)(o-) -= ( f P(e'(~r)
rBrDBdf2)fi = K(e)fl
ln,
BTDBdQ)fi = K(e)fi
(2.12) (2.12)
e
in in which which K(e) K(e) is is aa linear linear stiffness stiffness matrix. matrix. In In many many situations, situations, however, however, itit is is necessary necessary to to use non-linear or time-dependent stress-strain (constitutive) relations and in use non-linear or time-dependent stress-strain (constitutive) relations and in these these cases cases we l a) to l c). This we need need to to develop develop solution solution strategies strategies directly directly from from Eqs Egs (2.1 (2.11a) to (2.1 (2.11c). This will will be be considered considered further further in in detail detail in in later later chapters chapters for quite quite general general constitutive constitutive behaviour. behaviour. this stage stage we we simply simply need need to note note that that However, at this a0- == er(e) o-(e:)
(2.13) (2.13)
and that that the the functional functional relationship relationship can can be be very non-linear non-linear and occasionally occasionally non-unique. non-unique. Furthermore, Furthermore, it will will be be necessary necessary to use use a mixed mixed approach approach if if constraints, constraints, such such as near near incompressibility, incompressibility, are encountered. encountered. 22 We address this latter aspect aspect in Sec. 2.6; however, before before doing doing so we we consider consider the manner manner whereby whereby calculation calculation of of the finite element element arrays and solution solution of of the transient equations equations may be computed computed using numerical numerical methods. methods. ','i,i i',i',i'~:i!'~,i,'i'li,:,~': ,'~,~',::i',~i~,,i~::','i*',,'i,*,',i~,::~'~,'i,~i':~i i~iT'~'~"'~: 'i'~l'i~: i',ili~i,'~i',i~i'~',i',~i~,i!,i~:'~,i,i~':i~i'~i'~',i,i~,i',i,'~'~i,'i:~i iii'~~'(~ii',',:,i~'!~'i,i',:'i,~ii~,ii',':~i,i i':i,i':',:~i'~,'i,~ii ',i!'~i',i',~i',~,~',~''i,',',~',~ii!i'~i:~,',~i~,',i i'i~!i',i~ii~,i',i i'::i,~,i'i,'i,',i i',i'~',i:',i:~~',:i~',i:'~,~i',:'i,i',i'~,i'~,~,i','~i~ii',i!',i:i,'~,,',~~'i~,',~~,iiiii!','i,'i,~ii~'',i,:i~',~,,:'~i'i,'i,iii~' ','i:,',i~,','i,',i'~,',',',i',~i','i~',:,'~i,~'i',~',:':'i:f'~:,:,i,:::ii'i~!i:~',~i ,:(,i,i,':,::i~:~i:i','~,i',i!i':,i!'~',i'i,~':'i,'i,iiiii~,i~i'~,i',ii~i~,i~l i'i,~ii'~:~ii ',i',i'~,i'i:,i!i i i i!ii',',i i',:',i:iii i','i,iiiiiiiiii~~i,ii',i',i',',',i:'i~i:i,ii i'~i ili',iii~,i~' ,i'l,i',iii!'i,iii',i',iiil'~i!',i',i',iii'~,i~,ii~,'i~',il,i~,i',i',i',i~i,'i~:,i,i'~i:,ii~'i,'i:~i:i'~,i!i:',,i~:izi':i',i',iiii',i',i',i',i',':i
2.3 Numerical integration - quadrature
element arrays are most conveniently The integrations needed to compute the finite element performed numerically by quadrature. 22'3'9 ,3,9 Many forms of quadrature formulas exist; expressions is Gauss-Legendre Gauss-Legendre quadrahowever, the most accurate for polynomial expressions ture. IO 1~ Gauss-Legendre Gauss-Legendre quadrature tables are generally tabulated over the range of of - 1 < ~~ < 1 (hence our main reason for also choosing many shape funccoordinates -1 tion on this interval). Gaussian quadrature integrates a function as
(d f)
1 fl
2n
d~ 2n f ) f (~) d~d~ =-- ~ y n~n f(~j f )(~j) Wj wj + + 00 (d2,, _I f(~) den 1
1
(2.14a)
j=l
wj is a weight. Thus, an where ~j~j are the points where the function is evaluated and Wj n-point formula integrates exactly a polynomial of order 2n - 1. Table 2.1 presents the location of points and weights for the first five members of the family. Integrations over multi-dimensional domains may be performed by products of the one-dimensional formula. Thus, in two dimensions we use (with 17~7== - ~2)~2)
LLf(~, 1
1
~
f (~, n) ~7)d{ d~ dn d~7 =
t. t, j=l k=l
ff«j. (~j, n,) r/k) W tvjj w, wk
(2.14b) (2. 14b)
and and in in three dimensions (with (~ == ~3)~3)
111111 f(~, I
1
I
1
I n n
1
()d~d17d(
f(~, 17, r/, 0 d~ d~dr =--
~f; j=l
n
7~k, (/) ~l) Wj Wj Wk W k W/ 11)l ~ ff(~j' (~j, 17k'
k=l l=1
are exact exact when polynomials in in any any direction direction are are less than order order 2n. 2n. which are
(2.14c)
Numerical integration - quadrature Table 2.1 ~"~=1
Gaussian quadrature abscissae and weights for f_l I f ( ~ ) d ~
--
f (~j)wj. j
Order
~j
wj 2
n=l
1
0
n=2
1
+ 1/v/'3
1
2
-1/4~
1 5/9
n=3
n -- 4
1
+ ~/-0--.6
2
0
8/9
3
- ~/-O.-.-~
5/9
1 4-,,/((3 4- a ) / 7 )
0.5 - 1 / ( 3 a )
2
4- ~/((3 - a ) / 7 )
0.5 4- 1 / ( 3 a )
3
- ~/((3 - a ) / 7 )
0.5 4- 1 / ( 3 a )
4
- ~/((3 4- a ) / 7 )
0.5 - 1 / ( 3 a )
1
+ v/b
2
+~/-d
n = 5
3
0
4
- ~
5
- v/b
a=
((5c - 3)d/b) ((3-5b)d/c) 2-
44~.8
a -- v / l 1 2 0 b = (70 4- a ) / 1 2 6 c = (70 - a ) / 1 2 6
2(wl + / 0 2 )
5b)d/c) ((5c - 3)d/b) ((3 -
d = 1/(15(c - b))
Volume integrals
The problem remains to transform our integrals from the element region ~'2e to the gaussian range - 1 < ~ < 1. The determinant of J appearing in Eq. (2.6a) is used to transform the volume element from the cartesian coordinates to the natural coordinates as
dxldx2dx3 = d e t J d~l d~2 d~3 = j(~l, ~2, ~3) d~l d~2d~3
(2.15)
in which detJ - j must be positive to maintain a correct volume element. Using the above type of transformation, integrals of finite element arrays are given by f ( x ) d ~ = f~ f ( , ) j ( , ) d Q
(2.16)
e
where f is the function f written in terms of the parent coordinates ~, D denotes the range of parent coordinates for the dimension of problem considered and j (~) is the appropriate Jacobian transformation for the coordinate system considered. For the various problem classes these are given by: Problem type Three-dimensional: Plane strain: Plane stress: Axisymmetric:
n - domain d~l d~2 d~3 d~l d~2 d~l d~2 d~l d(2
j
-
Jacobian
J (~1' ~2' ~3) J (~1, ~2) h3 j (~l, ~2) 27rXl j (~1, ~2)
where for two-dimensional problems
I
0Xl
J (~1, ~2) ~"
det
0x2
OXl 0x2
>0
(2.17)
23
24 Galerkin Galerkinmethod of approximation approximation --irreducible mixed forms forms irreducible and mixed
The permissible is chosen so that The minimum number number of of quadrature quadrature points permissible 1. Elements Elements which have constant jacobians jacobians j are exactly integrated; or element stiffness matrix has full rank. 2. The resulting element
In either The first either case, the consistency part part of of the patch test must also be satisfied. 22 The criterion criterion may be used only for linear materials materials in small strain (such as discussed in this chapter). The non-linear problems. The second is applicable applicable to both linear and non-linear problems. Use of the quadrature than that satisfying the above is called reduced quadrature next lower order quadrature and generally should be avoided.* When integrated using 'full' order When some terms are integrated and some with 'reduced' method is referred 'reduced' order quadrature quadrature the method referred to as selective reduced 2•3 integration. 2,3 The above form of natural coordinates coordinates ~~ assumes that each finite element element is a line, The a quadrilateral quadrilateral or a hexahedron. hexahedron. For other shapes, such as a triangle or a tetrahedron, tetrahedron, For other shapes, appropriate changes changes are made coordinates and integration formula used appropriate made for the natural coordinates A).2.3.9 (see also Appendix Appendix A ) . 2'3'9
Surface integrals integrals Surface
element surfaces surfaces and this is most easily It is also necessary necessary to compute integrals integrals over element accomplished by considering considering df dF as a vector oriented in the direction normal to the accomplished surface. For three-dimensional three-dimensional problems problems we form the vector product product Ox Ox &x &x ndf d~1 n dr" = - dr dr = = -8 0s xx -8 ~ d~ 1 d~2 d~2 ~I ~2 = (VI (Vl x V2) v2) d~1 d~l d~2 d~ 2 = -- V Vn d~l d~2 d(2 = n d~1
(2.18) (2.18)
where ~I~1 and ~2~2 are parent coordinates for the surface surface element element and x denotes a vector where parent coordinates cross product. Surfaces for two-dimensional problems described in terms of ';1 ~1 only and Surfaces problems may be described Ox/O~2 by e3 (the ( t h e unit normal to the plane plane of of deformation). for this case we replace replace &x/a~2 If If necessary, the surface differential differential may be computed computed from T ) 1/2 dr = - ((VTVn)1/2 d~, d~2 d~2 df Vn Vn d~1
(2.19)
iiiiiiiMiii~iiiiiiiiii~iliiii~ii~ii~iiii ~iMi~i~i~i~ii~iiii~ii~i~i~i~i~ii~i~ii~i~i!iiiiiiiiiiiiiiiiii!i~i~i~i~iiiiiiiiiiiiiiiMii~iiiiiiiiiiiiiii~i~ii~i~i~i~iii~iiiiiiiiiiiiii~iii~iiiiiiii~iiiii~iii!iiiiiiiii~iiiiiiii~iii~iii~iiiiii~ii~i~%i!~i!iiiiiiii!~!iii~iiiii~i!iii~iiiiiiiiiiiiiiiiii!i~iiiiii iiii!ii~i~i~i iii~iiiiiii!i!iiiiiiiiiiiiiiiiiiiiii!iiiiiiiiiiTiiTiikiiii!ili !iiiiiiiiikliiikiiiii ii!iiiiiiiiiiiiiiiiiiiiiiiiiiiliiiiiiiiii!iiiii iiiiiiiiiiii!iii~':iiii! ~ iiiiiiii!iii :~ i~: ':~.......i~ i i........: ~:ii~-k~i ,,~iiii~~,,,i~ i.......~i!ill'~'~ ! ........i ,iiilJ ~i......i~..... iiiii!!!:i~ ,!': ii~ :i~: l~'~i!i !:!~ ~iii~'~i~~': iiii!i!iiii ........~i~'........i :~i !~........i........ !ik~:~ii!iiiiiii~iiiiiiiii!i!i!i!ii~iii~i!~!iii~i!ii!i~i!i!i!iii~!~!!ii!i~i!i~i!i~ii~i
2.4 Non-linear transient and steady-state problems
To obtain a set of algebraic equations for transient transient problems problems we introduce introduce a discrete approximation in time. We write the approximation approximation to the solution as U(tn+l) ~~
U(tn+l)
U U nn++,; l;
U(tn+d ~~ VVn+1 and UU(tn+l) ~~ an+1 U(tn+l) n+l (tn+l) an+l
where discrete variables variables is omitted for simplicity. Thus, the equilibrium equilibrium where the tilde on discrete equation (2.IIa) (2.1 la) at each discrete time tn+1 tn+l may be written in a residual form as 0
(2.20a)
Pn+l -- /~, B Trtrn+a d~ = - - P(un+d P(Un+l) P O'n+1 dQ n+1 ==
(2.20b)
~n+l = fn+l -- M
where where
1
a n + l -- P n + l =
a n y explicit e x p l i c i t codes c o d e s uuse s e reduced r e d u c e d qquadrature uadrature in combination w i t h so-called s o - c a l l e d hhour-glass our-glass stabilization. 4 4,11 •J I •* M Many in combination with stabilization.
Non-linear transient and steady-state problems 25 Here we have indicated that P can be expressed in terms of the displacement alone. This is correct for elastic materials but with inelastic behaviour the material model will depend on the solution variables in a more general form. In Chapter 4 we will show that most constitutive models may be given in terms of increments of u. Thus the above assumed form will need only a minor modification that does not significantly affect the following discussion. For transient problems we apply the GN22 method 2 (which is identical to the Newmark procedure ~2 except for the manner parameters are defined) to equations with second derivatives in time. The GN22 method relates the discrete displacements, velocities, and accelerations at tn+l to those at tn by the formulas Iln+ 1 =
U n
-q-- At Vn +
Yn+l = Yn + (1
1 ~(1 -/~2)At2an
-4-
1 ~/32At2an+l =
IJn+ 1 +
1 ~/32At2an+l
- / 3 l) At an +/31 At an+l - - Cgn+l + ~1 At an+l
(2.21)
in which At = tn+l -- tn is a time increment and fin+l, ~n+l are values depending only on the solution at t,. This one-step form is very desirable as it allows the At to change from one step to the next without introducing any complications (although very large changes should always be avoided). The two parameters/31 and/32 are selected to control accuracy and stability. The transient problem is now obtained for each time tn+l by solving the non-linear equation set (2.20a) and the pair of linear equations with scalar coefficients (2.21). It is possible to take as the basic unknown any one of the three variables at time tn+l (i.e., Un+l, Vn+l or an+l). Using Eq. (2.21) the non-linear equation may then be given in terms of a single unknown.
2.4.1 Explicit GN22 method A very convenient choice is to take/32 - 0 and select an+l as the primary unknown. Using Eq. (2.211) we immediately obtain ~ n + l "-- Un+l
and thus from Eq. (2.20a) M an+l - - fn+l -- P ( ~ )
may be solved directly for an+l. The velocity Yn+l is then obtained from Eq. (2.212). This leads to a so-called explicit scheme since only linear equations are solved. If the M matrix is diagonal 2 (or lumped) the solution for an+l is trivial and the problem can be considered solved since
I
1/Mll
M-l_
..
1/Mmml
where m is the total number of equations in the problem. However, explicit schemes are only conditionally stable with At < Atcrit, where Atcrit is related to the smallest
26
Galerkin method of approximation -irreducible and mixed forms time it takes for 'wave propagation' across any element or, alternatively, the highest 'frequency' in the finite element mesh. 2 Thus a solution by an explicit scheme may require many thousands of time steps to cover a specified time interval. For many transient problems, and indeed for most static (steady state) problems, it is often more efficient to deal with implicit methods for which much larger time steps may be used.
2.4.2 Implicit GN22 method In an implicit method it is convenient to use U~+l as the basic variable and to calculate Vn+l and an+l using Eq. (2.21). With this form we merely set Vn+l = an+l -- 0 to consider a quasi-static problem.* The equation system (2.20a) now can be written as 2c /32At2 M [Un+ 1 - a n + l ] -- Pn+l -- 0
~IJ(Un+l) -- fn+l
(2.22)
where c = 1 for transient problems and c = 0 for quasi-static ones. The solution to this set of equations requires an iterative process when any of the terms is non-linear. We shall discuss various non-linear calculation processes in some detail in Chapter 3; however, we note here that Newton's method t forms the basis of most practical schemes. In this method an iteration is written as ~ lI/k+ l ,~ k k n+l It/n+ 1 "t- dffffn+ 1 -- 0
(2.23a)
where, for fn+l independent of deformation, the increment of Eq. (2.22) is given by dffffkn+ 1 - - -
/32At2
M-
0un+----~n+l
The displacement increment is computed from k
A n + l dUn+ l :
k l ffff n+
(2.24a)
and the solution is updated using lak+l k n+l - - Un+l -~" dUkn+l ak+l n+l :
2 r k+l / 3 2 A t 2 [Un+l -- I]n+l]
u k+l - " ~rn+l "~- /31 A t
(2.24b)
a~++11
An initial iterate may be taken as zero or, more appropriately, as the converged solution from the last time step. Accordingly, a n1+ 1 - - a n
(2.25a)
* A quasi-static problem may be time or load path dependent; however, inertia effects are not included. t Often also called the Newton-Raphson method. See reference 13 for a discussion on the history of the method. ~t Note that an italic 'd' is used for a solution increment and an upright 'd' for a differential.
Non-linear transient and steady-state problems 27 in which a quantity without the superscript k denotes a converged value. For transient problems initial velocities and accelerations are given by 2
1 __ [U n _ i~n+l ] a n + l - - /~2 A t 2 1 1 Yn+l - - Vn+l -~-/~1 A t an+l
(2.25b)
Iteration continues until a convergence criterion of the form II kn+ 1 II _< e II~I'n+ 1 1 II
(2.26)
or similar is satisfied for some small tolerance e. A good practice when all terms of the Newton method are accurately computed is to assume the tolerance at half machine precision. Thus, if the machine can compute to about 16 digits of accuracy, selection of e = 10 -8 is appropriate. Additional discussion on selection of appropriate convergence criteria is presented in Chapter 3. A common choice of parameters is/31 =/32 = 1/2 which is also known as the 'trapezoidal integration rule'. The derivative of P appearing in Eq. (2.23b) is computed for each element from Eq. (2.1 lc2) as 0p(e) I k - - - ~ n+l
BrDkrBd~ - Kkr
(2.27)
e
We note that the above relation is similar but not identical to that of linear elasticity. Here DkT is the tangent modulus matrix for the stress-strain relation (which may or may not be unique but generally is related to deformations in a non-linear manner) and K r is the tangent stiffness matrix. Various forms of non-linear elasticity have in fact been used in the present context and here we present a simple approach in which we define a strain energy density, W, as a function of e W = W ( e ) "-" W ( e i j )
and we note that this definition gives us immediately cr =
OW 0e
(2.28)
If the nature of the function W is known the tangent modulus Dkr becomes 0trlk _ 02Wlk -Dk = ~ n+l 0 ~ 0 S n+l For problems in which path dependence is involved to compute O'n_t_1 (viz. Chapter 4) it is necessary to keep track of the total increment during the solution step from tn to tn+l and write tlk+l (2.29a) n+l - - U n + A U. nk+l +l with A U n1+ 1 -- 0 The total increment can be accumulated using the solution increments as .k+l
. k+l
k
k
m u n + 1 = Un+ 1 -- U n - - m U n + 1 "q- dun+ 1
(2.29b)
In an implicit scheme it is desirable to use the displacement from the last iteration to compute both A and ~I, - especially when inelastic material behaviour or large strains are considered.
28 Galerkin Galerkin method method of of approximation approximation -irreducible - irreducible and and mixed mixed forms forms 28
2.4.3 Generalized mid-point mid-point implicit implicit form 2.4.3 An alternative alternative form form to to that that just just discussed discussed satisfies satisfies the the balance balance of of momentum momentum equation equation An at at an an intermediate intermediate time time between between tn t n and and tn+l. tn+l' In In this this form form we we interpolate interpolate the the variables variables aas s
Un+o~ Un+a -=-
(1 (l - ce) a) Un Un + + ceaUU nn++11
Vn+a "-= (1 (1 Yn+o~
a) Vn Vn -q+ ceaVY nn++1l - ce)
(2.30) (2.30)
an+a -= (1 (l -- Ce)an a) an + + O~ aa an+a an+l n+1
and write write momentum momentum balance balance as and ~I'n+~ - fn+~ - M an+c~ - Pn+c~ "-- 0
=
(2.31) (2.31)
=
We We select select the the parameters parameters in the the GN22 GN22 algorithm algorithm as/31 as /31 =/32 /32 = / 3/3 and, and, thus, thus, obtain obtain the the simple form simple Un+ 1 -- Un +
1 At
(Vn+l -- u
--
1
~ At (v n + Yn+l)
(1 -/3)
(2.32) (2.32) an - F / ~ a n + l
Selecting now c~ equation in the form Selecting a -=/ 3f3 gives the the momentum momentum equation tTffn+o~ - - fn+o~ - - ~
1
At
M
(u
- - Vn) - -
Pn+~ - 0
(2.33)
which is the form utilized by Simo et at. al.** as part of of an energy-momentum conserving 15 14,15 a = / 3f3 - 1/2 1/2 method. 14 • For a linear elastic problem it is easy to show that choosing a If non-linear elastic forms are used will conserve energy during free motion (i.e. f = 0). If with these values, it is necessary to modify the manner by which acrn+l/2 n +l/2 is computed 16 We shall address this more in Chapters 5 and 6 to preserve the conservation property. 16 when we consider finite deformation forms and hyperelastic constitutive models. Un+l as the The solution of the above form of the balance equation may still use Un+l primary variable. The only modification is the appearance of the parameter parameter a in terms Pn+a. arising from linearizations of P n+a'
= =
tliitii!tilitiiiiiiiiiQ•it!•i!i•t•i!•i!•i!•i@•i!i•
i•i•i;i•ii•i•i•i!i•iit•it•iti!it•it!i!iti•i•i•ii;
i!i!ititi•i•iti•i!ii•ii•i•i•i!it!•i
il!iiii!i~i~i!i@i~ iiii~i~i~itiiii~i~i~i!iii~i~i~i!~i~i!iii~i~i~iii~i~i~!~i!ii!i!i~ii!ii~i~i!i~!i~i~!~
2.5 Boundary conditions: non-linear problems
In constructing a solution from a variational (weak) form, boundary conditions are classified into two categories: natural conditions that are satisfied by the variational special considerations; and essential conditions for which modifications form without special to the solution process must be made to make the variational form valid. For example, in an irreducible displacement method the traction boundary condition is a natural form condition is an an essential form and and must be imposed and the displacement boundary condition separately. separately. Simo et et ai. al. did did not not interpolate interpolate the the inertia inertia term term and, and, thus, thus, needed needed different different parameters parameters to to obtain obtain the the conservation conservation •* Simo property. property.
Boundary conditions: non-linear problems 29
2,5.1 Displacement (essential) condition The specification of a boundary condition for displacement is given by Eq. (1.7). In a finite element calculation the usual procedure to specify a displacement boundary condition merely assigns the value at a node as (~.la) i
-- Ui(Xa)
(2.34)
where (fia)i is the value at node a in the direction i, as shown for a two-dimensional case in Fig. 2.2. Here we note that the condition is imposed on the finite element approximation to the boundary Fuh and not the true boundary Pu. As the mesh is refined near the boundary the two converge, generally at a rate equal to or higher than errors from other approximations. Imposing a specified displacement condition may be implemented in several different ways. For example, consider the linear static problem given by
Igll K12] Ul < (u2}:{ h }
(2.35)
in which the condition Ul -- fil is to be imposed. 1. Impose the condition by replacing the first equation (associated with 6ui in a weak form) by the boundary condition giving
1 K0221{Ul} u2 _ (~1 which yields the desired solution. This method is not efficient if K is symmetric. However, by writing the system as
I0 K22 0 ] {Ul} { Ul } u2 = t " 2 - K z l u l
(2.36a)
the problem again becomes symmetric. 2. A second approach is to perform the modification as above and eliminate all equations for which values are known. Accordingly, we then have K22u2 = f2 - K21 ~1
Fu
[u=~]
Fig. 2.2 Boundary conditions for specified displacements.
(2.36b)
Fu
30
Galerkinmethod of approximation -irreducible and mixed forms together with the known condition/'/1 -- fi 1o This approach leads to a final set of equations with a minimum number of unknowns and is the one adopted in FEAPpv. 3. A third method uses a 'penalty' approach 2 in which the equations are given as kll K21
fk11 fi
K12] Ul K22J { u 2 } -
~
f2 1}
(2.36C)
in which kll -- c~K11 where c~ > > 1. This method is very easy to implement but requires selection of an appropriate value of c~. For simple point constraints, such as considered for u l = ill, a choice of c~ = 106 to 108 usually is adequate. When transient non-linear problems are encountered the imposition of displacement boundary conditions becomes slightly more involved. First, the boundary condition needs to be implemented on the incremental equations. This requires computation of initial values for the displacement, velocity and acceleration variables. As noted above there are two basic forms to consider: the explicit form and the implicit form.
Non-linear explicit problems
The explicit form is straightforward if velocity terms do not appear in the equilibrium equation. Here, for the GN22 algorithm,/32 = 0 and the value of the displacement at tn+l is obtained from Eq. (2.211) including those for the boundary Fu. Next we solve M an+l --- fn+l -- P(o'n+l)
(2.37)
for the new acceleration, with M given by a diagonal form. The velocity is then computed with the known acceleration using Eq. (2.212). By employing a diagonal M, there is no coupling of the acceleration between boundary and non-boundary nodes. If the velocity appears explicitly in the equilibrium equation an iterative strategy can be adopted where Vn+ 11 is taken as Vn and a 'trial' value of the acceleration is computed. Computing the velocity from the trial acceleration and performing one more iteration yields results which are adequate. Here it may be necessary to devise an expression for the boundary velocity updates to maintain high accuracy in final results.
Non-linear implicit problems
In an implicit form using the GN22 algorithm both /~1 and/32 are non-zero. If the time increment At is zero both the displacement and the velocity do not change [viz. Eq. (2.21)] and the new acceleration is determined from Eq. (2.37) - accounting only for any instantaneous change in fn+l. When At > 0, Eq. (2.24a) is used to impose the constraint Un+1 -- fin+l. In the first iteration we obtain du 1+1 = dUn+l -1 from Eqs (2.24bl) and (2.25a) such that Un+l e -- Un+l. This increment of the displacement boundary condition is employed in the incremental form
All AI2I A=I A22]
d~l } du2
"- {~12}
(2,38)
during the first iteration only. In the above the set Ul are associated with known displacements of boundary nodes and set u2 with the 'unknown' displacements. Any of the methods described above for the linear static problem may be used to obtain the solution.
Boundary conditions: non-linear problems 31
2.5.2 Traction condition The application of a traction is a 'natural' variational boundary condition and does not affect the active nodal displacements at a boundary- it only affects the applied nodal force condition. The imposition of a non-zero traction on the boundary requires an integration over the surface of each element. Thus for a typical node a as shown in Fig. 2.3 it is necessary to evaluate the integral
fa-- f e
(2.39) t
where e ranges over all elements belonging to 1-'t that include node a (e.g. for the two-dimensional case shown in Fig. 2.3 this is the element above and the element below node a). Of course if t is zero no evaluation of the integral is required.
Pressure loading
One important example is the application of a normal 'pressure' to a surface. Here the traction is given by
i - p,,n where Pn is the specified normal pressure (taken as positive when in tension) and n is a unit outward normal to the boundary rt, see Fig. 2.4. In this case Eq. (2.39) becomes
fa = Z fF Na [:?nIld r e
(2.40a)
t
Using Eq. (2.18) the computation of pressure loading is given by
fa -- ~ /O Na(~) Pn(~) (Vl X
(2.40b)
Y2) d F ]
e
where 121 = d~l d~2 and each element integral is performed on the natural coordinate system directly. For two-dimensional problems the surface shape functions are given h
rt
t=t]
Fig. 2.3 Boundaryconditions for specified traction.
Ft
fa
32
Galerkin method of approximation -irreducible and mixed forms
Fig. 2.4 Normal to surface.
by Na (~1),
1-"1--- d~ 1 and we use u ~
for plane strain for plane stress for axisymmetry
e3, h3 e3, 2"n X 1 e3,
where e3 is the unit normal vector to the plane of deformation.
2.5.3 Mixed displacement/traction condition The treatment of a mixed condition in which some displacement components are specified together with some traction components often requires a change in the nodal parameters. For example, a shaft with axis in the x3 direction and radius R that rotates inside a bearing (without friction or gaps) requires un = u r ( g ) -- 0 and to(R) = t z ( R ) = 0 (where the coordinate origin is placed at the centre of the shaft). In this case it is necessary to transform the degrees of freedom at each node on the boundary of the shaft such that
{(~l)a } [COS 0a Ua "-" (fi2)a "-" si 0 a
-
sin Oa
COS0 a 0
i]{ora} (blO)a
--
La ~la'
(2.41)
(~lz)a
This transformation is then applied to a residual as Ra, = LTRa
(2.42a)
and to the mass and stiffness as Ma'b' - L T M a b Lb; Ma,c -- LTMac; Mcb' ---- Mcb Lb Ka'b' -- L T K a b Lb; Ka,c -- L aT Kac; Kcb' -- Kcb Lb
(2.42b)
where a and b belong to transformed nodes and c to a node which retains its original orientation. It is usually convenient to perform these transformations on each individual element; however, if desired they can be applied to the assembled arrays. Once the transformation is performed, each individual displacement and traction condition may be imposed as described above.
Mixed or irreducible forms
2.6 Mixed or irreducible forms The evaluation of the stiffness given by Eq. (2.12) was cast entirely in terms of of the so-called displacement formulation which indeed is extensively used in many finite mixed finite eleelement solutions. However, on some occasions it is convenient to use mixedfinite ment forms and these are especially necessary when constraints such as (near) incompressibility arise. It has been frequently noted that certain constitutive laws, such as those of viscoelasticity and associative plasticity that we will discuss in Chapter 4, the material behaves in a nearly incompressible manner. For such problems a reformulation is necessary. On such occasions we have two choices of formulation. We can have the variables u and p (where p is the mean stress) as a two-fieldformulation two-field formulation or we can have the variables u, p and c:eov (where c:e~v is the volume change) as a three-field formulation (e.g. see reference 2 for more details). Here several alternatives are available and the matter of which we use may depend on the form of the constitutive equation employed. For situations where changes in volume affect only the pressure the two-field form can be easily used. However, for problems in which the response may become coupled between the deviatoric and mean components of stress and strain the three-field formulations lead to much simpler forms from which to develop a finite element model. To illustrate this point we present a general three-field mixed formulation and show in detail how such coupled effects can be easily included without any change change to the previous discussion on solving non-linear problems. The development also serves as a basis for the development of an extended form which permits the treatment of finite Chapter 5. deformation problems. This extension will be presented presented in Chapter
Deviatoric and components Deviatoric and mean mean stress and and strain components
The treatment of nearly incompressible materials is most easily considered by splitting the stress and strain into their deviatoric (isochoric) and mean parts. Accordingly, we define the mean stress (pressure) as P
--
1
5
1
[fill ~- 0"22 -~- 0"33] --- g O'ii
(2.43a)
and the deviator stress as (ad)ij - oij p (O'd)ij = -'- aij CTij -(~ijP
(2.43b)
(~ij is the Kronecker delta function where oij
1; (~ij --
0;
forii = j for 7~ j for i i=
Similarly, we define the mean strain (volume change) as /~v -- [ e l l -~- E'22 "~- 6"331 = eii
(2.44a)
and the deviator strain as (ed)ij
-- eij -- -~1 (~ij l?'v
(2.44b)
Note that the placement 1/3 factor appears in both, but at different locations in placement of the 1/3 the expressions.
33
34 Galerkinmethod of approximation -irreducible and mixed forms
A three-field mixed method for general constitutive models
In order to develop a mixed form for use with constitutive models in which mean and deviatoric effects can be coupled we define mean and deviatoric matrix operators given by 1 1 0 0 0] r and Id = I - - I gmm r , m-(2.45)
[1
respectively, where I is the identity matrix. The strains may now be expressed in a mixed form as e -- ld(SU)
1
+ 5 m
eo
(2.46a)
where the first term is the deviatoric part and the second the mean part. Similarly, the stresses may now be expressed in a mixed form as or = ld # + m p
(2.46b)
where 6" is the set of stresses deduced directly from the strains, incremental strains, or strain rates, depending on the particular constitutive model form. For the present we shall denote this stress by 6" = tr(e) (2.47) and we note that it is not necessary to split the model into mean and deviatoric parts. The weak form (variational Galerkin equations) for the case including transients is now given by
t
f~eo
[1mr*-
el dQ
0
(2.48)
f a 3 p [m r (Su) - eo] d a = 0 Introducing finite element approximations to the variables as u ~ fi = Nufi,
p ,~ p -
Npp
and
ev ~ ~v = Nolo
and similar approximations to virtual quantities as ~u ~ d;fi = Nu6fi,
6p ~ t~/~ -- Npt~p
and
&o ~ 3~v = No~o
the strain in an element becomes e = IdBfi+
1
~ mN~ ~
(2.49)
in which B is the standard strain--displacement matrix given in Eq. (2.9a). Similarly, the stresses in each element may be computed by using tr - Id t~ + m Np p where again ~ are stresses computed as in Eq. (2.47) in terms of the strains e.
(2.50)
Mixed or irreducible forms
Substituting the element stress and strain expressions from Eqs (2.49) and (2.50) into Eq. (2.48) we obtain the set of finite element equations oo
P+Mfi=f Pp - Kvp
p= 0
(2.51)
--K~Tg~ + Keu fi = 0
where f p - In Bror dr2' K~p
T T,, Pp - ~1 L Nom crdf2
Kpu = f~ NpTmTBdr2
f~ NrNp dr2,
(2.52)
t
If the pressure and volumetric strain approximations are taken locally in each element and N~ = Np it is possible to solve the second and third equation of (2.51) in each element individually. Noting that the array K~p is now symmetric positive definite, we may always write these as = K~Ipp ~ = K~plKeu~ = W~
(2.53)
The mixed strain in each element may now be computed as e-where
[ ; I E ldB+
mBo f i =
Id
~m
Bo fi
Bo = NoW
(2.54) (2.55)
defines a mixedform of the volumetric strain--displacement equations. From the above results it is possible to write the vector P in the alternative forms ~7'18 lnT T1J 6,dS2 = f~ [B T B T] [ l imT d] ~dfl P = f~ BTcrdfl - f~ [BTId+ ~n~m (2.56) Based on this result we observe that it is not necessary to compute the true mixed stress except when reporting final results. This is particularly important when we consider the effects of other material models in Chapter 4. The last step in the process is the computation of the tangent for the equations. This is straightforward using forms given by Eq. (2.47) where we obtain
d6"- OTdr Use of Eq. (2.54) to express the incremental mixed strains then gives Kr=
[B r
BT]
~r
Dr lid
89
(2.57a)
35
36
Galerkin method of approximation -irreducible and mixed forms
It should be noted that construction of a modified modulus term given by I)r -- / l m r L~ld
]
l)r[ld
[ Idi)rld
[ 89
89
3ldl)rm ]
[I)ll
I)12]
d 9 m r l ) r m j = [I)21 DEEJ
(2.57b)
requires very few operations because of the sparsity and form of the arrays Id and m. Consequently, the multiplication by the coefficient matrices B and B~ in this form is far more efficient than constructing a l] as 2 = IdB d- ~1 m B~
(2.58)
and computing the tangent from
KT -- ~ l~rl)rl~ d ~
(2.59)
In this form I~ has few zero terms which accounts for the difference in effort.
Example 2.1 Linear elastic tangent
As an example we consider a linear elastic material with the constitutive equation expressed as (2.60) o" = (K m m r + 2G I0) e in which
2 0 Io-~
1
0 0
0 0 0
0
0
0 0 0000
2
0
0
0 0 0
(2.61)
1 0 ! 0 1 0 0
accounts for the transformations used to define the strain. The incremental form is identical to Eq. (2.60) with o., e replaced by do', de. The above form for the mixed element is valid for use with many different linear and non-linear constitutive models. In Chapter 4 we consider stress-strain behaviour modelled by viscoelasticity, classical plasticity, and generalized plasticity formulations. Each of these forms can lead to situations in which a nearly incompressible response is required and for many examples included in this book we shall use the above mixed formulation. Here two basic forms of finite element approximations are considered: a four-node quadrilateral or an eight-node brick isoparametric element with constant interpolation in each element for one-term approximations to No and Np by unity; and a nine-node quadrilateral or a 27-node brick isoparametric element with linear interpolation for Np and No.* Accordingly, for the latter class of elements in two dimensions we use
Np--Nv---[1
~ r/]
or
[1
x
y]
* Formulationsusingthe eight-nodequadrilateraland 20-nodebrick serendipityelementsmay alsobe constructed; however, these elements do not fully satisfy the mixed patch test (see reference 2).
Non-linear quasi-harmonic field problems 37 and in three dimensions Np--N~=
[1
~ r/ ~]
or
[1
x
y
z]
The elements created by this process may be used to solve a wide range of problems in solid mechanics, as we shall illustrate in later chapters of this volume.
~~~[iii[i[iii!!~!~i[i~iii~i!i!i!iii!ii~i~i~i~i!ii!i!i[i!i}ii~i~i~i!i!i!i!ii~!~i~!~i!i!i~i!i!i!i!iii!i!~i!i~!~!iiiiii!i!i!i!i~!!ii~ii~i~i~iiiii!~iiiii!i!i!iii~!i~iii!ii~i!iiii~!i!~!~iii~i~i~i!i!iii!i~i~ii~i!i~iii~i!i~ii!
In subsequent chapters we shall touch upon non-linear problems in the context of inelastic constitutive equations for solids, plates, and shells and in geometric effects arising from finite deformation. Non-linear effects can also be considered for various fluid mechanics situations (e.g. see reference 19). However, non-linearity occurs in many other problems and in these the techniques described in this chapter are still universally applicable. An example of such situations is the quasi-harmonic equation which is encountered in many fields of engineering. Here we consider a simple quasiharmonic problem given by (e.g. heat conduction)
pc~b + V r q - a ( r
= 0
(2.62)
with suitable boundary conditions. Such a form may be used to solve problems ranging from temperature response in solids, seepage in porous media, magnetic effects in solids, and potential fluid flow. In the above, q is a flux and quite generally this can be written as q - q(r V r - - k(r V r 1 6 2 or, after linearization,
dq = - k~162- k l d ( V r
where kO =
(~qi 0r
and
klj--
Oqi Or
The source term Q(r also can introduce non-linearity. A discretization based on Galerkin procedures gives after integration by parts of the q term the problem
,H - j~ ~r p c ~pdf2 - J~ (V ~r r q df2 (2.63) -~6r162
6r q
and is still valid if q and/or Q (and indeed the boundary conditions) are dependent on r or its derivatives. Introducing the interpolations r
N~b(t)
and
~r
N~(b
(2.64)
a discretized form is given as ~I, - f((b) - CO - Pq ((b) - 0
(2.65a)
38
Galerkin method of approximation -irreducible and mixed forms where C-
~NrpcNdf~
pq __ _/,., (~rN)r q dr2
(2.65b)
q
Equation (2.65a) may be solved following similar procedures described above. For instance, just as we did with GN22, we can now use GN 11 as ~ ~n+l -- ~n + (1 - 0 ) ~ n .Ji-Off)n+ 1
(2.66)
Once again we have the choice of using ~b.+~ or ~n+~ as the primary solution variable. To this extent the process of solving transient problems follows the same lines as those described in the previous section and need not be further discussed here. We note again that the use of q~.+~ as the chosen variable will allow the solution method to be applied to static (steady state) problems in which the first term of Eq. (2.62) becomes zero.
iii~i!iii~iiiiiii~i~ii~iiii~ii~ii~iiii!i!iiiiiiiiiii~i~;i~ii~iii!i~i~!!i~iii~iii~i~!i!ii!!~J~!~ii!~!i!ii!ii~!i~i~i1iii~i~i~i~iiiiiiii~iii~iiii~i~i!ii~!i~iiii~i~!i~i!~:~!!iiiiiii!~!~i~i~i~i~i~i~!~!~:~!iiii~:~i!i~iii~i~i~ In this section we report results of some transient problems of structural mechanics as well as field problems. As we mentioned earlier, we usually will not consider transient behaviour in later parts of this book as the solution process for transients essentially follows the path described above.
Transient heat conduction
The governing equation for this set of physical problems is discussed in the previous section, with ~bbeing the temperature T now [Eq. (2.62)]. Non-linearity clearly can arise from the specific heat, c, thermal conductivity, k, and source, Q, being temperature dependent or from a radiation boundary condition aT k-- = -c~(TOn
To) n
(2.67)
with n # 1. Here c~ is a convective heat transfer coefficient and To is an ambient external temperature. We shall show two examples to illustrate the above. The first concerns the freezing of ground in which the latent heat of freezing is represented by varying the material properties with temperature in a narrow zone, as shown in Fig. 2.5. Further, in the transition from the fluid to the frozen state a variation in conductivity occurs. We now thus have a problem in which both matrices C and P [Eq. (2.65b)] are variable, and solution in Fig. 2.6 illustrates the progression of a freezing front which was derived by using the three-point (Lees) algorithm 2~ with C - Cn and P = Pn. A computational feature of some significance arises in this problem as values of the specific heat become very high in the transition zone and in time stepping can be missed
Typical examples of transient non-linear calculations !
:z: f5 CL
k(T)
pc(T)
!
---,
i
i
i
i i I
I
To
~-- 2AT
r~
Fig. 2.5 Estimation of thermophysical properties in phase change problems. The latent heat effect is approximated by a large capacity over a small temperature interval 2A T.
if the temperature step s t r a d d l e s the freezing point. To avoid this difficulty and keep the heat balance correct the concept of enthalpy is introduced, defining H =
f0Tpc d T
(2.68)
Now, whenever a change of temperature is considered, an appropriate value of pc is calculated that gives the correct change of H. The heat conduction problem involving phase change is of considerable importance in welding and casting technology. Some very useful finite element solutions of these T... 289K
~o,
,4
P~ss~ of frozen zone.
l
.,~ d I 1 :l .# /
gi
.4
.E,
=,2
,2 -6
(al
+
0
8
x(m)
Fig. 2.6 Freezing of a moist soil (sand).
20
8
~)
,(m)
10
12
14
39
40
Galerkin method of approximation -irreducible and mixed forms problems have been obtained. 22 Further elaboration of the procedure described above is given in reference. 23 The second non-linear example concerns the problem of spontaneous ignition. 24 We will discuss the steady state case of this problem in Chapter 4 and now will be concerned only with transient behaviour. Here the heat generated depends on the temperature Q = ~Ser
(2.69)
and the situation can become physically unstable with the computed temperature rising continuously to extreme values. In Fig. 2.7 we show a transient solution of a sphere at an initial temperature of T = 290 K immersed in a bath of 500 K. The solution is given for two values of the parameter ~ with k = pc = 1, and the non-linearity is now so severe that a full Newton iterative solution in each time increment is necessary. For the larger value of ~ the temperature increases to an 'infinite' value in afinite time and the time interval for the computation had to be changed continuously to account for this. The finite time for this point to be reached is known as the induction time and is shown in Fig. 2.7 for various values of ~. The question of changing the time interval during the computation has not been discussed in detail, but clearly this must be done quite frequently to avoid large changes of the unknown function which will result in inaccuracies.
Structural dynamics
Here the examples concern dynamic structural transients with material and geometric non-linearity. A highly non-linear geometrical and material non-linearity generally occurs. Neglecting damping forces, Eq. (2.11 a) can be explicitly solved in an efficient manner. If the explicit computation is pursued to the point when steady state conditions are approached, that is, until a = v ~ 0, the solution to a static non-linear problem is obtained. This type of technique is frequently efficient as an alternative to the methods described above and in Chapter 3 has been applied successfully in the context of finite differences under the name of 'dynamic relaxation' for the solution of non-linear static problems. 25 Two examples of explicit dynamic analysis will be given here. The first problem, illustrated in Plate 3, is a large three-dimensional problem and its solution was obtained with the use of an explicit dynamic scheme. In such a case implicit schemes would be totally inapplicable and indeed the explicit code provides a very efficient solution of the crash problem shown. It must, however, be recognized that such final solutions are not necessarily unique. As a second example Fig. 2.8 shows a typical crash analysis of a motor vehicle carried out by similar means.
Earthquake response of soil- structures
The interaction of the soil skeleton or matrix with the water contained in the pores is of extreme importance in earthquake engineering and here again solution of transient non-linear equations is necessary. As in the mixed problem which we referred to earlier, the variables include displacement, and the pore pressure in the fluid p.
Typical examples of transient non-linear calculations
10
~crit
8
1
2
(a) 700
a=2
700
600 v
10
20
40
8=16
600
t = 330, At = 160
L_
4
v
= 500
= 500
E
E
400
400
300
300
(b) Fig. 2.7 Reactive sphere. Transient temperature behaviour for ignition (~ -- 16) and non-ignition (~ --- 2) cases: (a)ind_uction time versus Frank-K_amenetskii parameter; temperature profiles; (b) temperature profiles for ignition (6 - 16) and non-ignition (6 - 2) transient behaviour of a reactive sphere.
We have in fact shown a comparison between some centrifuge results and computations elsewhere (viz. Chapter 18 of reference 2). These illustrate the development of the pore pressure arising from a particular form of the constitutive relation assumed. Many such examples and indeed the full theory are given in reference 26 and in Fig. 2.9 we show an example of comparison of calculations and a centrifuge model presented at a 1993 workshop known as VELACS. 27 This figure shows the displacements of a big retaining wall after the passage of an earthquake, which were measured in the centrifuge and also calculated.
41
42
Galerkin method of approximation - irreducible and mixed forms
Fig. 2.8 Crash analysis: (a) mesh at t == 0 ms; (b) mesh at t == 20 ms; (c) mesh at t == 40 ms.
Concluding Concluding remarks remarks 43
...
45.0 45.0
_...~ 5.0 5.0 .1 .1.
o0.5 . ~~li"'l·.~"'I~.t- =
17.o 17.0
't
....
~ CBI
ACCIO ACC,0
...
_ ~l ' _ _ACC12 L ~.t. ' ~ACC11 c" .i.ll. ~
~';m
PP~,6
PPT2 P"~T5 I • ACC9 PPT5
PPT3 PPT3
PPT2
~ac~,,,
....
,
= "~=1
backfill
lI!!IlI ACCS.-:l
...
z Z ,~x x
~
PPT4 PPT4 lI!!IlI
ACC3
r
it
"'r_:.1
20.0 -,,20:::·0'-
~, ~o,~
~5
lVDTl LVDT1 ~ -'=- T
I
/
model 5.0 water I wall w='m~176176 w=o, /
~~~ ~
~o 2.0
~ ~
;ACC2 ~;CCc, Acel
P
.~n 50
0.5 925.~ 0.35 0.35 0.5
I• . • ,0~o~ - .~=~
.0.20
V
1~1+
--.(------------- r-"
:0.35 • :'5' .., ............. 1~:~ ~.~o ,
Water saturated sand : Water saturate~sanOl i
Cambridge-test Cambridge-test
L~s 0.75 Water W ater
5.0
2.0 2.0
Jr
- - Initial boundary Initial boundary Boundary after after earthquate earthquate test test .- .-.-. -. .- - Boundary
0.5 0.5
Location 12.0, 7.0) = (25.0, (25.0, 12.0. 7.0) m m Location = LVDT1 LVDT1
~" 0.4 v:[0.4 r 'E r~ 0.3 0.3 E (9 Gl 0 co ~ 0.2 0.2
0.5 0.5
I Location -= (25.0, 12.0,7.0) 12.0, 7.0) m horizontal (LVDT1) horizontal
E 0.4 0.4 p
'E 8~ 0.3 0.3
L .~
|.~Gl o.e /
Q. a. r
'"0 .0.1 ~Ci 1 f 0
~ 0.2 ~. a. (n '" 0.1 F , ~ Ci ~o.1
ridge-test i
I
55
I
Time Time (s) (s)
10 10
15 15
00 I ~ 9 00
I 55
LVDT1 DT1 Prediction Prediction
Time (s)
I 10 10
15 15
Fig. 2.9 Retaining Retaining wall wall subjected subjected to earthquake earthquake excitation' excitation: comparison comparison of experiment experiment (centrifuge) (centrifuge) and 26 calculations. calculations.26 ~iii~iii~i~i~i i~i~i~i~%!~ii~%iiii~i~~i~ii!ii~!~%ii~ii~i~i~~i~i!ii~ii~ii~i~ii!i!i!~i~i~%i i~ii!!ii~ii~!ii~!iiiiiii~i:i~!i!iili~!i!i!i!ilii!~i!li!i!iiiiiiiiii!iil l ~ iiii!iliiiiiiiiiiliii~':i~:'iii!iiiiiiii!iiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiii!iiiii!i!ii!i!iiiiiiiiiiiiiiii!iiii!iiiiiiiiiiiiiiiiiiiiii iiiiiiiiii!iiiiiiiii!iiiiiiiiii!iiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiii!iiiii!iiiil iii!iiiiiiiiiiiiiiii!!iiiiii!ii!iiiiii!ii!il!il!iii!iliiiiiiiiiiiiiiiiiiii!iiiiiiii i!ii!i!iiiii!iiiii
2.9 Concluding remarks
!ii!ii!ii!ii!iii!!iii!iiiiiii!ii!iii! i!iiiii!ii!i !i!i!ii•i!ii!!i;i•i!i!!iii!i!!!!!!i!i!!!ii!i:i!i•i!i!•!!ii!!i!iii! !!ii!i!! i•!!!•!i ii!ii!iii !•iii!i iiiiiii,!!::,,~,~!! !iiiiiiiiiiiii!!i!iiii!i!iii!ili!!!ii!i!!!iiiiiii!i!iii!!!iiii !!!i!iiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiiii!!ii!i!iiiiiiiii! ii!!!!!!!i!ii!iiii iii!iii!ii!;!ii!i;iiil;;;i;i~ii~~i!iii!!iii!il!;!;i;;;;~ili;i~i;!i:!~ili~!ii~ii;ili!i!!!;!~i;;~i!i!i!;iii!i!i!i!i!iii!i!i!;!!!i~!; !!i;i~!i!~!~!;!~i~ii;~iii;iii ili; iiiiii~~i~i~i~i;i~i;i;;;;ii;!;;iiii;iiiiiiiiiiili;i;ilili;ii!;i;
this chapter chapter we we have have summarized summarized the the basic basic steps steps needed needed to solve solve a general general smallsmallIn this strain the quasi-harmonic quasi-harmonic field problem. problem. Only Only a strain solid solid mechanics mechanics problem problem as well well as the standard standard Newton Newton solution solution method method has has been been mentioned mentioned to to solve solve the the resulting resulting non-linear non-linear algebraic algebraic problem. problem. For For problems problems which which include include non-linear non-linear behaviour behaviour there there are are many many
44 Galerkin irreducible and mixed Galerkinmethod method of approximation approximation --irreducible mixed forms forms
situations where additional situations where additional solution solution strategies are required. required. In the next chapter chapter we problems. In will consider consider some some basic schemes schemes for solving solving such such non-linear non-linear algebraic algebraic problems. subsequent chapters chapters we shall address address some some of of these context of of particular subsequent these in the the context particular problems problems classes. classes. The reader will note except in the example example solutions, solutions, we have not discussed The reader note that, except not discussed problems solution problems in which which large strains occur. We can note here, however, that the solution described above remains The parts change are associated associated with strategy described remains valid. The parts that that change with the effects of of finite deformation deformation and and the manner manner in which computing which these affect the the computing of stresses, the stress-divergence stress-divergence term and the resulting tangent moduli of stresses, term and resulting tangent moduli and and stiffness. As these aspects involve more advanced we have have deferred treatment of As involve more advanced concepts concepts we deferred the treatment of finite strain problems problems to later later chapters where we we will address basic formulations formulations and chapters where address basic applications. applications.
References Vestn. equilibrium of rods and plates. Vestn. 1. B.G. Galerkin. Series solution of some problems in elastic equilibrium 19:897-908, 1915. Inzh. Tech., Tech., 19:897-908, Zhu. The Finite Element Method: Its Basis and Funda2. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. Butterworth-Heinemann, Oxford, 6th edition, 2005. mentals. Butterworth-Heinemann, T.J.R. Hughes. The Finite Element Method: Linear Static and Dynamic Analysis. Dover Publi3. TJ.R. cations, New York, York, 2000. W.K. Liu and B. Moran. Nonlinear Finite Elements for Continua and Structures. 4. T. Belytschko, W.K. John Wiley & & Sons, Chichester, 2000. 5. B. Cockburn, G.E. Karniadakis and Chi-Wang Shu. Theory, Shu. Discontinuous Galerkin Methods: Theory, Computation and Applications. Springer-Verlag, Berlin, 2000. 6. E.L. Wilson, R.L. Taylor, w.P. Doherty and J. Ghaboussi. Incompatible displacement models. Taylor, W.P. editor, Numerical and Computer Methods in Structural Mechanics, pages S.T. Fenves et al., editor, In ST. 43-57. Academic Press, New York, York, 1973. 7. G. Strang and GJ. G.J. Fix. An Analysis of of the Finite Element Method. Prentice-Hall, Englewood Cliffs, NJ, 1973. 8. J.C. Simo and M.S. Rifai. A class of mixed assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 29:1595-1638, 1990. Witt. Concepts andApplications and Applications ofFinite of Finite Element 9. R.D. Cook, D.S. D .S. Malkus, M.E. Plesha and R.J. RJ. Witt. Analysis. John Wiley & & Sons, New York, 4th edition, 2001. of Mathematical Functions. Dover PubliI.A. Stegun, editors. Handbook of 10. M. Abramowitz and LA. cations, New York, York, 1965. II. Frasier. Treatment of hour glass patterns in low order finite 11. D. Kosloff and G.A. Frasier. finite element codes. International Journal for Numerical Analysis Methods in Geomechanics, 2:57-72,1978. 2:57-72, 1978. Journalfor 12. N. Newmark. A method of computation for structural dynamics. J. Engineering Mechanics, ASCE, 85:67-94, 1959. 13. N. Bicanic Bi6ani6 and K.w. K.W. Johnson. Who was 'Raphson'? International Journal for Numerical 14:148-152, 1979. Methods in Engineering, 14:148-152, 14. J.C. Simo and N. Tarnow. Tarnow. The discrete energy-momentum method. Conserving algorithm for ftlr Mathematik und Physik, 43:757-793, 1992. nonlinear elastodynamics. Zeitschrift Zeitschriftfur J.C. Simo and N. Tarnow. Tarnow. Exact energy-momentum conserving algorithms and symplectic 15. lC. schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100:63-116, 100:63-116, 1992.
References 45 16. J.C. Simo and O. Gonz~ilez. Recent results on the numerical integration of infinite dimensional hamiltonian systems. In Recent Developments in Finite Element Analysis. CIMNE, Barcelona, Spain, 1994. 17. T.J.R. Hughes. Generalization of selective integration procedures to anisotropic and non-linear media. International Journal for Numerical Methods in Engineering, 15:1413-1418, 1980. 18. J.C. Simo and T.J.R. Hughes. On the variational foundations of assumed strain methods. J. Appl. Mech., 53(1):51-54, 1986. 19. O.C. Zienkiewicz, R.L. Taylor and P. Nithiarasu. The Finite Element Methodfor Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 20. M. Lees. A linear three level difference scheme for quasilinear parabolic equations. Maths. Comp., 20:516--622, 1966. 21. G. Comini, S. Del Guidice, R.W. Lewis and O.C. Zienkiewicz. Finite element solution of nonlinear conduction problems with special reference to phase change. International Journal for Numerical Methods in Engineering, 8:613-624, 1974. 22. H.D. Hibbitt and P.V. Marqal. Numerical thermo-mechanical model for the welding and subsequent loading of a fabricated structure. Computers and Structures, 3:1145-1174, 1973. 23. K. Morgan, R.W. Lewis and O.C. Zienkiewicz. An improved algorithm for heat convection problems with phase change. International Journal for Numerical Methods in Engineering, 12:1191-1195, 1978. 24. C.A. Anderson and O.C. Zienkiewicz. Spontaneous ignition: finite element solutions for steady and transient conditions. Trans ASME, J. Heat Transfer, pages 398-404, 1974. 25. J.R.H. Otter, E. Cassel and R.E. Hobbs. Dynamic relaxation. Proc. Inst. Civ. Eng., 35:633-656, 1966. 26. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, B.A. Schrefler and T. Shiomi. Computational Geomechanics: With Special Reference to Earthquake Engineering. John Wiley & Sons, Chichester, 1999. 27. K. Arulanandan and R.F. Scott, editors. Proceedings of VELACS Symposium, Rotterdam, 1993. Balkema.
iii iiiiii i iiiiiiiiiii i i i !i iii Solution of non-linear Solution of non-linear algebraic equations algebraic equations i!i!ii~i~i~~ii~iii i~i!~ii~i~i~iiiii~iii~ i~ii~i~'i'~i~ili!ii i!i~iiiii! iiii~!~i!i!ii!i!i!i ii~ii~ilili!i!i iiiiiiiiiiiiiiiiiiiii:':' 'iiii!iiiiiliiiiiiii!i!i!!!i iiiiiiiii~iiiiiiiiiiiiiiiiii~i~i~i i!iiiiii i~iiiiii iiliiiiiiiiiiii~iiii!i!il li iiiiiii iiiiii!iiiiiiiiiiiiil iii~i~iliiiili i!iii!i il iiiiiiiiiiiiiili!ililiiiii~iiiiiiiiiiiiiiiliiiiiil iiiiiiiii!ii i~iiif!ilii~iiiiiii i !!ii!iiiiii ili~iiiiiii! iiiiHiiii ~i~i!ii!iiiiiiiiiiiilii!iii~ii~~ii~ii~!iiliii iiiiiiii i ii!~ii~!i~!~i i ~iiiii~i!iiiliiiiii i~ili ii~iiiiii iiiii~i~i~i!i~i!iiii liiili~iiiiiiiii i iili~ii~ii~i!iiii!ii ii!iii~iiiiiiil iiii!iii~iiiiiiiil !i~~iii!~i~i i!iliiiiii!i !i~iiiiiiiii i ~!~!~i iiili!ili~iiiilil iiii ii i iii!ili!!iiiiiiiii iiiiilili!iii!~iiiiiliii ~iliiiiiiii i iiiiii i~i~i~iiii!il i!iliiiiiiiiiiil i~iiiiiiil i iiiiiiiiii ilili!i!iii~i!i~iiiiiiili i~ilHi !iiiiilii iliii
3.1 Introduction
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In the solution linear problems by a finitebyelement alwaysweneed to solve In the of solution of linear problems a finite method elementwe method always need to solve a set of simultaneous algebraic algebraic equationsequations of the form a set of simultaneous of the form Ku = f
Ku=f
(3.1)
(3.1)
Provided Provided the coefficient matrix is matrix non-singular the solution these equations is unique.is unique. the coefficient is non-singular thetosolution to these equations In the solution non-linear problemsproblems we will always obtain a set of algebraic equations;equations; of non-linear we will always obtain a set ofalgebraic In theof solution however, however, they generally will be non-linear. For example, in Chapterin2Chapter we obtained the they generally will be non-linear. For example, 2 we obtained the set (2.22)set at (2.22) each discrete tn+l.time Here, we Here, consider the genetic wewhich we at eachtime discrete tn +l. we consider theproblem generic which problem indicate as indicate as ~I/n+l "-- ~I/(Un+l) : fn+l -- =P(un+l) (3.2) W n +l = W(U f n + 1 - P(Un +l) (3.2) n +l) where u~+l is the set isofthe discretization parameters, f,,+l a vector is which independent where Un +l set of discretization parameters, f n +) which a vector is independent of the parameters and P a vector on the parameters. These equations may of the parameters and P dependent a vector dependent on the parameters. These equations may have multiple [i.e. more[i.e. thanmore one set Eq. (3.2)]. Thus, if a have solutions multiple solutions thanofu~+l one setmay OfUsatisfy +l may satisfy Eq. (3.2)]. Thus, if a n solution issolution achieved it may notitnecessarily be the solution is achieved may not necessarily be the sought. solutionPhysical sought. insight Physicalinto insight into the naturethe of nature the problem usually, incremental approaches from known of the and, problem and,small-step usually, small-step incremental approaches from known solutions solutions are essential to obtain to realistic Such increments are indeedarealways are essential obtain answers. realistic answers. Such increments indeed always required required if the problem is transient, if the constitutive law relating andstress strainand strain if the problem is transient, if the constitutive law stress relating is path dependent and/or if and/or the load-displacement path has path bifurcations or multiple is path dependent if the load-displacement has bifurcations or multiple branches branches at certainatload levels. certain load levels. The general from a nearby Theproblem general should problemalways shouldstarts always starts from asolution nearby at solution at u = u~, U=~ Un,= 0,
-- fn Wn f=0,
(3.3)
(3.3)
(3.4)
(3.4)
(3.5)
(3.5)
and oftenand arises from changes the forcing to often arises from in changes in thefunction forcing fn function f n to fn+l = fnf+ Afn+l n + 1 = fn
+ b.fn + 1
The determination of the change The determination of theAun+~ changesuch b.Uthat n +l such that Un+ 1 -- U n +
mUn+
1
will be thewill objective and generally the increments of Afn+l will ben +kept reasonably small be the objective and generally the increments of b.f 1 will be kept reasonably small so that path dependence can be followed. Further, such incremental procedures will so that path dependence can be followed. Further, such incremental procedures will
Iterative Iterative techniques techniques 47 47 f f~ PP
Softening range P=f
v
u
u
Fig. 3.1 Possibility Possibilityof multiple solutions.
be be useful in avoiding excessive numbers of of iterations iterations and in following the physically correct correct path. In Fig. 3.1 we show a typical non-uniqueness which may occur occur if the function ~I'n+l parameter u,+l un +! uniformly n +! decreases and subsequently increases as the parameter increases. It is clear that to follow the path ilf Af,+l n +! will have both positive and negative complete computation computation process. process. signs during a complete It is possible possible to obtain solutions in a single increment increment only in the case of mild nondependence), that is, with linearity (and no path dependence),
w
fn = 0,
Af,+l = f,+l
(3.6)
approaches and on particular The literature on general solution approaches particular applications is extenpossible to encompass fully all the variants which sive and, in a single chapter, it is not possible have been introduced. However, we shall attempt to give a comprehensive picture by outlining first the general general solution procedures. procedures. In later chapters we shall focus on procedures associated with rate-independent material non-linearity (plasticity), rate-dependent material non-linearity (creep and viscoplasticity), some non-linear field problems, large displacements and other special examples.
3.2 Iterative techniques 3.2.1 3.2.1 General General remarks remarks (3.2)-(3.5) cannot be approached directly The solution of the problem posed by Eqs (3.2)-(3.5) and some form of iteration will always be required. We shall concentrate here on procedures in which repeated solution of linear equations (i.e. iteration) of the form id ii ii i K K d Uunn++ 11 = --- rr nn ++ 11
(3.7) (3.7)
in which a superscript i indicates the iteration number. In these a solution increment i 1 is computed. Direct (Gaussian) elimination techniques or iterative methods can dun+ dU~+1
48
Solutionof non-linear algebraic equations be used to solve the linear equations associated with each iteration. 1-3 However, the application of an iterative solution method may prove to be more economical, and in later chapters we shall frequently refer to such possibilities although they have not been fully explored. Many of the iterative techniques currently used to solve non-linear problems originated by intuitive application of physical reasoning. However, each of such techniques has a direct association with methods in numerical analysis, and in what follows we shall use the nomenclature generally accepted in texts on this subject. 2'4-7 Although we state each algorithm for a set of non-linear algebraic equations, we shall illustrate each procedure by using a single scalar equation. This, though useful from a pedagogical viewpoint, is dangerous as convergence of problems with numerous degrees of freedom may depart from the simple pattern in a single equation.
3.2.2 Newton's method Newton's method is the most rapidly convergent process for solutions of problems in which only one evaluation of 9 is made in each iteration. Of course, this assumes that the initial solution is within the zone of attraction and, thus, divergence does not occur. Indeed, Newton's method is the only process described here in which the asymptotic rate of convergence is quadratic. The method is often called the Newton-Raphson method as it appears to have been simultaneously derived by Newton and Raphson, and an interesting history of its origins is given in reference 8. In this iterative method we note that, to the first order, Eq. (3.2) can be approximated as
(__..~) i
duin+l -- 0
ltX/~,Un+l ) ( . i+1,~ ~,~ ~ii(Uin+l) +
(3.8)
n+l
Here the iteration counter i usually starts by assuming 1
(3.9)
Un+ 1 -- a n
in which a n is a converged solution at a previous load level or time step. The jacobian matrix (or in structural terms the stiffness matrix) corresponding to a tangent direction is given by OP OffJ KT = ~ -Ou (3.10) Equation (3.8) gives immediately the iterative correction as i
i
KTdUn+ 1 -
i
~IIn+ 1
or
dUin+ 1 __ ( K ~ ) -1 ~IIn+ i 1
(3.11)
A series of successive approximations gives di+l i n+l -- Un+l -~-
i
dUn+l
i = U n q- A n n + 1
(3.12)
Iterative techniques
T
P
fn,1
i
. . . . . . . . . . . . . . . . . . lit1+ I
t.
_
~ Q I I I ! ! ! ! !
i !
'
'
:
',
i i
! i
"'
,'!
.~ ,
1
,
l! ,'L.
i
I i
AU ~
! i I
,.
I II
i ! I
4.,
,L,d§
v
u
Fig. 3.2 Newton's method.
where
i i AUn+ 1 = y~
dUnk+l
(3.13)
k=l
The process is illustrated in Fig. 3.2 and shows the very rapid convergence that can be achieved. i 1 is perhaps not obvious The need for the introduction of the total increment AUn+ here but in fact it is essential if the solution process is path dependent, as we shall see in Chapter 4 for some non-linear constitutive equations of solids. The Newton process, despite its rapid convergence, has some negative features: 1. a new KT matrix has to be computed at each iteration; 2. if direct solution for Eq. (3.11) is used the matrix needs to be factored at each iteration; 3. on some occasions the tangent matrix is symmetric at a solution state but unsymmetric otherwise (e.g. in some schemes for integrating large rotation parameters 9 or non-associated plasticity). In these cases an unsymmetric solver is needed in general. Some of these drawbacks are absent in alternative procedures, although generally then a quadratic asymptotic rate of convergence is lost.
3.2.3 Modified Newton's method This method uses essentially the same algorithm as the Newton process but replaces the variable jacobian matrix K~ by a constant approximation
K~ ~ giving in place of Eq. (3.11),
I~ T
i i dUn+l -- KT 1ffJ,+l
(3.14) (3.15)
49
50
Solution of non-linear algebraic equations f u
'n.,--'T-ill-
2 .......
|
....
-
A~
', ....
"i ,"
~
AUn~
_
ur,
P
: '
~1
J,
Ill
v
un~.~ Un2+, u3+~
U
(a) fb u
1
Q
[n+l
Atn
14/1+1
i~t11 ~-i i
1~/12', ; rkl3
AUn'~i i i
! !
|
i
i
i
',
Un
,11 , ! Un + 1 U2 + 1
.....
U
(b)
Fig. 3.3 The modified Newton method: (a) with initial tangent in increment; (b) with initial problem tangent. Many possible choices exist here. For instance, KT can be chosen as the matrix corresponding to the first iteration K~ [as shown in Fig. 3.3(a)] or may even be one corresponding to some previous time step or load increment K ~ [as shown in Fig. 3.3(b)]. In the context of solving problems in solid mechanics the method is also known as the stress transfer or initial stress method. Alternatively, the approximation can be chosen every few iterations as I~T = K~ where j < i. Obviously, the procedure generally will converge at a slower rate (generally a norm of the residual @ has linear asymptotic convergence instead of the quadratic one in the full Newton method) but some of the difficulties mentioned above for the Newton process disappear. However, some new difficulties can also arise as this method fails to converge when the tangent used has opposite 'slope' to the one at the current solution (e.g. as shown by regions with different slopes in Fig. 3.1). Frequently the 'zone of
Iterative techniques 51 attraction' for the modified process is increased and previously divergent approaches can be made to converge, albeit slowly. Many variants of this process can be used and symmetric solvers often can be employed when a symmetric form of KT is chosen.
3.2.4 Incremental-secant or quasi-Newton methods Once the first iteration of the preceding section has been established giving ] 1 dUn+ 1 --" K~ -1~I/n+ 1
(3 16)
a secant 'slope' can be found, as shown in Fig. 3.4, such that
dunl+l = (K2) -1 (~I/l+l- tI/n2+l )
(3.17)
This 'slope' can now be used to establish u,2 by using
2 -1 dUn+, 2 1 -- (Ks It/n+
(3.18)
Quite generally, one could write in place of Eq. (3.18) for i > 1, now dropping subscripts, d u i - - (K~) -1 ~I/i (3.19) where (K~) -1 is determined so that dui-1
__ (K~) -1 (~i/i-1-
ii/i)__
(K~) -I ,,y/-1
(3.20)
For the scalar system illustrated in Fig. 3.4 the determination of K~ is trivial and, as shown, the convergence is much more rapid than in the modified Newton process (generally a super-linear asymptotic convergence rate is achieved for a norm of the residual).
1 Af n
.
.
K~
~ln. 1
1, l .
.
.
~,,
' !
.
.
.
i-"
i ! ! .
,,,..-i-.~
' AUn1 ! ~
i ! .
L__
OUn i
! ~
,r,
i ! !
i
!
!
!
V
i
Un
Fig. 3.4 The secant method starting from a K~ prediction.
v
U
52 Solution of non-linear algebraic equations For systems with more than one degree of freedom the determination of K~ or its inverse is more difficult and is not unique. Many different forms of the matrix K~ can satisfy relation (3.1) and, as expected, many alternatives are used in practice. All of these use some form of updating of a previously determined matrix or of its inverse in a manner that satisfies identically Eq. (3.20). Some such updates preserve the matrix symmetry whereas others do not. Any of the methods which begin with a symmetric tangent can avoid the difficulty of non-symmetric matrix forms that arise in the Newton process and yet achieve a faster convergence than is possible in the modified Newton procedures. Such secant update methods appear to stem from ideas introduced first by Davidonl~ and developed later by others. Dennis and More 11 survey the field extensively, while Matthies and Strang 12 appear to be the first to use the procedures in the finite element context. Further work and assessment of the performance of various update procedures is available in references 13-16. The BFGS update 11 (named after Broyden, Fletcher, Goldfarb and Shanno) and the DFP update 11 (Davidon, Fletcher and Powell) preserve matrix symmetry and positive definiteness and both are widely used. We summarize below a step of the BFGS update for the inverse, which can be written as
(Ki) -1 = (I + wiu T) (Ki-I) -1 (I + SriwT)
(3.21)
where I is an identity matrix and
u
(dui-1)T"Y i-1 ] ffffi-
[1-- d (ui ) T lt~i-1 1
w i : du(i_l)T,.yi_ 1
l _
~i
(3.22)
d u i- 1
where "7 is defined by Eq. (3.20). Some algebra will readily verify that substitution of Eqs (3.21) and (3.22) into Eq. (3.20) results in an identity. Further, the form of Eq. (3.21) guarantees preservation of the symmetry of the original matrix. The nature of the update does not preserve any sparsity in the original matrix. For this reason it is convenient at every iteration to return to the original (sparse) matrix K 1, used in the first iteration and to reapply the multiplication of Eq. (3.21) through all previous iterations. This gives the algorithm in the form
i
T ~i/i
hi = 1-I (I + v i i i )
j=2
b2 = (K~)-' bl
i-2 dui : I X j=o
(3.23)
(I + Wi_jvTi_j)bE
This necessitates the storage of the vectors v j and w j for all previous iterations and their successive multiplications. Further details on the operations are described well in reference 12.
Iterative techniques
~n+l --
.............
,,,.
,
:,,
3
4
v,_. ~.v,.. ,
~
~
1,.,' ivL,
,+,1+,
17/i
i i i i i i
I Un
U
Fig. 3.5 Direct (or Picard)iteration. When the number of iterations is large (i > 15) the efficiency of the update decreases as a result of incipient instability. Various procedures are open at this stage, the most effective being the recomputation and factorization of a tangent matrix at the current solution estimate and restarting the process. Another possibility is to disregard all the previous updates and return to the original matrix K 1. Such a procedure was first suggested by Crisfield 13'17'18in the finite element context and is illustrated in Fig. 3.5. It is seen to be convergent at a slightly slower rate but avoids totally the stability difficulties previously encountered and reduces the storage and number of operations needed. Obviously any of the secant update methods can be used here. The procedure of Fig. 3.5 is identical to that generally known as direct (or Picard) iteration 4 and is particularly useful in the solution of non-linear problems which can be written as 9 (u) -- f - K(u)u = 0 (3.24) 1 In such a case un+ 1 -- un is taken and the iteration proceeds as
= [I,:min+,)] -' fn+,
(3.25)
3.2.5 Line search procedures- acceleration of convergence All the iterative methods of the preceding section have an identical structure described by Eqs (3.11)-(3.13) in which various approximations to the Newton matrix K~ are used. For all of these an iterative vector is determined and the new value of the unknowns found as iJti+l i i n+l - - U n + l "1- d U n + l
(3.26)
53
54
Solution of non-linear algebraic equations starting from 1
Un+l~Un
in which u, is the known (converged) solution at the previous time step or load level. The objective is to achieve the reduction of oi+~ =,+1 to zero, although this is not always easily achieved by any of the procedures described even in the scalar example illustrated. To get a solution approximately satisfying such a scalar non-linear problem would have been in fact easier by simply evaluating the scalar dfiri+l :": n+l for various values of U,+l and by suitable interpolation arriving at the required answer. For multi-degree-of-freedom systems such an approach is obviously not possible unless some scalar norm of the residual is considered. One possible approach is to write ui+l,j n+l
i i - - U n + l "~- T]i,jdUn+l
(3.27)
and determine the step size rli,j so that a projection of the residual on the search direction i 1 is made zero. We could define this projection as du,+
Gi,j ~ (duin+l) Tltl;i+l'j :=n+l where
l.~i + l,j n+l
~ ~I/ (Uin+l +
(3.28)
7]i,0
Tli,jdUin),
=
1
Here, of course, other norms of the residual could be used. This process is known as a line search, and ~Ti.jcan conveniently be obtained by using a regulafalsi (or secant) procedure as illustrated in Fig. 3.6. An obvious disadvantage of a line search is the need for several evaluations of ~ . However, the acceleration of the overall convergence can be remarkable when applied to modified or quasi-Newton methods. Indeed, line search is also useful in the full Newton method by making the radius of attraction larger. A compromise frequently used ]2 is to undertake the search only if + l, J Gi, 0 > 8 (gain+l) T ~jri =,,+] (3.29) where the tolerance e is set between 0.5 and 0.8. This means that if the iteration process directly resulted in a reduction of the residual to e or less of its original value a line search is not used.
GI
~
G,, L.,
E I"
(a)
x
~1
rli, 1 qi.2
"1
~ "1
I~
rli, 1 qi.2
(b)
Fig. 3.6 Regula falsi applied to line search: (a) extrapolation; (b) interpolation.
._1 "1
Iterative techniques 55
3.2.6 'Softening' behaviour and displacement control In applying the preceding to load control problems we have implicitly assumed that the iteration is associated with positive increments of the forcing vector, f, in Eq. (3.4). In some structural problems this is a set of loads that can be assumed to be proportional to each other, so that one can write Afn+l
-- A / ~ n + l f 0
(3.30)
In many problems the situation will arise that no solution exists above a certain maximum value of f and that the real solution is a 'softening' branch, as shown in Fig. 3.1. In such cases AAn+I will need to be negative unless the problem can be recast as one in which the forcing can be applied by displacement control. In a simple case of a single load it is easy to recast the general formulation to increments of a single prescribed displacement and much effort has gone into such solutions. 13'19-25 In all the successful approaches of incrementation of AAn+I the original problem of Eq. (3.2) is rewritten as the solution of ~I~n+l ~ ~ n + l f 0 -- P(u~+l) = 0
with U.+l = u. + A u . + l
(3.31)
and '~n+l "-" '~n + A / ~ n + l
being included as variables in any increment. Now an additional equation (constraint) needs tobe provided to solve for the extra variable AAn+l. This additional equation can take various forms. Riks 2~ assumes that in each increment AuT+IAun+I + AA2f~f0- A/2 (3.32) where A l i s a prescribed 'length' in the space of n + 1 dimensions. Crisfield 13'26 provides a more natural control on displacements, requiting that T 1Au,,+l = AI 2 Au~+ (3.33) These so-called arc-length and spherical path controls are but some of the possible constraints. Direct addition of the constraint Eq. (3.32) or (3.33) to the system of Eqs (3.31) is now possible and the previously described iterative methods could again be used. However, the 'tangent' equation system would always lose its symmetry so an alternative procedure is generally used. We note that for a given iteration i we can write quite generally the solution as
lt~in+1
i
i
--- )~n+lf0 __ P ( U n + l )
~I/i+l i n+l ~ ~Itn+l +
i If 0 dan+
- K Ti d U ni + l
(3.34)
The solution increment for u may now be given as dUn+l i f0] i __ (K~) -1 [~I/in+l ..1_ d,~n+l i ,,i i ,-i d U n + 1 -- d U n + 1 q- dAn+ldun+ 1
(3.35)
56
Solution of non-linear algebraic equations where dUn+ -'i 1 .__ (K~) -1 itI/ni + 1 ,,i dUn+,-
(3.36)
f0
Now an additional equation is cast using the constraint. Thus, for instance, with Eq. (3.3 3) we have (AUin~ll -~-
duin+l) T (AUin+ll
duin+l) =
+
AI 2
(3.37)
where Au~ll is defined by Eq. (3.13). On substitution of Eq. (3.35) into Eq. (3.37) a i (which quadratic equation is available for the solution of the remaining unknown dA,+l may well turn out to be negative). Additional details may be found in references 13 and 26. A procedure suggested by Bergan 22'25is somewhat different from those just described. Here a fixed load increment AAn+I is first assumed and any of the previously introi 1. Now a new duced iterative procedures are used for calculating the increment dun+ increment AAn+1 is calculated so that it minimizes a norm of the residual *
(AAn+lf0-
pi+l
T
pi+l)]
I n+l)(AAn+lf0
-- I n+l
__ At2
(3.38)
The result is thus computed from dAl 2 dAA*n+l
=0
and yields the solution AA~+I =
f0rv~+l " n+l fTfo
(3.39)
This quantity may again well be negative, requiting a load decrease, and it indeed results in a rapid residual reduction in all cases, but precise control of displacement magnitudes becomes more difficult. The interpretation of the Bergan method in a one-dimensional example, shown in Fig. 3.7, is illuminating. Here it gives the exact a n s w e r s - with a displacement control, the magnitude of which is determined by the initial AAn+~ assumed to be the slope KT used in the first iteration.
3.2.7 Convergence criteria In all the iterative processes described the numerical solution is only approximately achieved and some tolerance limits have to be set to terminate the iteration. Since finite precision arithmetic is used in all computer calculations, one can never achieve a better solution than the round-off limit of the calculations. Frequently, the criteria used involve a norm of the displacement parameter changes i [[dUn+l II or, more logically, that of the residuals II~in+~II. In the latter case the limit can often be expressed as some tolerance of the norm of forces Ilfn+~II- Thus, we may require that i II'I'n+~ II 5_ ellfn+l II (3.40)
Iterative techniques 57
| !
T
I//" IA :, i ! i i I i i i i
i
!
"
r
U
Fig. 3.7 One-dimensionalinterpretation of the Bergan procedure. where e is chosen as a small number, and
II'I'll = ('I'T~I ') 1/2
(3.41)
Other alternatives exist for choosing the comparison norm, and another option is to use the residual of the first iteration as a basis. Thus, [l~/n+l II
_< ell~I'n~+~II
(3.42)
The error due to the incomplete solution of the discrete non-linear equations is of course additive to the error of the discretization that we frequently measure in the energy n o r m . 27 It is possible therefore to use the same norm for bounding of the iteration process. We could, as a third option, require that the error in the energy norm satisfy
d E i _ (dUn,+T1~in+ 1)1/2 __< e (dul~l li/n+l ) 1 1/2
(3.43)
< edE 1 In each of the above forms, problem types exist where the fight-hand-side norm is zero. Thus a fourth form, which is quite general, is to compute the norm of the element residuals. If the problem residual is obtained as a sum over elements as
kX/n+1 = ~
~3e+l
(3.44)
e
where e denotes an individual element and ~3 e the residual from each element, we can express the convergence criterion as
IIlI/n+l i
e II II _< ell~n+l
(3.45)
where
[[~)e+l ]]- ~
11(~3e+l) i II e
(3.46)
58 Solutionof non-linear algebraic equations Once a criterion is selected the problem still remains to choose an appropriate value for e. In cases where a full Newton scheme is used (and thus asymptotic quadratic convergence should occur) the tolerance may be chosen at half the machine precision. Thus if the precision of calculations is about 16 digits one may choose e = 10 -8 since quadratic convergence assures that the next residual (in the absence of round-off) would achieve full precision. For modified or quasi-Newton schemes such asymptotic rates are not assured, necessitating more iterations to achieve high precision. In these cases it is common practice by some to use much larger tolerance values (say 0.01 to 0.001). However, for problems where large numbers of steps are taken, instability in the solution may occur if the convergence tolerance is too large. We recommend therefore that whenever practical a tolerance of half machine precision be used.
ii!i!iii!~iiii!iii!!!i!iiii ~':~'~i~iiiiiiiiiii! ~ ~i~i{i~i!~i!i~!~i!i!~i~!~!!~i~ii~i~i~ii~i~i~i!~!~i~ii!i!~i~!~i~i~i~ii!i~!i !i!i!~ii~i!!~ii!ii~!~i!~i~i~!i!i!~i!i~i!~{ii~i~i!~!i!i~i~!i~i!~!!ii~i!i!i~ii!!ii~iii~{!!~iii~ii!~!iiiiii!i~ii~{~!!i~i!~!i!~i~{!i!i!i!~i~!iiii~!iii~ii!~i!ii!~!iiii!iii~i!!ii!!i!ii!!iiiii~ii~ii~iii!~!i!i~!~i!ii~iii~i!~iii{~iii~iii~i~i~i!~!i!iiiiii~i~iii~:~!i!~!iiiiii!i~i!i!~!~ ~i~i~!!i!iiii~i!ii~!ii~ii ~iiiiiiiii~ii!~iiiiiiiii~!iiiii~i!~i!ii~i~!~!~!i!i~i!iii{iiiiiiii!i{!!i!iiiii~iiiiii!i{!iii!ii{!~ii!~iii!~iii~ii!iiii~i~iiii~:~i~!!i!!ii!~ii~!~!i~i!i~!!:!iiiii!iiiii ~ii~ii~i~i!i~ii~i~i~i~i!ii!~!~i~i~i!~iiii~i~ii~!~!i!i~i~i~i~i!~!i{~i~ii~i~ iiiiiiiiili!ii~iliiiiiiiiiiiii! .:~iiiiiili!-:iiiiii!iii!~iii~:~i ,- : i ~ : .,~,!, ~ iiiili :~....... i >. i .... .~i .::iiii!~!iii i :i: ...... i: , :i' : :~ ,~ :i iiiii~.. :i : !ilii ,~:. ~ ,:: i!iii i" :~ ~, :i........ i: i .:~:ii!!iiiiiii!ii!ii!iiiii~ii!iliiiiiiiiiiiiiiii~ili
The various iterative methods described provide an essential tool kit for the solution of non-linear problems in which finite element discretization has been used. The precise choice of the optimal methodology is problem dependent and although many comparative solution cost studies have been published 12'17'28the differences are often marginal. There is little doubt, however, that exact Newton processes (with line search) should be used when convergence is difficult to achieve. Also the advantage of symmetric update matrices in the quasi-Newton procedures frequently make these a very economical candidate. When non-symmetric tangent moduli exist it may be better to consider one of the non-symmetric updates, for example a Broyden method. 13'29 We have not discussed in the preceding direct iterative methods such as the various conjugate direction methods 3~ or dynamic relaxation methods in which an explicit dynamic transient analysis (see Chapter 2) is carried out to achieve a steady-state solution. 35'36 These forms are often characterized by" 1. a diagonal or very sparse form of the matrix used in computing trial increments du (and hence very low cost of an iteration) and 2. a significant number of total iterations and hence evaluations of the residual ~I,. These opposing trends imply that such methods offer the potential to solve large problems efficiently. However, to date such general solution procedures are effective only in certain problems. 37 One final remark concerns the size of increments Af or AA to be adopted. First, it is clear that small increments reduce the total number of iterations required per computational step, and in many applications automatic guidance on the size of the increment to preserve a (nearly) constant number of iterations is needed. Here such processes as the use of the 'current stiffness parameter' introduced by Bergan 22 can be effective. Second, if the behaviour is path dependent (e.g. as in plasticity-type constitutive laws) the use of small increments is desirable to preserve accuracy in solution changes. In this context, we have already emphasized the need for calculating such changes by i always using the accumulated AUn+ 1 change and not in adding changes arising from i each iterative dUn+1 step in an increment.
General remarks-incremental and rate methods 59
Third, if only a single Newton iteration is used in each increment of A)~ then the procedure is equivalent to the solution of a standard rate problem incrementally by direct forward integration. Here we note that if Eq. (3.2) is rewritten as P(u) = )~f0
(3.47)
we can, on differentiation with respect to )~, obtain dP du
= f0
(3.48)
= KT~f0
(3.49)
du d/k
and write this as du d--~
Incrementally, this may be written in an explicit form by using a Euler method as mUn_t_ 1 -- m / ~ K T l f 0
(3.50)
This direct integration is illustrated in Fig. 3.8 and can frequently be divergent as well as being only conditionally stable as a result of the Euler explicit method used. Obviously, other methods can be used to improve accuracy and stability. These include Euler implicit schemes and Runge-Kutta procedures.
t(;r
Possible divergence
\
sw
r
U
Fig. 3.8 Direct integration procedure.
60
Solution of non-linear algebraic equations
1. G. Strang. Linear Algebra and its Application. Academic Press, New York, 1976. 2. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997. 3. R.M. Ferencz and T.J.R. Hughes. Iterative finite element solutions in nonlinear solid mechanics. In RG. Ciarlet and J.L. Lions, editors, Handbook of Numerical Analysis, volume III, pages 3-178. Elsevier Science Publisher BV, 1999. 4. A. Ralston. A First Course in Numerical Analysis. McGraw-Hill, New York, 1965. 5. L. Collatz. The Numerical Treatment of Differential Equations. Springer, Berlin, 1966. 6. G. Dahlquist and/~. Bj6rck. Numerical Methods. Prentice-Hall, Englewood Cliffs, NJ, 1974. Reprinted by Dover, New York, 2003. 7. H.R. Schwarz. Numerical Analysis. John Wiley & Sons, Chichester, 1989. 8. N. Bidanid and K.W. Johnson. Who was 'Raphson'? International Journal for Numerical Methods in Engineering, 14:148-152, 1979. 9. J.C. Simo and L. Vu-Quoc. A three-dimensional finite strain rod model. Part II: Geometric and computational aspects. Computer Methods in Applied Mechanics and Engineering, 58:79-116, 1986. 10. W.C. Davidon. Variable metric method for minimization. Technical Report ANL-5990, Argonne National Laboratory, 1959. 11. J.E. Dennis and J. More. Quasi-Newton methods - motivation and theory. SIAM Rev., 19:46-89, 1977. 12. H. Matthies and G. Strang. The solution of nonlinear finite element equations. International Journal for Numerical Methods in Engineering, 14:1613-1626, 1979. 13. M.A. Crisfield. Non-linear Finite Element Analysis of Solids and Structures, volume 1. John Wiley & Sons, Chichester, 1991. 14. M.A. Crisfield. Non-linear Finite Element Analysis of Solids and Structures, volume 2. John Wiley & Sons, Chichester, 1997. 15. K.-J. Bathe and A.R Cimento. Some practical procedures for the solution of nonlinear finite element equations, cmame, 22:59-85, 1980. 16. M. Geradin, S. Idelsohn and M. Hogge. Computational strategies for the solution of large nonlinear problems via quasi-Newton methods. Computers and Structures, 13:73-81, 1981. 17. M.A. Crisfield. Finite element analysis for combined material and geometric nonlinearity. In W. Wunderlich, E. Stein and K.-J. Bathe, editors, Nonlinear Finite Element Analysis in Structural Mechanics. Springer-Verlag, Berlin, 1981. 18. M.A. Crisfield. A fast incremental/iterative solution procedure that handles 'snap through'. Computers and Structures, 13:55-62, 1981. 19. T.H.H. Pian and R Tong. Variational formulation of finite displacement analysis. In Syrup. on High Speed Electronic Computation of Structures, Liege, 1970. 20. O.C. Zienkiewicz. Incremental displacement in non-linear analysis. International Journal for Numerical Methods in Engineering, 3:587-592, 1971. 21. E. Riks. An incremental approach to the solution of snapping and buckling problems. International Journal of Solids and Structures, 15:529-551, 1979. 22. RG. Bergan. Solution algorithms for nonlinear structural problems. In Int. Conf. on Engineering Applications of the Finite Element Method, pages 13.1-13.39. Computas, 1979. 23. J.L. Batoz and G. Dhatt. Incremental displacement algorithms for nonlinear problems. International Journal for Numerical Methods in Engineering, 14:1261-1266, 1979. 24. E. Ramm. Strategies for tracing nonlinear response near limit points. In W. Wunderlich, E. Stein and K.-J. Bathe, editors, Nonlinear Finite Element Analysis in Structural Mechanics, pages 63-89. Springer-Verlag, Berlin, 1981.
References 61 25. E Bergan. Solution by iteration in displacement and load spaces. In W. Wundeflich, E. Stein and K.-J. Bathe, editors, Nonlinear Finite Element Analysis in Structural Mechanics. SpringerVerlag, Berlin, 1981. 26. M.A. Crisfield. Incremental/iterative solution procedures for nonlinear structural analysis. In C. Taylor, E. Hinton, D.R.J. Owen and E. Ofiate, editors, Numerical Methods for Nonlinear Problems. Pineridge Press, Swansea, 1980. 27. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 28. A. Pica and E. Hinton. The quasi-Newton BFGS method in the large deflection analysis of plates. In C. Taylor, E. Hinton, D.R.J. Owen and E. Ofiate, editors, Numerical Methods for Nonlinear Problems. Pineridge Press, Swansea, 1980. 29. C.G. Broyden. Quasi-Newton methods and their application to function minimization. Math. Comp., 21:368-381, 1967. 30. M. Hestenes and E. Stiefel. Method of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand., 49:409-436, 1954. 31. R. Fletcher and C.M. Reeves. Function minimization by conjugate gradients. The Computer Journal, 7:149-154, 1964. 32. E. Polak. Computational Methods in Optimization. A Unified Approach. Academic Press, London, 1971. 33. B.M. Irons and A.E Elsawaf. The conjugate Newton algorithm for solving finite element equations. In K.-J. Bathe, J.T. Oden and W. Wunderlich, editors, Proc. U.S.-German Symp. on Formulations and Algorithms in Finite Element Analysis, pages 656-672, Cambridge, Mass., 1977. MIT Press. 34. M. Papadrakakis and P. Ghionis. Conjugate gradient algorithms in nonlinear structural analysis problems. Computer Methods in Applied Mechanics and Engineering, 59:11-27, 1986. 35. J.R.H. Otter, E. Cassel and R.E. Hobbs. Dynamic relaxation. Proc. Inst. Civ. Eng., 35:633-656, 1966. 36. O.C. Zienkiewicz and R. L6hner. Accelerated relaxation or direct solution? Future prospects for FEM. International Journal for Numerical Methods in Engineering, 21:1-11, 1986. 37. M. Adams. Parallel multigrid algorithms for unstructured 3D large deformation elasticity and plasticity finite element problems. Technical Report UCB//CSD-99-1036, University of California, Berkeley, 1999.
Inelastic and non-linear materials 4.1 Introduction In Chapter 2 we presented a framework for solving general problems in solid mechanics. In this chapter we consider several classical models for describing the behaviour of engineering materials. Each model we describe is given in a strain-driven form in which a strain or strain increment obtained from each finite element solution step is used to compute the stress needed to evaluate the internal force, BTudQ, as well as a tangent modulus matrix, or its approximation, for use in constructing the tangent stiffness matrix. Quite generally in the study of small deformation and inelastic materials (and indeed in some forms applied to large deformation) the strain (or strain rate) or the stress is assumed to split into an additive sum of parts. We can write this as
in which we shall generally assume that the elastic part is given by the linear model
in which D is the matrix of elastic moduli. In the following sections we shall consider the problems of viscoelasticity, plasticity, and general creep in quite general form. By using these general types it is possible to present numerical solutions which accurately predict many physical phenomena. We begin with viscoelasticity, where we illustrate the manner in which we shall address the solution of problems given in a rate or differential form. This rate form of course assumes time dependence and all viscoelastic phenomena are indeed transient, with time playing an important part. We shall follow this section with a description of plasticity models in which time does not explicitly arise and the problems are time independent. However, we shall introduce for convenience a rate description of the behaviour. This is adopted to allow use of the same kind of algorithms for all forms discussed in this chapter.
Viscoelasticity-history dependence of deformation 63
Viscoelastic phenomena are characterized by the fact that the rate at which inelastic strains develop depends not only on the current state of stress and strain but, in general, on the full history of their development. Thus, to determine the increment of inelastic strain over a given time interval (or time step) it is necessary to know the state of stress and strain at all preceding times. In the computation process these can in fact be obtained and in principle the problem presents little theoretical difficulty. Practical limitations appear immediately, however, that each computation point must retain this history information-thus leading to very large storage demands. In the context of linear viscoelasticity, means of overcoming this limitation were introduced by Zienkiewicz et al. l and White. 2 Extensions to include thermal effects were also included in some of this early work. 3 Further considerations which extend this approach are also discussed in earlier editions of this book. 4,5
4.2.1 Linear models for viscoelasticity The representation of a constitutive equation for linear viscoelasticity may be given in the form of either a differential equation or an integral equation. 6,7 In a differential model the constitutive equation may be written as a linear elastic part with an added series of partial strains q. Accordingly, we write M
tr(t) -- Doe(t) + Z
Dmq(m)(t)
(4.4)
m=l
where for a linear model the partial strains are solutions of the first-order differential equations (l (m) "q- Tmq (m) -- ~ (4.5) with Tm a constant matrix of reciprocal relaxation times and Do, Dm constant moduli matrices. The presence of a split of stress as given by Eq. (4.2) is immediately evident in the above. Each of the forms in Eq. (4.5) represents an elastic response in series with a viscous response and is known as a Maxwell model. In terms of a springdashpot model, a representation for the Maxwell material is shown in Fig. 4.1(a) for a single stress component. Thus, the sum given by Eq. (4.4) describes a generalized
w
(a)
(b)
Fig. 4.1 Spring-dashpot models for linear viscoelasticity: (a) Maxwell element; (b) Kelvin element.
64
Inelasticand non-linear materials
Maxwell solid in which several elements are assembled in a parallel form and the Do term becomes a spring alone. In an integral form the stress-strain behaviour may be written in a convolution form as o" = D(t)e(0) +
f0 t D(t -
t')~- 7 dt'
(4.6)
where components of D(t) are relaxation moduli functions. Inverse relations may be given where the differential model is expressed as M e(t) = Jotr(t) + Z Jmr(m)(t)
(4.7)
m=l
where for a linear model the partial stresses r are solutions of r(m) "l- Vm r(m) : O"
(4.8)
in which Vm are constant reciprocal retardation time parameters and J0, Jm constant compliance ones (i.e. reciprocal moduli). Each partial stress corresponds to a solution in which a linear elastic and a viscous response are combined in parallel to describe a Kelvin model as shown in Fig. 4.1(b). The total model thus is a generalized Kelvin solid. In an integral form the strain-stress constitutive relation may be written as 00"7 dt' e -- J(t)tr(0) + f0 t J(t - t')~-
(4.9)
where J(t) are known as creep compliance functions. The parameters in the two forms of the model are related. For example, the creep compliances and relaxation moduli are related through 0D 0J7 d t ' = I J(t)D(0) + f0 t J(t - t')-0~ d t ' = D(t)J(0) + fot D(t - t')~-
(4.10)
as may easily be shown by applying, for example, Laplace transform theory to Eqs (4.6) and (4.9). The above forms hold for isotropic and anisotropic linear viscoelastic materials. Solutions may be obtained by using standard numerical techniques to solve the constant coefficient differential or integral equations. Here we will proceed to describe a solution for the isotropic case where specific numerical schemes are presented. Generalization of the methods to the anisotropic case may be constructed by using a similar approach and is left as an exercise to the reader.
4.2.2 Isotropic models To describe in more detail the ideas presented above we consider here isotropic models where we split the stress as o" -- s + m p
with
1 Ttr p - ~m
(4.11)
Viscoelasticity- history dependence of deformation where s is the stress deviator, p is the mean (pressure) stress and, for a three-dimensional state of stress, In is given in Eq. (2.45). Similarly, a split of strain is expressed as e = e + ~ Ira0
with
0 - mTe
(4.12)
where e is the strain deviator and 0 is the volume change. In the presentation given here, for simplicity we restrict the viscoelastic response to deviatoric parts and assume pressure-volume response is given by the linear elastic model p = KO (4.13) where K is an elastic bulk modulus. A generalization to include viscoelastic behaviour in this component also may be easily performed by using the method described below for deviatoric components.
Differential equation model The deviatoric part may be stated as differential equation models or in the form of integral equations as described above. In the differential equation model the constitutive equation may be written as s = 2G
#mq(m)
#0 e +
(4.14)
m=l
in which ~m are dimensionless parameters satisfying M
~m "--" 1
with
#m > 0
(4.15)
m=0
and dimensionless partial deviatoric strains q(m) are obtained by solving q(m) + l q ( m ) '~m
__ 6
(4.16)
in which )k m a r e relaxation times. This form of the representation is again a generalized Maxwell model (a set of Maxwell models in parallel). Each differential equation set may be solved numerically using a one-step time integration method [e.g., GN 11 in Sec. 2.7, Eq. (2.66)]. 8,9 To solve numerically we first define a set of discrete points, tk, at which we wish to obtain the solution. For a time tn+l we assume the solution at all previous points up to tn are known. Using a simple single-step method the solution for each partial stress is given by:
(OAt)-(m)((1-O)At) 1+
'~m J q n + l
--
1 --
Am
-(m)
tl n
+ e n + l - - en
(4.17)
in which At = t,+l - tn. We note that this form of the solution is given directly in a strain-driven form. Accordingly, given the strain from any finite element solution step we can immediately compute the stresses by using Eqs (4.13), (4.14) and (4.17) in Eqs (4.11) and (4.12).
65
66
Inelasticand non-linear materials
Inserting the above into a Newton-type solution strategy requires the computation of the tangent moduli. The tangent moduli for the viscoelastic model are deduced from
O0"n+1
OqSn._bl
OPn+l
DTIn+I ---- O~n+1 = O~n+l+ m ~O~n+1
(4.18)
The tangent part for the volumetric term is elastic and given by
Op,,+l
OPn+l OOn+1 = Kmm T 00n+l O~n+l
m ~
= m ~
O~n+l
(4.19)
Similarly, the tangent part for the deviatoric term is deduced from Eq. (4.17) as
OSn+l O~n+l
M #o -{- E
-- O~n+l Oqen+l -- 2G O~n+l
OSn+l
m=l
la
~m 1+
-~m
(4.20)
where la is defined in Eq. (2.45). Using the above, tangent moduli are expressed as M
DTI,,+I -- K m m T + 2G
#0+~( m=l
/~0/~t)
ld
(4.21)
1 ..+- --~m J
and we note that the only difference from a linear elastic material is the replacement of the elastic shear modulus by the viscoelastic term M
G --+ G m=l
l+-~m
This relation is independent of stress and strain and hence when it is used with a Newton scheme it converges in one iteration (i.e. the residual of a second iteration is numerically zero). The set of first-order differential equations (4.16) may be integrated exactly for specified strains, e. The integral for each term is given by q~m)(t) --
f
t oo
Oe exp [--(t -- t')/Am] -~7 dt'
(4.22)
An advantage to the differential equation form, however, is that it may be extended to include aging or other non-linear effects by making the parameters time or solution dependent. The exact solution to the differential equations for such a situation will then involve integrating factors, leading to more involved expressions. In the following parts of this section we consider the integral equation form and its numerical solution for linear viscoelastic behaviour. Models and their solutions for more general cases are left as an exercise for the reader.
Viscoelasticity-history
dependence
of d e f o r m a t i o n
Integral equation model
The integral equation form for the deviatoric stresses is expressed in terms of a relaxation modulus function which is defined by an idealized experiment in which, at time zero (t = 0), a specimen is subjected to suddenly applied and constant strain, e0, and the stress response, s(t), is measured. For a linear material a unique relation is obtained which is independent of the magnitude of the applied strain. This relation may be written as s(t) = 2G(t)e0 (4.23) where G(t) is defined as the shear relaxation modulus function. A typical relaxation function is shown in Fig. 4.2. The function is shown on a logarithmic time scale since typical materials have time effects which cover wide ranges in time. Using linearity and superposition for an arbitrary state of strain yields the integral equation specified as s(t) =
(4.24)
2G(t - t')-~, dt'
We note that the above form is a generalization to the Maxwell material. However, the integral equation form may be specialized to the generalized Maxwell model by assuming the shear relaxation modulus function in a Prony series form #m exp(--t/,Xm)
G(t) = G #o +
(4.25)
m--1
where the #m satisfy Eq. (4.15).
Solution to integral equation with Prony series
The solution to the viscoelastic model is performed for a set of discrete points tk. Thus, again assuming that all solutions are available up to time t~, we desire to compute the next step for time tn+l. Solution of the general form would require summation over
1.0 ,ww.,,
0.8
C
, .o ..
~ 0.6 C
o
..=,
x
0.4
i .
(b
n"
0.2
0 -5
I I -4-3-2
I
Fig. 4.2 Typical viscoelastic relaxation function.
I I I -1 0 1 log time t
I 2
I 3
I 4
5
67
68
Inelastic and non-linear materials
all previous time steps for each new time; however, by using the generalized Maxwell model we may reduce the solution to a recursion formula in which each new solution is computed by a simple update of the previous solution. We will consider a special case of the generalized Maxwell material in which the number of terms M is equal to 1 [which defines a standard linear solid, Fig. 4.3(a)]. The addition of more terms is easily performed from the one-term solution. Accordingly, we take G(t) = G [#0 + #1 exp(-t/A1)] (4.26) where #0 + #1 = 1. For the standard solid only a limited range of time can be considered, as can be observed from Fig. 4.3(b) for the model given by
G(t) = G[0.15 + 0.85 e x p ( - t ) ] To consider a wider range it is necessary to use terms in which the '~m cover the total time by using at least one term for each decade of time (a decade being one unit on the lOgl0 time scale). Substitution of Eq. (4.26) into Eq. (4.24) yields
$(t) = 2G
[~0 + ~1
exp(-(t - t')/)~l)] ~ dt'
(4.27)
which may be split and expressed as s(t) - 2G#oe(t) + 2G#l
e x p ( - ( t - t')/)~l)~-; dt'
(4.28/
oo
= 2G[#oe(t ) + ~lq(1)(t)] where we note that q(l~ is identical to the form given in Eq. (4.22). Thus use of a Prony series for G(t) is identical to solving the differential equation model exactly. In applications involving a linear viscoelastic model, it is usually assumed that the material is undisturbed until a time identified as zero. At time zero a strain may be
q~ 1.0
tO
0.8
~ 0.6 tO .~
0------
0.4
n." 0.2 0 I I -5 -4-3-2
(a)
I
I I i -1 0 1 log time t
I 2
(b)
Fig. 4.3 Standard linear viscoelastic solid: (a) model for standard solid; (b) relaxation function.
I 3
I 4
5
Viscoelasticity- history dependence of deformation suddenly applied and then varied over subsequent time. To evaluate a solution at time tn+l the integral representation for the model may be simplified by dividing the integral into t.+l
O-
tn
(.) dt' =
(.) dt' +
J --00
(.1 dt' +
t.+l
(.) dt' +
O0
(.) dt'
(4.29)
J tn
In each analysis considered here the material is assumed to be unstrained before the time denoted as zero. Thus, the first term on the right-hand side is zero, the second term includes a jump term associated with e0 at time zero, and the last two terms cover the subsequent history of strain. The result of this separation when applied to Eq. (4.27) gives the recursion 3 q(1) n+l - - e x p ( - A t / A 1 ) q ~ 1) + Aq (1) (4.30) where f
t~+l
Aq (1) --
Oe
exp[-(tn+l - t')/Az]~- 7 dt'
(4.31)
d tn
and q 0
(4.50)
The particular case of Q = F is known as associative plasticity. When this relation is not satisfied the plasticity is non-associative. In what follows this more general form will be considered initially (reductions to the associative case follow by simple substitution of Q = F). The satisfaction of the normality rule for the associative case is essential for proving so-called upper and lower bound theorems of plasticity as well as uniqueness. In the non-associative case the upper and lower bounds do not exist and indeed it is not certain that the solutions are always unique. This does not prevent the validity of nonassociated rules as it is well known that in frictional materials, for instance, uniqueness is seldom achieved but the existence of friction cannot be denied.
4.3.3 Hardening/softening rules
Isotropic hardening
The parameters ~ and ~ must also be determined from rate equations and define hardening (or softening) of the plastic behaviour of the material. The evolution of ~, governing the size of the yield surface, is commonly related to the rate of plastic work or directly to the consistency parameter. If related to the rate of plastic work ~ has dimensions of stress and a relation of the type = tr Tk p = Y (~)k p
(4.51)
is used to match behaviour to a uniaxial tension or compression result. The slope
OY
A -- ~ 0~
(4.52)
provides a modulus defining instantaneous isotropic hardening. In the second approach ~ is dimensionless (e.g. an accumulated plastic strain 16) and is related directly to the consistency parameter using = [(_~p)T~p] 1/2 = ~[aXtra,tr]~/2
(4.53)
A constitutive equation is then introduced to match uniaxial results. For example, a simple linear form is given by
O'y(/'C) - -
O'y 0 31-
Hi ol~
where Hio is a constant isotropic hardening modulus.
Kinematic hardening
A classical procedure to represent kinematic hardening was introduced by Prager 26 and modified by Ziegler. 27 Here the stress in each yield surface is replaced by a linear relation in terms of a 'back stress' ~ as m
= tr- ~
(4.54)
Classical time-independent plasticity theory 77 with the yield function now given as F ( g - ~, t~) = F (5_, t~) = 0
(4.55)
during plastic behaviour. We note that with this approach derivatives of the yield surface differ only by a sign and are given by
F,r
F_~ =-F,_~
(4.56)
m
Accordingly, the yield surface will now translate, and if isotropic hardening is present will also expand or contract, during plastic loading. A rate equation may be specified most directly by introducing a conjugate work variable 13 from which the hardening parameter ~ is deduced by using a hardening potential 7-/. This may be stated as
- -7-t,~
(4.57)
which is completely analogous to use of an elastic energy to relate r and e . A rate equation may be expressed now as /3 = j~Q,_~
(4.58)
It is immediately obvious that here also we have two possibilities. Using Q in the above expression defines a non-associative hardening, whereas replacing Q by F would give an associative hardening. Thus for a fully associative model we require that F be used to define both the plastic potential and the hardening. In such a case the relations of plasticity also may be deduced by using the principle of maximum plastic dissipation. 15'16'28"29A quadratic form for the hardening potential may be adopted and written as _ 7 - t - ~1f-.ITH _k/3 (4.59) in which H~ is assumed to be an invertible set of constant hardening parameters. Now 13 may be eliminated to give the simple rate form
~_-
-~I'Ik
OQ O~
-
-
AH Q,,, -k
_
(4.60)
Use of a linear shift in relation (4.54) simplifies this, noting Eq. (4.56), to -- ,~H k Q,r
(4.61)
In our subsequent discussion we Shall usually assume a general quadratic model for both elastic and hardening potentials. For a more general treatment the reader is referred to references 16 and 30. Another approach to kinematic hardening was introduced by Armstrong and Frederick 31 and provides a means of retaining smoother transitions from elastic to inelastic behaviour during cyclic loading. Here the hardening is given as
= A[Hk Q,~- HNL~]
(4.62)
78
Inelastic and non-linear materials
Applications of this approach are presented by Chaboche 32'33 and numerical comparisons to a simpler approach using a generalized plasticity model 34'35 are given by Auricchio and Taylor. 36 Many other approaches have been proposed to represent classical hardening behaviour and the reader is referred to the literature for additional information and discussion. 21-23'37-39 A physical procedure utilizing directly the finite element method is available to obtain both ideal plasticity and hardening. Here several ideal plasticity components, each with different yield stress, are put in series and it will be found that both hardening and softening behaviour can be obtained easily retaining the properties so far described. This approach was named by many authors as an 'overlay' model 4~ and by others is described as a 'sublayer' model. There are of course many other possibilities to define change in surfaces during the process of loading and unloading. Here frictional soils present one of the most difficult materials to model and for the non-associative case we find it convenient to use the generalized plasticity method described in Sec. 4.6.
4.3.4 Plastic stress-strain relations To construct a constitutive model for plasticity, the strains are assumed to be divisible into elastic and plastic parts given as g_ = s__e + s__p
(4.63)
For linear elastic behaviour, the elastic strains are related to stresses by a symmetric 9 • 9 matrix of constants D__. Differentiating Eq. (4.63) and incorporating the plastic relation (4.50) we obtain - - I ) - l l ~ " -t- ,~a,ff (4.64) The plastic strain (rate) will occur only if the 'elastic' stress changes &__e__ DS
(4.65)
tend to put the stress outside the yield surface, that is, is in the plastic loading direction. If, on the other hand, this stress change is such that unloading occurs then of course no plastic straining will be present, as illustrated for the one-dimensional case in Fig. 4.6. The test of the above relation is therefore crucial in differentiating between loading and unloading operations and underlines the importance of the straining path in computing stress changes. When plastic loading is occurring the stresses are on the yield surface given by Eq. (4.44). Differentiating this we can therefore write
OF
P -- ~ O ' x 00"x
OF
"JI-~ ( T y O0"y
-~-''"-t"
OF ~
~x "Jr-
OF ~
~y -t"''" "JI-
OF
~ --- 0
or
P -- F ~
+ F~-
Hi ~ = 0
(4.66)
in which we make the substitution
OF
Hi,~ = - 0----~-/~= - F,~t~ where Hi denotes an isotropic hardening modulus.
(4.67)
Classical time-independent plasticity theory 79 For the case where kinematic hardening is introduced, using Eq. (4.54) we can substitute Eq. (4.61) and modify Eq. (4.64) to De = ~_+ (D D_ + I-Ik)J~Q,r
(4.68)
Similarly, introducing Eq. (4.56) into Eq. (4.66) we obtain (4.69)
P - F T ~ - Hi~ -- 0
Equations (4.68) and (4.69) now can be written in matrix form as (D__+ H__k)Q,_~] -Hi {~}
(4.70)
The indeterminate constant ~ can now be eliminated (taking care not to multiply or divide by Hi or ~ which are zero in ideal plasticity). To accomplish the elimination we solve the first set of Eq. (4.70) for ~, giving ~ = D e - (_D_+ H__s)Q,~J~
and substitute into the second, yielding the expression FTDe - [Hi + F,T(D__+ H__~)Q,r
= 0
Equation (4.64) now results in an explicit expansion that determines the stress changes in terms of imposed strain changes. Using Eq. (4.43) this may now be reduced to a form in which only six independent components are present and expressed as* ,k
9
O" - - Dep~:
(4.71)
and 1 Dep = pTD--P - H* pTDQ'-r FTDp = D-
1
(4.72)
H.pTD__Q,sF~_DP_~
where n*--
H i -Jr-F T (I)-t--a_Hk) Q,g__
The elasto-plastic matrix De*p takes the place of the elasticity matrix D T in a continuum rate formulation. We note that in the absence of kinematic hardening it is possible to make reductions to the six-component form for all the computations at the very beginning. However, the manner in which the back stress enters the computation is not the same as that for the plastic strain and would be necessary to scale the two differently to make the general reduction. Thus, for the developments reported here we prefer to * We shall show this step in more detail below for the J2 plasticity model. In general, however, the final result involves only the usual form of the D matrix and six independent components from the derivative of the yield function.
80
Inelasticand non-linear materials
carry out all calculations using the full nine-component form (or, in the case of plane stress, to follow a four-component form) and make final reductions using Eq. (4.72). For a generalization of the above concepts to a yield surface possessing 'comers' where Q,_~is indeterminate, the reader is referred to the work of Koiter ~9or the multiple surface treatments in Simo and Hughes. 16 An alternative procedure exists here simply by smoothing the comers. We shall refer to it later in the context of the Mohr-Coulomb surface often used in geomechanics and the procedure can be applied to any form of yield surface. The continuum elasto-plastic matrix is symmetric only when plasticity is associative and when kinematic hardening is symmetric. In general, non-associative materials present stability difficulties, and special care is needed to use them effectively. Similar difficulties occur if the hardening moduli are negative which, in fact, leads to a softening behaviour. This is addressed further in Sec. 4.11 and later for large strain in Sec. 6.7.2. The elasto-plastic matrix given above is defined even for ideal plasticity when Hi and H_H_kare zero. Direct use of the continuum tangent in an incremental finite element context where the rates are approximated by
e~+~At .~ Aen+~
and
b'n+~At
~
Aorn+ 1
was first made by Yamada et al. 42 and Zienkiewicz et al. 43 However, this approach does not give quadratic convergence when used in the Newton scheme. For the associative case we can introduce a discrete time integration algorithm in order to develop an exact (numerically consistent) tangent which does produce quadratic convergence when used in the Newton iterative algorithm.
We have emphasized that with the use of iterative procedures within a particular increment of loading, it is important always to compute the stresses as k
O'n+ 1 :
(4.73)
or n + mOr~n
corresponding to the total change in displacement parameters A Uknand hence the total strain change k
AS nk--BAUkn
Auk -- ~-'~duin
(4.74)
i=O
which has accumulated in all previous iterations within the step. This point is of considerable importance as constitutive models with path dependence (viz. plasticitytype models) have different responses for loading and unloading. If a decision on loading/unloading is based on the increment dUkn erroneous results will be obtained. Such decisions must always be performed with respect to the total increment A ukn. In terms of the elasto-plastic modulus matrix given by Eq. (4.72) this means that the stresses have to be integrated as
Computation of stress increments 81 k O'n+ 1 - - Orn "{-
f
zxe,k * I)ep de
(4.75)
do
incorporating into D*ep the dependence on variables in a manner corresponding to a linear increase of Aekn (or Au~). Here, of course, all other rate equations have to be suitably integrated, though this generally presents little additional difficulty. Various procedures for integration of Eq. (4.75) have been adopted and can be classified into explicit and implicit categories.
4.4.1 Explicit methods In explicit procedures either a direct integration process is used or some form of the Runge-Kutta process is adopted, an In the former the known increment Ae nk is subdivided into m intervals and the integral of Eq. (4.75) is replaced by direct summation, writing m-1
AO~n -- % ~ I)(n+j/m)Aekn m
(4.76)
j=0
where I)(n+j/m ) denotes the tangent matrix computed for stresses and hardening parameters updated from the previous increment in the sum. This procedure, originally introduced in reference 45 and described in detail in references 46 and 47, is known as subincrementation. Its accuracy increases with the number of subincrements, m, used. In general it is difficult a priori to decide on this number, and accuracy of prediction is not easy to determine. Such integration will generally result in the stress change departing from the yield surface by some margin. In problems such as those of ideal plasticity where the yield surface forms a meaningful limit a proportional scaling of stresses (or return map) has been practised frequently to obtain stresses which are on the yield surface at all times. 47,48In this process the effects of integrating the evolution equation for hardening must also be treated. A more precise explicit procedure is provided by use of a Runge-Kutta method. Here, first an increment of Ae/2 is applied in a single-step explicit manner to obtain Aorn+l/2
--
1 . -~DnAe n
(4.77)
using the initial elasto-plastic matrix. This increment of stress (and corresponding tCn+~/2) is evaluated to compute l)n+l/2 and finally we evaluate mor n -- Dn+l/zAen
(4.78)
This process has a second-order accuracy and, in addition, can give an estimate of errors incurred as mor n - 2Aorn+l/2 (4.79) If such stress errors exceed a certain norm the size of the increment can be reduced. This approach is particularly useful for integration of non-associative models or models without yield functions where 'tangent' matrices are simply evaluated (see Sec. 4.6).
82
Inelastic and non-linear materials
4.4.2 Implicit methods: return map algorithm The integration of Eq. (4.75) can, of course, be written in an implicit form. For instance, we could write in place of Eq. (4.75), during each iteration k, that Ao~n+l = [(1 -- O)Dn+ODn~l] km~n+1
(4.80)
where here D~, denotes the value of the tangential matrix at the beginning of the time *,k step and Dn+ 1 the current estimate to the tangential matrix at the end of the step. This non-linear equation set could be solved by any of the procedures previously described; however, derivatives of the tangent matrix are quite complex and in any case a serious error is committed in the approximate form of Eq. (4.80). Further, there is no guarantee that the stresses do not depart from the yield surface.
Return map algorithm
In 1964 a very simple algorithm was introduced simultaneously by Maenchen and Sacks 49 and by Wilkins. 5~ This algorithm uses a two-step process to compute the new stress and was originally implemented in an explicit time integration form, thus requiring no explicit construction of an elasto-plasfic tangent matrix; however, later its versatility and robustness was demonstrated for implicit solutions. 5~'52 The steps of the algorithm are: 1. Perform a predictor step in which the entire increment of strain (for the present discussion we omit the iteration counter k for simplicity)
is used to compute trial stresses (denoted by superscript TR) assuming elastic behaviour. Accordingly, o',a~+, = D (~_,+1 - ~_P,) (4.81) where only an elastic modulus D_Dis _ required. 2. Evaluate the yield function in terms of the trial stress and the values of the plastic parameters at the previous time: F(cr a~ ~ , , t~) --- ' - -
< O, > 0,
elastic plastic
(4.82)
(a) For an elastic value of F set the current stress to the trial value, accordingly TR oon+I - - o o n + I ,
---.~n+l = ~ n
and
B;n+ 1 :
B;n
(b) For a plastic state solve a discretized set of plasticity rate equations (namely, using any appropriate time integration method as described, for example, in reference 8) such that the final value of Fn+l is zero. A plastic correction can be most easily developed by returning to the original Eq. (4.64) and writing the relation for stress increment as moo n = D (m~__ n - - A~__Pn)
(4.83)
Computation of stress increments 83 Now integrating the plastic strain relation (4.50) using a form similar to that in Eq. (4.80) yields Ae~ = AA[(1 - o ) a , g l n -I- o a , g l n + l ]
(4.84)
where AA represents an approximation to the change in consistency parameter over the time increment. Kinematic hardening is included by integrating Eq. (4.60) as m__.~ n =
(4.85)
-AAI-Ik[(1 - O)Q,~ln -i- OQ,~ln+l]
Finally, during the plastic solution we enforce F,,+I = 0
(4.86)
thus ensuring that final values at tn+l satisfy the yield condition exactly. The above solution process is particularly simple for 0 = 1 (backward difference or Euler implicit) and now, eliminating A_e~, we can write the above non-linear system in residual form Ricr - - m e n Rixo ri
-- I ) - l m o . i n
--Hkl
AIr n ~
_
AAa,zlin+,
A A Q , xo i
~
_]n+l
__ ~ F/+I
and seek solutions which satisfy R / - 0, Ri~ = 0 and r i - O. Any of the general iterative schemes described in Chapter 3 can now be used. In particular, the full Newton process is convenient. Noting that A ~ is treated here as a specified constant (actually, the Aekn from the current finite element solution), we can write, on linearization,
["
-1 + AA Q,,~__z~ AA Q , ~__n~ AAQ,,~ I-I~ 1 + A A Q , ~ F T ~ ,tr
F T '~
Q ,z a,,, -I~i
--
li / / / / d~ i
n+l
dro i d-~ i
R__i
=
Ri re
(4.87)
where Hi is the same hardening parameter as that obtained in Eq. (4.67). Some complexity is introduced by the presence of the second derivatives of Q in Eq. (4.87) and the term may be omitted for simplicity (although at the expense of asymptotic quadratic convergence in the Newton iteration). Analytical forms of such second derivatives are available for frequently used potential surfaces. 16'30'51-53Appendix A also presents results for second derivatives of stress invariants. It is important to note that the requirement that/7,+1 = - r i [Eq. (4.87)] ensures that the r i residual measures precisely the departure from the yield surface. This measure is not available for any of the tangential forms if Oep is adopted. For the solution it is only necessary to compute d)~ i and update as i
AA / -- ~ j=0
dA j
(4.88)
84
Inelastic and non-linear materials
This solution process can be done in precisely the same way as was done in establishing Eq. (4.72). Thus, a solution may be constructed by defining the following:
V__F=
F o. , V_.Q=
i..l k 1 + A
Q,trS
(4.89) Q ,~--~
and expressing Eq. (4.87) as d ~_ i
= _A_IR _ i -- - A1- - ~ A - I ~ T Q i [ ( ~ _ _ F i ) T A - 1 R
i - r i]
where A* -
(4.90) (4.91)
Hi + (V__Fi)TA__-IV___Qi
Immediately, we observe that at convergence R i -- 0 and r i -- 0, thus, here we obtain a zero stress increment. At this point we have computed a stress state ~n+l which satisfies the yield condition exactly. However, this stress, when substituted back into the finite element residual [e.g. Eq. (2.20a) or (2.51)], may not satisfy the equilibrium condition and it is now necessary to compute a new iteration k and obtain a new strain increment d~_kn from which the process is repeated. We note that inserting this new increment into Eq. (4.87) will again result in a non-zero value for R~, but that R~ and r remain zero until subsequent iterations. Thus, Eq. (4.90) provides directly the ~, required tangent matrix l)ep from d= ~ }_ {dcr
[A-1 - ~1 1 a *A - V__ Q ( V F__) T A -_ 13 { d g } =
[D--.ep " ] { d g }
(4.92)
Thus, we find the tangent matrix ~ep is obtained from the upper diagonal block of Eq. (4.92). We note that this development also follows exactly the procedure for computing Dep in Eq. (4.72). At this stage the terms may once again be reduced to their six-component form using P as indicated in Eq. (4.42). Some remarks on the above algorithm are in order: 1. For non-associative plasticity (namely, Q r F) the return direction is n o t normal to the yield surface. In this case no solution may exist for some strain increments (in general, arbitrary selection of F and Q forms in non-associative plasticity does not assure stability) and the iteration process will not converge. 2. For associative plasticity the normality principle is valid, requiting a convex yield surface. In this case the above iteration process always converges for a hardening material. 3. Convergence of the finite element equations may not always occur if more than one quadrature point changes from elastic to plastic or from plastic to elastic in subsequent iterations. Based on these comments it is evident that no universal method exists that can be used with the many alternatives which can occur in practice. In the next several sections we illustrate some formulations which employ the alternatives we have discussed above.
Isotropic plasticity models 85
iiiii~iiii~i1ii4iii5i~i|i1i~iiiiii~iiiii~iiiiiiiiP|~ ii,i'i~iiiiiP|~S~|~ ii~iiiiii~Iiiiiii~iiiiiiiiiiiiii~i~i~iii,i~~iiiiii,mD~i~ ~i~2~ i~i~"i'i~i~~ ~i~i~i~i~~i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~i ~~~~~~~~~~~~~~ i~i~~i~~~~i ~~~i~i~i~i~~i ~~~i~i~~ii~i~~i~~~~~~~~~~~~~~~~~ We consider here some simple cases for isotropic plasticity-type models in which both a yield function and a flow rule are used. For an isotropic material linear elastic response may be expressed by moduli defined with two parameters. Here we shall assume these to be the bulk and shear moduli, as used previously in the viscoelastic section (Sec. 4.2). Accordingly, the stress at any discrete time tn+l is computed from elastic strains in matrix form as O ' n + 1 "-" P n + l m
-k- Sn+ l - - K m m e e + l
-+-
2 G ( I - gmmlT)~n+le (4.93)
- - D-D-(gn+l - ~--Pn+l)
where the elastic modulus matrix for an isotropic material is given in the simple form D__= K m m T + 2 G ( I - 3mm v)
(4.94)
and I is the 9 x 9 identity matrix and in is the nine-component matrix m--[1
1
1 0
0
0
0
0
0] T
Using Eqs (4.42) and (4.43) immediately reduces the above to D-
K m m x + 2G(I0 - 3mm x)
(4.95)
The above relation yields the stress at the current time provided we know the current total strain and the current plastic strain values. The total strain is available from the finite element equations using the current value of nodal displacements, and the plastic strain is assumed to be computed with use of one of the algorithms given above. In the discussion to follow we consider relations for various classical yield surfaces.
4.5.1 Isotropic yield surfaces The general procedures outlined in the previous section allow determination of the tangent matrices for almost any yield surface applicable in practice. For an isotropic material all functions can be represented in terms of the three stress invariants:* I1 =
oi i ~
2J2
-- sijsji
3J3
=
SijSjkSki
raTtY
(4.96)
- - s T s - - IS[ 2 --
dets
where we can observe that definition of all the invariants is most easily performed in indicial notation. One useful form of these invariants for use in yield functions is given by 45 3Crm - -
I1
wit * Appendix B presents a summary of invariants and their derivatives.
71"
6-
~0~
71"
-6
-
-
(4.97)
86 Inelasticand non-linear materials Using these definitions the surface for several classical yield conditions can be given as"
1. Tresca: F = 2# cos 0 - Y (~) = 0
(4.98)
2. Huber-von Mises: F = ~/-2~-
Y(~;) = I_sl-
Y(~;) = 0
(4.99)
Both conditions 1 and 2 are well verified in metal plasticity. For soils, concrete and other 'frictional' materials the Mohr-Coulomb or Drucker-Prager surfaces is frequently used. 54 3. Mohr--Coulomb: F
=
om
sin ~b+ #
(
1
cos 0
,r
sin ~bsin 0
)
- c cos ~b = 0
(4.100)
where c(n) and qS(t~) are the cohesion and the angle of friction, respectively, which can depend on an isotropic strain hardening parameter ~. 4. Drucker-Prager: F = 3o/(/s;)ffm + # - K(~) = 0 (4.101) where o/=
2 sin ~b ~/~(3 - sin qS)
K=
6 cos ~b ~/3(3 - sin ~b)
and again c and 05 can depend on a strain hardening parameter. These forms lead to a convenient definition of the gradients F,z or Q,_~, irrespective of whether the surface is used as a yield condition or a flow potential. Thus we can always write F_r = F r
0~,n 0# 00 0rr + F,~ ~ + F,o 0rr
(4.102)
and upon noting that
O# Oor
O# OJ2 OJ2 0o"
1 OJ2 2~/~
oo
oo o12
oo oj3
= tan30
1 OJ3
1jz OJ2 ]
~J3-~ - g
(4.103)
-~j
Alternatively, we can always write"
O0"m OJ2 OJ3 F'O" = F'o'm OCT + F J2 ~ + F'J3 0~ r
(4.104)
which can be put into a matrix form as shown in Appendix B. The values of the three derivatives with respect to the invariants are shown in Table 4.1 for the various yield surfaces mentioned. The form of the various yield surfaces given above is shown with respect to the principal stress space in Fig. 4.8, though many more elaborate ones have been developed, particularly for soil (geomechanics) problems. 55-57
Isotropic plasticity models 87 "(73
Drucker-Prager ~ > 0 .G1= 02 = ($3
cot o__M~ses
~/3C
- - O ~ n
"~3 ~/3c cot~
Mohr-Coulomb ~ > 0 01 = G2 = ($3
/~
--02
Tr'~esca ---02
1
"~1
(a)
(b)
Fig. 4.8 Isotropic yield surfaces in principal stress space: (a) Drucker-Prager and von Mises; (b) Mohr-Coulomb and Trexa.
4.5.2 -/2 model with isotropic and kinematic hardening
(PrandtI-Reuss equations)
As noted in Table 4.1 a particularly simple form results if we assume the yield function involves only the second invariant of the deviatoric stresses J2. Here we present a more detailed discussion of results obtained by using an associated form and the return map algorithm. Since the yield function involves deviatoric quantifies only we can initially make all the calculations in terms of these. Accordingly, the elastic deviatoric stress-strain relation is given as _s = 2GeZ - 2G (g - e_p)
(4.105)
Continuum rate form
Before constructing the return map solution we first consider the form of the plasticity equations in rate form for this simple model. The plastic deviatoric strain rates are deduced from eZ
9O F
= A--~- = J~F,_s
(4.106)
Including the effects of isotropic and kinematic hardening the Huber-von Mises yield function may be expressed as F
Is- El-
~Y(~)
- 0
(4.107)
in which ~ are back stresses from kinematic hardening and t~ is an isotropic hardening parameter. We assume linear isotropic hardening given by* Y ( n ) - u + Hie;
(4.108)
* More general forms of hardening may be approximated by piecewise linear segments, thus making the present formulation quite general.
88
Inelastic and non-linear materials Table 4.1 Invariant derivatives for various yield conditions Yield condition
F.crm
~/-~F, J2
J2 F, J3
Tresca
0
2 cos 0(1 + tan 0 tan 30)
Huber-von Mises
0
4c3
4 ~ sin 0 cos 30 0
Mohr-Coulomb
89c o s 0 [ 1 + tan 0 sin 30
sin
L
"1
~/3 sin 0 + sin 4~cos 0 2 cos 30
+_!_1 sin O(tan 3 0 - tan 0)[ Drucker-Prager
3a'
1
1
0
Here a rate of ~ is computed from a norm of the plastic strains, by using Eq. (4.53), as /c - ~ , ~
(4.109)
in which the factor v/2-/3 is introduced to match uniaxial behaviour given by Eq. (4.108). On differentiation of F it will be found that OF
0s_
=
OF
=n
where
-
n-
-
s-
-
Is-_
ro
-
l
(4.110)
Using the above, the plastic strains are given by
An
eP -
(4.111)
and, when substituted into a rate-of-stress relation, yield s_"= 2G [e_"- ,~n_]
(4.112)
A rate form for the kinematic hardening is taken as 2
(4.113)
P = n__. T (s_-k__) - 52 H i ~
(4.114)
The rate of the yield function becomes
and when combined with the other rate equations gives the expression for the plastic consistency parameter as (noting that with the nine-component form nTn = 1) A = GnTe
where G*=G+:
l ( n i --I- Hk)
(4.115) (4.116)
Substitution of Eq. (4.115) into Eq. (4.112), and using Eq. (4.42) to reduce to the six-component form, gives the rate form for stress-strain deviators as s -- 2G Io - G---~nnT /~
(4.117)
Isotropic plasticity models We note that for perfect plasticity Hi = Hk = 0 leading to G~ G* = 1 and, thus, the elastic-plastic tangent for this special case is also here obtained. Use of Eq. (4.117) in the rate form of Eq. (4.93) gives the final continuum elasticplastic tangent I)ep KmmT + 2G I i 0 . l m m T 2G nn T1 * -(4.118) 3 G* This then establishes the well-known Prandtl-Reuss stress-strain relations generalized for linear isotropic and kinematic hardening.
Incremental return map form
The return map form for the equations is established by using a backward (Euler implicit) difference form as described previously (see Sec. 4.4.2). Omitting the subscript on the n + 1 quantities the plastic strain equation becomes, using Eqs (4.106) and (4.110), gP - gPn + AA_n (4.119) and the accumulated (effective) plastic strain ~; = ~;n +
AA
(4.120)
Thus, now the discrete constitutive equation is s -- 2G (e_ - _ep) the kinematic hardening is 2
(4.121)
HkAAn
(4.122)
and the yield function is F = Is_- El _ ~ / 3ny2 _ g2Hi AA
(4.123)
where Yn -- Yo -t- ,v/2/ 3 t% . The trial stress, which establishes whether plastic behaviour occurs, is given by s TR = 2G ( e - e__p)
(4.124)
which for situations where plasticity occurs permits the final stress to be given as s - s_TR - 2GAAn__
(4.125)
Using the definition of n__,we may now combine the stress and kinematic hardening relations as I s - ~ l l l - IsT R - ~ n l n T R - ( 2 G + g 2 Hk) AAn (4.126) and noting from this that we must have 111T R - -
(4.127)
n
we may solve the yield function directly for the consistency parameter as 16'36 AA -where G* is given by Eq. (4.116).
[swR -
---~n I -
2G*
~/~-/3
Yn
(4.128)
89
90 Inelasticand non-linear materials We can also easily establish the relations for the consistent tangent matrix for this J2 model. From Eqs (4.121) and (4.119) we obtain the incremental expression ds_ = 2G [ d g - _ndA- AAdn__]
(4.129)
The increment of relation (4.127) gives 16
d n = d n rR= -
-
2G
[l-nn ride
I s - _ ~ l
- -
(4.130)
-
and we have have and from from Eq. Eq. (4.128) (4.128) we d)~ = ur nrde .
~
(4.131)
m
Substitution into Eq. (4.129) gives the consistent tangent matrix ds - 2G
I(
1 --
2GAA
]sT R __ ----~n I
)I_(G
2GAA
G*
-
1-STR -- ---~n ]
)nnT]de
(4.132)
This may now be expressed in terms of the total strains, combined with the elastic volumetric term and reduced to six-components to give Dep
= Kmm T + 2G
1 --
-~mm)-
IsTR __ ----.~nI
G*
l_sTR -- -~-~n I
nnT
(4.133) We here note also that when A)~ = 0 the tangent for the return map becomes the continuum tangent, thus establishing consistency of form.
4.5.3 J2 plane stress The discussion in the previous part of this section may be applied to solve problems in plane strain, axisymmetry, and general three-dimensional behaviour. In plane strain and axisymmetric problems it is only necessary to note that some strain components are zero. For problems in plane stress, however, it is necessary to modify the algorithm to achieve an efficient solution process. In a plane stress process only the four stresses O'x, O'y, 7"xy and Ty x need be considered. When considering deviatoric components, however, there are five components, sx, Sy, Sz, Sxy a n d Syx. The deviators may be expressed in terms of the independent stresses as Sx
2
Sy s =
Sz
Sxy Syx
-
1
-~
-I
-
-1
2
0
0
0
~
- 1 0 0 0 3 0
0
0
ax Oy
= Ps_~
(4.134)
7"xy TYX
The Huber-von Mises yield function may be written as F = [(~ - ~)TPsrPs(cr - ~)] 1/2 _ ~y2 (~) < 0
(4.135)
Isotropic plasticity models 91 Expanding Eq. (4.135) gives the plane stress yield function F = [9x2 -
ffxffy
+ 9y2 + 1.5(g'xZy+ ff2x)] 1/2_ Y(tc) < 0
(4.136)
where
g'x ~ O'x -- ~x,
~y - - O'y ~ I'f,y,
~yx - " 7"yx -- I'~yx
~xy - - Txy -- I'~xy ,
(4.137)
define stresses which are shifted by the kinematic hardening back stress. Plastic strain rates may now be computed by direct differentiation of the yield function, giving kx ~Z--
~F,o"
"~
2
~y ~xy
~ -- 2~1
- 1
--i
kyx
0
0
Crx
02 03 i
O'y Txy
0
7"yx
0
.
= i_~1As~
(4.138)
where As = PsTPs. Similarly, the rate of the back stress for the kinematic hardening case is given by 1
~/~ =
As~r
nk--
(4.139)
--
The elastic components are computed by using the plane stress relation. Accordingly, for a plastic step the constitution is given by = D_D(_e _ - ~P)
(4.140)
where for isotropic behaviour
~
E 1 -u2
1
u
0
u
1 0 0
0 1
0
u 0
0 0 1-
(4.141)
with E the modulus of elasticity and u the Poisson's ratio. We note that for a J2 model the volumetric plastic strain must always be zero; consequently, we can complete the determination of plastic strains at any instant by using ezp - -exp - ePy (4.142) This may be combined with the elastic strain given by ee =
/]
E (ax + O-y)
(4.143)
to compute the total strain ez and, thus, the thickness change. The solution process now follows the procedures given for the general return mapping case. A procedure which utilizes a spectral transformation on the elastic and plastic parts is given in references 16 and 52. The process given there is more elegant but lacks the clarity of working directly with the stress and plastic strain increments.
92
Inelastic and non-linear materials
Plastic behaviour characterized by irreversibility of stress paths and the development of permanent strain changes after a stress cycle can be described in a variety of ways. One form of such description has been given in Sec. 4.3. Another general method is presented here.
4.6.1 Non-associative case- frictional materials This approach assumes a priori the existence of a rate process which may be written directly as -- D*e (4.144) in which the matrix D__*depends not only on the stress o" and the state of parameters ~, but also on the direction of the applied stress (or strain) rate ~r (or ~).58 A slightly less ambitious description arises if we accept the dependence of D* only on two directions those of loading and unloading. If in the general stress space we specify a 'loading' direction by a unit vector n_ given at every point (and also depending on the state parameters e~), as shown in Fig. 4.9, we can describe plastic loading and unloading by the sign of the projection n T&. Thus
-
nT&
~ > 0 for loading L< 0 for unloading
(4.145)
while nTb " = 0 is a neutral direction in which only elastic straining occurs. One can now write quite generally that f D___[e_" for loading = I,D~k for unloading where the matrices D__[and D__~depend only on the state described by tr and ~. 0'2
,•
(load)
6 (unload)
Fig. 4.9 Loading and unloading directions in stress space.
(4.146)
Generalized plasticity 93 The specification of D___[and I)~ must be such that in the neutral direction of the stress increment ~ the strain rates corresponding to this are equal. Thus we require ~"
~
~r
(DL)
--1
"
O" = O~l-l____~
when n T& = 0
(4.147)
A general way to achieve this end is to write 1
(D~) -1 ~ D -1 --['- ~LngL aT
1
( D u ) -1 -- D -1 -~- -~ungU aT
and
(4.148)
where D__is the elastic matrix, ng L and ng U are arbitrary unit stress vectors for loading and unloading directions, and HL and Hu are appropriate plastic moduli which in general depend on cr and ~. The value of the tangent matrices D[ and D~ can be obtained by direct inversion if HL/u ~ O, but more generally can be deduced following procedures given in Sec. 4.3.4 or can be written directly using the Sherman-Morrison-Woodbury f o r m u l a 59 as" D[ - D
H~ = HL + nXDngL
H~DngLnTD
(4.149)
This form resembles Eq. (4.72) and indeed its derivation is almost identical. We note further that (D[) -1 is now well behaved for HL zero and a form identical to that of perfect plasticity is represented. Of course, a similar process is used to obtain D__~. This simple and general description of generalized plasticity was introduced by Mr6z and Z i e n k i e w i c z . 6~ It allows: 1. the full model to be specified by a direct prescription of n, ng and H for loading and unloading at any point of the stress space; 2. existence of plasticity in both loading and unloading directions; 3. relative simplicity for description of experimental results when these are complex and when the existence of a yield surface of the kind encountered in ideal plasticity is uncertain. For the above reasons the generalized plasticity forms have proved useful in describing the complex behaviour of soils. 62'62-65 Here other descriptions using various interpolations of n and moduli form a unique yield surface, known as bounding surface plasticity models, are indeed particular forms of the above generalization and have proved to be useful. 66 Classical plasticity is indeed a special case of the generalized models. Here the yield surface may be used to define a unit normal vector as 1
n__= tr b_r, jr,-T,,_11/F_~ 2
(4.150)
and the plastic potential may be used to define 1
ng = r,t~g.~g -,T ,-,,_,1/2 J Q.z
(4.151)
where once again some care must be exercised in defining the matrix notation. Substitution of such values for the unit vectors into Eq. (4.149) will of course retrieve
94
Inelastic and non-linear materials
Experimental
100
a_
Computational model
100
v
50
50 L_
0
0 0
0
L_
"~ -50
.~ -50 >
>
a-100 O
~
0
~
I 200
250
a -100 nt~
j
v
50 0
100 ...,.
o "~ -50 "N
._~ -50 > r
L
a-1~176
I
!
I
I
-6 -4 -2
0 Axial strain (%)
250
Mean effective pressure p' (kPa)
Mean effective pressure p' (kPa) t~ 100
0
i 2
(D
I a -100 -" 0 - 8
I I I -6 -4 -2 0 Axial strain (%)
I 2
Fig. 4.10 A generalizedplasticity model describinga very complex path, and comparisonwith experimental data. Undrainedtwo-way cyclicloading of Nigata sand.67(Note that in an undrainedsoil test the fluid restrains all volumetricstrains,and pore pressuresdevelop;see reference68.)
the original form of Eq. (4.72). However, interpretation of generalized plasticity in classical terms is more difficult. The success of generalized plasticity in practical applications has allowed many complex phenomena of soil dynamics to be solved. 69'70 We shall refer to such applications later but in Fig. 4.10 we show how complex cyclic response with plastic loading and unloading can be followed. While we have specified initially the loading and unloading directions in terms of the total stress rate ~ this definition ceases to apply when strain softening occurs and the plastic modulus H becomes negative. It is therefore more convenient to check the loading or unloading direction by the elastic stress increment cre of Eq. (4.65) and to specify nT&, ~ > 0 for loading L < 0 for unloading
(4.152)
This, of course, becomes identical to the previous definition of loading and unloading in the case of hardening.
4.6.2 Associative case- J2 generalized plasticity Another modification to the classical rate-independent approach is one in which the transition from an elastic to a fully plastic solution is accomplished with a smooth
Some examples of plastic computation 95 transition. This approach is useful in improving the match with experimental data for cyclic loading. A particularly simple form applicable to the J2 model was introduced by Lubliner. 34'35 In this approach, the yield function is modified to a rate form directly and is expressed as h(F)l~-A =0 (4.153) where h(F) is given by the function
h(F) =
F ( f l - F)~ + H/3
(4.154)
in which H = Hi + Hk, and ~, 3 are two positive parameters with dimension of stress. In particular, fl is a distance between a limit plastic state and the current radius of the yield surface, and d; is a parameter controlling the approach to the limit state with increasing accumulated plastic strain. On discretization and combination with the return map algorithm a rate-independent process is evident and again only minor modifications to the algorithm presented previously are necessary. A full description of the steps involved is given by Auricchio and Taylor.36 Their paper also includes a development for the non-linear kinematic hardening model given in Eq. (4.62). In the case where the yield function is associative (i.e. F = Q) the use of the non-linear kinematic hardening model leads to an unsymmetric tangent stiffness when used with the return map algorithm. On the other hand, the generalized plasticity model is fully symmetric for this case. In the next section we present further discussion on the use of generalized plasticity to model the behaviour of frictional materials. In general, these involve use of nonassociative models where the retum map algorithm cannot be used effectively.
The finite element discretization technique in plasticity problems follows precisely the same procedures as those of corresponding elasticity problems. Any of the elements already discussed can be used for problems in plane stress; however, for plane strain, axisymmetry, and three-dimensional problems it is usually necessary to use elements which perform well in constrained situations such as encountered for near incompressibility. For this latter class of problems use of mixed elements is generally recommended, although elements and constitutive forms that permit use of reduced integration may also be used. The use of mixed elements is especially important in metal plasticity as the Hubervon Mises flow rule does not permit any volume changes. As the extent of plasticity spreads at the collapse load the deformation becomes nearly incompressible, and with conventional (fully integrated) displacement elements the system locks and a true collapse load cannot be obtained. 71'72 Finally, we should remark that the possibility of solving plastic problems is not limited to a displacement and mixed formulation alone. Equilibrium fields form a suitable vehicle, 73-75 but owing to their convenient and easy interpretation displacement and mixed forms are most commonly used.
96
Inelastic and non-linear materials
4.7.1 Perforated plate- plane stress solutions Figure 4.11 shows the configuration and the division into simple triangular and quadrilateral elements. In this example plane stress conditions are assumed and solution is obtained for both ideal plasticity and strain hardening. This problem was studied experimentally by Theocaris and Marketos 76 and was first analysed using finite element methods by Marcal and King 77 and Zienkiewicz et al. 43 (See reference 5 for discussion on these early solutions.) The von Mises criterion is used and, in the case of strain hardening, a constant slope of the uniaxial hardening curve, H, is taken. Data for the problem, from reference 76, are E = 7000 kg/mm 2, H = 225 kg/mm 2 and tZy = 24.3 kg/mm 2. Poisson's ratio is not given but is here taken as in reference 43 as v, = 0.3. To match a configuration considered in the experimental study a strip with 200 mm width and 360 mm length containing a central hole of 200 mm diameter. Using symmetry only one quadrant is discretized as shown in Fig. 4.11. Displacement boundary restraints are imposed for normal components on symmetry boundaries and the top boundary. Sliding is permitted, to impose the necessary zero tangential traction boundary condition. Loading is applied by a uniform non-zero normal displacement (a)
(b)
[/IZ~
(c)
(d)
Fig. 4.11 Perforated plane stress tension strip: mesh used and development of plastic zones at loads of 0.55, 0.66, 0.75, 0.84, 0.92, 0.98, 1.02 times O-y. (a) T3 triangles; (b) plastic zone spread; (c) Q4 quadrilaterals; (d) Q9 quadrilaterals.
Some examples of plastic computation 97
1.0 0.8
G)
E
0.6
0.4 0.2
rU 0
Ex0ermenta I
0.5
I
I
1.0
I
I
I
I
I
I
1.5 2.0 2.5 3.0 3.5 4.0 4.5 E (ey/Oy)
5.0
Fig. 4.12 Perforated plane stress tension strip: load deformation for strain hardening case (H = 225 kg/mm2). with equal increments. Displacement elements of type T3, Q4, and Q9 are used with the same nodal layout. Results for the three elements are nearly the same, with the extent of plastic zones indicated for various loads in Fig. 4.11 obtained using the Q4 element. The load--deformation characteristics of the problem are shown in Fig. 4.12 and compared to experimental results. The strain ~y is the peak value occurring at the hole boundary. This plane stress problem is relatively insensitive to element type and load increment size. Indeed, doubling the number of elements resulted in small changes of all essential quantities.
4.7.2 Perforated plate- plane strain solutions .........................................................................................................................................................
-=-
. ...........................................................................................................................................................................................................................
The problem described above is now analysed assuming a plane strain situation. Data are the same as for the plane stress case except the lateral boundaries are also restrained to create a zero normal displacement boundary condition. This increases the confinement on the mesh and shows more clearly the locking condition cited previously. In Fig. 4.13 we plot the resultant axial load for each load step in the solution. Figure 4.13(a) shows results for the displacement model using T3, Q4, and Q9 elements and it is evident that the T3 and Q4 elements result in an erroneous increasing resultant load after the fully plastic state has developed. The Q9 element shows a clear limit state and indicates that higher order elements are less prone to locking (even though we have shown that for the fully incompressible state the Q9 displacement element will lock!). Figure 4.13(b) presents the same results for the Q4/1 and Q9/3 mixed elements and both give a clear limit load after the fully plastic state is reached.
4.7.3 Steel pressure vessel This final example, for which test results obtained by Dinno and Gill TM are available, illustrates a practical application, and the objectives are twofold. First, we show that this
98
Inelastic
and non-linear
materials
problem which can really be described as a thin shell can be adequately represented by a limit number (53) of isoparametric quadratic elements. Indeed, this model simulates well both the overall behaviour and the local stress concentration effects [Fig. 4.14(a)]. Second, this problem is loaded by an internal pressure and a solution is performed up to the 'collapse' point (where, because there is no hardening, the strains increase without limit) by incrementing the pressure rather than displacement. A comparison of calculated and measured deflections in Fig. 4.14(b) shows how well the objectives are achieved.
2500
2000 "o 1500
m
O ..=,, .m X
< 1000 .............. T 3 - Displ. . . . . . . . Q 4 - Displ.
500 0
i 0
Q 9 - Displ. I
5
I
10
I
15
I
I
I
I
I
I
20 25 30 Step number
35
40
45
50
20 25 30 Step number
35
40
45
50
(a)
2500 2000 ~5oo
~ 1000 500 0
0
5
10
15
(b)
Fig. 4.13 Limit load behaviour for plane strain perforated strip: (a) displacement (displ.) formulation results; (b) mixed formulation results.
Some examples of plastic computation E = 29120000 Ib/in 2 1.60inl
v=0.3 O'y = 40540 Ib/in 2 No strain hardening
1000
I
I
I
-~1
, __~ ~_0_0.125in . __~ 2~8125 in E_~ D
900 o
760
0.25 in
9
i
,
1
I!
L
8.687in
p
J
r!
1000 1080
0.545
1400
..
1000 -a ~- 800
,., . .
,,,.,
......
"
"
"
"
" "
"
"
"
" "
Finite element analysis
t.D
=
/
Experimental results - Dinno and Gill 78
1200
r~
-i
Contours for plastic zone at different pressures (Ib/in 2)
(a)
z..
j
600
r~
n
400 200 0
(b)
0
I 10
I I I 20 30 40 Vertical deflection of point A (x 10 -3 in)
I 50
Fig. 4.14 Steel pressure vessel: (a) element subdivision and spread of plastic zones; (b) vertical deflection at point A with increasing pressure.
99
100
Inelastic and non-linear materials
The phenomenon of 'creep' is manifested by a time-dependent deformation under a constant stress. Indeed the viscoelastic behaviour described in Sec. 4.2 is a particular model for linear creep. Here we shall deal with some non-linear models. Thus, in addition to an instantaneous strain, the material develops creep strains, e c, which generally increase with duration of loading. The constitutive law of creep will usually be of a form in which the rate of creep strain is defined as some function of stresses and the total creep strains (ec), that is, kc -
0e c
Ot
=/3(or, e c)
(4.155)
If we consider the instantaneous strains are elastic (ee), the total strain can be written again in an additive form as : ~e _~_ ~c (4.156) with (4.157)
~e _. n - l o .
where we neglect any initial (thermal) strains or initial (residual) stresses. A special case of this form was considered for linear viscoelasticity in Sec. 4.2. Here we consider a more general non-linear approach commonly used in modelling behaviour of metals at elevated temperatures and in modelling creep in cementitious materials. We can again use any of the time integration schemes considered above and approximate the constitutive equations in a form similar to that used in plasticity as o'.+1 -- D (r - eC+l) c c ~n+l -- ~n -Jl" Att~n+O
(4.158)
where t~n+O is calculated as i~n+O --- (1 -- O)jO n +
0t~n+ 1
On eliminating Ae c we have simply a non-linear equation Rn+l
~
~n+l
D-10rn+ 1
--
__ ~c
__
At/3.+ 0 : 0
(4.159)
The system of equations can be solved iteratively using, say, the Newton procedure. Starting from some initial guess, say Crn+l = ~r. and an increment of strain is given by the finite element process, the general iterative/incremental solution can be written as R i+1 - - 0 -
Ri -
( D -1 +
AtCn+l)dOrin+ 1
(4.160)
where Cn+l =
~
n+O
(4.161) n+l
Solving this set of equations until the residual R is zero we obtain a set of stresses o'.+ and tangent matrix Dn+ 1 --= [ D - l + Atfn+l] -1 (4.162)
Basic formulation of creep problems 101 which may once again be used to perform any needed iterations on the finite element equilibrium equations. The iterative computation that follows is very similar to that used in plasticity, but here At is an actual time and the solution becomes rate dependent. While in plasticity we have generally used implicit (backward difference) procedures; here many simple alternatives are possible. In particular, two schemes with a single iterative step are popular.
4.8.1 Fully explicit solutions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
"Initial strain" procedure: o - 0
Here, from Eqs (4.161) and (4.162) we see that Cn+l - 0
and
Dn+ 1 = D
(4.163)
~rn+l = D[en+l - e c - At/3n]
(4.164)
Thus, from Eq. (4.159) we obtain
which may be used in Eq. (2.1 lc2) of Chapter 2 to satisfy a discretized equilibrium equation. We note that this form will lead to a standard elastic stiffness matrix. This, of course, is equivalent to evaluating the increment of creep strain from the initial stress values at each time tn and is exceedingly simple to calculate. While the process has been popular since the earliest days of finite elements 79-81 it is obviously less accurate for a finite step than other alternatives. Of course accuracy will improve if small time steps are used in such calculations. Further, if the time step is too large, unstable results will be obtained. Thus it is necessary for At _< Atcrit
(4.165)
where Atcrit is determined in a suitable manner (see, for example, Chapter 17 in reference 8). A 'rule of thumb' that proves quite effective in practice is that the increment of creep strain should not exceed one half the total elastic strain 82 A t [tOnt~n]
~_~ ~
(4.166)
Fully explicit process with modified stiffness: 1/2 ey
(4.181)
For unloading from any plastic point the material behaves elastically as shown. One possible solution is, of course, that in which all elements yield identically. Plotting the applied stress versus the elongation strain ~ = u/L the material behaviour curve is simply obtained identically as shown in Fig. 4.23(b) (h/L = 1). However, it is equally possible that after reaching the maximum stress O'y only one element (probably one with infinitesimally smaller yield stress owing to computer round-off) continues into the plastic range while all the others unload elastically. The total elongation strain is now given by u cr h ( c r - Or) = -- (4.182) L
E
LH
Non-uniqueness and localization in elasto-plastic deformations and as h tends to 0 then ~ tends to cr/E. Clearly, a multitude of solutions is possible for any arbitrary element subdivision and in this trivial example a unique finite element solution is impossible (with localization to a single element always occurring). Further, the above simple 'thought experiment' points to another unacceptable paradox implying the inadmissibility of the softening model specified with constant softening modulus. The difficulties are as follows. 1. The behaviour seems to depend on the size (h) of the subdivision chosen (also called a mesh sensitive result). Clearly this is unacceptable physically. 2. If the element size falls below a value given by h = H L / E only a catastrophic, brittle, behaviour is possible without involving an unacceptable energy gain. Similar difficulties can arise with non-associated plasticity which exhibits occasionally an effectively strain softening behaviour in some circumstances (see reference 130). The computational difficulties can be overcome to some extent by introducing viscoplasticity as a start to any computation. Such regularization was introduced as early as 197495 and was considered seriously by De Borst and co-authors. TM However, most of the difficulties remain as steady state is approached. The problem remains a serious line of research but two possible alternative treatments have emerged. The first of these is physically difficult to accept but is very effective in practice. This is the concept of properties which are labelled as non-local. In such an approach the softening modulus is made dependent on the element size. Many authors have contributed here, with the earliest being Bazant and co-workers. 125'126 Other relevant references are 131 and 132. The second approach, that of a concentrated discontinuity, is more elegant but, we believe, computationally more difficult. It was first suggested by Simo, Oliver and Armero in 1993133 and extended in later publications. 134-137 Both approaches allow strain and indeed displacement discontinuities to develop following the brittle failure behaviour on which we have already remarked. In the numerical application this limit is approached as element size decreases or alternatively when stress singularities, such as comers, trigger this type of behaviour. In the second approach, continuous plastic behaviour is not permitted and all action is concentrated on discontinuity lines which have to be suitably placed. A particular form of the non-local approach is illustrated in Fig. 4.24. Here we examine in detail a unit width of an element in which the displacement discontinuity is approximated. In the examples which we shall consider later this discontinuity is a slip one with the 'failure' being modelled as shown. However, an identical approach has been used to model strain softening behaviour of concrete in cracking. 125'126 The most basic form of non-local behaviour assumes that the work (or energy) expended in achieving the discontinuity must be the same whatever the dimension h of the element. This work is equal to O'ye y h ~
~ H = -~ryAU
(4.183)
If this work is to be identical in all highly strained elements we will require that H
= constant h Such a requirement is easy to apply in an adaptive refinement process.
(4.184)
115
116
Inelastic and non-linear materials .
,==C> a ==4:>
1
---f-
Ul_._.~~U2 (a)
ul/]u --~ ~ 12
h---~(~
Localized failure
(b)
/
a/Ik
t.,
Work dissipated in failure
per unit volume .....
1
y/H
=-I
EH
r
8
(c) Fig. 4.24 Illustration of a non-local approach (work dissipation in failure is assumed to be constant for all elements): (a) an element in which localization is considered; (b)localization; (c) stress-strain curve showing work dissipated in failure.
At this stage we can comment on the concentrated discontinuity approach of Bazant and co-workers. ~25'126In this we shall simply assume that the displacement increment of Eq. (4.183), that is, A U, is permitted to occur only on a discontinuity line and that its magnitude is strictly related to the energy density previously specified in Eq. (4.183). After considering the effects of large deformation we shall show in Sec. 6.7.2 how a very effective treatment and capture of discontinuity can be made adaptively.
Non-linearity may arise in many problems beyond those of solid mechanics, but the techniques described in this chapter are still universally applicable. Here we shall look again at one class of problems which is govemed by the quasi-harmonic field equations of Chapter 2.
Non-linear quasi-harmonic field problems 117 In some formulations it is assumed that q = -k(4~)V4~
(4.185)
which gives, then (with use of definitions from Sec. 2.6), Pq - H(~D)~
(4.186)
H - 3~ ( V N ) T k ( 0 ) V N dr2
(4.187)
where now H has the familiar form
In this form the general non-linear problem may be solved by direct iteration methods; however, as these often fail to converge it is frequently necessary to use a scheme for which a tangential matrix to 9 is required, as presented in Sec. 3.2.4 [see Eq. (3.25)]. The tangent for the form given by Eq. (4.185) is generally unsymmetric; however, special forms can be devised which lead to symmetry. 138 In many physical problems, however, the values of k in Eq. (4.185) depend on the absolute value of the gradient of Vq~, that is, V-
4(W0)TV0 dk
(4.188)
dV In such cases, we can write HT -where A-
OH(O)~b
= H + A
f J ( V N ) T [(V~b)Tk'Vq~] VN, d~2
(4.189)
(4.190)
and symmetry is preserved. Situations of this kind arise in seepage flow where the permeability is dependent on the absolute value of the flow velocity, 14~ in magnetic fields, 139'142-144 where magnetic response is a function of the absolute field strength, in slightly compressible fluid flow, and indeed in many other physical situations. 145 Figure 4.25 from reference 139 illustrates a typical non-linear magnetic field solution. While many more interesting problems could be quoted we conclude with one in which the only non-linearity is that due to the heat generation term Q [see Chapter 2, Eq. (2.69)]. This particular problem of spontaneous ignition, in which Q depends exponentially on the temperature, serves to illustrate the point about the possibility of multiple solutions and indeed the non-existence of any solution in certain non-linear cases. 146 Taking k = 1 and Q = ~ exp q~, we examine an elliptic domain in Fig. 4.26. For various values of 5, a Newton iteration is used to obtain a solution, and we find that no convergence (and indeed no solution) exists when ~ > ~crit exists; above the critical
118 Inelastic and non-linear materials
'
~Ii.
.
'
),..) ..\I
,
9
9 " -
9
-
"-,
-., ~.
9
. -
.
.
9
!i :.. '. ..;::::
..... '
,9
.
'. : : ,',,-_7...'.." ".~.. ~ ~ f :, t," ""-'~"' 9 ,., ~j~ 0.'~ It, l,
,
,.,~
','.'..~,.
.~..,,:.\.,:,.
9 z.-"..,
: : : "" ~t . . . . . .
-..,.. ,Irlq~.,.l,!.:;il-l-ll,. ,/
":,,l~:;,.
,',~'.,::..
9 "
,~:,,',..:.~: ..,
9
,.'
,,
~.
.
. . . . . . . .
...',
9
",,-.'..
9
.
.
..-
,: ..
9-
" ",
. . . .
,-.-
...
o
"..-.-"-;.:-
9
t,.:.:,:"
-
",,..
.
9
.','.,
,,:::.:. :':':,
9
li, :. ,'.: :-.~
.
": : " ' ' L . ' "
6 9
,
.
...:
I.. "' .'-: ':~'::' !.:': -
.
-'"
'
9 "
" i 9 . '1
Fig. 4.25 Magnetic field in a six-pole magnet with non-linearity owing to saturation, i39
value of c~the temperature rises indefinitely and spontaneous ignition of the material occurs. For values below this, two solutions are possible and the starting point of the iteration determines which one is in fact obtained. This last point illustrates that an insight into the problem is, in non-linear solutions, even more important than elsewhere.
~i~ii~!~iii~i~!~i~!~!~i~iii~i~!~!~ii~iii~i~i~i!~ii!~!~i!~!~!~i~!~i~iii~i~i~ii~i~i~i~i~ii~i!i~i~!ii~iii!~iiii~!i~iiiii~iii~iiiiii~!~ii~i~iiii~!~iii~i~i~i~i~i~i~i~ii~ill~ii!iilli!~i~i~!~i~ii~i~!~i!i~iiiii~iiiii!i~iiiiii~!
In this chapter we have considered a number of classical constitutive equations together with numerical algorithms which permit their inclusion in the formulations discussed in Chapter 2. These permit the solution to a wide range of practical problems in solid mechanics and geomechanics. The possibilities for models of real materials is endless
Concluding remarks
k 5.0 ~ B
4.0 I,...
~_ 3.0 E 0 2.0
Y~k ( ( (
lc
1.0
2' ( ( (
C
0; 0; < 0;
No contact Contact Penetration
(7.2)
193
194
Treatment of constraints - contact and tied interfaces
(a)
(b)
(c) Fig. 7.3 Contact between semicircular discs: node-node solution. (a) Undeformed mesh; (b) deformed mesh; (c) vertical stress contours.
We note that penetration can exist for any solution iteration in which the constraint condition is not imposed. Thus, the next step is to insert a constraint condition for any nodal pair (element) in which the gap g is negative or zero (here some tolerance may be necessary to define 'zero'). There are many approaches which can be used to insert a constraint. Here we discuss use of a Lagrange multiplier form, penalty approaches, and an augmented Lagrangian approach. 7'36
7.2.2 Contact models
Lagrange multiplier form
A Lagrange multiplier approach is given simply by multiplying the gap condition given in Eq. (7.1) by the multiplier. Accordingly, we can write for each nodal pair for which a contact constraint is assigned a variational term I-Ic = f r tr~(xS- xm) dr' ~,~ )~ng
(7.3)
c
where tr is the surface traction, x s is the position on the surface of the slave body, x m is the position on the surface of the master body,/~n is a Lagrange multiplier force and g is the gap given by Eq. (7.1). We then add the first variation of Plc to the variational
Node-node contact: Hertzian contact 195
equations being used to solve the problem. The first variation to Eq. (7.3) is given as
(~/~7 (~)kn]
(~II c -- (~)kn g "11-( (~as2 - (~a ~ ) ,,~n -- [(~a ~
--'~n g
(7.4)
and thus we identify "~n as a 'force' applied to each node to prevent penetration. Linearization of Eq. (7.4) produces a tangent matrix term for use in a Newton solution process. The final tangent and residual for the nodal contact element may be written as
0 l
0 --I
--
~d/~7 [,d,~n
=
,~n -g
(7.5)
and is accumulated into the global equations in a manner identical to any finite element assembly process. It is evident that the equations in this form introduce a new unknown for each contact pair. Also, as for any Lagrange multiplier approach, the equations have a zero diagonal for each multiplier term, thus, special care is needed in the solution process to avoid division by the zero diagonal. Of course in a contact state, one could select one of the parameters, say ~ , as a primary variable and directly satisfy the gap constraint by making ~ ' - :~. This approach is called c o n s t r a i n t e l i m i n a t i o n and may be used to reduce the number of overall unknowns. In the simple frictionless node-to-node contact case it is simple to implement as no transformations are needed to write the constraint equation. In a general case, however, the approach can become quite cumbersome and it is often simpler to use the Lagrange multiplier form directly or to consider other related approaches. If the global tangent matrix has its non-zero sparse structure defined for the case when all the specified contact elements are active (e.g. the tangent matrix defined by Eq. (7.5) can be inserted without adding new non-zero terms) then a full contact analysis may be performed using Eq. (7.5) when g < 0 and using the alternate tangent matrix and residual
[i0i] 0 0
~dt/~n td,~n
{0}
=
0 0
(7.6)
for nodal pairs when g > 0. However, if a large number of possible contact pairs are inactive (i.e. g > 0) it is more efficient to recompute the sparse structure of the global tangent matrix to just accommodate the active contact pairs (i.e. those for which g < 0). This step can be performed by determining all the active pairs prior to computing the tangent arrays.
Perturbed Lagrangian
The problem related to the zero diagonal may be resolved by considering a p e r t u r b e d L a g r a n g i a n form where 1
I-l c -- ,~n g - - ~
2
)k n
(7.7)
in which t~ is a parameter to be selected. As t~ --+ oo the perturbed Lagrangian method 9converges to the same functional as the standard Lagrange multiplier method. The first
196 Treatment of constraints- contact and tied interfaces
variation of 1-Ic becomes (~l"I c - -
(~)k n
(1) g -
- )k n /s
+
[(~/~
-
(~/42n1,~ n
(7.8)
and again we identify A, as a 'force' applied to each node to prevent penetration. Linearization of Eq. (7.4) produces a tangent matrix term for use in a Newton solution process. The final tangent and residual for the nodal contact element may be written as
i0 0 ll/d / { 0
0
dt/~'
1 -1
-1/t~
"~n - g Jr- )kn/~
--
d)~n
}
(7.9)
which is added into the equations in a manner identical to the Lagrange multiplier form. It is also possible to eliminate/~n directly from Eq. (7.8) giving /~n - -
/'~
Substitution into Eq. (7.9) and eliminating
I ~-~
(7.10)
g -- t~ (Y~ - Y~)
dan gives the reduced form
-~]~duS21=l
/~n
(7.11)
In a perturbed Lagrangian approach the final gap will not be zero but becomes a small number depending on the value of the parameter ~ selected. Thus, the advantage of the perturbed Lagrangian method is somewhat offset by a need to identify a value of the parameter that gives an acceptable answer. Indeed, in a complex problem this is not a trivial task, especially for problems involving contact between structural elements (e.g. rods, plates, or shells) and solid elements. This can be avoided in part by modifying Eq. (7.9) to read 0
0
1 -1
dti~'
-1/~
perturbed tangent
=
d)~n
A,
(7.12)
-g
Here this form is called a method and is a combination of the perturbed Lagrangian tangent matrix with the Lagrange multiplier residual. As such it is not a consistent linearization of any functional and there is some loss in convergence rate in solving the overall non-linear problem. Moreover, it is not possible to directly solve for )k n in each element and an iterative update must be used with
d/~n -- ~ (g + da~2- daT) /~ n oo yields exact satisfaction of the constraint. Use of a large kj and variation with respect to )~j give
E 1 ]
~%j C j - ~ ) ~ j
=0
(8.48)
and may easily be solved for the Lagrange multiplier as
)~j = kjCj
(8.49)
which when substituted back into Eq. (8.47) gives the classical form
nj -
kj
[cj
]2
(8.50)
The reader will recognize that Eq. (8.47) is a mixed problem, whereas Eq. (8.50) is irreducible. An augmented Lagrangian form is also possible following the procedures described in reference 28 and used in Chapter 7 for contact problems.
Multibody coupling by joints 239
8.5.2 Rotation constraints A second kind of constraint that needs to be considered relates to rotations. We have already observed in Fig. 8.3 that the disc is free to rotate around only one axis. Accordingly, constraints must be imposed which limit this type of motion. This may be accomplished by constructing an orthogonal set of unit vectors V I in the reference configuration and tracking the orientation of the deformed set of axes for each body as V~ c) --
~ilA(C)Vl
for c = a, b v~Vj - 5ij
(8.51)
A rotational constraint which imposes that axis i of body a remains perpendicular to axis j of body b may then be written as (v!a))Tv t
(b) - j
vT(A(a))TA(b)vj
-- 0
(8.52)
Example 8.1: Revolute joint
As an example, consider the situation shown for the disc in Fig. 8.3 and define the axis of rotation in the reference configuration by the Cartesian unit vectors EI (i.e. V I = EI). If we let the disc be body a and the arm body b the set of constraints can be written as (where v3 is axis of rotation) X (a) __ X (c)
Cj =
~,*l[w(a)~Tw(b)J "3 = 0
(8.53)
l'w(a)~Tw(b) k*2 I *3
and included in a formulation using a Lagrange multiplier form
rlj = )~jT Cj
(8.54)
The modifications to the finite element equations are obtained by appending the variation and linearization of Eq. (8.54) to the usual equilibrium equations. Here five Lagrange multipliers are involved to impose the three translational constraints (spherical joint) and the angle constraints for the rotating disc. The set of constraints is known as a revolute joint, e
8.5.3 Library of joints Translational and rotational constraints may be combined in many forms to develop different types of constraints between rigid bodies. For the development it is necessary to have only the three types of constraints described above. Namely, the spherical joint, a single translational constraint, and a single rotational constraint. Once these are available it is possible to combine them to form classical constraint joints and here the reader is referred to the literature for the many kinds commonly encountered. 2,4,7,35 The only situation that requires special mention is the case when a series of rigid bodies is connected together to form a closed loop. In this case the method given above can lead to situations in which some of the joints are redundant. Using Lagrange multipliers this implies the resulting tangent matrix will be singular and, thus, one
240
Pseudo-rigid and rigid-flexible bodies cannot obtain solutions. Here a penalty method provides a viable method to circumvent this problem. The penalty method introduces elastic deformation in the joints and in this way removes the singular problem. If necessary an augmented Lagrangian method can be used to keep the deformation in the joint within required small tolerances. An alternative to this is to extract the closed loop rigid equations from the problem and use singular valued decomposition 36 to identify the redundant equations. These may then be removed by constructing a pseudo-inverse for the tangent matrix of the closed loop. This method has been used successfully by Chen to solve single loop problems. 35
'iii!iiiiiii iiiiiiiiiiii'ii,i'~i~ ,i~:~:',~::~::~iiiiiii'i,~:~i : :::~~i~,i,i:~,':ii':~iiii'i',i{iiii::~::::~'~'i~ii'~,iiiii::::~i~i'i~il~,ii,i~','i~,'i',i~',il!ili'~,i',iliiiiiii! i i:,i:,iiiiiiii~,~,',i~i{i'i,{!'!,!~,i ~!',iiiiiiiiii 'i,i ~""i~',~{'~,'!i~~:~ !i'i:~i',~'i~i'i i'~,i'~i,iiii'i!i'i,'i ':,,i'i,~i~,'il~,i!iiiiliiiiiiii' i i~i~,i',i!'i,:'iii'',',:iii!iii~'~'':iiili',iii',ii!'iii'~i',i',!i~,:,il,i',:{:i''i,~,i,i:ii!:i'',:,iii",','i, iii'!~,:',:ii',ii'"i~,,'~ili:,l'i,'ii'i:{~iiiiii'','i:,,',i!!,i~,i:i~:,'i:':,i:ii:i:,:i:,i::iif'~,i,'!',!i!'ii~'i,i'::,'i:~',,i'::',!,!iii!ii{i:i:,i!iii:,!:,i!':!:i{::i','i:i"~,ii'iiii!':i'~,'~,:!,i~:,i':i',i:i',,i','!'~,,i'i:!i~,,',',ii:: i"i'':,~,i i'!i,'','!~ii',i:,i' : ii',i'::,{:iiii'i':,,',i!:,:,i:,ii:!ii:i:' '~!~,i::ii':i':~ !i~i:'i :i,i','':iii',i,,'~~!: ':ii': i'
8.6.1 Rotating disc As a first example we consider a problem for the rotating disc on a rigid arm which is attached to a deformable base as shown in Fig. 8.4. The finite element model is constructed from four-node displacement elements in which a St Venant-Kirchhoff material model is used for the elastic part. The elastic properties in the model are E = 10 000 and z~ = 0.25, with a uniform mass density P0 = 5 throughout. The disc and arm are made rigid by using the procedures described in this chapter. The disc is attached to the arm by means of a revolute joint with the constraints imposed using the Lagrange multiplier method. The rigid arm is constrained to the elastic support by using the local Lagrange multiplier method described in Sec. 8.4. The problem is excited by a constant vertical load applied at the revolute joint and a torque applied to spin the disc. Each load is applied for the first 10 units of time. The mesh and configuration are shown in Fig. 8.4(a). Deformed positions of the model are shown at 2.5 unit intervals of time in Fig. 8.4(b)-(h). A marker element shows the position of the rotating disc. The displacements at the revolute joint and the radial exterior point at the marker element location are shown in Fig. 8.5.
8.6.2 Beam with attached mass ....................................................................................
--: .....................................................................................................................................................................................
As a second example we consider an elastic cantilever beam with an attached end mass of rectangular shape. The beam is excited by a horizontal load applied at the top as a triangular pulse for two units of time. The rigid mass is attached to the top of the beam by using the Lagrange multiplier method described in Sec. 8.4 and here it is necessary to constrain both the translation and the rotation parameters of the beam. The beam is three dimensional and has an elastic modulus of E = 100 000 and a moment of inertia in both directions of I1~ = 122 = 12. The beam mass density is low, with a value of /90 = 0.02. The tip mass is a cube with side lengths 4 units and mass density/90 = 1. The shape of the beam at several instants of time is shown in Fig. 8.6 and it is clear that large translation and rotation is occurring and also that the rigid block is correctly following a constrained rigid body motion.
Numerical examples 241
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 8.4 Rigid-flexible model for spinning disc: (a) problem definition, solutions at time; (b) t = 2.5 units; (c) t - 5.0 units; (d) t - 7.5 units; (e) t -- 10.0 units; (f) t - 12.5 units; (g) t - 15.0 units; (h) t - 17.5 units.
242
Pseudo-rigid and rigid-flexible bodies
15, -,r
0.5
E o
0
r 03
-0.5
A
A
l!
h
1 I I---Horizontall 10~"'Vertical I
A
".......",..-'I .- ::, ,,:I,- 1 0
4_~,,-
L
---Vertical J _ . -1 10 20 0 Time (a)
30
40
-15 -20
/""%
-.
'
0
10
'
20 Time (b)
30
40
Fig. 8.5 Displacements for rigid-flexible model for spinning disc. Displacement at: (a) revolute; (b) disc rim.
(a)
(c)
(b)
(e)
(f)
(g)
(d)
(h)
(i)
Fig. 8.6 Cantilever with tip mass: (a) t -- 2 units; (b) t -- 4 units; (c) t -- 6 units; (d) t - 10 units; (e) t -- 12 units; (f) t - 14 units; (g) t - 16 units; (h) t - 18 units; (i) t - 20 units.
1. H. Cohen and R.G. Muncaster. The Theory of Pseudo-rigid Bodies. Springer, New York, 1988. 2. A.A. Shabana. Dynamics of Multibody Systems. John Wiley & Sons, New York, 1989. 3. J.M. Solberg and P. Papadopoulos. A simple finite element-based framework for the analysis of elastic pseudo-rigid bodies. International Journal for Numerical Methods in Engineering, 45:1297-1314, 1999. 4. D.J. Benson and J.O. Hallquist. A simple rigid body algorithm for structural dynamics programs. International Journal for Numerical Methods in Engineering, 22:723-749, 1986.
References 243 5. R.A. Wehage and E.J. Haug. Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems. Journal of Mechanical Design, 104:247-255, 1982. 6. A. Cardona and M. Geradin. Beam finite element nonlinear theory with finite rotations. International Journal for Numerical Methods in Engineering, 26:2403-2438, 1988. 7. A. Cardona, M. Geradin and D.B. Doan. Rigid and flexible joint modelling in multibody dynamics using finite elements. Computer Methods in Applied Mechanics and Engineering, 89:395418,91. 8. J.C. Simo and K. Wong. Unconditionally stable algorithms for rigid body dynamics that exactly conserve energy and momentum. International Journal for Numerical Methods in Engineering, 31:19-52, 1991. [Addendum: 33:1321-1323, (1992).] 9. H.T. Clark and D.S. Kang. Application of penalty constraints for multibody dynamics of large space structures. Advances in the Astronautical Sciences, 79:511-530, 1992. 10. G.M. Hulbert. Explicit momentum conserving algorithms for rigid body dynamics. Computers and Structures, 44:1291-1303, 1992. 11. M. Geradin, D.B. Doan and I. Klapka. MECANO: a finite element software for flexible multibody analysis. Vehicle System Dynamics, 22:87-90, 1993. Supplement issue. 12. S.N. Atluri and A. Cazzani. Rotations in computational solid mechanics. Archives of Computational Methods in Engineering, 2:49-138, 1995. 13. O.A. Bauchau, G. Damilano and N.J. Theron. Numerical integration of nonlinear elastic multibody systems. International Journal for Numerical Methods in Engineering, 38:2727-2751, 1995. 14. J.A.C. Ambr6sio. Dynamics of structures undergoing gross motion and nonlinear deformations: a multibody approach. Computers and Structures, 59:1001-1012, 1996. 15. R.L. Huston. Multibody dynamics since 1990. Applied Mechanics Reviews, 49:$35-$40, 1996. 16. O.A. Bauchau and N.J. Theron. Energy decaying scheme for non-linear beam models. Computer Methods in Applied Mechanics and Engineering, 134:37-56, 1996. 17. O.A. Bauchau and N.J. Theron. Energy decaying scheme for non-linear elastic multi-body systems. Computers and Structures, 59:317-331, 1996. 18. C. Bottasso and M. Borri. Energy preserving/decaying schemes for nonlinear beam dynamics using the helicoidal approximation. Computer Methods in Applied Mechanics and Engineering, 143:393-415, 1997. 19. O.A. Bauchau. Computational schemes for flexible, nonlinear multi-body systems. Multibody System Dynamics, 2:169-222, 1998. 20. O.A. Bauchau and C.L. Bottasso. On the design of energy preserving and decaying schemes for flexible nonlinear multi-body systems. Computer Methods in Applied Mechanics and Engineering, 169:61-79, 1999. 21. O.A. Bauchau and T. Joo. Computational schemes for non-linear elasto-dynamics. International Journal for Numerical Methods in Engineering, 45:693-719, 1999. 22. G.-H. Shi. Block System Modelling by Discontinuous Deformation Analysis. Computational Mechanics Publications, Southampton, 1993. 23. E.G. Petocz. Formulation and analysis of stable time-stepping algorithms for contact problems. PhD thesis, Department of Mechanical Engineering, Stanford University, Stanford, California, 1998. 24. M.E. Gurtin. An Introduction to Continuum Mechanics. Academic Press, New York, 1981. 25. L.E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, NJ, 1969. 26. J. Bonet and R.D. Wood. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge, 1997. ISBN 0-521-57272-X. 27. J.C. Garcia Orden and J.M. Goicolea. Dynamic analysis of rigid and deformable multibody systems with penalty methods and energy-momentum schemes. Computer Methods in Applied Mechanics and Engineering, 188:789-804, 2000.
244 Pseudo-rigidand rigid-flexible bodies 28. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 29. E.T. Wittaker. A Treatise on Analytical Dynamics. Dover Publications, New York, 1944. 30. J.H. Argyris and D.W. Scharpf. Finite elements in time and space. Nuclear Engineering and Design, 10:456--469, 1969. 31. A. Ibrahimbegovic and M. A1 Mikdad. Finite rotations in dynamics of beams and implicit timestepping schemes. International Journal for Numerical Methods in Engineering, 41:781-814, 1998. 32. J.C. Simo. On a stress resultant geometrically exact shell model. Part VII: Shell intersections with 5/6 DOF finite element formulations. Computer Methods in Applied Mechanics and Engineering, 108:319-339, 1993. 33. P. Betsch, F. Gruttmann and E. Stein. A 4-node finite shell element for the implementation of general hyperelastic 3d-elasticity at finite strains. Computer Methods in Applied Mechanics and Engineering, 130:57-79, 1996. 34. H. Goldstein. Classical Mechanics. Addison-Wesley, Reading, 2nd edition, 1980. 35. A.J. Chen. Energy-momentum conserving methods for three dimensional dynamic nonlinear multibody systems. PhD thesis, Department of Mechanical Engineering, Stanford University, Stanford, California, 1998. (Also SUDMC Report 98--01.) 36. G.H. Golub and C.F. Van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore MD, 3rd edition, 1996.
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In the previous chapters we have considered the solution of solid mechanics problems from the view of a continuum. A computational framework which allows for some level of displacement field discontinuity to be represented a priori might be better suited to model particular phenomena, for example the behaviour of jointed rock or the granular material flow in silos. The treatment of these classes of problems is more naturally related to discrete element methods, in which distinctly separate material regions interacting with other discrete elements in some way are considered. A number of more complex models for both solid materials as well as for contact interactions have been formulated in the context of a discrete element methodology, with successful applications in many fields of science and engineering. Moreover, most media are discontinuous at some level of observation (nano, micro, meso, macro), where the continuum assumptions cease to apply. This happens when the scale of the problem becomes similar to the characteristic length scale of the associated material structure and surface interaction laws between bodies or particles are invoked instead of a homogenized continuum constitutive law. The computational modelling of inherently discontinuous media requires the discrete nature of discontinuities to be taken into account. Discontinuities can be either pre-existing (e.g. joints, bedding planes, interfaces, planes of weakness, construction joints) or they evolve (e.g. in the case of cohesive frictional materials, where the growth and coalescence of micro cracks eventually appear in a form of a macro-crack). Many structures, structural systems or structural components comprise discrete discontinuities, appearing either in a highly regular or structured manner or they are of a heterogeneous nature. An obvious example of structured discontinua is brick masonry or jointed rock structures where the displacement discontinuities commonly occur at block interfaces without necessarily rendering structures unsafe. Such problems are best considered with discrete element methods. The term discrete element methods will here be understood to comprise different techniques suitable for a simulation of dynamic behaviour of systems of multiple rigid, simply deformable (pseudo-rigid) or fully deformable separated bodies of simplified or arbitrary shapes, subject to continuous changes in the contact stares and varying contact forces, which in turn influence the subsequent movement of the bodies. Such problems * This chapter was contributed by Professor Nenad Bidanid, University of Glasgow, UK.
246 Discreteelement methods
are non-smooth in space (separate bodies) and in time (jumps in velocities upon collisions) and the unilateral constraints (non-penetrability) need to be considered. A system of bodies changes its position continuously under the action of external forces and interaction forces between bodies, which may eventually lead to a steady-state configuration, once static equilibrium is achieved. For rigid bodies, the contact interaction law is the only constitutive law considered, while the continuum constitutive law (e.g. elasticity, plasticity, damage, fracturing) needs to be included for deformable bodies. Computational modelling of multi-body contacts (both the contact detection and contact resolution) represents the dominant feature in discrete element methods, as the number of bodies considered may be very large. If the number of potential contact surfaces is relatively small (e.g. non-linear finite element analysis of contact problems) it is convenient to define groups of nodes, segments or surfaces which belong to a possible contact set a priori. These geometric attributes can then be continuously checked against one another and the kinematic resolution can be treated in a very rigorous manner. Bodies which are possibly in contact may be internally discretized by finite elements (Fig. 9.1), and their material behaviour can essentially be of any complexity (viz. Chapter 5). The category of discrete element methods specifically refers to simulations involving a large number of bodies where the contact locations and conditions cannot be defined in advance and need to be continuously updated as the solution progresses. Discrete element methods are most frequently applied to macroscopically discrete system of bodies (jointed rock, granular flow) but have also been successfully utilized in a microscopic setting, where very simple interaction laws between individual particles provide the material behaviour observed at a homogenized, macroscopic level. The discrete element method is most commonly defined 1 as a computational modelling framework which 1. allows finite displacements and rotations of discrete bodies, including complete detachment; 2. recognizes new contacts automatically, as the calculation progresses.
Fig. 9.1 System of rigid or deformable bodies, discretization of bodies into finite elements, continuously changing configurations and possible fragmentation.
Early DEM formulations There exist many methods (e.g. DEM discrete element method, RBSM rigid block spring method, DDA discontinuous deformation analysis, DEM/FEM combined discrete/finite elements, NSCD non-smooth contact dynamics), which belong to a broad family of discrete element methods. 2-4 Although these methods appear under different names and each of them is developing in its own fight, there are many unifying aspects and a more general framework is emerging, which allows for an equivalence between these apparently different methodologies to be recognized. Possible classification may be based on the manner these methods address: (a) detection of contacts, (b) treatment of contacts (rigid, deformable), (c) deformability (constitutive law) of bodies in contact (rigid, deformable, elastic, elasto-plastic, etc.), (d) large displacements and large rotations, (e) number (small or large) and/or distribution (loose or dense packing) of interacting bodies considered, (f) consideration of the model boundaries, (g) possible subsequent fracturing or fragmentation and (h) time-stepping integration schemes (explicit, implicit). Discrete element methods are also used for problems where the discrete nature of the emerging discontinuities needs to be taken into account. Application ranges from modelling problems of a discontinuous behaviour a priori (granular and particulate materials, silo flow, sediment transport, jointed rocks, stone or brick masonry) to problems where the modelling of transition from a continuum to a discontinuum is more important. Increased complexity of different discontinuous models is achieved by incorporating the deformability of solid material and/or by more complex contact interaction laws, as well as by the introduction of some failure or fracturing criteria controlling the solid material behaviour and the emergence of new discontinuities.
The initial formulation of the discrete element method, originally termed distinct element method or DEM, 5 was based on the assumption of rigid circular bodies in two dimensions with deformable contacts. The overall solution scheme for the DEM is straightforward, typically formulated in an explicit time-stepping format. Movement of bodies is driven by external forces (Fig. 9.2) and varying contact forces (normal and tangential contact forces, proportional to current overlap and viscous contact forces proportional to the relative velocities in the normal and tangential direction). The method considers each body in turn and at any given time determines all forces (external or contact) acting at it. Out of balance forces (or moments) induce accelerations (translational or rotational), which then determine the movement of that body during the next time step. The simplest computational sequence for the DEM (often formulated in an explicit 'leap-frog' format, see Table 9.1) typically proceeds by solving the equations of motion of a given discrete element and updating contact force histories as a consequence of contacts between different discrete elements and/or resulting from contacts with model boundaries. The Rigid Bodies Spring Model, RBSM, 7 was proposed early as a generalized limit plastic analysis framework. Solid structures are assumed to be assemblies of rigid blocks, interconnected by discrete deformable interfaces with distributed (elastic) normal and tangential springs.
247
248
Discrete element methods
F3
ma
~. =/~kn
F2 (a)
F4 (b)
Fig. 9.2 (a) Discrete element bodies (particles)in contact, giving rise to axial and tangential contact forces. Force magnitudes related to the relative normal and tangential velocity and to relative normal and tangential velocity at the contact point; (b) arbitrary particle shapes as assemblies of clustered particles of simple shapes.
The stiffness matrix is obtained by considering rigid bodies to be connected by distributed normal and tangential springs with stiffness values k, and ks per unit length, respectively. The rigid displacement field within an arbitrary two-dimensional block is expressed in terms of the centroid displacements and rotation (u, v, O)r. Centroid degree offreedoms (Ui, 1)/, Oi) T and (uj, vj, Oj) r ofthe two neighbouring blocks (Fig. 9.3) with centroids located at (x ~ yO) and (x ~ yO), respectively, define independently the displacements at a common interface point P
~C I I
Di
C
Fig. 9.3 Two rigid blocks with an elastic interface contact in RBSM.
Early DEM formulations Table 9.1
Simple discrete element method algorithm (after Cundall and Strack 6)
ny (y~,y2) \ ei - - y i - x i - - ( c o s a D -ti -- (sin a, -- cos a)
sin
a)
Body centre (Xl, X2), (Yl, Y2)
1 ,.
~,, ( ,
Translational velocity xi, j;i
xi, 0 = const
Rate of rotation
"~i, O" = const
I
n-1
n
i
At
Ox,Oy
Time
n+l
n+2
A. F O R C E - D I S P L A C E M E N T LAW (1) Relative velocities (2) Relative displacements
J~i -- (Jci - Yi) - (Ox gx -[-Oyey)ti h : X i e i , k -- Xiti An = nat, As = sat
(3) Contact force increments
A F . = kn(An +/~h), A F s = k s ( A s +/3k)
(4) Total forces (5) Check for slip
F . = Fn-~ +
AFn, Fs =
F~"-~ +
AFs
Fs = min ( Fs , C + Fn tan q~)
If not converged go to (1), else Mx -- ~-~FxRx, My -- ~ - ] F y R y
(6) Compute moments B. EQUATIONS OF MOTION (1) Assume force and moment constant over At (2) Acceleration (3) Velocity
A t = (t n+l/2 -- t n-l~2) m Jci -- Y]~ Fi, I 0 : ~]~ Mi jcn+l/2 __ ion-l~2 0/+1/2 i "~- J(i At, 0 n-1/2 Jr- OAt At : (t n+l - t n)
(4) Assume velocities constant over At (5) Displacements (6) Rotation
~7+' = x~ + ~7+'/:at on+ 1 _. On _[_ on+ 1/2 A t
If not converged go to A (6), else Next time step: Go to A
C. TIME I N C R E M E N T
Up
--
Qpu
v;] U-
[Ui
1 QP-
Ui
0
i 1 0
0
0i
Hj
_ ( y ~ _ yO) (x~ - x ~
Vj
Oj] T
oo 0 1 o
0 o 1
o ]
0 -(y~-y~ (x~-x~
(9.1)
249
250 Discreteelement methods and the relative displacements at the location P are expressed as
10]
-- MUp = M R Q p u -- B u with M = [ ; 1
0 -1
0
and Up : l~Up - R Q p u
(9.2)
after Up is projected fJp aligned to the local coordinate system along the interface. The constitutive relation in plane stress can be written as of=Dr with D -
~r=
[
O'n,
ks
7"s
IT
h (1 - 2u) (1 + u)
E ks= h(1 + u )
(9.3)
where h is the sum of shortest distances between the two block centroids to the contact line. This projected distance h is also used to evaluate approximate normal and shear strain components through ep-
I l[ n] 1
% e
-h
6s = h r P
(9.4)
Applying the virtual work principle along the interface leads to f=
[~sBrDBds]u-Ku
(9.5)
Generalization to the three-dimensional situation is straightforward and the method can clearly be interpreted as a similar method to the finite element method with joint (interface) elements of zero thickness, the only difference coming from the assumption that the overall elastic behaviour is represented only by the distributed stiffness springs along interfaces. Given some criterion, the contact springs may be deactivated, so that an interface becomes a discontinuity and the progressive failure is modelled by following evolving discontinuities, through cracks and/or slipping at the interfaces between rigid blocks.
A principal algorithmic issue of the discrete element method represents a detection of bodies in contact followed by the evaluation of the contact forces (both a magnitude and a direction) emanating from the contact. The contact detection problem generally can be stated as one of finding a contact or overlap of a given contactor body with a number of bodies from a target set of N bodies in R" space but the strategies for contact detection are intimately related to the geometric characterization and topological attributes of interacting bodies. If the interacting bodies are of very simple geometry (e.g. circular in two dimensions or spherical in three dimensions) an algorithmic check for a possible overlap is simple and the definition of the tangential contact plane is unambiguous. Rigid bodies of more complex shapes can be approximated by forming convenient clusters of rigidly connected circular or spherical shapes, while the contact detection
Contact detection =
"
NG
J=i
0
h
I=i
I
..........
1..._
cell location for each body i= (int)[(xm/h) + 1]j= (int)[(Ym/k) + 1] Xm, Ym coordinates body centroid
Fig. 9.4 Hashing or binning algorithm for simple particle shapes and clustered particles. and resolution of the tangential contact plane remain the same as for individual bodies. However, this may be a crude approximation, and when interacting bodies of arbitrary geometry are considered, the algorithmic complexity of the contact detection and the associated definition of the contact plane between the two bodies increase significantly. The efficiency of these algorithms is crucial, as the conceptually simple procedure to test the possibility of contact of a body with all other bodies at every time step becomes highly uneconomical, once the number of bodies becomes large. Contact search algorithms are typically based on so-called body-based search or a space-based search. In the former, only the space in the vicinity of the specified discrete element is searched (and the search repeated only after a number of time steps), whereas the latter implies a subdivision of the total searching space into a number of overlapping
windows. For arbitrary geometric shapes, most algorithms typically employ a two phase strategy, where the bodies are first approximated by simpler geometric constructs (bounding boxes or bounding spheres) which encircle the actual body and a list of possible contact pairs is established via an efficient global neighbour or region search algorithm. This is then followed by a detailed local contact resolution phase, where the potential contact pairs are examined by considering the actual body geometries. This phase is strongly linked with the manner that the geometry of actual bodies is characterized.
9.3.1 Global neighbour or region search An example of the region search represents the boxing algorithm, g also referred to as hashing or binning algorithm.
251
252
Discrete element methods
The entire computational domain is typically subdivided into regular cells, and a list of bodies overlapping a given cell is established via the contact detection of square (2D) or cube (3D) regions. The contact resolution phase for a given body then comprises a detailed check for possible contact with all bodies which share the same cell and the check is usually extended to a list of bodies associated with neighbouring cells (i.e. 8 cells in two dimensions, 26 cells in three dimensions). The relationship between the cell size and the maximum size of a body is important for the overall efficiency. If the cell size is large compared to the body size, the initial search is fast, but many bodies are listed as potential contact pairs and the contact resolution phase is thereby extensive. However, if the cell size is small, the initial search is computationally more demanding, but this results in a smaller number of potential contact pairs and consequently a faster contact resolution phase. A balance is reached with cell sizes that are approximately of the size of the largest body in the system. Efficient contact detection algorithms and powerful data representation concepts are often borrowed from other disciplines, notably computer graphics, 9 with compact data representation techniques to describe the current geometric position of a discrete element - e.g. nodes, sides or faces. The decomposition of the computational space and various cell data representation for a large number of contactor objects (binary tree, quad tree, direct evidence, combination of direct evidence, rooted trees, alternating data trees) are usually adopted. 8'1~ Algorithmic issues and details of the associated data structures are quite involved and there is a non-linear relationship between the number of cells and the total number of bodies. Linear complexity contact detection algorithms are desirable and have been shown to be essential for simulations involving a very large number of bodies - e.g. the NBS (no binary search) algorithm for bodies of similar sizes has a total contact detection time proportional to the total number of bodies, irrespective of the particle packing density. 11 For simple shapes very efficient data structures map a minimum set of parameters which uniquely define a domain in Rn into a representative point in an associated R2n space (Fig. 9.5)- for example, a one-dimensional segment (a-b) is mapped into a representative point in two-dimensional space, with coordinates (a, b), or a two-dimensional rectangle of a size ( X m i n - - Xmax) and (Ymin -- Ymax) is mapped to a representative point in a four-dimensional space (Xmin, Ymin, Xmax, Xmax). Alternative representation schemes are also possible, e.g. by characterizing a rectangular domain in R 2 by the starting point coordinates (Xmin, Ymin) and the two rectangle sizes (hx, hy) followed by a mapping into an associated R 4 space (X~n, Ymin,hx, hy). As the representation of the physical domain is reduced to a point, region search algorithms are more efficient in the mapped R 2n spaces than in the physical R n space.
9.3.2 Contact resolution After the list of potential contact pairs is established through the global neighbour search, a detailed contact resolution algorithm is required, which will in turn depend on the way the detailed body geometries are defined. The contact resolution phase searches through potential pairs and if the actual contact is established, the algorithm needs to define the orientation of the contact plane, so that a local (n, t, s) coordinate
Contact detection Body 2 :
i
::Body 1
!
9
,
1 , , , ,
, , , , ,
I
~s
Body 3
i
X2S XlE
X3S X2E
X3E
in 2D space Body 1 ~ [x 1s, Xl E] Body 2 ~ [X2s,X2E] Body 3 ~ [X3s,X3E] XEnd
,.~
~
X3E X2E
tnd
X~E
XlS
X2S
X3S
Xstart
~ ............... - - ~ . - - o P4 P2 ~
P6 '" ..................
iI
o
ms
x"
in 4D space Body 1 ~ [P1,P2] Body 1 ~ [P1,P2] Body 1 -~ [P1,P2] Fig. 9.5 Mapping of a segment from one-dimensional space into a point in an associated two-dimensional space and mapping of a box in two dimensions into an associated four-dimensional space 9
system can be determined and the conditions for impenetrability or sliding can be properly applied. Body geometry characterization can be categorized 16 into three main groups: (a) polygon or polyhedron representation, (b) implicit continuous function representation (elliptical or general superquadrics) and (c) discrete function representation (DFR). Polygonal representation in two dimensions defines a body in terms of comers and edges, and there are a number of algorithms to determine an intersection of two
253
254
Discreteelement methods
Fig. 9.6 Definition of the contact plane: a unique definition for the corner to edge case (a), the edge to edge case (b) and an ambiguous situation for the corner-to-corner (c) contact problem (after Hogue16).
coplanar polygons. Convex polygons simplify the algorithm, as concave comers imply a possibility of multiple contact points. There are no difficulties to define the orientation of the contact plane when considering the comer-to-edge or the edge-to-edge contact, as the contact plane normal is uniquely defined by the edge normal. Difficulties arise when a comer-to-comer contact needs to be resolved (Fig. 9.6), as the orientation of the contact plane is not uniquely defined. This problem can be regularized by the rounding of comers 17 to ensure continuous changes of the contact plane outer normals. The generalization in three dimensions can be realized by the common plane concept, 17 which 'hovers' between the two bodies coming into a comer-to-comer contact, and the actual orientation of this common plane is found by a local optimization problem of maximizing the gap between the plane and a set of closest comers. Another possible procedure (restricted to a two-dimensional situation) utilizes an optimum triangularization of the space between the polygons, 18 whereby a collapse of a triangle indicates an occurrence of contact. A continuous implicit function representation of bodies, e.g. elliptical particles in two dimensions, 19'2~ellipsoids in three dimensions, 21'22 or superquadrics (Fig. 9.7) in two and three dimensions, 23-25 provides an opportunity to employ a simple analytical check (i.e. inside-outside) to identify whether a given point lies inside or on the boundary (~b(x, y) < 0) or outside (~b(x, y) > 0) of the body where ~(x , y)= (x) ~' + (b) & a
1
(9.6)
Unlike a polygonal representation, it is now significantly more difficult to solve for a complete intersection of overlapping superquadrics, and the solution is normally found by discretizing one of the surfaces into facets and nodes, and the contact for a specific node on one body can be verified through the inside-outside analytical check with respect to the functional representation of the other body. A discrete functional representation (DFR) describes the body boundary with a parametric function in one parameter. The concept of the DFR 26 replaces the continuous implicit function representation of bodies by the set of pre-evaluated function values on a background grid for the inside-outside check, which is used as an algorithmic look-up table. As such discrete function values at the grid nodes can be arbitrary, a grid (or cage) of cells can also be used to model an arbitrarily shaped body - including bodies with holes. The DFR concept in contact detection is illustrated through the polar DFR descriptor in two dimensions 27 (Fig. 9.8) where, following the global region search for possible neighbours, the local contact is established by transforming the local coordinates of the approaching comer Pi of a body i into the polar coordinates
Contact detection
r
84
Fig. 9.7 Superquadrics in three dimensions (reproduced from Hogue~6).
~
Ri
Nj
Fig. 9.8 Contact detection and the polar discrete functional representation (DFR) of bodies' geometry.
255
256
Discreteelement methods
of the other body Pij and checking if an intersection between the segments (Oj P/J) and (Mj Nj) can be found.
In the case of rigid bodies of simple shapes, an event-by-event simulation strategy can be applied in order to strictly satisfy the condition of impenetrability. In such cases, if the contact time is considered to be infinitely short, time instants of collisions can be calculated exactly and momentum exchange methodologies used to determine post-impact velocities, with possible energy loss accounted for by a coefficient of restitution or by friction losses. In either case there are non-smooth step changes and reversals in velocities. Such methodologies are used for molecular dynamics (MD) simulations with a very large number of particles. Although the event-driven algorithms work well for loose (gas-like) assemblies of particles, for dense configurations these lead to an effective solution locking or inelastic collapse, 28 manifested in critically slow simulations.
9.4.1 Regularization of non-smooth contact In the case of deformable bodies the contact time is finite and contact forces vary for the duration of the contact. In computational simulations it is necessary to regularize the non-smooth nature for the impenetrability and friction conditions. Constraints on impenetrability during the contact between the two bodies f2c and f2t require that the gap between them must be non-negative. In frictionless and cohesionless contacts, only a compressive interaction force Fn exists and this interaction force vanishes for an inactive contact described by the Signorini condition g > 0. The infinitely steep 'non-smooth' graph (viz. Fig. 9.9) is regularized by assuming that the interaction force Fn is a function of the gap violation and is replaced by a penalty formulation, with a linear or non-linear penalty coefficient. Non-smooth relations also exist if the interaction law considers a tangential friction force Fs, related and opposed to the relative sliding velocity k. For a Coulomb friction law, there exists a threshold tangential force, proportional to the normal interaction force Fs = # Fn, before any sliding can occur, corresponding again to an infinitely steep graph.
9.4.2 Contact constraints between bodies Once the contact between discrete elements is detected as a geometric overlap, the actual contact forces have to be evaluated, which are then used for the subsequent motion of the discrete elements controlled by the governing dynamic equilibrium equations. Contact forces follow from an imposition of contact constraints at contacting points. In variational formulations, a constraint functional FIc can therefore be added to the functional of the unconstrained system using a penalty function, Lagrange multiplier, augmented Lagrangian or perturbed Lagrangian form, as discussed in Sec. 7.2.
Contact constraints and boundary conditions 257
Fn
S
g i
g>-O
Fn>-O
Fng>- O
~Fn
f (gm k
g v
F =-kg >_0
Fn = f (gm) >_ 0
~=llFn
~-s=~F,
~ =/IFn
P,=~F,
IIF~II~~F~
F~=~
i IIF~II---~Fn
Fig. 9.9 Non-smooth treatment of normal contact, regularized treatment of normal contact (linear and nonlinear penalty term) and non-smooth and regularized treatment of frictional contact.
258 Discreteelement methods
'',,
%
%%%
n
t
g = (x i - x j ) . n < 0
g= (xi-xj).n>
O
Fi 9. 9.10 Determination of the contact surface and its local n - t coordinate system.
Most discrete element formulations utilize the penalty function concept. The information about the position and the orientation normal n of the contact surface (Fig. 9.10) as well as the current geometric overlap or penetration of contactor objects is used to establish the direction and the intensity of the contact forces between contactor objects at any given time. The impenetrability condition is formulated through the gap function g = [x i - xJ] 9n _< 0, which defines the relative displacement in the normal direction Un = [ u i - u j ] 9 n and the tangential direction ut = [ u i - u j ] 9 t. The resolution of the total contact traction tr into normal and tangential components t~ = (tc. n)n + (t~. t)t = t~' n + t [ t is then integrated over the contact surface to obtain the normal F, and tangential component Ft of the contact force. In the case of a comer-to-comer contact, no rigorous analytical solution exists, and rounding of comers for arbitrary shaped bodies leads to an approximate Hertzian solution. In the case of a non-frictional contact (i.e. normal contact force only) an elegant resolution to the comer-to-corner problem in two dimensions is provided by the contact energy potential algorithm. 29 Contact energy W is assumed to be a function of the overlap area between the two bodies W (A) and the contact force is oriented in the direction which corresponds to the highest rate of reduction of the overlap area A. As the overlap area is relative to bodies f2i and f2j, it can be expressed as a function of position of the comer point Xp and the rotational angle 0 with respect to the starting reference frame. The procedure (Fig. 9.11) fumishes a robust, unambiguous orientation of the contact plane in the two-dimensional comer-to-comer contact case, running through the intersection points g and h, and the contact force over the contact surface bw is applied through the reference contact point shifted by a distance d = Mo/11Fn11from the comer P, where the force Fn and the moment Mo are defined from the contact energy potential as
Fn=
OW(A) OA(xp, O) OW(A) OA(xp, O) and Mo= OA Oxp OA O0
Contact constraints and boundary conditions
A(Xp, O)
bw=
" ..... g
;
h
Fig. 9.11 Corner-to-cornercontact, based on energy potential (after Feng and Owen29). Different choices for the potential function (Table 9.2) are capable of reproducing various traditional models for contact forces.
9.4.3 Contact constraints on model boundaries Treatment of model boundaries represents an important aspect in the discrete element methods. Boundaries can be formulated either as real physical boundaries, or through virtual constraints. In the so-called periodic boundary, often adopted in the DEM analysis of granular media, a virtual constraint implies that the particles (or bodies) 'exiting' on one side of the computational domain with a certain velocity are reintroduced on the other side, with the same, now 'incoming', velocity. The use of periodic boundaries excludes capturing of any localization phenomena. In cases of particle assemblies, flexible and hydrostatic boundaries have also been employed. A flexible boundary 3~ framework is equivalent to physically 'stringing' together particles on the perimeter of the discrete element assembly, forming a boundary network. Flexible boundaries are mostly used to simulate the controlled stress boundary condition t Y i j = t7 C , where the traction vector tim _ _ Am crijnjC m is distributed over the centroids of a triangular facet Am, that is formed by three boundary particles. The hydrostatic boundary 31 can be interpreted as a virtual wall of pressurized fluid imagined to surround granular material particles and the desired hydrostatic pressure at the centroid of the intersection area between the particle and virtual wall. Table 9.2 Contact energy potential functions (after Feng and Owen 29)
W (A )
IIF.II
Hertz-type form
kn A 2 kn A 3/2
kn bw kn A 1/2bw
General power form
1 A mkn
Linear form
m
kn Am-1 bw
259
260 Discreteelement methods
The simplest form of a physical boundary representation in two dimensions is given by the definition of line segments (often referred to as 'walls' in the DEM context), and the kinematics of the contact between the particle and the wall is again resolved in the penalty format. Frequently, individual particles are declared as immovable, thereby creating an efficient way of representing and characterizing a rigid boundary, without any changes in the contact detection algorithm. An interesting idea is the finite wall (FW) method, 32'33 with the boundary surface triangulated into a number of rigid planar elements, which are then represented by inscribed circles and subsequently used in the contact detection analysis between the particles and the boundary.
Consideration of deformability increases the complexity for the analysis of multibody systems, where the bodies represent fully deformable media or belong to a class of constrained media, corresponding to restricted deformability. The deformability of individual discrete elements was initially dealt with by subdividing the body into triangular constant strain elements, 34 which can be identified as an early precursor of today's combined finite/discrete element modelling. Further developments of the discrete element methods include simple body deformability, so that a displacement at any point within a simply deformable element can be expressed by U i = U 0 "-["~.)ijXj + EijX J, where u ~ are the rigid body displacements of the element centroid, ~gj, Eij are the rotation and strain tensor respectively and x~ represent local coordinates of the point, relative to element centroid. It should be noted that this deformability statement implies a spatially constant deformation gradient that is equivalent to the class of pseudo-rigid bodies 35 discussed in Chapter 8. Displacements of body centroids follow from balance equations for the translation of the centre of mass in direction ~-~i Fi -" nit, and the rotation about the centre of mass ~-~i M/c - I ~Ji. The simply deformable discrete elements 17 comprise generalized strain modes ek, independent of the rigid body modes mk)~ = ~ - ~r~, where m ~ is the generalized mass, ~rk is a generalized applied stresses and crk/ generalized internal stresses, corresponding to strain modes. The discrete element deformation field (displacements relative to the centroid) can also be expanded in terms of the eigenmodes of the generalized eigenvalue problem, associated with the discrete element stiffness and mass matrix, giving rise to the Modal Expansion Discrete Element Method, 36'37 where the 'deformability' equations are uncoupled (due to the orthogonality of eigenvectors) and appear as modal equations written with respect to a non-inertial frame of reference. In the practical realization and implementation of the discrete element methodology the deformability of a discrete element of an arbitrary shape is either described by an intemal division into finite elements (discrete finite elements and/or combined finite/discrete elements) or by the polynomial expansion of a given order for the displacement field (discontinuous deformations analysis).
9.5.1 Combined finite/discrete element method A combination of discrete and finite elements was employed in the discrete finite element approach by Ghaboussi. 38 The combined finite/discrete element approach of
Block deformability 261 Munjiza e t al. 39'40 introduces bodies deformability through finite element expansion of the displacement field within a discrete element, while the contact between the discrete elements is solved again in an explicit transient dynamic setting. The overall algorithmic framework for a combined finite element/discrete element framework (Table 9.3) proceeds by solving the balance equations, while updating force histories as a consequence of contacts between different discrete element domains, internally subdivided into finite elements. A combined finite element/discrete element method is easily extended to problems comprising progressive fracturing and fragmentation, as complex constitutive relations can be utilized within discrete elements. Large displacements and rotations of discrete domains, internally discretized by finite elements, have been formulated in a generalized updated Lagrangian (UL) format by Barbosa and G h a b o u s s i . 41 The contact forces were obtained using a penalty form (concentrated and distributed contacts) and transformed into equivalent nodal forces on the finite element mesh. The equations of motion for each of the deformable discrete elements (assuming also a presence of mass proportional damping (C = ctMt)) are then expressed as MtU
+ o~MtU .
fetxt .+ fctont.
fitnt.
fetxt _1_ ftont
~-~/
Ja
t T t0"k df2k (Bk)
(9.7)
In the computational solution Barbosa and Ghaboussi 41 used a central difference scheme for integrating the incremental updated Lagrangian formulation, which neglects the non-linear part of the stress-strain relationship. Evaluation of the internal force vector fi
nt
At-
~-~~ k
~+At
(Btk+At)T--t+Atdak ~
at the new time station t + At recognizes the continuous changing of the configuration, as the Cauchy stress at t + At cannot be evaluated by simply adding a stress increment Table 9.3 Pseudo code for the combined discrete/finite element method, small displacement analysis, including material non-linearity (after Petrinid 12) (1) Increment from the time station t -- tn current displacement state Un external load vector, contact forces -nFeXt, FnC ~ ~-ext internal force, e.g. Finnt = ff2 BT O'n df2 (2) Solve for the displacement increment from M / i n + F int - - 1 ~ ext --n --n tin+l/2 = M -l(~'extn - Fint)n At + iln-1/2 Un+l = Un + iln+l/2At for an explicit time-stepping scheme (3) Compute the strain increment A~n+l = f(Aun+l) (4) Check the total stress predictor O'*+1 = 0"n + D A~Tn+l against a failure criterion, e.g. hardening plasticity qS(0"n+l, N) = 0 (4) Compute inelastic strain increment 'A~-inel e.g. using an associated plastic flow rule - " " n + 1' A ~inel (5) Update stress state 0"n+l = D(A~n+I - "'~'n+l ) (6) Establish contact states between discrete element domains at tn+l and the associated contact forces Fn+ 1 (7) n ~ n + l ,
go to step (1)
262 Discreteelement methods due to straining of the material to the Cauchy stress at t and the effects of the rigid body rotation on the components of the Cauchy stress tensor need to be accounted for [viz. incremental form (6.90b) or Jaumann-Zaremba rate form (6.109)]. For inelastic analyses, care needs be given to the objectivity of the adopted constitutive law. Advanced combined discrete/finite element frameworks 42 include a rigorous treatment of changes in configuration, evaluation of the deformation gradient and the objective stress measures.
9.5.2 Discontinuous deformation analysis The discontinuous deformation analysis (DDA) employs a general polynomial approximation of the displacement field superimposed over the centroid movement for each discrete body. 43'44 The DDA development followed the formulation of the Keyblock Theory 45 and was used for simulating the behaviour of a jointed deformable rock. Blocks of arbitrary shapes with convex or concave boundaries, including holes, are considered. The original framework under the name the 'DDA method' comprises a number of distinct features: (a) the assumption of the order of the displacement approximation field over the whole block domain, (b) the derivation of the incremental equilibrium equations on the basis of the minimization of potential energy, (c) the block interface constitutive law (Mohr-Coulomb) with tension cutoff and (d) use of a special implicit time-stepping algorithm. The original implementation has been expanded and modified by many other researchers. The method is realized in an incremental form and it deals with the large displacements and deformations as an accumulation of small displacements and deformations. The issue of inaccuracies when large rotations are occurring has been recognized and several partial remedies have been proposed. 46'47 The DDA method represents an alternative way of introducing solid deformability into the discrete element framework, and block sliding and separation is considered along predetermined discontinuity planes at block boundaries. The initial formulation was restricted to simply deformable blocks (constant strain state over the entire block of arbitrary shapes in two dimensions, similar to a constant deformation gradient as in pseudo-rigid bodies, Fig. 9.12). The first-order polynomial displacement field for the block [u ' v] ir 1st is equivalent to the three displacement components of the block centroid, augmented by the displacement field with a clear physical meaning of three constant strain states, i.e. it represents a constrained medium capable only of sustaining a spatially constant displacement gradient. The six deformation variables are denoted by the block deformation vector D~st.
[;]
_ [0 st 9
nZlst =
-
L.J 1st
[u0
0 1
--(Y--Yo) (x - Xo)
1)o d/)0 ~x
(X--Xo) 0
Cy 7xy]ir
0 (Y - Yo)
(y--yo)/2] Dilst (x - Xo)/2]i (9.8)
Tilst Dilst
An increase of block deformability characterization can be achieved by increasing the order of the displacement polynomial used, leading to a correspondingly larger number
Block deformability 263
Di = [ u, v, q~, ex, % exy]
ii
~ao ~
bo
81 bl
",
[;1:[,o ~ ; o o~; ~~ ~ x
,.
I
82
:Ox~~o ~I ~
y
~o,
53
a4 b4 85 .55_
Fig. 9.12 Deformation variables for the first- and second-order polynomial approximation in discontinuous deformation analysis.
of block deformation variables (higher-order DDA, where higher-order strain fields are added for blocks of arbitrary shape). By increasing the order of the displacement field, the block medium becomes less constrained and gradually moves towards full deformability. A second-order approximation for the block displacement field requires 12 deformation variables, which can again be given a recognizable physical meaning. The formulation implies a linearly varying displacement gradient across the block domain and the deformation parameters comprise the centroid displacements and rotation, strain tensor components at the centroid, as well as the spatial gradients of the strain tensor components, i.e.
D~nd--[U0
Vo
0 ----C x
dp0
0
~y
0
")/xy ~x,x
~y,x
~xy,x
~x,y
~y,y
'ffxy,y ] iT
(9.9) For a higher-order approximation of order n for the block displacement field, 48 the spatial distribution of the deformation gradient is of the order (n - 1) and a clear physical interpretation of the deformation variables is no longer plausible. The generalized block deformation variables are defined as U
=
dl at- d3x + dsy + d7x2 ~- d9xy + dllY 2 -+-... + dr_l xn + . . . + dm_lY n d2 4- d4x + d6y + d8x2 + dloxy + dl2y 2 + . . . + drx n + . . . + dmy n
Iul i
i
-
kJ
i
T.thD.t h
i
Dnth
d3
.
(9.10)
nth
Once the block displacement field is approximated with a finite number of generalized deformation variables, the associated block strain and block stress field can be expressed
264
Discrete
element
methods
in a manner similar to finite elements as
ei -- BiDi Eiei -- EiBiDi
ori =
(9.11)
For a system of N blocks, all block deformation variables (n variables per block, depending on the order of the approximation) are assembled into a set of system deformation variables (N x n) and the incremental equilibrium equations are derived from the minimization of the total potential energy FI comprising contributions from the block strain energy Fie, energy from the external concentrated and distributed loads FI p, Flq, interblock contact energy FIe, block initial stress energy H~, as well as the energy associated with an imposition of displacement boundary conditions using a penalty approach Fib I-I = He ~- lip if- I-Iq d- 1-Ic d-- I'I~r q- lib
(9.12)
Components of the stiffness matrix and the load vector are obtained by the usual process of the minimization of the potential energy
Kii -- f~i BTEiBi dr2
(9.13)
The global system stiffness matrix (Fig. 9.13) contains (n x n) submatrices Kii and non-zero submatrices Kij are included when the blocks i and j are in active contact and D comprises displacement variables of all blocks considered in the system. The interblock contact conditions of impenetrability and Mohr-Coulomb friction can be interpreted as block displacement constraints, which are algorithmically reduced to an interaction problem between a vertex of one block with the edge of another block. Denoting the increments of deformation variables of the two blocks in contact by I)i and Dj respectively, the penetration of the vertex in the direction normal to the block edge can be expressed as a function of these deformation increments and various algorithmic approaches can be adopted for an implicit imposition of the impenetrability condition (Fig. 9.14). Ifthepenaltyformat is adopted to impose contact constraints, additional terms appear both in the global DDA stiffness matrix as well as in the RHS load vector (Table 9.4),
Kll
K21 9
Kr,,~
Mlm
D1
fl
K12
"'"
i
"'"
:
"
"
Kn~2
"'"
r~,,,
Dr,
fr,,
K22
.."
K
K2r,,
D2
D
f2
=
Fig. 9.13 Assembly process in the DDA analysis.
f
Block deformability
C
~i
V
Fig. 9.14 Non-Penetrationand frictional contact constraint in DDA, point-to-edge contact.
Table 9.4 Additional terms in the DDA stiffness matrix and the load vectors as a result of contact between bodies i and j Normal non-penetration constraint Kii : Kii + pHi HT T Kij - H i G j
fi = fi - p C Hi
Kji ~-- G j n f
fj -- fj - p C G j
Kjj = Kjj + pGjGj T
Frictional constraint Kii -- Kii Jr p n f r H f r'T Kij "- U f r G f r'T
fi -- fi - p c f r n f r
Kji = G f r a f r,T
fj ~- fj - p C fr G f r
_l-~fr f~ fr, T
K j j -- K j j at- lJ~J j IJ j
and the terms differ depending on the nature of the constraint (normal gap or frictional constraint). The DDA method typically adopts an implicit algorithm and uses so-called openclose iterations which proceed until the global equilibrium is satisfied (norm of the outof-balance forces within some tolerance) and a near zero penetration condition satisfied at all active contact positions. For the normal gap condition convergence implies that the identified set of contacts does not change between iterations, whereas for the frictional constraint it implies that the changes in the location of the projected contact point remains within a given tolerance. For complex block shapes, the convergence may be very slow, as both activation and deactivation of contacts during the iteration process are possible.
9.5.3 Block fracturing and fragmentation Consideration of block deformability allows for a more precise determination of stress states throughout the discrete element. Block interface and through-block fracturing
265
266
Discrete element methods
and fragmentation were introduced to the discrete element method in the form of brittle fracturing. 49-52 Later models recognized the need to regularize the strain softening response for quasi-brittle materials, prior to eventual separation through cracking and/or shear slip). Constitutive models adopted for a pre-fragmentation stage are often based on concepts of continuum damage mechanics, regularized strain softening plasticity formulations or are formulated using some higher-order continuum theory. 53 Computational issues for discrete element methods, when more complex formulations for inelastic constitutive models are adopted, are the same as in the continuum FEM context computational inelasticity, where the consistent linearization procedure has affected the application of plasticity of fracture criteria in the DEM context. Moreover, every time a partial fracture takes place, the discrete element changes its overall geometry while a complete fracture leads to a creation of two or more discrete elements, there is a need for automatic remeshing of the newly obtained domains (Fig. 9.15). An unstructured remeshing technique can be applied, where the new mesh orientation and density is decided upon based on the distribution of the residual material strength, or some other state variable. 39 An additional algorithmic problem arises upon separation, as the 'book-keeping' of neighbours and updating of the discrete element list are needed whenever a new (partial or complete) failure occurs. In addition, there is also a need to transfer state variables (plastic strains, equivalent plastic strain, dissipated energy, damage variables) from the original deformable discrete element to the newly created deformable discrete elements. Fragmentation frameworks are also used with DEM implementations which consider clusters of particles bonded together to represent a solid body of a complex shape. The bond stiffness terms are derived on the basis of equivalent continuum strain energy. 54'55 In these lattice-like models of solids, fracturing (Fig. 9.16) is introduced through breakage of interparticle lattice bonds, which may be treated as a simple normal bond (truss elements) and/or parallel bond (beam elements), which can be seen as a microscopic representation of the Cosserat continuum. Bond failure is typically considered on the basis of limited strength, but some softening lattice models for quasi-brittle material have also considered a gradual reduction of strength, i.e. softening, before a complete
\
Fracture orientation J
/
/
N2
'
Remeshing with through-element fracture
Remeshing with interelement fracture
Fig. 9.15 Elementand node-based local remeshing algorithm in a two-dimensional context (Munjiza et al.;39 Feng and 0wen29), following the weighted local residual strength concept.
Time integration for discrete element methods 267
!
,!.!,!-..
l
i~,.
.
..:!!! 8 4
..~.. ~
Fig. 9.16 Fracturing of the notched concrete beam modelled by jointed particulate assembly, with normal contact bond of limited strength (Itasca PFC 2DS6).
breakage of the bond takes place. Despite use of very simple bond failure rules, bonded particulate assemblies have been shown to reproduce macroscopic manifestations of softening, dilation, and progressive fracturing. Simulations using bonded particle assemblies typically use an explicit time-stepping scheme, i.e. the overall stiffness matrix is never assembled and steady-state solutions are obtained through dynamic relaxation (local or global damping by viscous or kinetic means). A jointed particulate medium is very similar to the more recent developments in lattice models for heterogeneous fracturing media. 57-59 Adaptive continuum/discontinuum strategies are also envisaged, where the discrete elements are adaptively introduced into a continuum model, if and when conditions for fracturing are met. Discrete element methods are also coupled with fluid flow methods and a number of combined finite/discrete element simulations of various multi-field problems have been reported.
Governing balance equations are integrated in time using a time-stepping scheme, such as the GN22 method discussed in Chapter 2 and in reference 60. Both the usual DEM and the DDA time-stepping schemes can be interpreted as members of the GN22 family of algorithms and only the issues pertaining to their use in a discrete element context will be discussed here. The traditional DEM framework implies a conditionally stable explicit time-stepping scheme. Local numerical dissipation is sometimes introduced to avoid any artificial increase in the contact energy. 61 Such modified temporal operators do not affect the size of the critical time step, and the choice of the actual computational time step is controlled by accuracy requirements resulting from the numerical energy dissipation added to the system. An energy balance check is desired, for the purpose of monitoring possible creation of spurious energy, as well as monitoring energy dissipation during fracturing, where the inclusion of softening imposes severe limits on the admissible time step. When inertial effects are omitted in the DDA context, the resulting system of equilibrium equations may be singular when blocks are separated or have insufficient
268
Discreteelement methods
constraints. The regularity of the system stiffness matrix is restored by adding very soft spring stiffness to the block centroid deformation variables. Such modification is not necessary when inertial effects are included. The DDA framework typically utilizes a particular type of the generalized collocation time integration scheme 62 M
Stn+ol "t- C Xn+c~ + K Xn+o~ - -
f.+~
(1 - a) Stn 4- O/Stn+l fn+~ -- (1 -- a) fn + afn+l 1 (aAt)2[(1 _/32) St. +/32 Stn-t-1]
Stn+a
--
(9.14)
Xn+~ -- Xn + a At[(1 - ill) Stn +/~1Stn+l] which, for the case a - 1, leads to a recursive algorithm 2 M + 2/~1 C] Xn 2 M + 2/31 C + K] fl2At [/~2At i /~2At Xn+l =fn+l+ [fl2At2
1
M+ ( 2/31-1) C]xn+ [(~2-1)
/31
M+At(~22-1)
CI
_st"
(9.15)
A specific choice of the time integration parameters /~1 - - /~2 - - 1 represents an implicit, unconditionally stable scheme. The DDA time-stepping algorithm is also referred to as Right Riemann in accelerations, 63 as both the new displacement and velocity depend only on the acceleration at the end of the time increment as Xn+1 - -
Xn
Xn+l - - Xn
-~- At xn + g1 A t 2 St.+1 "-t- At Stn+ 1
(9.16)
which may also be expressed as 1 Xn+ 1 - - X n -~- ~
1
At [~:. +
Xn+l]
(9.17)
~t.+l - A-t [/~.+l -/~n]
For this choice of time integration parameters the coefficient matrix associated with the acceleration vector St. vanishes, hence
i+xTC+I
xT M+ TC Xn+
i+C
Xn+l= L+I
(9.18)
The effective stiffness matrix (( includes the inertia and damping terms and the effective load vector fn+l accounts for the velocity at the start of the increment. Thus, the next solution Xn+l is obtained from (9.19) Xn+ 1 ~ l ~ - l ? n + 1 Using Eq. (9.17) the next rate of deformation vector (velocity) then equals 2 X~+l = ~ [X.+l - x.] - ~:.
(9.20)
Time 1,4
0.8
'
t
I
I
I
I
methods I
I
atio
_
~u~ 0.4
t_
.~ 0 . 6 -
0.3 0.2
0.4t
0.20.01
J
element
0.5
~ 9 0.8-
0
I
discrete
0.6
1
"0
Q..
I
for
0.7
1.2=
integration
'
,
,I
O. 1
,
,
,I
,
1
\
0.1 ,
~I--
10
,
,
,I
1 O0
,
,
,
1000
0 ~,
0
i
1
i
2
i
3
i
4
i
i
5 6 f~
i
7
i
8
i
9 10
Fig. 9.17 Spectralradius and algorithmic damping for the DDA time integration scheme (generalized Newmark scheme GN22/~i = ~2 = I) (reproduced from Doolin and Sitar63).
which is then used for the start of the next time increment. We note that in this form it is never necessary to compute the acceleration during the solution process. If necessary, the acceleration may be computed to interpret the inertial effects in the solution. The effective stiffness matrix is regular due to the presence of inertia terms and the block separation can be accounted for without a need to add artificial springs to block centroid variables, even when blocks are separated. The time-stepping procedure can also be used to obtain a steady-state solution through a dynamic relaxation process. The steady-state solution can be obtained by the use of the so-called kinetic damping, i.e. by simply setting all block velocities to zero at the beginning of every increment. From the analysis of spectral radii for the above time integrator (Fig. 9.17), it is clear that the scheme is associated with a very substantial numerical damping, 63'64 which is otherwise absent in the central difference scheme. Moreover, predictor-corrector or even predictor-multiple corrector schemes are adopted in simulations of granular materials, 65 in order to capture the high frequency events, like the collapse of arching or avalanching. The discrete element method simulations are computationally intensive and there is a need for the development of parallel processing procedures. The explicit time integration procedure is a naturally concurrent process and can be implemented on parallel hardware in a highly efficient form. 66'67 Much of the computational effort devoted to contact detection and the search algorithms, which is formulated principally for sequential implementation, has to be restructured to suit parallel computation. Efficient solution procedures are employed to minimize communication requirements and maintain a load balanced processor configuration. Parallel implementations of DEM are typically made on both workstation clusters as well as multi-processor workstations. In this way the effort of porting software from sequential to parallel hardware is usually minimized. The program development often utilizes the sequential programming language with the organization of data structure suited to multi-processor workstation environment, where every processor independently performs the same basic DEM algorithm on a subdomain and only communicates with other subdomains via interface data. In view of the large mass of time dependent data produced by a typical FEM/DEM simulation, it is essential that some means be available of continually visualizing the solution process. Visualization is
269
270
Discreteelement methods
particularly important in displaying the transition from a continuous to a discontinuous state. Visual representation is also used to monitor energy balance, as many discretization concepts both in space (stiffness, mass, fracturing, fragmentation, contact) and time can contribute to spurious energy imbalances.
ii~liJ~ili~i i~i~ i ililiU':~"~i. .i.i.!.i.ii;i.iiii.iiiiii;iilJiJii~, ~ ~'iJ~i~'i:~' i~ii~ililiiiii~il iii~iiliiiii~i~~i~i~i~i;i~i~i i !i~i i i ililJiiii~iiiiiiiiiiiiii!i! i i iJi~i~'i:::::!~!~i :ii!iiii~i~iil~ii?~ :ii~iiiiii!i i ~iiiiiilJi~i i~i~i~iiii!ii~iiiiiiiiiiii ~i~iiiiiiiii i ii~iiiii i ~'~'~iiii~ii~~iliiililil;iiiii!iiiiiiiiii!iiii~ i~iiiiilii!~iiiiiiiiii!i!iiii ii!i E~i~iiii iJi!i~i~i~!~d~iii~ iiiiliiiiiliii~iii~iiii iiiiiiiii~i i~ii~iii!ii!i i i':':i::i i i i ~!il:i~::i:'~i~'!i :':i'ii~iiii!i iiii~iiiiii!iiiii;iiii i~i ~i~iiiiiiiiii~ii!i ! iliiiiiiili i i~ilil ii~!iiliii~ i~~i~iiii'i:':i'~i!;iii i iiiii~ii!i ~ iiiiiidii!iilil~i~iiiii:' l:':iiiiiiiiiii~ ~ i~ii~i!if:~:':'~iiiliiiliiiiiiliiii~i~i;iiii~iilii~ l iiiii!iiiiiliiiiiii~i!iliiiii~~ii~i~i~!ii~!iilii~!~~i iii~i~!~i~!!~ ',!',i',!':i~i'~i',:~ii '~i~',!i~','i~i:~,"!~i~i~!i~:~i~!~i~i~iii~i~i'!,~i'~~i',i~i'i,~~ii~i!~i'~i~',i',i~',i:'i~i~i~i~,i~i,~ii~,i'i,~!i~'i~,i~',i',~i~i'~,!',i!'i~!'~!~, ,i'~,i,~'i~i',i~',ii'i!i~',~i~',i,'i!~,!ii',i',i~,i',ii~i~, ~,ii'~'~ii'~,,i',i'~,iili'~i~, '~'~il:i',i~i',i'i',i~,,!iii',iil!',i!ii!i'~i':iii~,!i~,~i~:i' i','~i~'~/i~,,iii!i~ ii!~,',',i}i'~,,i'!},~i~',i'i',~,i,',ii'i',i!',i'!,iili',i~,~,!i:,',i'~iii'i',i',~i'i:',iiii~,~:~,i~,i',i'i,ii!'',i,i'~i:,ili'iii,',:,i'i,i~,i!'',i!'~'~'/~i',i'~'~:,'~,i ~,~,~,~,~i~::~',i',i:!i':,~',~i~,~i~:,!,~~,',~,','~i':',~,i~,~,~~:~:~/~:~i~iii~ii~iii~i!~:~iii~::~ii~ ,',',i~i',':i',ii~,~/,',i ,,~i',':',',',i',~,',~i',ii~'~,',',~'~i':',',',,~'/:'i,'::', ~''~:
There are several other modelling frameworks which have been proposed as variations of the existing discrete element methods, e.g. the modified distinct element method (MDEM). 68 It is also interesting to observe a convergence of methodologies designed to deal with a transition from a continuum to a discontinuum. For example, in a continuum setting, the pre-existing macro discontinuities are traditionally accounted for through the use of interface (or joint) elements, 34 which may be used to model crack formation as well as shearing at joints or predetermined planes of weakness. 69 Joint planes are assumed to be of a discrete nature, their location and orientation are either given, or are fixed once they are formed. The term discrete cracking is adopted, as opposed to the term smeared cracking, where the localized failure at an integration point level is considered in a locally averaged sense. Natural extensions are interface constitutive models which account for a combination of cracking and Coulomb friction, which have been frequently formulated as two parameter failure surfaces in the context of computational plasticity. The non-smooth contact dynamics method (NSCDM) 7~ is closely related to both the combined FEM/DEM and the DDA, but it comprises significant differences, as the unilateral Signorini condition and the dry friction Mohr-Coulomb model are employed without resorting to contact regularization. For multi-body contact, with jumps in the velocity field, it is not possible to define the acceleration as a usual derivative of a smooth function and the non-smooth time discretized form of dynamic equations is obtained by integrating the balance equations, so that the velocities become the primary unknowns
ftn+l M (/r
-- Zk.) = J
tn
Xn+~ = x~ +
ftn+l f (t, X,/~) dt +
t,~+l
fd tn
r dt dtn
(9.21)
~: dt
The force impulse on the RHS is split into two parts, where the first part t,t"+' f(t, x, x) dt (which excludes the contact forces) is considered continuous, whereas the second part
f
t=+, r dt
(representing contact force contributions to the total impulse) is replaced by the mean value impulse 1 ft.+l = r dt rn+l "~ Jt,~
Unifying aspects of discrete element methods 271 over the finite time interval. In physical terms this implies that the actual contact interaction history is only accounted for in an averaged sense over the time interval. This has a consequence that the fine details of the contact history are discarded, which are either impossible to characterize due to insufficient data available, or inconsequential in terms of their effect on the overall behaviour of the multi-body system. Different time-stepping algorithms may be adopted and typically a low-order implicit time-stepping scheme is used. Resolution for contact kinematics then considers the relationship between the relative velocities of contacting bodies and the mean value impulse at discrete contact locations ]Krel -- Xrel-free "q- A t
K~rn+l
(9.22)
where the first part represents the 'free' relative velocity (without the influence of contact forces) and the second part comprises 'corrective' velocities emanating from contacts. The actual algorithm is realized in a similar manner to the closest point projection stress return schemes in computational plasticity- a 'free predictor' for relative velocities is followed by iterations to obtain 'iterative corrector' values for the mean value impulses, such that the inequality constraints (both Signorini and MohrCoulomb) are satisfied in a similar manner as the plastic consistency condition is iteratively satisfied in computational plasticity algorithms. The admissible domains in contact problems are generally non-convex and it is argued that it is necessary to treat the contact forces in a fully implicit manner, whereas other forces can be considered either explicitly or implicitly, 73 leading to either implicit/implicit or explicit/implicit algorithms.
There are many similarities in the apparently different discrete element methodologies, which partly stem from the fact that a given methodology is often presented as a package of specific choices, e.g. comprising a choice for the manner the bodies' deformability is treated, a way the governing equations are arrived at, a specific time integration scheme or a special way of dealing with unilateral constraints. For example, the DDA framework is often perceived as a methodology which specifically deals with simply deformable (constant strain) domains of arbitrary shapes, uses a very special case of implicit time-stepping scheme and treats contact constraints through a series of openclose iterations. Conversely, the DEM framework is often presented as a methodology for rigid particles of simple shapes, where the balance equations are integrated by an explicit scheme, with a penalty format for contacts. These perceived restrictions seem arbitrary, as there is nothing wrong, for example, in integrating the DEM equations with an implicit scheme or solving the DDA equations by an explicit method. It is also interesting to observe that the choice of a contact detection scheme was never perceived fixed and many contact detection algorithms have been applied. It can be observed that the major developments for all discrete element methods have primarily been associated with a characterization of bodies' enhanced deformability. While the higher-order DDA increases the order of polynomial approximation for the displacement field, other attributes of the method remain the same. In the same spirit
272
Discrete element methods
the combined finite/discrete element again enhances the description of deformability, while the other algorithmic features remain the same. It is therefore perhaps most appropriate to leave all other attributes aside and concentrate on the characterization of bodies' deformability and the treatment of geometrically non-linear motions as the basis for any comparison or equivalence and a potential identification of common, unifying concepts of different discrete element methodologies. In that context it is interesting to return to the background of the theory of pseudo-rigid bodies, 35 as discussed in Chapter 8. As in all discrete element methodologies, the theory is concerned with large-scale motions of deformable bodies. It was presented simultaneously as a generalization of the classical rigid body mechanics for bodies with some added deformability, as well as a 'restriction' (or coarse version in the terminology of reference 35) of a fully deformable continuum. Clearly, a hierarchy of theories emerges, depending on the degree of deformability added to the rigid body or on the class of restrictions introduced to the fully deformable continuum body. This is where the relationship between the theory of pseudo-rigid bodies and the discrete element methods provides a powerful link that allows for a rational comparison of different assumptions and an assessment of consequences of approximations adopted. In that spirit, the equivalence of the equations of motion between the elastic pseudorigid bodies and constant strain finite element approximations was confirmed and it was argued TM that the pseudo-rigid bodies can be viewed as a generalization of the low-order DDA in the regime of finite (as opposed to stepwise linearized as in DDA) kinematics. The crucial equivalence stems from the spatially homogeneous deformation gradient present in both cases, which can again be viewed as an enrichment of deformability given to the rigid body, or as a restriction imposed to the fully deformable continuum. The availability of a rational theory has an additional benefit by fully explaining manifestations of excessive volume changes for large rotations, discussed earlier in Sec. 9.7. Similarly, the higher-order DDA and higher-order pseudo-rigid bodies 75 can be unified in the sense that the deformation gradient F varies linearly over the body, i.e. F = F ~1) (X - X0) + F ~~
(9.23)
where F ~~ is the deformation gradient at X - X0 and F ~1) is the derivative of the deformation gradient with respect to X, assumed constant. Clearly, if F ~1) - 0, the simple pseudo-rigid body (or the lower-order DDA) is recovered. The extension to general higher-order approximations is plausible and again can be interpreted as enrichments of deformability or restrictions of the fully deformable continuum. Another rational interpretation of discrete element methods was developed based on Cosserat points theory 76 which moves away from algorithmic differences and concentrates on similarities between the DEM and DDA methods when they are recast using Hamilton's principle of least action.
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There is a degree of commonality of novel ideas in terms of describing the block deformability in discrete element methods and novel developments in the continuumbased techniques. The numerical manifold method of Shi 77 and Chen et al. 78 advocates
References 273 similar ideas to the ones advocated in the meshless 79 or the partition of unity methods 8~ in dealing with emerging discontinuities. Similar to the meshless methods, the manifold method identifies a cover displacement function Ci and the cover weighting function wi, where the geometry of the actual blocks ~"2 i is utilized for numerical integration purposes over a background grid. The treatment of any emerging discontinuities is envisaged by introducing the concept of effective cover regions, where there is a need to introduce n independent covers, if a cover intersects n disconnected domains. These concepts point to a range of possibilities in the simulation of progressive discontinuities in quasi-brittle materials. Discrete element methods increasingly appear in formulations and applications in multi-field or multi-physics problems, in particular in the area of the coupled fluid flow in discontinuous, jointed media. These discontinuous modelling frameworks are promising especially in the context of fragmentation and in the microscopic simulation of the behaviour of heterogeneous materials, where simple constitutive laws at the macro, meso or nano level directly generate manifestations of complex macroscopic behaviour, such as plasticity or fracture. 81 Increased computing power and efficient contact detection algorithms will not only allow modelling of progressive fracturing of continua and a transition to discontinuum including a fragmented state but will also encourage the development of discrete micro-structural models where an internal length scale may be intrinsically incorporated into the model. Moreover the large-scale simulations with adaptive multi-scale material models are increasingly feasible, where different regions are accounted for at a different scale of observation. Some of these aspects are considered in the next chapter.
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In the previous chapters we considered the domain to be a continuum, a rigid multibody system or a set of discrete elements. In the study of continuum problems we developed finite element approximations based on approximation of the displacement, stress and strain fields at each point in the domain. While such approximation is general there are instances when it is difficult to obtain viable solutions economically. Many such situations arise when one or two dimensions of the domain are small compared to the others. For example, when two dimensions are small we have a very slender cross-section which is translated along a one-dimensional axis as shown in Fig. 10.1. Such a form is herein called a rod and consists of a member which carries axial, shear, moment and torsion force resultants. When one dimension is small compared to the other two we have either a plate theory for initially flat surfaces or a shell theory for general curved surfaces. In this chapter we consider the behaviour of rods. Plate and shell problems will be considered in subsequent chapters. Bending of rods is generally associated with a beam theory such as the classical Euler-Bernoulli theory studied in introductory strength of materials. ~-3 If one attempts to model a rod with a standard three-dimensional finite element model there are two aspects which give difficulty. One is purely numerical and associated with large roundoff errors when attempting to solve the simultaneous equations. 4 The other is a new form of locking in interactions between bending, shear and axial behaviour when loworder elements are used. Often a much more economical solution is to use a structural mechanics approach in which the problem is formulated as a one-dimensional problem along the axis of the rod. Using this approach and appropriate interpolation forms one can avoid numerical difficulties associated with round-off and locking. In this chapter we consider approximation for two classical rod theories. The first combines the Euler-Bernoulli theory of bending with axial and torsion theories. In this theory the deformation field is restricted to axial, torsion and bending strains. The application of the Euler-Bernoulli theory is usually restricted to situations where dimensions along the axis of the rod are at least ten times those of the transverse dimensions. In the second form we consider the Timoshenko theory of bending together with the axial and torsion theories. The Timoshenko theory adds a transverse shear deformation to the other strains and is applicable when the length to cross-section dimensions are above five (when smaller than this the continuum theory becomes viable).
Governing equations 279
Fig. 10.1 Slenderrod. Care should always be used when measuring the length parameter. Distances between sudden changes in cross-section or loads should be considered. Also, when transient effects are included the frequency content of the solution will establish which natural 'modes' are active and the length between modal zeros also needs to be considered. Use of the Timoshenko theory can lead to 'locking' effects when the theory is applied to cases where the Euler-Bernoulli theory also could be used. However, it is desirable to have a single formulation which remains valid throughout the range of length to cross-section considerations and for this the Timoshenko theory should be used. In this chapter we show how finite element approximations, which are free from locking effects, can be developed for the Timoshenko theory. These are also useful when we consider plate and shell problems in the later chapters. Indeed one of our goals for presenting the rod theory is to develop an understanding of the delicate nature of 'locking' when structural mechanics formulations, including those of plate and shell problems, are solved by finite element methods.
We consider a straight rod with the axis of definition in the x direction and the crosssection A in the y - z plane (Fig. 10.1).* In the study of rods we will assume that the primary stress components are the normal stress Ox and shear stresses 7"xy , Txz acting on each cross-section. The remaining stresses, while they can exist, are of less importance and their effects are either ignored or are included as applied boundary loads to the rod as indicated next.
10.2.1 Equilibrium equations We first consider the static behaviour of a rod where the local balance of momentum equations at each point of a body are expressed as * In the study of structural mechanics formulations we shall revert to the notation where coordinates are denoted by x, y, z. A similar type of notation will also be introduced for displacements, strains and stresses.
280
Structural mechanics problems in one dimension- rods
O~x
Or,~
O-r=,
-~x + - ~ y + - ~ z + b x = 0
+N+W+b,,-o 0r
Orv~.
(lO.la)
Oo-~.
and
T~y = "
. .
o
.
,
0.02 0
. .
~-0.061" .
,
.
.
.
-0.04 I. . . . .
,
.
;,, r,5,'" .
'~ . \\ .
j
;f-t . . . . . . . . . .
.
,~ .
f.,
-
1
Fig. 10.11 theory.
1.2
-
-, ,
.
.
.
.
.
.
.
. ,
.
.
.
.
.
.
.
,
,
,
1.6
1.8
Solution for tapered
0
.
. . . . .
,
1.4
x/L (c) Moment
.
.....
beam using
2
-0.2
1
". . . . . ,. . . . . . . . .
1.2
i - - -e'- --" i,,-"- -
"
~
-
,
1.6
1.8
-, . . . . . . . . . . . . . . . . . . . . .
1.4
x/L
~
(d) Shear
irreducible
(displacement)and
mixed solutions -Timoshenko
2
Finite element solution: Timoshenko rods 315
tions have error in all quantifies, although those for the displacement and slope are quite small.
10.5.3 Discrete Kirchhoff constraints .................................................................
- ............
- .....................
- .....................
-.........
=
................
=
........
. ..........
~ -:
.....................................................
. ......................
. ..............
-~- .....................................
: ......................................
- ........................................................................
In the previous discussion we have considered formulations for the Euler-Bernoulli and Timoshenko theories of beams. Using appropriate interpolations in the Timoshenko theory we have shown that it is possible to obtain a viable solution for situations in which the Euler-Bemoulli theory is accurate. We consider here an alternative to the previous approach by considering a reduced form of the functional for the Timoshenko theory to directly solve the thin beam case. In this form we use an irreducible form for the bending strains, set the constitutive behaviour for shear to zero, and introduce constraints to enforce zero shear strains. Initially this approach was applied to the study of thin plates based on the Kirchhoff theory and, thus, the approach is termed a discrete Kirchhoff method. In this section we illustrate the approach by considering a beam example. The reduced functional is given by
(~I'I ( u3, Oy ) -- fL ~O~Oy l~/Iy ( X.y ( Oy ) ) d x (10.95)
-- (~OyI~Y) IOMy --
OSz =
and is valid if we introduce appropriate constraints to satisfy (discretely)
Ow -~- Oy Ox
= 0
(10.96)
To solve the problem posed by Eqs (10.95) and (10.96) we can 1. approximate w and Oy by independent interpolations of Co continuity as w - Nw'~'
and
Oy = N00
(10.97)
2. impose a discrete approximation to the constraint of Eq. (10.96) and solve the problem resulting from substitution of Eq. (10.96) into Eq. (10.95) by either discrete elimination, use of suitable Lagrangian multipliers, or penalty procedures. In the application of the so-called discrete Kirchhoff constraints, Eq. (10.96) is approximated by point (or subdomain) collocation and direct elimination is used to reduce the number of nodal parameters. Of course, other means of imposing the constraints could be used with identical effect and we shall return to these in the next chapter. However, direct elimination is advantageous in reducing the final total number of variables and can be used effectively.
Example 10.12 One-dimensionalbeam example
We illustrate the process to impose discrete constraints on a simple, one-dimensional, example of a beam shown in Fig. 10.12. In this, initially the displacements and rotations
316
Structural mechanics problems in one dimension- rods
./-
0
Constraint
/x,
1
0
~a
(3
A
3
2
~13
0
(3
1
2
Fig. 10.12 A beam element with independent, Lagrangian, interpolation of w and Oy with constraint OwlOx + Oy - 0 applied at points x.
are taken as determined by a quadratic interpolation of an identical kind and we write in place of Eq. (10.97),
IWlOy
3
- ~ . ~ Na
a-1
Ill)aloa
(10.98)
where a are the three element nodes. The constraint is now applied by point collocation at coordinates x~ and x~ of the beam; that is, we require that at these points
Ow -~x + Oy - 0
(10.99)
This can be written by using the interpolation of Eq. (10.98) as two simultaneous equations
3
3
a=l
a=l
ENa(Xa)Wa-~L~Na(xa)Oa-O 3
E
a=l
where
3
N~a(Xfl)Ul)a-'1"Y~
[
N~ (x~) -- N~ (x) x=x~
a=l
and
Equations (10.100) can be used to eliminate we have
3 a=l
Aa
/tOa/ 0a
--
0 where Aa
(10.100)
Ua(xfl)Oa= 0
aN/[
N" (x~) - ~
x=x~
tO3 and 03. Writing Eqs (10.100) explicitly
- [Na'(Xa)'
[Na,(X•) '
Na(xa) 1 Na(xe) j
(10.101)
Substitution of the above into Eq. (10.98) results directly in shape functions from which the centre node has been eliminated, that is,
2 113 1~a ( 0y / - - E ( l~)a 0a } a=l
(10.102)
Forms without rotation parameters
with Na -- Na I - N3 A31Aa; a - 1, 2
where I is a 2 x 2 identity matrix. If these functions are used for the beam, we arrive at an element that is convergent. Indeed, in the particular case where x~ and x 9 are chosen to coincide with the two Gauss quadrature points the element stiffness coincides with that given by a displacement formulation involving a cubic w interpolation as described above for the irreducible form. In fact, the agreement is exact for a uniform beam.
It is possible to formulate the Timoshenko beam theory without direct use of rotation parameters. Such an approach has advantages for problems with large rotations where use of rotation parameters leads to introduction of trigonometric functions (e.g. see Chapter 17). Here we consider the case of a straight beam in two dimensions where each element is defined by coordinates at the two ends. Starting from a four-node rectangular element in which the origin of a local Cartesian coordinate system passes through the centroid of the element we may write interpolations as (Fig. 10.13)
x -- Ni(~, T]).~i 7I- Nj(~, ~?)2j + Nk(~, ~7)2k + NI(~, ?]).~l y = Ni(~, ?])Yi -']- Nj(~, ?])yj -Jl" N~(~, rl)Yk + NI(~, ~])Yl
(10.103)
in which Ni, etc. are the usual four-node bilinear shape functions. Noting the rectangular form of the element, these interpolations may be rewritten in terms of alternative parameters [Fig. 10.13(b)] as x = N1 (~)21 + N2 (~)3~2
y-~~7[Nl(~)t'~ +
N2(~)t'2]
(10.104)
where shape functions are NI(~) = ~1 (1
-
~)
,
N2(~)
--
~1 (1 + ~)
(10.105)
and new nodal parameters are related to the original ones through 1 -~1 = ~(3~i + X2--~~l(3~j +
-~l) and t'l -" Yl - Yk 2k) and t"2 -- Yk -- Yj
(10.106)
Since the element is rectangular t'l = t'2 = t" [Fig. 10.13(b)]; however, the above interpolations can be generalized easily to elements which are tapered. We can now use isoparametric concepts to write the displacement field for the element as
E
r~t u -- N1 (~) /~1 -~- ~ A/~I v -- NI(~)
1~1 + ~ A I ~ I
]
[
rF "~ N2 (~) /~2 "~ ~ A/~2 + N2(~)
1~2 -~- ~ A I ~ 2
1
(10.107)
317
318
Structural mechanics problems in one dimension - rods
(u., w.)
'I
I
i
(a)
i-o,
t
~.--x J
o
(,~u2, ~xv2)
(b)
(,aUn, ,aVn)
on
,,
l
i
(Un, Vn)
(c) Fig. 10.13 One-dimensional bending of planar beams: (a) geometry, Q4 element; (b) geometry, no rotation parameters; (c) joining elements with different thickness. in which/-is a 'thickness' parameter chosen to permit elements of different cross-section to be joined at a common node n [Fig. 10.13(c)]. It is evident that the above interpolations are identical to those originally written for the quadrilateral element. Only the parameters are different. Based on results for the incompressible problem, we also know the element will not perform well in bending situations because of 'shear locking', especially when the aspect ratio of the element length to depth becomes very large. In order to improve the behaviour we introduce a three-field approximation by using the enhanced strain concept described in reference 6. Accordingly, the mixed strain approximation will be taken as
~7
Cy =
Ni'x 0
%y
0
0
-~Ni'xO 1 -Ni t
0 Nix,
10
/~i
t Ni
ma i Ui
~]t I ~-~Ni,xj
mVi
Jr-
I0~
~] ~/~12
0 0 0 7] 0
~3
0
~4
(lO.lO8)
where fli are parameters of the enhanced strains. 6 The remainder of the development is straightforward and is left as an exercise for the reader. We do note that here it is not necessary to use a constitutive equation which has been reduced to give zero stress in the through-thickness (y) direction. By including additional enhanced terms in the
Moment resisting frames thickness direction one may use the three-dimensional constitutive equations directly. Such developments have been pursued for plate and shell applications. 34-37 We note that while the above form can be used for fiat surfaces and easily extended for smoothly curved surfaces it has difficulties when 'kinks' or multiple branches are encountered as then there is no unique 'thickness' direction. Thus, considerable additional work remains to be done to make this a generally viable approach.
The formulation given above may be used to solve moment resisting flame problems such as the plane frame shown in Fig. 10.14(a). Using a global coordinate system x' = (x', y', z') a transformation between the local frame x used in the above description of axial, bending and torsion is given by [Fig. 10.14(b)] x = Lx'
(10.109)
where L is an array of direction cosines. The same form may be used to transform the degrees of freedom for the problem. In a general three-dimensional setting we have
where Ua - -
Ua
,
/0"" / 0~ Oyo
with similar relations for 8' and 0'. ZI
>>y'
///
//
///
//
(a) Two-dimensional frame
(b) Coordinate system
Fig. 10.14 Momentresisting frame and coordinate transformation.
319
320
Structural mechanics problems in one dimension - rods
Using the above transformation, the residuals for each element are transformed by
and the tangent matrix by
K;,,
Koo j
{si}{s:} o] kKo. KooJ Lr
(10.112)
In this chapter we have summarized the basic steps needed to formulate and solve rod problems by the finite element method. We have focused attention on the development of accurate interpolation of the variables which, in certain circumstances, give exact solutions at interelement nodes. In addition, for the Timoshenko beam theory, we have shown how to obtain interpolation functions which do not 'lock' in applications to thin beam problems. This latter aspect will be exploited in the solution of plate and shell problems in the next chapters.
1. G. Wempner. Mechanics of Solids, with Applications to Thin Bodies. McGraw-Hill, New York, 1973. 2. E.P. Popov and T.A. Balan. Engineering Mechanics of Solids. Prentice-Hall, Upper Saddle River, NJ, 2nd edition, 1999. 3. F.P. Beer, Jr., E.R. Johnson and W.E. Clausen. Mechanics of Materials. McGraw-Hill, Boston, 3rd edition, 2002. 4. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997. 5. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 6. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 7. S. Klinkel and S. Govindjee. Anisotropic bending-torsion coupling for warping in a non-linear beam. Computational Mechanics, 31:78-87, 2003. 8. S.P. Timoshenko and S. Woinowski-Krieger. Theory of Plates and Shells. McGraw-Hill, New York, 2nd edition, 1959. 9. P. Tong. Exact solution of certain problems by the finite element method. Journal of AIAA, 7:179-180, 1969. 10. M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions. Dover Publications, New York, 1965. 11. V. Ciampi and U Carlesimo. A nonlinear beam element for seismic analysis of structures. In Proc. European Conference on Earthquake Engineering, pages 73-80, Lisbon, June 1986.
References 321 12. E. Spacone, V. Ciampi and EC. Filippou. Mixed formulation of nonlinear beam finite element. Computers and Structures, 58:71-83, 1995. 13. E. Spacone, F.C. Filippou and EE Taucer. Fiber beam-column model for non-linear analysis of R/C frames. 1. Formulation. Earthquake Engineering and Structural Dynamics, 25:711-725, 1996. 14. E. Spacone, F.C. Filippou and EE Taucer. Fiber beam-column model for non-linear analysis of R/C frames. 2. Applications. Earthquake Engineering and Structural Dynamics, 25:727-742, 1996. 15. A. Neuenhofer and F.C. Filippou. Evaluation of nonlinear frame finite element models. J. Structural Engineering, ASCE, 123:958-966, 1997. 16. A. Neuenhofer and EC. Filippou. Geometrically nonlinear flexibility-based frame finite element. J. Structural Engineering, ASCE, 124:704-711, 1998. 17. M. Petrangeli and V. Ciampi. Equilibrium based iterative solution for the non-linear beam problem. International Journal for Numerical Methods in Engineering, 40:423-437, 1997. 18. M. Petrangeli, P.E. Pinto and V. Ciampi. Fiber element for cyclic bending and shear of RC structures. I. Theory. J. Engineering Mechanics, ASCE, 125:994-1001, 1999. 19. M. Schultz and EC. Filippou. Non-linear spatial Timoshenko beam element with curvature interpolation. International Journal for Numerical Methods in Engineering, 50:761-785,2001. 20. M. Schultz and EC. Filippou. Generalized warping torsion formulation. J. Structural Engineering, ASCE, 124:339-347, 1998. 21. A. Ayoub and EC. Filippou. Nonlinear finite-element analysis of RC shear panels and walls. J. Structural Engineering, ASCE, 124:298-308, 1998. 22. A. Ayoub and F.C. Filippou. Mixed formulation of bond-slip problems under cyclic loads. J. Structural Engineering, ASCE, 125(ST6):661-671, 1999. 23. A. Ayoub and F.C. Filippou. Mixed formulation of nonlinear steel-concrete composite beam element. J. Structural Engineering, ASCE, 126:371-381, 2000. 24. R.L. Taylor, EC. Filippou and A. Saritas. Finite element solution of beam problems. Computational Mechanics, 31, 2003. 25. K.D. Hjelmstad and E. Taciroglu. Mixed variational methods for finite element analysis of geometrically non-linear, inelastic Bernoulli-Euler beams. Communications in Numerical Methods of Engineering, 19:809-832, 2003. 26. M.A. Crisfield. Non-linear Finite Element Analysis of Solids and Structures, volume 2. John Wiley & Sons, Chichester, 1997. 27. B. Fraeijs de Veubeke. Displacement and equilibrium models in finite element method. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 9, pages 145-197. John Wiley & Sons, Chichester, 1965. 28. A. Tessler and S.B. Dong. On a hierarchy of conforming Timoshenko beam elements. Computers and Structures, 14:335-344, 1981. 29. M.A. Crisfield. Finite Elements and Solution Procedures for Structural Analysis, Vol. 1, Linear Analysis. Pineridge Press, Swansea, 1986. 30. H.K. Stolarski, N. Carpenter and T. Belytschko. Bending and shear mode decomposition in C O structural elements. Journal of Structural Mechanics, ASCE, 11(2): 153-176, 1983. 31. J.C. Simo and M.S. Rifai. A class of mixed assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 29:1595-1638, 1990. 32. U. Andelfinger, E. Ramm and D. Roehl. 2d- and 3d-enhanced assumed strain elements and their application in plasticity. In D. Owen, E. Ofiate and E. Hinton, editors, Proceedings of the 4th International Conference on Computational Plasticity, pages 1997-2007. Pineridge Press, Swansea, 1992. 33. M. Bischoff, E. Ramm and D. Braess. A class of equivalent enhanced assumed strain and hybrid stress finite elements. Computational Mechanics, 22:443-449, 1999.
322
Structural mechanics problems in one dimension- rods 34. N. Btichter, E. Ramm and D. Roehl. Three-dimensional extension of non-linear shell formulations based on the enhanced assumed strain concept. International Journal for Numerical Methods in Engineering, 37:2551-2568, 1994. 35. M. Braun, M. Bischoff and E. Ramm. Nonlinear shell formulations for complete threedimensional constitutive laws include composites and laminates. Computational Mechanics, 15:1-18, 1994. 36. P. Betsch, F. Gruttmann and E. Stein. A 4-node finite shell element for the implementation of general hyperelastic 3d-elasticity at finite strains. Computer Methods in Applied Mechanics and Engineering, 130:57-79, 1996. 37. M. Bischoff and E. Ramm. Shear deformable shell elements for large strains and rotations. International Journal for Numerical Methods in Engineering, 40:4427--4449, 1997.
Plate bending approximation: thin (Kirchhoff) plates and
continuity requirements
iiiiii~,iil!ii{i',~ii~~ili~i::~,'::'~iii{~{i!~:~::'~'~ill!!',!'i'~i{~i'ii'~;i~~:,~,i:~~''i!~,!iiliii{i!ii~,!iilii"~::'~i:!i:i~il,~i~ili!il~iiiii:::::: ~'i~'~i!i'~',iiii~,',iii!i~i!i~,i:,~i',i:,i~'i:::':i~!'~i'ii{~iiii!ii'~i~iiiiiiiiiii',!'i::::::,i::::iiii~i ~,i~,ii''~,i'~'~,'i'~,'~,i'{,!i'~!~,!iiiiii'i~~iii',i i~,}ili i',~,i~i,~'i}~,il'~iiiiiiiii!iii'i',,il'~'~i!iilii',iiii}iilii'~i'i,'i,iii',iiii}~ii'~i'~~,i',i~'il,i'~'i~!'~ii'iil,i!i','~iiiiiiiiiiiiiiiiiiiiiil~,'~iii~iililii',~,i ~'i~i'~'~i,:i{iii iili~{i'iiii,i~,i'i',,ili'i~i~i'~i'!!i,~i;ili'~,i~i;ii',i~i~'~,i'~,i~,i'',i~iii''~,i'i',i~i'~i~'~iliiii{i!i{iili'~il'i,i',iiiii'!ii',i,'li:,i', i!i',ii!ii' ', i,'i'~',i ',~' !i,!'i',i,'~'~i{,':i'ii'~:i?!'i{,~''~,'{i,'~'~!'i!ii,!',ii'lii,i!ii!iii!il}i'{~','~i,{i{i',!ii!i~iil','~i!ii',iii~,',i'i,l'~i 'i'~i,lilii!i!li! !'}i,i'i,'~',~i'~i~i{~,i~~{~, '~~,i~i{~~i~i,'~'',~i'~,'~i'~i{i~,','~'~i'~i!',i',i~{'i~i'i!'~'i!i'~,'i,i!i'~~i'i~i''i~,'~i'ii,i',!ii'~~ii'ii,i''~,i!'~,',!
The subject of bending of plates and indeed its extension to shells was one of the first to which the finite element method was applied in the early 1960s. At that time the various difficulties that were to be encountered were not fully appreciated and for this reason the topic remains one in which research is active to the present day. Although the subject is of direct interest only to applied mechanicians and structural engineers there is much that has more general applicability, and many of the procedures which we shall introduce can be directly translated to other fields of application. Plates and shells are but a particular form of a three-dimensional solid, the treatment of which presents no theoretical difficulties, at least in the case of elasticity. However, the thickness of such structures (denoted throughout this and later chapters as t) is very small when compared with other dimensions, and complete three-dimensional numerical treatment is not only costly but in addition often leads to serious numerical ill-conditioning problems. To ease the solution, even long before numerical approaches became possible, several classical assumptions regarding the behaviour of such structures were introduced. Clearly, such assumptions result in a series of approximations. Thus numerical treatment will, in general, concern itself with the approximation to an already approximate theory (or mathematical model), the validity of which is restricted. On occasion we shall point out the shortcomings of the original assumptions, and indeed modify these as necessary or convenient. This can be done simply because now we are granted more freedom than that which existed in the 'pre-computer' era. The thin plate theory is based on the assumptions formalized by Kirchhoff in 1850,1 and indeed his name is often associated with this theory, though an early version was presented by Sophie Germain in 1811. 2-4 A relaxation of the assumptions was made by Reissner in 19455 and in a slightly different manner by Mindlin 6 in 1951. These modified theories extend the field of application of the theory to thick plates and we shall associate this name with the Reissner-Mindlin postulates. It turns out that the thick plate theory is simpler to implement in the finite element method, though in the early days of analytical treatment it presented more difficulties. As it is more convenient to introduce first the thick plate theory and by imposition of
324
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements additional assumptions to limit it to thin plate theory we shall follow this path in the present chapter. However, when discussing numerical solutions we shall reverse the process and follow the historical procedure of dealing with the thin plate situations first in this chapter. The extension to thick plates and to what turns out always to be a mixed formulation will be the subject of Chapter 12. In the thin plate theory it is possible to represent the state of deformation by one quantity w, the lateral displacement of the middle plane of the plate. Thus we find that thin plates share some of the same characteristics as the Euler-Bemoulli beam theory considered in the previous chapter. Clearly, such a formulation is irreducible. The achievement of this irreducible form introduces second derivatives of w in the strain definition and continuity conditions between elements have now to be imposed not only on this quantity but also on its derivatives (C1 continuity). This is to ensure that the plate remains continuous and does not 'kink'.* Thus at nodes on element interfaces it will always be necessary to use both the value of w and its slopes (first derivatives of w) to impose continuity, again this is similar to the treatment of Euler-Bemoulli beam theory. Determination of suitable shape functions is now much more complex than those needed for Co continuity or for beams. Indeed, as complete slope continuity is required on the interfaces between various elements, the mathematical and computational difficulties often rise disproportionately fast. It is, however, relatively simple to obtain shape functions which, while preserving continuity of w, may violate its slope continuity between elements, though normally not at a node where such continuity is imposed, t If such chosen functions satisfy the 'patch test' then convergence will still be found. 7 The first part of this chapter will be concemed with such 'non-conforming' or 'incompatible' shape functions. In later parts new functions will be introduced by which continuity can be restored. The solution with such 'conforming' shape functions will now give bounds to the energy of the correct solution, but, on many occasions, will yield inferior accuracy to that achieved with non-conforming elements. Thus, for practical usage the methods of the first part of the chapter are often recommended. The shape functions for rectangular elements are the simplest to form for thin plates and will be introduced first. Shape functions for triangular and quadrilateral elements are more complex and will be introduced later for solutions of plates of arbitrary shape or, for that matter, for dealing with shell problems where such elements are essential. The problem of thin plates is associated with fourth-order differential equations leading to a potential energy function which contains second derivatives of the unknown function. It is characteristic of a large class of physical problems and, although the chapter concentrates on the structural problem, the reader will find that the procedures developed will also be equally applicable to any problem which is of fourth order. The difficulty of imposing C1 continuity on the shape functions has resulted in many alternative approaches to the problems in which this difficulty is side-stepped. Several possibilities exist. Two of the most important are:
* If 'kinking' occurs the second derivative or curvature becomes infinite and squares of infinite terms occur in the energy expression. t Later we show that even slope discontinuity at the node may be used.
The plate problem: thick and thin formulations 325 1. independent interpolation of rotations q~ and displacement w, imposing continuity as a special constraint, often applied at discrete points only; 2. the introduction of Lagrange multiplier variables or indeed other variables to avoid the necessity of C1 continuity. Both approaches fall into the class of mixed formulations and we shall discuss these briefly at the end of the chapter. However, a fuller statement of mixed approaches will be made in the next chapter where both thick and thin approximations will be dealt with simultaneously. i~!iiiiif~!ii!!i i!:i!i i ~ii~l~;~i:i ii~~'~'~!il:i!i~ ii i~i~l~'~~i!ii i iili ii'~~'~''~'~~"~~''i i i ii i i~i i iili i!ii~.!.i i~li iili i':~'~iil'i i!i i iiili!i~~ii!iiiii i ii!i!ii i!iil!i:~:~ ii~i i i!ii~ii~i~i i!iil~ii i~ii ie~i ili~!~,i!~i ,,~~,iii~i i!i!i,~i~,i~,iii~i~i i~ii!~iiii ~i ili~ii~i ii~iifi!ii!l!i i!ii!iiii~!~ ~i~~!ii,~,,, i ii~i !i!l~i~i i~iili ii,~~i,!!!i, !ii!~i i ~i i i!i i!~ii i i ii!ii i i ii i,l~,~i ~i~i!~i !~ii ii,~,i~li!,~iii ~i ii!ii~lili~i ii~i"l~i l'~ ,iii~i i i iil!!ii!i!i!ii i~iiliii~ii i iiil~i iili i ,!i~;~i,i~i i ili~i ii~ iiii~~i ii!i!i i~lii i iii iili ii!iliiii ii i!ii !i!ii i ii !i!i!!!ii!ii iii i ilii~!i ~i i iiii!i!li i~i
11.2.1 Governing equations The mechanics of plate action can be illustrated in one dimension by considering a plate of infinite extent in one dimension (here assumed the y) and considering equations similar to those developed for a beam in Sec. 10.2. Here we consider the problem of cylindrical bending of plates. 2 In this problem the plate is to be loaded and supported by conditions independent of y. In this case we may analyse a strip of unit width subjected to some stress resultants Mx, Px, and Sx, which denote x-direction bending moment, axial force and transverse shear force, respectively,* as shown in Fig. 11.1. For cross-sections that are originally normal to the middle plane of the plate we can use the approximation that at some distance from points of support or concentrated loads plane sections will remain plane during the deformation process. The postulate that sections normal to the middle plane remain plane during deformation is thus the first and most important assumption of the theory of plates (and indeed shells). To this is added the second assumption. This simply observes that the direct stresses in the normal direction, z, are small, that is, of the order of applied lateral load intensifies, q, and hence direct strains in that direction can be neglected. This 'inconsistency' in approximation is compensated for by assuming a plane stress condition in each lamina. With these two assumptions it is easy to see that the total state of deformation can be described by displacements u and w of the middle surface (z = 0) and a rotation ~bx of the normal (Fig. 11.1). Thus the local displacements in the directions of the x and z axes are taken as
Ul(X, z) = u(x) + Z~x(X)
u3(x, z) = w(x)
and
(11.1)
Immediately the strains in the x and z directions are available as
Ou
OUl Cx ~
tc)qX
"-
Ox
-t-Z~
OCx Ox (11.2)
Cz=0
Oul %Z=Oz+
C~ 3
Ox
=
0//3
Ox
+4~x
* Here we change our notation slightly from that used for beams in order to conform to the notation commonly used for plates.
326
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements I)X
qz
llllll ~
x
7 I A'
.A" I I
P~
I
~A' Fig. 11.1 Displacements and force resultants for cylindrical bending of a plate. The non-zero strains are identical to those obtained for bending of beams. For the cylindrical bending problem a state of linear elastic, plane stress for each lamina yields the stress-strain relations E
~rx - - 1 -
and
u a ex
~'xz = G % z
The stress resultants are obtained as Px --
f
t/2
On Crx d z = B
J -t/2 Sx =
ax Txz d z = n G t
d -t/2 Mx =
f
t/2
J -t/2
(O;x ) + C~x
(11.3a)
Od;x Crx z d z = D
OX
where B is the in-plane plate stiffness and D the bending stiffness for an isotropic elastic material and are computed from B =
Et 1 -
ua
and
D =
Et 3
12(1
-
ua)
(11.3b)
with u Poisson's ratio, E and G direct and shear elastic moduli, respectively.* Three equations of equilibrium complete the basic formulation. These equilibrium equations may be computed directly from a differential element of the plate or by integration over the thickness of the local equilibrium equations as was performed for * A constant ~ has been added here to account for the fact that the shear stresses are not constant across the section. A value of ~ = 5 / 6 is exact for a rectangular, homogeneous section and corresponds to a parabolic shear stress distribution.
The plate problem: thick and thin formulations 327 the beam in Sec. 10.2.1. Using the latter approach and assuming zero inertial forces we have for the axial resultant
it~2 i~O.x ~xz ] 0 ill2 d --t/2 "~X -Ji--~Z -Jr-bx dz = ~ .J-t/2 a~ dz Oex Ox
+
it~2 bx dz J -t/2
+ Txz
I ,/2
Txz
I = -t/2
0
+qx = 0 (11.4a)
where qx is an axial load similar to that obtained for the beam. Similarly, the shear resultant follows from
f,, r xzOo z ] .J--t~2 L OX -JI--~Z "JI-bz OSx Ox
oft/d-t~2
dz = -~x
~-~z dz +
d-t~2 bx dz
+ az
It/2 I--t/2 - crz
= 0
-t-qz = 0 (11.4b)
where the transverse loading qz arises from the body force and the resultant of the normal traction on the top and/or bottom surfaces. Finally, the moment equilibrium is deduced from
/t/2[OO.xOT.xz z + ,I-t~2 ~X ~ OMx Ox
+ bx
]
dz =
o/t/2 z Crx dz ~ J-t/2
-
it~2 rxz dz ,I-t~2
+
ill2 z bx dz ,I-t~2
- 0
Sx + m ~ = 0
(11.4c)
Generally, mx loads are not included in plate theory except as an artifice to introduce the d' Alembert inertial forces. As we found was true for beams, in the elastic case of a plate it is easy to see that the in-plane displacements and forces, u and Px, decouple from the other terms and the problem of lateral deformations can be dealt with separately. We shall thus only consider bending in the present chapter, returning to the combined problem, characteristic of shell behaviour, in later chapters. Equations (11.1)-(11.4c) are typical for thick plates, and the thin plate theory adds an additional assumption. This simply neglects the shear deformation and puts G = c~ (or %z = 0). Equation (11.3a) thus becomes Ow Ox
+ ~x = 0
(11.5)
This thin plate assumption is equivalent to stating that the normals to the middle plane remain normal to it during deformation and is the same as the Bernoulli-Euler assumption for thin beams considered in Chapter 10. The thin, constrained theory is very widely used in practice and proves adequate for a large number of structural problems, though, of course, should not be taken literally as the true behaviour near supports or where local load action is important and is three dimensional. In Fig. 11.2 we illustrate some of the boundary conditions imposed on plates and immediately note that the diagrammatic representations of simple support as a knife
328
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements Singularity j disregarded
(a) Built-in support (clamped) with u= v= w= 0, 0=0
(b) Free edge with M = 0, S=0(P=0)
Singularity
Rigid
disregarded
~/M=0 w=0
Real approximation
Conventional illustration
Plate " - - . ~ s ) _ (n)
or 0x = 0s = 0
w- 0
SS1 (soft support) SS2 (hard support)
(c) Simply supported condition Fig. 11.2 Support (end) conditions for a plate. Note: the conventionally illustrated simple support leads to infinite displacement - reality is different.
edge would lead to infinite displacements and stresses. Of course, if a rigid bracket is added in the manner shown this will alter the behaviour to that which we shall generally assume. The one-dimensional problem of plates and the introduction of thick and thin assumptions translate directly to the general theory of plates. In Fig. 11.3 we illustrate the extensions necessary and write, in place of Eq. (11.1) (assuming u0 and v0 to be zero), Ux -- z #Px(X, y)
Uy = Z ~y(X, y)
uz = w ( x , y)
(11.6)
where we note that displacement parameters are now functions of x and y. It is sometimes advantageous to replace q5x and qSyby rotations about the x and y coordinates in a manner used for the beam developments. Thus,
-~ {~
4,=TO
or
IOy}-[o oJ Ox
(11.7)
The plate problem: thick and thin formulations
~
z(~
x(u)
y(v)
, ,+
.I
t I
I
-
t
~,
'1 E
I'-'%
J \
I
J
/,~ ~''!
~,"
~.
0
~ rr
~
,
"
;
I I
I
I
,
I
"~ c~
!,
I
'~
§ ! .f, j I ~ ! I',,'1.,( . .,~I - / ,
~,
.," L .
/, ~r .~,
,
o
QJ
u'l
CL
u ro
Fig. 11.16 Cont. (c) clamped uniformly loaded square plate.
0
~
iI
if)
!
II
l
1
'
%
i
~9 i~ ..
_1~,~__. I
-
(%) ~,~ u! JoJ.~3
,,
gr %%
~"...! 0
(%) o,~ u! JoJJ3
0
T
(,O
O ,e'-
O'J
CO I~ (s
Ul
~
E:
rr v
u) ,.i..., e-
E
I-
u_
~D
Q
u
E
Q.
"o
c'-
"(:3 o) ,ca .i-, (.-
ca 4-,, ca~ L)
o
ca
-d
(3) (1,1 En ~D c-(1,) U '.-.q) CL
oL_
.c_
C: J
k i
(a)
r
x
i
io-"--
(b)
Fig. 11.21 The compatible functions of Fraeijs de Veubeke.13,16
(c)
M
362
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements W -- W a-~- W b-~- tOc
(11.65)
W a -- O~1 -~- O~2X -~---.--~ CelOY3
The second function w b is defined in a piecewise manner. In the lower triangle of Fig. 11.21(b) it is taken as zero; in the upper triangle a cubic expression with three constants merges with slope discontinuity into the field of the lower triangle. Thus, in
jkm, tOb -- O~llY '2 + OZl2Y'3 + OZl3X'y '2
(11.66)
in terms of the locally specified coordinates x' and y'. Similarly, for the third function, Fig. 11.21 (c), w e = 0 in the lower triangle, and in i mj we define
to c -'-Ctl4Y 'r2 -k-Cel5Y "3 -if-Ctl6x"y
'r2
(11.67)
The 16 external degrees of freedom are provided by 12 usual comer variables and four normal mid-side slopes and allow the 16 constants c~ to O~16to be found by inversion. Compatibility is assured and once again non-unique second derivatives arise at comers. Again it is possible to constrain the mid-side nodes if desired and thus obtain a 12 degree-of-freedom element. The expansion can be found explicitly, as shown by Fraeijs de Veubeke, and a useful element generated. 16 The element described above cannot be formulated if a comer of the quadrilateral is re-entrant. This is not a serious limitation but needs to be considered on occasion if such an element degenerates to a near triangular shape.
ii!iiii~HH~ iii~i~i~iiii~' iii~~i;~i!iiii~~ii~i~'i~~iii'~~i!iii ' ~~i~~ii ~i~iiiii!iii~iiiiiiiiiiiiliiiiiiiiiiiiiiiiiiiiiiii~ iiiiii~ ~'3iiiiiiiiii!iiii~i~iiiiiiiiiiiiii~i~ ~~i'li ~iii~iiii~!iiii~~i~ii~ii!i?i~!i~i!ii~i~i~i~iiiii~iiiiii~iiii ~i ~ i ~ ~i ii ~ ii ~ i ili i ii iHii i i i ~iiii~iiiii~ii~iiiii~iii~!i!i!i!i!ii~!!~!ii!i~ii~i~iiiii!i~i!!~!!!iiiiii~iiii!i!i!!~i!~iii~i~i!i~i~iii~iii~i~ii!i!ii~iiii!i!iiii~ii~i~iii~i!ii!~i~i!!i~i~i~i~i~i!i!i~i~ii~iii~!iiii!iii~i!iiii~ii~i~i!i!!~!~iiiii~i!iii~i!i The performance of some of the conforming elements discussed in Secs 11.10-11.12 is shown in the comparison graphs of Fig. 11.16. It should be noted that although monotonic convergence in energy norm is now guaranteed, by subdividing each mesh to obtain the next one, the conforming triangular elements of references 11 and 12 perform almost identically but are considerably stiffer and hence less accurate than many of the non-conforming elements previously cited. To overcome this inaccuracy a quasi-conforming or smoothed element was derived by Razzaque and Irons. 33'34 For the derivation of this element substitute shape functions
are used. The substitute functions are cubic functions (in area coordinates) so designed as to approximate in a least-square sense the singular functions e and their derivatives used to enforce continuity [see Eqs (11.58)-(11.64)], as shown in Fig. 11.22. The algebra involved is complex but a full subprogram for stiffness computations is available in reference 33. It is noted that this element performs very similarly to the simpler, non-conforming element previously derived for the triangle. It is interesting to observe that here the non-conforming element is developed by choice and not to avoid difficulties. Its validity, however, is established by patch tests.
Hermitian rectangle shape function 363 3
Discontinuity
...e,~o~oO~ ~
b_2 ~ ei
~'~176176 f~~~~"~O'q.
Zero boundary slope
Li L2L2 (1 + L,) ei = (Li+ Lj) (Li+ Lk)
E/* = 1/6 Li (2 Li- 1) (L i- 1)
Fig. 11.22 Least-square substitute cubic shape function e* in place of rational function e for plate bending triangles.
Conforming shape functions with additional degrees of freedom
With the rectangular element of Fig. 11.7 the specification of OZw/OxOy as a nodal parameter is always permissible as it does not involve 'excessive continuity'. It is easy to show that for such an element polynomial shape functions giving compatibility can be easily determined. A polynomial expansion involving 16 constants [equal to the number of nodal parameters Wa, (Ow/OX)a, (Ow/Oy)a and (omw/OxOy)a] could, for instance, be written
364
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements retaining terms that do not produce a higher-order variation of w or its normal slope along the sides. Many alternatives will be present here and some may not produce invertible C matrices [see Eq. (11.33)]. An alternative derivation uses Hermitian polynomials used for shape functions of the Euler-Bemoulli beam (see Sec. 10.4.1) which permit the writing down of suitable functions directly. It is easy to verify that the following shape functions
Na-
[H~a~176
H~~
H~a~)(x)H(a~
H~al)(x)H~al)(y)] (11.68)
correspond to the values of //3,
Ow
Ow
Ox'
Oy' OxOy'
OZw
specified at the comer nodes, taking successively unit values at node a and zero at other nodes. An element based on these shape functions has been developed by Bogner et al. 17 and used with success. Indeed it is the most accurate rectangular element available as indicated by results in Fig. 11.16. A development of this type of element to include continuity of higher derivatives is simple and outlined in reference 18. In their undistorted form the above elements are, as for all rectangles, of very limited applicability.
If continuity of higher derivatives than first is accepted at nodes (thus imposing a certain constraint on non-homogeneous material and discontinuous thickness situations as explained in Sec. 11.2.4), the generation of slope and deflection compatible elements presents less difficulty. Considering as nodal degrees of freedom tO,
Ow Ox '
Ow Oy '
OZw Ox2'
OZw OxOy '
OZw Oy2'
a triangular element will involve at least 18 degrees of freedom. However, a complete fifth-order polynomial contains 21 terms. If, therefore, we add three normal slopes at the mid-side as additional degrees of freedom a sufficient number of equations appear to exist for which the shape functions can be found with a complete quintic polynomial. Along any edge we have six quantities determining the variation of w (displacement, slopes, and curvature at comer nodes), that is, specifying a fifth-order variation. Thus, this is uniquely defined and therefore w is continuous between elements. Similarly, Ow/On is prescribed by five quantities and varies as a fourth-order polynomial. Again this is as required by the slope continuity between elements. If we write the complete quintic polynomial as* l/3 - - OZ1 + OL2X " 1 - " " " "1- Ct21Y 5
(11.69)
* For this derivation use of simple Cartesian coordinates is recommended in preference to area coordinates. Symmetry is assured as the polynomial is complete.
The 21 and 18 degree-of-freedom triangle 365 proceed along the lines of the argument used to develop the rectangle in Sec. 11.3 and write llO a - -
Ol21Ya5
- - OL2 - - ~ - . . . . q - O~20Ya 4
0Xo
ow I 0ya
OX 2
OL 1 -~- OL 2 X a -~- . . . -~-
i
- - Ce3 - + - ' ' ' - + -
5Ce21Y 4
= 2ot4 -+--..-+- 2ot19y3
and so on, and finally obtain an expression ~e __ CCI~
(11.70)
in which C is a 21 x 21 matrix. The only apparent difficulty in the process that the reader may experience in forming this is that of the definition of the normal slopes at the mid-side nodes. However, if one notes that ~ (11.71) Ow Ow + sin ~ bOw Oy On = cos q50x in which 4) is the angle of a particular side to the x axis, the manner of formulation becomes simple. It is not easy to determine an explicit inverse of C, and the stiffness expressions, etc., are evaluated as in Eqs (11.22)-(11.25) by a numerical inversion. The existence of the mid-side nodes with their single degree of freedom is an inconvenience. It is possible, however, to constrain these by allowing only a cubic variation of the normal slope along each triangle side. Now, explicitly, the matrix C and the degrees of freedom can be reduced to 18, giving an element illustrated in Fig. 11.19(e) with three comer nodes and 6 degrees of freedom at each node. Both of these elements were described in several independently derived publications appearing during 1968 and 1969. The 21 degree-of-freedom element was described independently by Argyris et al., 23 Bell, 19 B o s s h a r d , 22 and Visser, 24 listing the authors alphabetically. The reduced 18 degree-of-freedom version was developed by Argyris et al., 23 Bell, 19 Cowper et al., 21 and Irons. 14 An essentially similar, but more complicated, formulation has been developed by Butlin and Ford, 2~and mention of the element shape functions was made earlier by Withum 69 and Felippa. 7~ It is clear that many more elements of this type could be developed and indeed some are suggested in the above references. A very inclusive study is found in the work of Zenisek, 71 Peano, 72 and others. 73-75 However, it should always be borne in mind that all the elements discussed in this section involve an inconsistency when discontinuous variation of material properties occurs. Further, the existence of higherorder derivatives makes it more difficult to impose boundary conditions and indeed the simple interpretation of energy conjugates as 'nodal forces' is more complex. Thus, the engineer may still feel a justified preference for the more intuitive formulation
366
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements involving displacements and slopes only, despite the fact that very good accuracy is demonstrated in the references cited for the quartic and quintic elements.
Avoidance of continuity difficulties in mixed and constrained elements
Equations (11.10a)-(11.11 b) of this chapter provide for many possibilities to approximate both thick and thin plates by using mixed (i.e. reducible) forms. In these, more than one set of variables is approximated directly, and generally continuity requirements for such approximations can be of either C I or Co type. The options open are large and indeed so is the number of publications proposing various alternatives. We shall therefore limit the discussion to those that appear most useful. To avoid constant reference to the beginning of this chapter, the four governing equations (11.10a)-(11.1 l b) are rewritten below in their abbreviated form with dependent variable sets M, q~, S, and w" M - Ds
= 0
1
-S-4,-Vw =0 a Z~TM + S = 0
(11.72)
v T s q- q - - 0 in which a = t~ Gt. To these, of course, the appropriate boundary conditions can be added. For details of the operators, etc., the fuller forms previously quoted need to be consulted. Mixed forms that utilize direct approximations to all the four variables are not common. The most obvious set arises from elimination of the moments M, that is ~TI)~D
"-[- S -- 0
1 -S - ~b- Vw O~
= 0
(11.73)
v T s -[- q = 0 and is the basis of a formulation directly related to the three-dimensional elasticity consideration. This is so important that we shall devote Chapter 12 entirely to it, and, of course, there it can be used for both thick and thin plates. We shall, however, return to one of its derivations in Sec. 11.18. One of the earliest mixed approaches leaves the variables M and w to be approximated and eliminates S and ~b. The form given is restricted to thin plates and thus a = o~ is taken. We now can write for the first two of Eqs (11.72), D-1M + Z~Vw = 0
(11.74)
Mixed formulations- general remarks and for the last two of Eqs
(11.72), V TETM - q = 0
(11.75)
The approximation can now be made directly putting M
-
NM]~/I
and
w = Nw#
(11.76)
where M and ~ list the nodal (or other) parameters of the expansions, and NM and Nw are appropriate shape functions. The approximation equations can, as is well known, be made either via a suitable variational principle or directly in a weighted residual, Galerkin form, both leading to identical results. We choose here the latter, although the first presentations of this approximation by H e r r m a n n 76 and o t h e r s 52'77-84 all use the Hellinger-Reissner principle. A weak form from which the plate approximation may be deduced is given by
~51"I= f rM (D-1M + ff.,Vw) d~+ f ~Sw (VT ff.,TM - q) d~+~5I-lbt = O ( 1 1 . 7 7 ) where ~nbt describes appropriate boundary condition terms. weighting approximations 6M = NMrl~I
and
Using the Galerkin
5w = NwSff
(11.78)
gives on integration by parts the following equation set (11.79) where
A= ~
NTD-1NM dr2
fl =
(VNw) T
l~ns
dF
(11.80)
t
C = ~ (ENM)TX7Nw dr2
=
Nwq dr2 +
NwS~dF t
where /~n and t~ns are the prescribed boundary moments, and Sn is the prescribed boundary shear force. Immediately, it is evident that only Co continuity is required for both M and w interpolation,* and many forms of elements are therefore applicable. Of course, appropriate patch tests for the mixed formulation must be enforced 43 and this requires a necessary condition that
nm >__nw
(11.81)
where nm stands for the number of parameters describing the moment field and nw the number in the displacement field. * It should be observed that, if Co continuity to the whole M field is taken, excessive continuity will arise and it is usual to ensure the continuity of Mn and Mnsat interfaces only.
367
368
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements Many excellent elements have been developed by using this type of approximation, though their application is limited because of the difficulty of interconnection with other structures as well as the fact that the coefficient matrix in Eq. (11.79) is indefinite with many zero diagonal terms. Indeed, a similar fate is encountered in numerous 'equilibrium element' forms in which the moment (stress) field is chosen a priori in a manner satisfying Eq. (11.75). Here the research of Fraeijs de Veubeke 83 and others 3~ has to be noted. It must, however, be observed that the second of these elements 3~ is in fact identical to the mixed element developed by Herrmann 77 and Hellan 85 (see also reference 52).
ii......... il!iiii!iiiiiiiiiiiiii !::,!,~ i!ii!ilili~i~i~iiiiiiiii!i ~,,~!~ ii~i~iliiiii!iii!iiiil ~!i~i~iiiiiiii~~i~i~i~i~i~iiliiiii !!!,,,:H~,~!:~,,:!:p~,~,:!:, !iilili!iiii~i~iiiilii~ii~~il!iiliiiiiii!iii!i~~,~,il~iiiiiiiiiiiiiiiiiiT iiil!i!iil~~iiiiiliiii!iiii!iiiiif!i!iii~'~'~iii!ii!!iiliiiiiiiii!iiiiiiiiii!i!i!ii!i iiiiiii!i!iiii e iiililiill ! ~'~'~'iiililiiiiiiiiii , iiiiiii ~ iiiiiiiil t iiifill~i!iiiil!iiilliiii!iiiiiiiliiiii!ii~ii!ii!iiiii!i!ii!iiii........................ iiiiiiiili!ii!iii!iiiiiiiiiiiiiliii!iiliiii............................................................................................. i iiiiii!iiiililiiliiii!iiliiii!iii!iiliiiiiiiiiiiiiiili!ii!iliiiiiiiiliiiliiiiiiiliiiiiiiiiiiili!iliiiiiiliil!iiliiii!iiiiiiiiiiii !i!.!il. .ii.iiiii . .iiiliil . .ii.ii!iiii . .i!i!iil . .!i.ii!ii . .iiii.iilii . .iiiiiil . .il.iiiiiii . .il.iilil . .i!iii!iii . . .iiiilii . .il.ii!il.iil.!i............. .i.ili.liliiiiiiii....................................... iil~!iiii!iiii!iiliiiiiiiiiiiiiii!i~!i!i!~iiiiliiiiiiiliiiiilii!i!i!i ,..... ii~iiiliiiiiiiii!iil!!i,!ii~~iil~ii~ii~ii~
Hybrid elements are essentially mixed elements in which the field inside the element is defined by one set of parameters and the one on the element frame by another, as shown in Fig. 11.23. The latter are generally chosen to be of a type identical to other displacement models and thus can be readily incorporated in a general program and indeed used in conjunction with the standard displacement types we have already discussed. The internal parameters can be readily eliminated (being confined to a single element) and thus the difference from displacement forms are confined to the element subprogram. The original concept is attributable to Pian 86'87 who pioneered this approach, and today many variants of the procedures exist in the context of thin plate theory. 65,88-97 In the majority of approximations, an equilibrating stress field is assumed to be given by a number of suitable shape functions and unknown parameters. In others, a mixed stress field is taken in the interior. A more refined procedure, introduced by Jirousek, 65'97 assumes in the interior a series solution exactly satisfying all the differential equations involved for a homogeneous field. w and ~)wt;~nd e f i ~ on frame by usual connection
/
Interior field defined by independent parameters
Fig. 11.23 Hybridelements.
Singularity (crack)
Discrete Kirchhoff constraints 369
All procedures use a suitable linking of the interior parameters with those defined on the boundary by the 'frame parameters'. The procedures for doing this are described in Chapter 12 of reference 7 in the context of elasticity equations, and only a small change of variables is needed to adapt these to the present case. We leave this extension to the reader who can also consult appropriate references for details. Some remarks need to be made in the context of hybrid elements. Remark: The first is that the number of internal parameters, n~, must be at least as large as the number of frame parameters, nF, which describe the displacements, less the number of rigid body modes if singularity of the final (stiffness) matrix is to be avoided. Thus, we require that
n~ > n F - 3
(11.82)
for plates. Remark: The second remark is a simple statement that it is possible, but counterproductive, to introduce an excessive number of internal parameters that simply give a more exact solution to a 'wrong' problem in which the frame is constraining the interior of an element. Thus additional accuracy is not achieved overall. Remark: Most of the formulations are available for non-homogeneous plates (and hence non-linear problems). However, this is not true for the Trefftz-hybrid elements 65'97where an exact solution to the differential equation needs to be available for the element interior. Such solutions are not known for arbitrary non-homogeneous interiors and hence the procedure fails. However, for homogeneous problems the elements can be made much more accurate than any of the others and indeed allow a general polygonal element with singularities and/or internal boundaries to be developed by the use of special functions (see Fig. 11.23). Obviously, this advantage needs to be bome in mind.
A number of elements matching (or duplicating) the displacement method have been developed and the performance of some of the simpler ones is shown in Fig. 11.16. Indeed, it can be shown that many hybrid-type elements duplicate precisely the various incompatible elements that pass the convergence requirement. Thus, it is interesting to note that the triangle of Allman 96 gives precisely the same results as the 'smoothed' Razzaque element of references 33 and 34 or, indeed, the element of Sec. 11.5.
Another procedure for achieving excellent element performance is achieved as a constrained (mixed) element. Here it is convenient (though by no means essential) to use a variational principle to describe the first and third of Eqs (11.73). This can be written simply as the minimization of the functional n =
1/sTls o /wq +Hbt nimum
( z : o ) q ) ( z : o ) dr2 + ~
a
(11.83)
370
Plate bending approximation: thin (Kirchhoff) plates and C~ continuity requirements subject to the constraint that the second of Eqs (11.73) be satisfied, that is, 1
-S - O- Vw = 0
(11.84)
o~
We shall use this form for general thick plates in Chapter 12, but in the case of thin plates with which this chapter is concerned, we can specialize by putting c~ = cx~ and rewrite the above as
'L
I1 -- ~
(ff,,0)TD(s
dr2-
L
wq dr2 +
1-Ibt =
minimum
(11.85)
subject to 0 + ~Tw = 0
(11.86)
and we note that the explicit mention of shear forces S is no longer necessary. To solve the problem posed by Eqs (11.85) and (11.86) we can 1. approximate w and 0 by independent interpolations of Co continuity as w = Nw~r
and
0 = N00
(11.87)
2. impose a discrete approximation to the constraint of Eq. (11.86) and solve the minimization problem resulting from substitution of Eq. (11.87) into Eq. (11.85) by either discrete elimination, use of suitable Lagrangian multipliers, or penalty procedures. In the application of the so-called discrete Kirchhoff constraints, Eq. (11.86) is approximated by point (or subdomain) collocation and direct elimination is used to reduce the number of nodal parameters. Of course, the other means of imposing the constraints could be used with identical effect and we shall return to these in the next chapter. However, direct elimination is advantageous in reducing the final total number of variables and can be used effectively. The procedure for constructing the discrete Kirchhoff relations was presented in Sec. 10.5.3 and applied to beams. For two-dimensional plate elements the situation is a little more complex, but if we imagine x to coincide with the direction tangent to an element side, precisely identical elimination to that presented in Sec. 10.5.3 enforces complete compatibility along an element side when both gradients of w are specified at the ends. However, with discrete imposition of the constraints it is not clear a priori that convergence will always o c c u r - though, of course, one can argue heuristically that collocation applied in numerous directions should result in an acceptable element. Indeed, patch tests turn out to be satisfied by most elements in which the w interpolation (and hence the Ow/Os interpolation) have Co continuity. The constraints frequently applied in practice involve the use of line or subdomain collocation to increase their number (which must, of course, always be less than the
Rotation-free elements
number of remaining variables) and such additional constraint equations as Ir ~
+ ~bs e
lax--
L(~ L(~
+ ~x
e
I~2y =--
e
-1- ~y
ds = 0
) )
d~ = O
(11.88)
dr2 = 0
are frequently used. The algebra involved in the elimination is not always easy and the reader is referred to original references for details pertaining to each particular element. The concept of discrete Kirchhoff constraints was first introduced by Wempner et al., 98 Stricklin et al., 61 and Dhatt 62 in 1968-69, but it has been applied extensively since.
99-110
In particular, the 9 degree-of-freedom triangle 99'100 and the complex semi-loof element of Irons 102 are elements which have been successfully used. Figure 11.24 illustrates some of the possible types of quadrilateral elements achieved in these references. iliii!ii!iii! iiiiiiiiiiiii~,7,~~i::~'~::::i~i~i~!iii~ ~~:'~~':':~'~'!i,7ii~ii'~,!~~'~'~'::i!i!i7 ~~:~::'::':::4i!',iil','!,:,!iiiii!~,~~~':~:::::~iiii i iiiiii~il~7iiiii~,~:~:iiiiiiliiii!!i~,i!i~,i',i~:~i~ii~i :i iiiii~~i777 ~iiiiiilii~iii!i!i',iiF'~:~:i!i!!iiiiiii',!iiil ~,!ii',iii!!i!i !iliTi',iiililiii!iiii~iiii~,:i~!!iiii":"i !i ~iiiii'~i,~,i~iiili!'~'~77, ii iili~iii~:~i!ii'!,iiiiiiii',iii'~i',ii7ii'~,~:ill~i~iiii::~i,iiil:i~,ilii!i!iiiiiiiiiiiiiiiiiiiiiiii!i!iiiiiiiiiiiiiiiiiTii!iiii!',!iiiiiiiiiiiiiii!iilii7iiiiii!i!i!i! ii':i!iiiiii iiiii7i!ii'i,7iiiiiiiiiiiiiiiiii!iiiii!i!iiiiiiiiiiiiiiili iii iliiili!iliiii~,ii'i, ~!i!:, iiliiil7iiiiiiiiiiTiiT, iTii!i!!ii ii!iiiiiiii!ii 7iiiTiiiiiT!i!iiiii iiiiiiiiiiiii!7~i!ii!iliiii7iiii~,iii'~,,i'i,Ti7i~i ',i'~,i,'i!i'i!,7i:,iliiiii' ~ i,!iiii~ii,7~i~,1!ii!i7,',ii~i'1 , iiiiii~iiiiiiiiii7~iii~iiiiiiiiiiiiiiiiiii!7iiiiiiii~7iiii!ii~iiiiiii~7i~i!ii~iiiii~i~i~ii~i~ii7i7i~i~i:~iii~i7i~ii!i:~i~7~!i~i~!~ii!iiii!iiii~i~i~i~iiiiiiiii77iii~7iiiiiiiiiii~7i~ii~i~i~i~iiiiiiiiiii~i~i77i~iiiiiiiiiiii
It is possible to construct elements for thin plates in terms of transverse displacement parameters alone. Nay and Utku used quadratic displacement approximation and minimum potential energy to construct a least-square fit for an element configuration shown in Fig. 11.25(a). lll The element is non-conforming but passes the patch test and therefore is an admissible form. An alternative, mixed field, construction is given by Ofiate and Zfirate for a composite element constructed from linear interpolation on each triangle, ll2'll3 In this work a mixed variational principle is used together with a special approximation for the curvature. We summarize here the steps in the better approach. A three-field mixed variational form for a thin plate problem based on the Hu-Washizu functional may be written as
'/a
1-I -- -~
x T D x dA -
/a
[ ( E V ) w - X] dA - j a wq dA + l"Ibt
(11.89)
where now X and M are mixed variables to be approximated, (E~7)w are again second derivatives of displacement w given in Eq. (11.13) and integration is over the area of the plate middle surface. Variation of Eq. (11.89) with respect to X gives the discrete constitutive equation 31-Ix
/ dA
~XT [Dx - M] dA -- 0
(11.90)
e
where Ae is the domain of the patch for the element. Two alternatives for Ae are considered in reference 112 and named BPT and BPN as shown in Figs 11.25(a) and 11.25(b), respectively. For the BPT form the integration is taken over the area of the element 'abc' with area Ap and boundary Fp. For the type BPN integration is over
371
372
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements Virgin 0
Constrained "I
'
x2
x2
x2
x2
(~ .
,, o 24 DOF
(a) "
o
1 1
""
I
1 1
(b)
[~~
1 0 . 25 D O F
"
;" 1
c
1 .
.
J ' 16 D O F
~ 11
1
E:~
1 o.,
(c)
fl .
.
--.
;." 1
11 "
irons 102
t 16 DOF
[] 11
..
j'3
""
27 DOF
'i
(
' Lyons 103
. . . . . . '
c
'1
irons 102
"
c
16 DOF
Lyons 1~
~3 (d)
23 DOF xr 11
~
11
~
(e)
[] t
I
I
~,, 11•
11
27 DOF DOF o
12 DOF
~I
~
Degrees of freedom Nodal DOF [w, 0x, 0y] Nodal DOF [w] Nodal DOF [w, On] Nodal [On] Nodal DOF [On, 0s]
[]
I Irons 102 (semi-Ioof)
I
o
16 DOF
I
1 point constraint x 1, etc. 3 integral constraints .1"3,etc.
Fig. 11.24 A series of discrete Kirchhoff theory (DKT)-type elements of quadrilateral type.
the more complex area Aa with boundary Fa. Each, however, is simple to construct. Similarly, variation of Eq. (11.89) with respect to moment gives the discrete curvature relation ~l-IM -- / dA
~M T [(~7)w e
- ~i~] dA - 0
(11.91)
Rotation-free elements e
e
b
d (a)
(b)
Fig. 11.25 Elementsfor rotation-free thin plates: (a) patch for Nay and Utku procedureTM BPT triangle; and (b) patch for BPN triangle. 112
Finally, the equilibrium equations are obtained from the variation with respect to the displacement, and are expressed as
6nw - f
[(/2V)6w]TMdA-fa
6 w q d A + 6rlbt - O
e
(11.92)
e
A finite element approximation may be constructed in the standard manner by writing M--
NmI~/Ia,
x-
NXa~(.a and
ll) -- NW'wa
(11.93)
The simplest approximations are for N a = NaX = 1 and linear interpolation over each triangle for N~~ Equation (11.90) is easily evaluated; however, the other two integrals have apparent difficulty since a linear interpolation yields zero derivatives within each triangle. Indeed the curvature is now concentrated in the 'kinks' which occur between contiguous triangles. To obtain discrete approximations to the curvature changes an integration by parts (i.e. application of Green's theorem) is used to rewrite Eq. (11.91) as
fA ~ T ~ u dA + f r 6MTgVw dF -- 0 e
e
where g
0]
ny
-
Lny
(11.94)
(11 95)
nx
is a matrix of the direction cosines for an outward pointing normal vector n to the boundary F e and V w --
'
W,y
(11.96)
In these expressions l-'e is the part of the boundary within the area of integration Ae. Thus, for the element type BPT it is just the contour 1-'p as shown in Fig. 11.25(a).
373
374
Plate bending approximation: thin (Kirchhoff) plates and C~ continuity requirements For the element type BPN no slope discontinuity occurs on the boundary F a shown in Fig. 11.25(b); however, it is necessary to integrate along the half sides of each triangle within the patch bounded by Fa. The remainder of the derivation is now straightforward and the reader is referred to reference 112 for additional details and results. In this paper results are also presented for thin shells. We note that the type of element discussed in this section is quite different from those presented previously in that nodes exist outside the boundary of the element. Thus, the definition of an element and the assembly process are somewhat different. In addition, boundary conditions need some special treatments to include in a general manner. 112 Because of these differences we do not consider additional members in this family. We do note, however, that for explicit dynamic programs some advantages occur since no rotation parameters need be integrated. Results for thin shells subjected to impulsive loading are particularly noteworthy. 112
iiiiiiiii!!i~i!ii~ii~ii!ii~iiiiii~iiiiiiii~iiiii~i~iiiiiiiiiiii~iii~iiiii~iiiiiiii!~iiii~i~iii~iii~iiiiiiiii!i~iii~iiiiiiiiiiiii!i!ii~i~iiii~i~i~iiiii~ii~ii~i~iiiiii~iiiiii~iiiiii~%iiii~iiii~iiiiiii~iiiiiii~iiiiiiii~iiiiii~ii~i~ ii~ii!ii!~ i!~i~iiii!iii!i ili !~iii!i2Q ~iiiii!!i~ i ~li~ ~|~ C~!~~ i~~i~it~!ii~iih~h~| O UiF i~!!!~!~~iiiiiiii~~i~!iiiiiiii ii~iiii~iii~ii~iiiiii iiiiii~~i~i!~liiiii!ii~i~!i~iiii!iiiii~iii!iiii!i~i~i~i~i~i~iiiiiiii The preceding discussion has assumed the plate to be a linear elastic material. In many situations it is necessary to consider a more general constitutive behaviour in order to represent the physical problem correctly. For thin plates, only the bending and twisting moment are associated with deformations and are related to the local stresses through
M =
My Mxy
=
O-y z dz a -t/2
(11.97)
Txy
Any of the material models discussed in Chapter 4 which have symmetric stress behaviour with respect to strains may be used in plate analysis provided an appropriate plane stress form is available (either analytically or through iterative reduction of the three-dimensional equations). The symmetry is necessary to avoid the generation of in-plane force resultants - which are assumed to decouple from the bending behaviour. If such conditions do not exist it is necessary to use a shell formulation as described in Chapters 13-16. In practice two approaches are considered- one dealing with the individual lamina using local stress components ~rx, O'y a n d Txy and the other using plate resultant forces Mx, My and Mxy directly.
Numerical integration through thickness
The most direct approach is to use a plane stress form of the stress-strain relation and perform the through-thickness integration numerically. In order to capture the maximum stresses at the top and bottom of the plate it is best to use Gauss-Lobattotype quadrature formulae TM where integrals are approximated by 1
J[ f(()d~
1
N-1 f ( - 1 ) W 1 + Z f ( ~ n ) W ~ + f(1)WN
n=2
(11.98)
These formula differ from the typical Gaussian quadrature considered previously and use the end points on the interval directly. This allows computation of first yield to be
Inelastic material behaviour Table 11.4 Gauss-Lobatto quadrature points and weights
N
~,,
3
4-1.0
1/3
0.0
4/3
4 5
W.
4-1.0
1/6
4- 0q/0-~.2
5/6
4-1.0
0.1
4-~
49/90 0.0
6
64/90
4-1.0
1/15
q-~//~
0.6/[tl (1 - t o ) 2]
•
0.6/[t2 ( 1 - t o ) z]
to = V/ff
tl = (7 + 2to)/21 t2 = ( 7 - 2to)/21
more accurate. The values of the parameters ~n and Wn are given in Table 11.4 up to the six-point formula. Parameters for higher-order formulae may be found in reference 114. Noting that the strain components in plates [see Eq. (11.8a)] are asymmetric with respect to the middle surface of the plate and that the z coordinate is also asymmetric we can compute the plate resultants by evaluating only half the integral. Accordingly, we may use M =
My xy
= 2
Oy ,!o
z dz
(11.99)
Txy
and here a six-point formula or less will generally be sufficient to compute integrals. Equation (11.21) is replaced by the non-linear equation given as ~ ( w ) ----f -- L BTM dff2 = 0
(11.100)
The solution process (for a static case) may now proceed by using, for instance, a Newton scheme in which the t a n g e n t m o d u l i for the plate are obtained by using the tangent moduli for the stress components as dM =
dMy dMxy
=
2
dz
12Vdw
-
DT(LV)dw
(11.101)
ao
where D~ps)(z) is the tangent modulus matrix of a plane stress material model at each lamina level z, and DT is the resulting bending tangent stiffness matrix of the plate. The Newton iteration for the displacement increment is computed as K(Tk)d~(k) = ffj(k) with iterative updates
~Tv(k+1) _- ~TV(k) ~-
d~r (k)
(11.102) (11.103)
375
376
Plate bending approximation: thin (Kirchhoff) plates and C~ continuity requirements until a suitable convergence criterion is satisfied. previously defined for solids.
This follows precisely methods
Resultant constitutive models
A resultant yield function for plates with Huber-von Mises-type material is given by ~15 F(M)-
( M 2 + M 2 - M~M,, + 3M~,) - M2(tQ _< 0
(11.104)
where ~ is an 'isotropic' hardening parameter and Mu denotes a uniaxial yield moment and which for homogeneous plates is generally given by 1 Mu -- ~t2Cry(~)
(11.105)
in which or,, is the material uniaxial yield stress in tension (and compression). We observe that, in the absence of hardening, Mu is the moment that exists when the entire cross-section is at a yield stress. . . . . . . . . .].::,. .~.,~. .~:~U ~::~:~!:' 9 ]:"~::~...................... ~ i:~'i~:~N !iiiiiii!iii!i~ n iii!i!i ~i ii'::'~li'l~ '::iiii!ii !ili!ii::':'~ION :i~iiiiTi!!iiiiiiiiiiiiiiii!iiiiiiiiii!iiiii!iiiiiiiiiiiiiiii!iiii":' V ~ m ~i rk~ iii',:ii:i i ii::!iiiiii!',i',ii!!'wi!iiii!!iii ,!i,'!~ii!i!:'i~~ii',,i!i:,iilf~ ~i ! ,,i!ii,:iii!ii'~'~'iiii!ii~iii !iil!i!iiii ~i|~ iiiiiiiiilli~,:i~':'~'iiiiiiiiii:iiiii:i:i:i!i:!ii!iiiii!: !:i:!i!'i~,i:iiiii:iiii' : ~!iiii::~:!: ,~': i:'~'~i',:ii!:i:i~i!:!ii:ii :i:i!i:,!:ii:i:ii!i :iiii!ii!iiiiii:i: i:i'~i:ilii ii~i~ii:i:i:ii:i~ii~ii~i~,!:i'i~i~!:ii'!i~iiii~iii',iiiii!ii'~'ii'!:~i!ii!:~, :,i i!iiiiiiil ...... ::~,~i,, : i!:iiiiiii:::::::u,::i 7:!:!:!:ii::i iii i!'.......iili'.........~.........iiiil/-::~i..............iii~i 'i :i ::/~!:iiii~::il/:i,-~::: .....'i~ ?:!:',:!i'..... : 'iiiii'~i',u:,i ii:~: '::m:e i:R~~ ,~i ~:ii~i~iiii~ii~i~i~iii!iiiiiiii~iiiiiiiiiiiiii!iiii~iiii!:~!!~iiiii~:~i~:~::i::~:ii~:~i:~i!ii~:i~:i:~:::~i~:!~:~i~iiiii::ii!ii:~i~i~:i::i!
The extensive bibliography of this chapter outlining the numerous approaches capable of solving the problems of thin, Kirchhoff, plate flexure shows both the importance of the subject in structural engineering - particularly as a preliminary to shell analysis and the wide variety of possible approaches. Indeed, only part of the story is outlined here, as the next chapter, dealing with thick plate formulation, presents many practical alternatives of dealing with the same problem. We hope that the presentation, in addition to providing a guide to a particular problem area, is useful in its direct extension to other fields where governing equations lead to C1 continuity requirements. Users of practical computer programs will be faced with a problem of 'which element' is to be used to satisfy their needs. We have listed in Table 11.3 some of the more widely known simple elements and compared their performance in Fig. 11.16. The choice is not always unique, and much more will depend on preferences and indeed extensions desired. As will be seen in Chapter 13 for general shell problems, triangular elements are an optimal choice for many applications and configurations. Further, such elements are most easily incorporated if adaptive mesh generation is to be used for achieving errors of predetermined magnitude.
iilliiiiiiiiiiiiii! !iiiiiiilili!iliiiiiiiiiiiiiiiiiiiiiiiiiiiili!iiii !iiiiiilliiiiii!il !iiiiiiiiiiiii!iiiiiiil iiliiiii~iiiiiiiiiiiiiiiii iiiiiiiiiiiii~!iiiiiiiiiiiii! iiiii!iiiiiiiiiiiiiii !iiiiiiiiiii!iiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiii i!iiiiiiiiiiiiiiii!iiiiili!iiii!i! iiii!iiiiiiiii!!!iiiiiiii!iiiiiiiiiiiiii!iiiiiiii ii!iili!ii!ii!iiliilil!iii!iiiii!iiiiiiii iiiiiiiiiili~iiiiiiiiiii!iii~! i~i~iii!iiii~iiiiii~ii~~iii~ii~ii~i~~i~Jii~ i~ ~ii~i~i~i~i~i~i~iii~iii~ii~i~ iiiii~~i~i~i~i~i~i~i~i!~!~i ~i~!~!iiii!~~ii~i!ii!i~iiii~iiiii~ii!i !~!~!~ i~i ~i~ili~i~iii~i i!!ii~i~
1. G. Kirchhoff. Ober das Gleichqewicht und die Bewegung einer elastichen Scheibe. J. Reine und Angewandte Mathematik, 40:51-88, 1850. 2. S.P. Timoshenko and S. Woinowski-Krieger. Theory of Plates and Shells. McGraw-Hill, New York, 2nd edition, 1959. 3. L. Bucciarelly and N. Dworsky. Sophie Germain, an Essay on the History of Elasticity. Reidel, New York, 1980.
References 377 4. E. Reissner. Reflections on the theory of elastic plates. Appl. Mech. Rev., 38:1453-1464, 1985. 5. E. Reissner. The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech., 12:69-76, 1945. 6. R.D. Mindlin. Influence of rotatory inertia and shear in flexural motions of isotropic elastic plates. J. Appl. Mech., 18:31-38, 1951. 7. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 8. I. Babugka and T. Scapolla. Benchmark computation and performance evaluation for a rhombic plate bending problem. International Journal for Numerical Methods in Engineering, 28:155180, 1989. 9. I. Babugka. The stability of domains and the question of formulation of plate problems. Appl. Math., pages 463-467, 1962. 10. B.M. Irons and J.K. Draper. Inadequacy of nodal connections in a stiffness solution for plate bending. Journal of AIAA, 3:5, 1965. 11. R.W. Clough and J.L. Tocher. Finite element stiffness matrices for analysis of plate bending. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 515-545, Wright Patterson Air Force Base, Ohio, October 1966. 12. G.E Bazeley, Y.K. Cheung, B.M. Irons and O.C. Zienkiewicz. Triangular elements in bending -conforming and non-conforming solutions. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 547-576, Wright Patterson Air Force Base, Ohio, October 1966. 13. B. Fraeijs de Veubeke. Bending and stretching of plates. Special models for upper and lower bounds. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 863-886, Wright Patterson Air Force Base, Ohio, October 1966. 14. B.M. Irons. A conforming quartic triangular element for plate bending. International Journal for Numerical Methods in Engineering, 1:29-46, 1969. 15. R.W. Clough and C.A. Felippa. A refined quadrilateral element for analysis of plate bending. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, pages 399-440, Wright Patterson Air Force Base, Ohio, October 1968. 16. B. Fraeijs de Veubeke. A conforming finite element for plate bending. International Journal of Solids and Structures, 4:95-108, 1968. 17. EK. Bogner, R.L. Fox and L.A. Schmit. The generation of interelement-compatible stiffness and mass matrices by the use of interpolation formulae. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 397-443, Wright Patterson Air Force Base, Ohio, October 1966. 18. I.M. Smith and W. Duncan. The effectiveness of nodal continuities in finite element analysis of thin rectangular and skew plates in bending. International Journal for Numerical Methods in Engineering, 2:253-258, 1970. 19. K. Bell. A refined triangular plate bending element. International Journal for Numerical Methods in Engineering, 1:101-122, 1969. 20. G.A. Butlin and R. Ford. A compatible plate bending element. Technical Report 68-15, University of Leicester Engineering Department, 1968. 21. G.R. Cowper, E. Kosko, G.M. Lindberg and M.D. Olson. Formulation of a new triangular plate bending element. Trans. Canad. Aero-Space Inst., 1:86-90, 1968 (see also NRC Aero Report LR514, 1968). 22. W. Bosshard. Ein neues vollvertr~igliches endliches Element for Plattenbiegung. Mt. Ass. Bridge Struct. Eng. Bull., 28:27-40, 1968. 23. J.H. Argyris, I. Fried and D.W. Scharpf. The TUBA family of plate elements for the matrix displacement method. The Aeronaut. J., Roy. Aeronaut. Soc., 72:701-709, 1968. 24. W. Visser. The finite element method in deformation and heat conduction problems. Dr. Wiss. dissertation, Technische Hochschule, Delft, 1968.
378
Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements 25. B.M. Irons, J.G. Ergatoudis and O.C. Zienkiewicz. Comments on 'complete polynomial displacement fields for finite element method' (by P.C. Dunne). Trans. Roy. Aeronaut. Soc., 72:709, 1968. 26. O.C. Zienkiewicz and Y.K. Cheung. The finite element method for analysis of elastic isotropic and orthotropic slabs. Proc. Inst. Civ. Eng., 28:471-488, 1964. 27. R.W. Clough. The finite element method in structural mechanics. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 7. John Wiley & Sons, Chichester, 1965. 28. D.J. Dawe. Parallelogram element in the solution of rhombic cantilever plate problems. J. Strain Anal., 1:223-230, 1966. 29. J.H. Argyris. Continua and discontinua. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 11-189, Wright Patterson Air Force Base, Ohio, October 1966. 30. L.S.D. Morley. On the constant moment plate bending element. J. Strain Anal., 6:20-24, 1971. 31. R.D. Wood. A shape function routine for the constant moment triangular plate bending element. Engineering Computations, 1:189-198, 1984. 32. R. Narayanaswami. New triangular plate bending element with transverse shear flexibility. Journal of AIAA, 12:1761-1763, 1974. 33. A. Razzaque. Program for triangular bending element with derivative smoothing. International Journal for Numerical Methods in Engineering, 5:588-589, 1973. 34. B.M. Irons and A. Razzaque. Shape function formulation for elements other than displacement models. In C.A. Brebbia and H. Tottenham, editors, Proc. of the International Conference on Variational Methods in Engineering, volume II, pages 4/59-4/72. Southampton University Press, 1973. 35. J.E. Walz, R.E. Fulton and N.J. Cyrus. Accuracy and convergence of finite element approximations. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, pages 995-1027, Wright Patterson Air Force Base, Ohio, October 1968. 36. R.J. Melosh. Structural analysis of solids. J. Structural Engineering, ASCE, 4:205-223, August 1963. 37. A. Adini and R.W. Clough. Analysis of plate bending by the finite element method. Technical Report G-7337, Report to National Science Foundation USA, 1961. 38. Y.K. Cheung, I.P. King and O.C. Zienkiewicz. Slab bridges with arbitrary shape and support conditions. Proc. Inst. Civ. Eng., 40:9-36, 1968. 39. J.L. Tocher and K.K. Kapur. Comment on basis of derivation of matrices for direct stiffness method (by R. Melosh). Journal of AIAA, 3:1215-1216, 1965. 40. R.D. Henshell, D. Waiters and G.B. Warburton. A new family of curvilinear plate bending elements for vibration and stability. J. Sound Vibr., 20:327-343, 1972. 41. R.L. Taylor, O.C. Zienkiewicz, J.C. Simo and A.H.C. Chan. The patch test - a condition for assessing FEM convergence. International Journal for Numerical Methods in Engineering, 22:39-62, 1986. 42. O.C. Zienkiewicz, S. Qu, R.L. Taylor and S. Nakazawa. The patch test for mixed formulations. International Journal for Numerical Methods in Engineering, 23:1873-1883, 1986. 43. O.C. Zienkiewicz and D. Lefebvre. Three field mixed approximation and the plate bending problem. Comm. Appl. Num. Meth., 3:301-309, 1987. 44. P.G. Bergen and L. Hanssen. A new approach for deriving 'good' element stiffness matrices. In J.R. Whiteman, editor, The Mathematics of Finite Elements and Applications, pages 483-497. Academic Press, London, 1977. 45. R.V. Southwell. Relaxation Methods in Theoretical Physics. Clarendon Press, Oxford, 1st edition, 1946. 46. P.G. Bergan and M.K. Nygard. Finite elements with increased freedom in choosing shape functions. International Journal for Numerical Methods in Engineering, 20:643-663, 1984.
References 379 47. C.A. Felippa and EG. Bergan. A triangular plate bending element based on energy orthogonal free formulation. Computer Methods in Applied Mechanics and Engineering, 61:129-160, 1987. 48. A. Samuelsson. The global constant strain condition and the patch test. In R. Glowinski, E.Y. Rodin and O.C. Zienkiewicz, editors, Energy Methods in Finite Element Analysis, Chapter 3, pages 49-68. John Wiley & Sons, Chichester, 1979. 49. B. Specht. Modified shape functions for the three node plate bending element passing the patch test. International Journal for Numerical Methods in Engineering, 26:705-715, 1988. 50. T.J.R. Hughes. The Finite Element Method: Linear Static and Dynamic Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1987. 51. F. Kikuchi and Y. Ando. A new variational functional for the finite element method and its application to plate and shell problems. Nuclear Engineering and Design, 21(1):95-113, 1972. 52. L.S.D. Morley. The triangular equilibrium element in the solution of plate bending problems. Aero. Q., 19:149-169, 1968. 53. W. Ritz. Ober eine neue Methode zur Ltisung gewisser variationsproblem der mathematischen physik. Journal far die reine und angewandte Mathematik, 135:1-61, 1908. 54. B.G. Galerkin. Series solution of some problems in elastic equilibrium of rods and plates. Vestn. Inzh. Tech., 19:897-908, 1915. 55. H. Hencky. Der Spannungszustand in rechteckigen Platten. Technical ReportVI, 94 S, Miinchen, 1913. 56. H. Hencky. Der Spannungszustand in rechteckigen Platten. PhD thesis, Darmstadt, Published by R. Oldenbourg, Munich and Berlin, Germany, 1913. 57. I.A. Wojtaszak. The calculation of maximum deflection, moment, and shear for uniformly loaded rectangular plate with clamped edges. J. Applied Mechanics, ASME, 59:A173-A176, 1937. 58. R.L. Taylor and S. Govindjee. Solution of clamped rectangular plate problems. Communications in Numerical Methods of Engineering, 20:757-765, 2004. 59. H. Marcus. Die Theorie elastisher Geweve und ihre Anwendung auf die Berechnung biegsamer Platten. Springer, Berlin, 1932. 60. P. Ballesteros and S.L. Lee. Uniformly loaded rectangular plate supported at the comers. Int. J. Mech. Sci., 2:206-211, 1960. 61. J.A. Stricklin, W. Haisler, P. Tisdale and K. Gunderson. A rapidly converging triangle plate element. Journal of AIAA, 7:180-181, 1969. 62. G.S. Dhatt. Numerical analysis of thin shells by curved triangular elements based on discrete Kirchhoff hypotheses. In W.R. Rowan and R.M. Hackett, editors, Proc. Symp. on Applications of FEM in Civil Engineering, Vanderbilt University, Nashville, Tennessee, 1969. ASCE. 63. J.L. Batoz, K.-J. Bathe and L.W. Ho. A study of three-node triangular plate bending elements. International Journal for Numerical Methods in Engineering, 15:1771-1812, 1980. 64. M.M. Hrabok and T.M. Hrudey. A review and catalogue of plate bending finite elements. Computers and Structures, 19:479-495, 1984. 65. J. Jirousek and L. Guex. The hybrid-Trefftz finite element model and its application to plate bending. Int. J. Num. Meth. Eng., 23:651-693, 1986. 66. A. Razzaque. Finite element analysis of plates and shells. PhD thesis, Civil Engineering Department, University of Wales, Swansea, 1972. 67. E.L. Wilson. The static condensation algorithm. International Journal for Numerical Methods in Engineering, 8:1974, 199-203. 68. G. Sander. Bournes suprrieures et infrrieures dans l'analyse matricielle des plates en flexiontorsion. Bull. Soc. Royale des Sci. de Liege, 33:456--494, 1974. 69. D. Withum. Berechnung von Platten nach dem Ritzchen Verfahren mit Hilfe dreieckftirmiger Meshnetze. Technical report, Mittl. Inst. Statik, Technische Hochschule, Hanover, 1966.
380 Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements 70. C.A. Felippa. Refined finite element analysis of linear and non-linear two-dimensional structures. PhD dissertation, Department of Civil Engineering, SEMM, University of California, Berkeley, 1966. Also: SEL Report 66-22, Structures Materials Research Laboratory. 71. A. Zanisek. Interpolation polynomials on the triangle. International Journal for Numerical Methods in Engineering, 10:283-296, 1976. 72. A.G. Peano. Conforming approximation for Kirchhoff plates and shells. International Journal for Numerical Methods in Engineering, 14:1273-1291, 1979. 73. J.J. Grel. Construction of basic functions for numerical utilization of Ritz's method. Numerische Math., 12:435-447, 1968. 74. G. Birkhoff and L. Mansfield. Compatible triangular finite elements. J. Math. Anal. Appl., 47:531-553, 1974. 75. C.L. Lawson. Cl-compatible interpolation over a triangle. Technical Report RM 33-770, NASA Jet Propulsion Laboratory, Pasadena, California, 1976. 76. L.R. Herrmann. Finite element bending analysis of plates. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, pages 577-602, Wright Patterson Air Force Base, Ohio, 1965. 77. L.R. Herrmann. Finite element bending analysis of plates. J. Engineering Mechanics, ASCE, 94(EM5): 13-25, 1968. 78. W. Visser. A refined mixed type plate bending element. Journal of AIAA, 7:1801-1803, 1969. 79. J.C. Boot. On a problem arising from the derivation of finite element matrices using Reissner's principle. International Journal for Numerical Methods in Engineering, 12:1879-1882, 1978. 80. A. Chaterjee and A.V. Setlur. A mixed finite element formulation for plate problems. International Journal for Numerical Methods in Engineering, 4:67-84, 1972. 81. J.W. Harvey and S. Kelsey. Triangular plate bending elements with enforced compatibility. Journal of AIAA, 9:1023-1026, 1971. 82. B. Fraeijs de Veubeke and O.C. Zienkiewicz. Strain energy bounds in finite element analysis by slab analogy. J. Strain Anal., 2:265-271, 1967. 83. B. Fraeijs de Veubeke. An equilibrium model for plate bending. International Journal of Solids and Structures, 4:447-468, 1968. 84. J. Bron and G. Dhatt. Mixed quadrilateral elements for plate bending. Journal of AIAA, 10: 1359-1361, 1972. 85. K. Hellan. Analysis of elastic plates in flexure by a simplified finite element method. Technical Report Civ. Eng. Series 46, Acta Polytechnica Scandinavia, Trondheim, 1967. 86. T.H.H. Pian. Derivation of element stiffness matrices by assumed stress distribution. Journal of AIAA, 2:1332-1336, 1964. 87. T.H.H. Pian and P. Tong. Basis of finite element methods for solid continua. International Journal for Numerical Methods in Engineering, 1:3-28, 1969. 88. R.J. Allwood and G.M.M. Comes. A polygonal finite element for plate bending problems using the assumed stress approach. International Journal for Numerical Methods in Engineering, 1:135-160, 1969. 89. B.E. Greene, R.E. Jones, R.M. McLay and D.R. Strome. Generalized variational principles in the finite element method. Journal of AIAA, 7:1254-1260, 1969. 90. P. Tong. New displacement hybrid models for solid continua. International Journal for Numerical Methods in Engineering, 2:73-83, 1970. 91. B.K. Neale, R.D. Henshell and G. Edwards. Hybrid plate bending elements. J. Sound Vibr., 22:101-112, 1972. 92. R.D. Cook. Two hybrid elements for analysis of thick, thin and sandwich plates. International Journal for Numerical Methods in Engineering, 5:277-299, 1972. 93. C. Johnson. On the convergence of a mixed finite element method for plate bending problems. Num. Math., 21:43-62, 1973. 94. R.D. Cook and S.G. Ladkany. Observations regarding assumed-stress hybrid plate elements. International Journal for Numerical Methods in Engineering, 8:513-520, 1974.
References 381 95. I. Torbe and K. Church. A general quadrilateral plate element. International Journal for Numerical Methods in Engineering, 9:855-868, 1975. 96. D.J. Allman. A simple cubic displacement model for plate bending. International Journal for Numerical Methods in Engineering, 10:263-281, 1976. 97. J. Jirousek. Improvement of computational efficiency of the 9 dof triangular hybrid-Trefftz plate bending element. International Journal for Numerical Methods in Engineering, 23:2167-2168, 1986. (Letter to editor.) 98. G.A. Wempner, J.T. Oden and D.K. Cross. Finite element analysis of thin shells. Proc. Am. Soc. Civ. Eng., 94(EM6):1273-1294, 1968. 99. G. Dhatt. An efficient triangular shell element. Journal of AIAA, 8:2100-2102, 1970. 100. J.L. Batoz and G. Dhatt. Development of two simple shell elements. Journal of AIAA, 10: 237-238, 1972. 101. J.T. Baldwin, A. Razzaque and B.M. Irons. Shape function subroutine for an isoparametric thin plate element. International Journal for Numerical Methods in Engineering, 7:431-440, 1973. 102. B.M. Irons. The semi-Loof shell element. In D.G. Ashwell and R.H. Gallagher, editors, Finite Elements for Thin Shells and Curved Members, Chapter 11, pages 197-222. John Wiley & Sons, Chichester, 1976. 103. L.ER. Lyons. A general finite element system with special analysis of cellular structures. PhD thesis, Imperial College of Science and Technology, London, 1977. 104. R.A.E Martins and D.R.J. Owen. Thin plate semi-Loof element for structural analysis including stability and structural vibration. International Journal for Numerical Methods in Engineering, 12:1667-1676, 1978. 105. J.L. Batoz and M.B. Tahar. Evaluation of a new quadrilateral thin plate bending element. International Journal for Numerical Methods in Engineering, 18:1655-1677, 1982. 106. J.L. Batoz. An explicit formulation for an efficient triangular plate bending element. International Journal for Numerical Methods in Engineering, 18:1077-1089, 1982. 107. M.A. Crisfield. A new model thin plate bending element using shear constraints: A modified version of Lyons' element. Computer Methods in Applied Mechanics and Engineering, 38: 93-120, 1983. 108. M.A. Crisfield. A qualitative Mindlin element using shear constraints. Computers and Structures, 18:833-852, 1984. 109. M.A. Crisfield. Finite Elements and Solution Procedures for Structural Analysis, Vol. 1, Linear Analysis. Pineridge Press, Swansea, 1986. 110. G. Dhatt, L. Marcotte and Y. Matte. A new triangular discrete Kirchhoff plate-shell element. International Journal for Numerical Methods in Engineering, 23:453-470, 1986. 111. R.A. Nay and S. Utku. An alternative for the finite element method. Variational Methods in Engineering, 1, 1972. 112. E. Ofiate and E Z~rate. Rotation-free triangular plate and shell elements. International Journal for Numerical Methods in Engineering, 47:557-603, 2000. 113. E. Ofiate and G. Bugeda. A study of mesh optimality criteria in adaptive finite element analysis. Engineering Computations, 10:307-321, 1993. 114. M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions. Dover Publications, New York, 1965. 115. G.S. Shapiro. On yield surfaces for ideal plastic shells. In Problems of Continuum Mechanics, pages 414-418. SIAM, Philadelphia, 1961.
ii~i~i~i~ii!~i~:~i~i!~i!i~i~i~i~ii~i:i!i!ii~:~i~!i~i~i~i~.i~i~i~i~i~!~!i~
'Thick' Reissner-Mindlin platesirreducible and mixed formulations ':',
We the the
i ' ~?~i , i ~7?~z~=~i 2 ~ i~i~:~=~=~: i'=='~'==|~~dUC~|O=~i ~!~i~i~i~=~i=~:~i!~i~=""~' i~i~i~i~!~i~i~i!~i~,~' i~~ i~;i~ .......~ii ~ 2 ~ : ~ : ~ ~ = ~ = :~~~=~:~~:=~=:~=~=~ ~ i{ ~ {i ~ { iili ~ { ~ ~i ~ { ~ ~ i
i
have already introduced in Chapter 11 the full theory of thick plates from which thin plate, Kirchhoff, theory arises as the limiting case. In this chapter we shall show how the numerical solution of thick plates can easily be achieved and how, in limit, an alternative procedure for solving all problems of Chapter 11 appears. The development of approximations to the Timoshenko beam form discussed in Chapter 10 plays an important role in the development of viable solutions of the thick, ReissnerMindlin, theory that work in 'thin' plate applications. To ensure continuity we repeat below the governing equations [Eqs (11.10a)(11.1 l b), or Eqs (11.72)]. Referring to Fig. 11.3 of Chapter 11 and the text for definitions, we remark that all the equations could equally well be derived from full three-dimensional analysis of a flat and relatively thin portion of an elastic continuum illustrated in Fig. 12.1. All that it is now necessary to do is to assume that whatever form of the approximating shape functions in the xy plane those in the z direction are only linear. Further, it is assumed that Crz stress is zero, thus eliminating the effect of vertical strain.* The first approximations of this type were introduced quite early 1'2 and the elements then derived are exactly of the Reissner-Mindlin type discussed in Chapter 11. The equations from which we shall start and on which we shall base all subsequent discussion are the moment constitutive equation [see Eqs (11.10a) and (11.72)], M-
D/~t~ = 0
(12.1a)
where D is the matrix of bending rigidities, the shear constitutive equation [see Eqs (11.10c) and (11.72)] 1 -S OL
- q5-
Vw
= 0
(12.1b)
where c~ - nGt is the shear rigidity, the moment equilibrium (angular momentum) equation [see Eqs ( 11.1 lb) and ( 11.72)] ETM + S = 0 *
Reissner includes the effect of O'z in bending but, for simplicity, this is disregarded here.
(12.1c)
Introduction
i 55/
(w, ,~, %)
Fig. 12.1 An isoparametric three-dimensional element with linear interpolation in the transverse (thickness) direction and the 'thick' plate element.
and the shear equilibrium equation [see Eqs (11.1 la) and (11.72)]
~rTs + q = 0
(12.1d)
In the above the moments M, the transverse shear forces S, and the elastic matrices D are as defined in Chapter 11, and -0 s
0
0 0 ~yy
0
0
_Oy
Ox_
(12.2)
defines the strain-displacement operator on rotations qS, and its transpose the equilibrium operator on moments, M. Boundary conditions are of course imposed on w and ~b or the corresponding plate forces S,, M,, M,s in the manner discussed in Sec. 11.2.2. It is convenient to eliminate M from Eqs (12.1 a)-(12.1d) and write the system of three equations [Eqs (11.73)] as
LTDL~b + S = 0 1 --S - q~- ~'w = 0
OL
(12.3)
vTS+q =0
This equation system can serve as the basis on which a mixed discretization is builtor alternatively can be reduced further to yield an irreducible form. In Chapter 11 we
383
384 'Thick' Reissner-Mindlin plates-irreducible and mixed formulations have dealt with the irreducible form which is given by a fourth-order equation in terms of w alone and which could only serve for solution of thin plate problems, that is, when c~ = c~ [Eq. (11.14a)]. On the other hand, it is easy to derive an alternative irreducible form which is valid only if c~ ~ cx~. Thus, the shear forces can be eliminated yielding two equations: / ~ T D / ~ -4- ct (VW -4- ~b) -- 0 V v [c~ ( V w + qS)] + q -- 0
(12.4)
This is an irreducible system corresponding to minimization of the total potential energy
I'I : 7
(s
12
D/2th dr2 + ~
( V w -+- ~t~)T OL(Vl/) + ~ ) d r 2 (12.5)
/.
- / ~ w q dr2 + Flbt = minimum as can easily be verified. In the above the first term is simply the bending energy and the second the shear distortion energy [see Eq. (11.83)]. Clearly, this irreducible system is only possible when a ~ oo, but it can, obviously, be interpreted as a solution of the potential energy given by Eq. (11.83) for 'thin' plates with the constraint of Eq. (11.84) being imposed in a penalty manner with a being now a penalty parameter. Thus, as indeed is physically evident, the thin plate formulation is simply a limiting case of such analysis. We shall see that the penalty form can yield a satisfactory solution only when discretization of the corresponding mixed formulation satisfies the necessary convergence criteria. The thick plate form now permits independent specification of three conditions at each point of the boundary. The options which exist are: w (/)n q5s
or or or
Sn M,, M.~,
in which the subscript n refers to a normal direction to the boundary and s a tangential direction. Clearly, now there are many combinations of possible boundary conditions. A 'fixed' or 'clamped' situation exists when all three conditions are given by displacement components, which are generally zero, as
W= n
=0
and a free boundary when all conditions are the 'resultant' components
S.=Mn =Mns=O When we discuss the so-called simply supported conditions (see Sec. 11.2.2), we shall usually refer to the specification w = 0
and
M~ = M~s = 0
The irreducible formulation-reduced integration
385
as a 'soft' support, SS 1 (and indeed the most realistic support), and to w = 0
Mn = 0
and
q~ = 0
as a 'hard' support, SS2. The latter in fact replicates the thin plate assumptions and, incidentally, leads to some of the difficulties associated with it. Finally, there is an important difference between thin and thick plates when 'point' loads are involved. In the thin plate case the displacement w remains finite at locations where a point load is applied; however, for thick plates the presence of shearing deformation leads to an infinite displacement (as indeed three-dimensional elasticity theory also predicts). In finite element approximations one always predicts a finite displacement at point locations with the magnitude increasing without limit as a mesh is refined near the loads. Thus, it is meaningless to compare the deflections at point load locations for different element formulations and we will not do so in this chapter. It is, however, possible to compare the total strain energy for such situations and here we immediately observe that for cases in which a single point load is involved the displacement provides a direct measure for this quantity.
!:7i7i:~i~i:i~:~i~i7~i7i~!~!~{!i~ii~!~!~!i ...7.~i.~i.~.!~i i:.~:{i~i!{!i!~i~!i~i:~:i~ii~i~:~ii~i~i~i!i~:~i~i7~ii~i~::~:~;~i~i~;~i~i7~:~i!~:~:~:~:~:~.............. :!~:~::~:~:~7~:~::~!~::~::~+::~::~::~:~7:ii:~i~iii~::~::~i~::~:!i~i~i~ii!i!i!~!~!i!~ii7iii!i~7~:~ii!i!~:~i~i~i~i!i::~:~:~:~:~::~i~i~:~:~i~::~::~:~ii~i~::::~::~i~i~iiiii::::~::~i77:~i~i~i~7~!::~::!i~:::~i~!i!:::~::~i~ :~7i~:~ii~7!iii!i!i~!iii:~:~:i7~:~7i~7~
The procedures for discretizing Eq. (12.4) appear to be straightforward. However, we will find that the process is very sensitive. First, we consider standard isoparametric interpolation in which the two displacement variables are approximated by shape functions and parameters as 0 5 - No~b
and
w = Nw~
(12.6)
We recall that the rotation parameters 05 may be transformed into physical rotations about the coordinate axes, O, using Eq. (1 1.7). The parameters 0 are often more convenient for calculations and are essential in shell developments. T h e approximation equations are now obtained directly by the use of the total potential energy principle [Eq. (1 2.5)], the Galerkin process on the weak form, or by the use of virtual work expressions. Here we note that the appropriate generalized strain components, corresponding to the moments M and shear forces S, are r
--
~ b -
(s162
(12.7a)
and (12.7b)
e~ - V w + 05 - VNw~r + N4,~b We thus obtain the discretized problem
(L and
dO+ L N:~
=
386 'Thick' Reissner-Mindlin plates -irreducible and mixed formulations
or simply [Kww Kw,] { ~ } Kow K,,.]
= Kfi = (Kb + Ks) fi =
{f~}--f fo
(12.8a)
with
[: oo]
,,:
q)y, Ks
(12.8b)
[K;~ = [K~
K;o ] K~J
where the arrays are defined by b = fn (s Ko,
= L s1
dg2 K~,w = fn (VNw)T oLVN,,, dr2 K~,w = fn NTaVN"-' dr2 = (K~0)T
so aa
(12.8c)
and forces are given by --
Nw ~SndF
Nwq d ~ +
(12.8d)
s
fo-/
N~I~ dr m
where Sn is the prescribed shear on boundary Fs, and 1~ is the prescribed moment on boundary Fm. The formulation is straightforward and there is little to be said about it a priori. Since the form contains only first derivatives apparently any Co shape functions of a two-dimensional kind can be used to interpolate the two rotations and the lateral displacement. Figure 12.2 shows some rectangular (or with isoparametric distortion,
o Node with two rotation parameters ~) r-1 Node with one lateral displacement parameter w
QS Fig. 12.2 Someearlythick plate elements.
QL
QH
The irreducibleformulation-reduced integration quadrilateral) elements used in the early work. ~-3 All should, in principle, be convergent as Co continuity exists and constant strain states are available. In Fig. 12.3 we show what in fact happens with a fairly fine subdivision of quadratic serendipity and Lagrangian rectangles as the ratio of span to thickness, L / t , varies. Here L is a characteristic length of the plate and may be a side length, a loading length or a normal mode characteristic. We note that the magnitude of the coefficient a is best measured by the ratio of the bending to shear rigidities and we could assess its value in a non-dimensional form. Thus, for an isotropic material with a = G t this ratio becomes Et 3
(12.9)
~
Obviously, 'thick' and 'thin' behaviour therefore depends on the L / t ratio. Clamped edge 0.0016 0.0015 Exact thin plate 0.0014 Exact thin plate sol .00406 0.0013 ",'~,, solution 0.00127 0.0012 0.0011 0.0010 llfl I-''~--- ~lijj l I:~ ~ ~ J I III I i l,,~J i J 0.0009 102 103 104 lO 4 101 102 103 L/t L/t
Simply supported
0.0044 0.0043 k 0.0042 0.0041 0.0040 0.0039 0.0038 0.00371111
101
(a)
QS-R , QS-N . . . . . . .
0.0044
- •
2 x 2 Gaussian integration of all terms 3 x 3 Gaussian integration of all terms
Simply supported ..........
0.0043~~._ 0.0042 0.0041 0.0040 0.0039 0.0038 0.0037 I 101
(b)
Clamped edge
0.001!~6
Exact thin plate solution 0.00406
" "= " " '= =' " ' " "= " ' "= ="
'
0.001 0.001 0.001
'
Exact thin plate solution 0.00127 ~
0.0012 0.0011 I-0.00101-
9
""
/
I I I J~ ~ ~ I~m ~ ~
QS-R QS-N . . . . . . .
102
L/t
103
104
0.0009111 I 101
III
102
I
L/t
i Ill l l 103
104
2 x 2 Gaussian integration of all terms 3 x 3 Gaussian integration of all terms
Fig. 12.3 Performance of (a) quadratic serendipity (QS) and (b) Lagrangian (QL) elements with varying span-to-thickness L/t, ratios, uniform load on a square plate with 4 x 4 normal subdivisions in a quarter. R is reduced 2 x 2 quadrature and N is normal 3 x 3 quadrature.
387
388
'Thick' Reissner-Mindlin plates -irreducible and mixed formulations
It is immediately evident from Fig. 12.3 that, while the answers are quite good for smaller L / t ratios, the serendipity quadratic fully integrated elements (QS) rapidly depart from the thin plate solution, and in fact tend to zero results (locking) when this ratio becomes large. For Lagrangian quadratics (QL) the answers are better, but again as the plate tends to be thin they are on the small side. The reason for this 'locking' performance is similar to those considered for the nearly incompressible problem 4 and the Timoshenko beam problem (viz. Chapter 10). In the case of plates the shear constraint implied by the third of Eq. (12.3), and used to eliminate the shear resultant, is too strong if the terms in which this is involved are fully integrated. Indeed, we see that the effect is more pronounced in the serendipity element than in the Lagrangian one. In early work the problem was thus mitigated by using a reduced quadrature, either on all terms, which we label R in the figure, 5'6 or only on the offending shear terms selectively 7,8 (labelled S). The dramatic improvement in results is immediately noted. The use of reduced quadrature to develop a four-node plate element has been revisited recently by Gruttmann and Wagner using a stabilized form. 9 The same improvement in results is observed for linear quadrilaterals in which the full (exact) integration gives results that are totally unacceptable (as shown in Fig. 12.4), but where a reduced integration on the shear terms (single point) gives excellent performance, 1~ although a careful assessment of the element stiffness shows it to be rank deficient in an 'hourglass' mode in transverse displacements. (Reduced integration on all terms gives additional matrix singularity.) A remedy thus has been suggested; however, it is not universal. We note in Fig. 12.3 that even without reduction of integration order, Lagrangian elements perform better in the quadratic expansion. In cubic elements (Fig. 12.5), however, we note that (a) almost no change occurs when integration is 'reduced' and (b), again, Lagrangian-type elements perform very much better.
~"
0.0044 0.0043 0.0042 0.0041
#
o.oo4o-
Simply supported
0.0016 0.0015 0.0014 _ ~ 0.0013 0.0012 0.0011 0.0010
Clamped edge
m
Exact thin plate solution 0.00406 ~
0.0039 0.0038 0.0037 ~ t i I~ i i IIII 101 102 103 L/t L-R L-N . . . . . . .
l
104
0.0009
I I
101
Exact thin plate solution 0.00127
I l=~l
102
I
L/t
Iiii 103
I
2 x 2 flexure integration - 1 x 1 shear integration 2 x 2 integration of all terms- this gives poor results, and diverges rapidly as L/t increases
Fig. 12.4 Performanceof bilinear elementswith varyingspan-to-thickness,L/t, values.
10 4
The irreducible
0.0044~
~
~176176176 f
(a)
~
102
integration
Clamped edge
Simply supported
0.0043 0.0042 Exact thin plate 0.0041 I" ~,,...~solution 0.00406 0.0039 0.0038 0.0037 I I 101
formulation-reduced
_•
0"0016 I 0.0015 0.0014 0.0013
Exact thin plate solution 0.00127
0"0012f
Ill 103
0.0011 0.0010 I 0.0009 t t i Ill i~ Itl i I 101 104 102 103 104
L/t L/t QS-R 3 x 3 Gaussian integration of all terms QS-N . . . . . . . 4 x 4 Gaussian integration of all terms
0.0044 0.0043 ~-.a 0.0042
Simply supported
Clamped edge
Exact thin plate
0.0015, 0. 0010"0016 f,3 0.0014
Exact thin plate
"-,,._ solution 0.00127 0.0012 ~ 0.0041 0.0038 " ...... s.oI.ution0.00406 ~t, 0.0040 0"0012f 0.0011 0.0039
0.0010 0.0009 I i I ill i I ill i i 102 103 104 101 102 103 104 L/t L/t QL-R 3 x 3 Gaussian integration of all terms QL-N . . . . . . . 4 x 4 Gaussian integration of all terms
0.0037 101 (b)
lal
i ilJ i
t
Fig. 12.5 Performanceof cubic quadrilaterals: (a) serendipity (QS) and (b) Lagrangian (QL) with varying span-to-thickness, Lit, values.
In the late 1970s many heuristic arguments were advanced for devising better elements,l 1-14all making use of reduced integration concepts. Some of these perform quite well, for example the so-called 'heterosis' element of Hughes and Cohen ]1 illustrated in Fig. 12.3 (in which the serendipity-type interpolation is used on w and a Lagrangian one on qS), but all of the elements suggested in that era fail on some occasions, either locking or exhibiting singular behaviour. Thus such elements are not 'robust' and should not be used universally. A better explanation of their failure is needed and hence an understanding of how such elements could be designed. In the next section we shall address this problem by considering a mixed formulation. The reader will recognize here arguments used in the nearly incompressible problem in Chapter 2 and in reference 4 which led to a better understanding of the failure of some straightforward elasticity elements as incompressible behaviour was approached. The situation is completely parallel here.
389
390 'Thick' Reissner-Mindlin plates-irreducible and mixed formulations
i"i!~ii'iDi'~"i"'~ii~ ii"~'~i"i~'~'i~!i'~i"'~i'"i~i'!~i~'!'i~~ ii'~i'i~~! i'!'ii~i'il!i'~i~~ii~i~i~i i~iJ~: ~!~~i~i~i!~i !i~i!i~ii~!i!~i i~i!ii~i!i!i!i!i~!i~i'!i~i!i'ii~'th ii'~i9!i~ii~ki~ !i!i~!i~~i i~~ii~~i!ii!!iii!iP!~~i i~i,i~!ili~' ii~i!i~ii!~iii~ii~i~!i~ii!i~i!i~i~i~i~!........... ii~i~i i~i~i~!i................ i~i:~ii!ii~i 'i~ii!~!ii!iii~ii!iii!ii!i!ii!iiiiiii~iii!i~iii~ii!i!i!iiiii ........................................................
12.3.1 The approximation The problem of thick plates can, of course, be solved as a mixed one starting from Eq. (12.3) and approximating directly each of the variables w, q~ and S independently. Using Eq. (12.3), we construct a weak form as _~oSwlVTS+q] d ~ = 0 f~q~T [/~TD/~q~ + S] dff2
0
(12.10)
fa~sT[----c~Is+q~+VW] d f 2 = 0 We now write the independent approximations, using the standard Galerkin procedure, as
w = N,L,~r d;w = N w ~
q~ = N~,tp
and
d;q~ = N~q~
S = NsS
and
~S = N ~ S
(12.11)
though, of course, other interpolation forms can be used, as we shall note later. After appropriate integrations by parts of Eq. (12.10), we obtain the discrete symmetric equation system
E
KCb H r
~
=
f'~
(12.12)
where Kb : f~ (s162
D (Z2Nr
E = f~ N~VNw d~ (12.13) C - f~ N~N~ dr2 H -- - f N r l N s dr2 and where fw and fr are as defined in Eq. (12.8d). The above represents a typical three-field mixed problem of the type discussed in Sec. 10.5.1 of reference 4, which has to satisfy certain criteria for stability of approximation as the thin plate limit (which can now be solved exactly) is approached. For this limit we have a = c~ and H = 0 (12.14)
Mixed formulation for thick plates 391 In this limiting case it can readily be shown that necessary criteria of solution stability for any element assembly and boundary conditions are that nr + nw
n o + n~ > ns
or
C~p =
ns > nw
or
/3p ---- ~
and
ns
ns
nw
> 1
> 1
(12.15a)
(12.15b)
where n 0, n~ and nw are the number of q~, S and ~ parameters in Eqs (12.11). When the count condition is not satisfied then the equation system either will be singular or will lock. Equations (12.15a) and (12.15b) must be satisfied for the whole system but, in addition, they need to be satisfied for element patches if local instabilities and oscillations are to be avoided. 15-17 We remind the reader that Eqs (12.15a) are (12.15b) are necessary conditions; however, they are not sufficient conditions. It is always necessary to conduct consistency and stability tests to ensure the proposed element passes the complete mixed patch test. 4 The above criteria will, as we shall see later, help us to design suitable thick plate elements which show convergence to correct thin plate solutions.
12.3.2 Continuity requirements The approximation of the form given in Eqs (12.12) and (12.13) implies certain continuities. It is immediately evident that Co continuity is needed for rotation shape functions N o (as products of first derivatives are present in the approximation), but that either Nw or Ns can be discontinuous. In the form given in Eq. (12.13) a Co approximation for w is implied; however, after integration by parts a form for Co approximation of S results. Of course, physically only the component of S normal to boundaries should be continuous, as we noted also previously for moments in the mixed form discussed in Sec. 11.16. In all the early approximations discussed in the previous section, Co continuity was assumed for both tp and w variables, this being very easy to impose. We note that such continuity cannot be described as excessive (as no physical conditions are violated), but we shall show later that successful elements also can be generated with discontinuous w interpolation (which is indeed not motivated by physical considerations). For S it is obviously more convenient to use a completely discontinuous interpolation as then the shear can be eliminated at the element level and the final stiffness matrices written simply in standard q~, ,~ terms for element boundary nodes. We shall show later that some formulations permit a limited case where c~-1 is identically zero while others require it to be non-zero. The continuous interpolation of the normal component of S is, as stated above, physically correct in the absence of line or point loads. However, with such interpolation, elimination of S is not possible and the retention of such additional system variables is usually too costly to be used in practice and has so far not been adopted. However, we should note that an iterative solution process applicable to mixed forms can reduce substantially the cost of such additional variables. TM
392
'Thick' Reissner-Mindlin p l a t e s - i r r e d u c i b l e and mixed formulations (zx = two S variables) Irreducible - with shear integration at 2 x 2 Gauss points
=
Mixed - discontinuous shear interpolation with shear nodes at 2 x 2 Gauss points
Fig. 12.6 Equivalence of mixed form and reduced shear integration in quadratic serendipity rectangle.
12.3.3 Equivalence of mixed forms with discontinuous S
interpolation and reduced (selective) integration
An equivalence of penalized mixed forms with discontinuous interpolation of the constraint variable and of the corresponding irreducible forms with the same penalty variable may be demonstrated following work of Malkus and Hughes for incompressible problems. 19Indeed, an exactly analogous proof can be used for the present case, and we leave the details of this to the reader; however, below we summarize some equivalencies that result. Thus, for instance, we consider a serendipity quadrilateral, shown in Fig. 12.6 (a), in which integration of shear terms (involving c~) is made at four Gauss points (i.e. 2 • 2 reduced quadrature) in an irreducible formulation [see Eqs (12.8a)-(12.8d)], we find that the answers are identical to a mixed form in which the S variables are given by a bilinear interpolation from nodes placed at the same Gauss points. This result can also be argued from the limitation principle first given by Fraeijs de Veubeke. 2~ This states that if the mixed form in which the stress is independently interpolated is precisely capable of reproducing the stress variation which is given in a corresponding irreducible form then the analysis results will be identical. It is clear that the four Gauss points at which the shear stress is sampled can only define a bilinear variation and thus the identity applies here. The equivalence of reduced integration with the mixed discontinuous interpolation of S will be useful in our discussion to point out reasons why many elements mentioned in the previous section failed. However, in practice, it will be found equally convenient (and often more effective) to use the mixed interpolation explicitly and eliminate the S variables by element-level condensation rather than to use special integration rules. Moreover, in problems where the material properties lead to coupling between bending and shear response (e.g. elastic-plastic behaviour) use of selective reduced integration is not convenient. It must also be pointed out that the equivalence fails if c~varies within an element or indeed if the isoparametric mapping implies different interpolations. In such cases the mixed procedures are generally more accurate. ii!ii i iiiiiiiiiiliziil~i~i i'~'ii'iii~ii! ~i~'~'~'~'~,iiliiiiiiii~i !i i~'~':~i, iiiiiii iiiiii!i'~,',~,',~"i~'i'ii:::'i!iiiil ii!iiiiilii~ilii~i~iii~i iiii!ii~i i i i ihiiiiiiiii~!J~i~i i!ii i i i i~::'::ii iiiiiiiiii i iiilii :i~~,iiii i i i i iiiiiiiiiiiiii!~iiiiiiiiii~i ~:::'ii i i i i i i i i i i i iiii~iiiiil~ii~i~ iiiiiiii i i! i i !iii~::'ili !iiiiiiiiiiii i ~J~iiiii i i i i i!~i i ~:"::~!iiiiili ~iliii~diili!i~i~i !Eiiiili~iiiiiii ~!i!i~~:~:: i:iii~iiiiiiiiiiiiiiiiili ~i i i iilili i iiiiiiiiii~iiii~iii i~:'i:::iiii~li i !iiiii~iiiiiiiili i~i i i iiiili~iiiiiii il~iiiiiiiiiiiiii~i~i~il!iii!!iiiii!i~iiiiiili~iililiiiliiiiiiiiiliiiiiiiiiilililiililililiiii~iiii~iiiiiliiiliiiiiili!iiiiiiiiii!iilii i i i i i i~i~iiiiili i ~il>ii!i:!i!i!iilii ::ii i i i i:i i i i !i ~i i~i!ii iilili i ::ii!i i i ii!iiiiiiiiiiii i i i! i !i!iiii!iiiiiiiiiiiiiiiiiiiiiiiiii! ~i iiiiili::::!iiiiiiiiiiii::iiiiiiiiiiii::iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii i i :i!i::i!iiii iiiii!iiiiiiiii:i;iiiiiiiii i iiii ii !i il i ii i~i!ii~i~i~i!i~i::::: :i!i:ii:i i i iii!!:i:~ii i!i:il ii ii i!i i i!i i!:i~:!i!i!~ii!i~i!i:ii !ii l:ilili: i i:iiii!iiii :ili :iii i!~i!ii:ii:l!i i li!iii i i i i !i i ii~ii!i i iiiil::iiiiiii i i :!::!:!::::~iiiii i!i iiiiiiiiiiliiiiliiil ii!iliii ~iii:ii:i i i i i i!!iiiiiiiiiiliiiliili!iiiiii i i i::ii!i !ii ii!il i!i ii ii i i i!:i:i i i:::i:iiiii i i i i i::ii i !i i i i i!::ii !iiii i i::::iililiiiiil!ii !i::i!ili!i !ii::!ii!iiiiiiiiiiili !i i i!::ii i!i i i i ?:iiiiiiilli ii
12.4.1 Why elements fail The nature and application of the patch test have changed considerably since its early introduction. As shown in references 15-17 and 21-25 this test can prove, in addition
The patch test for plate bending elements to consistency requirements (which were initially the only item tested), the stability of the approximation by requiting that for a patch consisting of an assembly of one or more elements the stiffness matrices are non-singular whatever the boundary conditions imposed. To be absolutely sure of such non-singularity the test must, at the final stage, be performed numerically. However, we find that the 'count' conditions given in Eqs (12.15a) and (12.15b) are necessary for avoiding such non-singularity. Frequently, they also prove sufficient and make the numerical test only a final confirmation. ]6'17 We shall demonstrate how the simple application of such counts immediately indicates which elements fail and which have a chance of success. Indeed, it is easy to show why the original quadratic serendipity element with reduced integration (QS-R) is not robust. In Fig. 12.7 we consider this element in a single-element and four-element patch subjected to so-called constrained boundary conditions, in which all displacements on the external boundary of the patch are prescribed and a relaxed boundary condition in which only three displacements (conveniently two 0s and one w) eliminate the rigid body modes. To ease the presentation of this figure, as well as in subsequent tests, we shall simply quote the values of C~p and/3p parameters as defined in Eqs (12.15a) and (12.15b) with subscript replaced by C or R to denote the constrained or relaxed tests, respectively. The symbol F will be given to any failure to satisfy the necessary condition. In the tests of Fig. 12.7 we note that both patch tests fail with the parameter c~c being less than 1, and hence the elements will lock under certain circumstances (or show a singularity in the evaluation of S). A failure in the relaxed tests generally
Relaxed
Constrained
(a)
(xc = 0/8 (F)
0f,R = (24-3)/8 .- 21/8
I~c = 810
JIB = 8/(8-1) = 8/7
Boundaries constrained I F Failure I Only 3 DOF
Boundaries constrained
~ constrained 0CC = 15/32
(F)
E E
A
A
A
A
A
A
Z&.
A
A
A
A
Z~
Z~
A
A A
B
on this boundary
~ R = (63-3)/32 = 60/32
13R=
E (b)
Fig. 12.7 'Constrained' and 'relaxed' patch test/count for serendipity (quadrilateral). (In the C test all boundary displacements are fixed. In the R test only three boundary displacements are fixed, eliminating rigid body modes.) (a) Single-element test; (b) four-element test.
393
394 'Thick' Reissner-Mindlin plates -irreducible and mixed formulations
predicts a singularity in the final stiffness matrix of the assembly, and this is also where frequently computational failures have been observed. As the mixed and reduced integration elements are identical in this case we see immediately why the element fails in the problem of Fig. 12.3 (more severely under clamped conditions). Indeed, it is clear why in general the performance of Lagrangiantype elements is better as it adds further degrees of freedom to increase nr (and also nw unless heterosis-type interpolation is used), ll In Table 12.1 we show a list of the C~p and/3p values for single- and four-element patches of various rectangles, and again we note that none of these satisfies completely the necessary requirements, and therefore none can be considered robust. However, it is interesting to note that the elements closest to satisfaction of the count perform best, and this explains why the heterosis elements 26 are quite successful and indeed why the Lagrangian cubic is nearly robust and often is used with s u c c e s s . 27 Of course, similar approximation and counts can be made for various triangular elements. We list some typical and obvious ones, together with patch test counts, in the first part of Table 12.2. Again, none perform adequately and all will result in locking and spurious modes in finite element applications. We should note again that the failure of the patch test (with regard to stability) means that under some circumstances the element will fail. However, in many problems a reasonable performance can still be obtained and non-singularity observed in its performance, providing consistency is, of course, also satisfied.
Numerical patch test
While the 'count' condition of Eqs (12.15a) and (12.15b) is a necessary one for stability of patches, on occasion singularity (and hence instability) can still arise even with its satisfaction. For this reason numerical tests should always be conducted ascertaining the rank sufficiency of the stiffness matrices and also testing the consistency condition. In Chapter 9 of reference 4 we discussed in detail the consistency test for irreducible forms in which a single variable set u occurred. It was found that with a second-order operator the discrete equations should satisfy at least the solution corresponding to a linear field u exactly, thus giving constant strains (or first derivatives) and stresses. For the mixed equation set [Eq. (12.3)] again the lowest-order exact solution that has to be satisfied corresponds to: 1. constant values of moments or Z~q5and hence a linear th field; 2. linear w field; 3. constant S field. The exact solutions for which plate elements commonly are tested and where full satisfaction of nodal equations is required consist of: 1. arbitrary constant M fields and arbitrary linear q~ fields with zero shear forces (S = 0); here a quadratic w form is assumed still yielding an exact finite element solution; 2. constant S and linear w fields yielding a constant q~ field. The solution requires a distributed couple on the fight-hand side of the first of Eq. (12.3) and this was not included in the original formulation. A simple procedure is to disregard the satisfaction of the moment equilibrium in this test. This may be done simply by inserting a very large value of the bending rigidity D.
The patch test for plate bending elements 395 Table 12.1
Quadrilateral mixed elements: patch count Single-element patch
Element
Four-element patch
Reference
ac
/~c
~R
~R
~C
~C
~R
Q4S15,8,1o
0
2
9
2
3
8
24
8
(F)
(F)
15 3"2 (F)
32 -5-
60 32
32 20
32 "9-
72 32
32 20
23 3-'2
32 "5-
68 3-'2
32 2"-0
27 7-2
72 -if"
9672
7232
I
] I
(F)
Q8S45
0 i (F)
8 0
21 "-if"
8 7
Q9S47'8
3 g
8 T
16 "8"
8 8
(F)
Q9HS411
2
(F) Q 12S95
0 Yg
(F)
8 0
15 T
8 7
18 "0"
23 1-8
18 1--i
(F)
Q 16S927
27 32
(F)
(F)
12 1-8
18 4
45 1--8
18 ~
75 72
72 2-5
150 7-2-
72 5-0
Q 4 S 1B 128 Q 4 S 1B 1L29.3~
2 2
2 0
11 2
2 3
!1 T
8 T
32 'if
8
r~A ~ O ze'~Dor31 , Ol.L,
4
4 i
13 T
4 ~
19 1-6
16 T
40 1-6
16 -g
(F) II
[ ] I ~
* I
(F)
F Failure to satisfy necessary conditions.
12.4.2 Design of some useful elements The simple patch count test indicates how elements could be designed to pass it, and thus avoid the singularity (instability). Equation (12.15b) is always trivial to satisfy for elements in which S is interpolated independently in each element. In a single-element test it will be necessary to restrain at least one ,~' degree of freedom to prevent rigid body translations. Thus, the m i n i m u m number of terms which can be included in S for each element is always one less than the number of,~, parameters in each element. As patches with more than one element are constructed the number of w parameters will increase proportionally with the number of nodes and the number of shear constraints increase by the number of elements. For both quadrilateral and triangular elements the requirement that ns > nw - 1 f o r no b o u n d a r y restraints ensures that Eq. (12.15b) is satisfied on all patches for both constrained and relaxed boundary conditions. Failure to satisfy
396
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations Table 12.2
Triangular mixed elements: patch count Single-element patch
Element
Six-element patch
Reference
aC
tiC
O~R
fiR
OLC
tiC
O~R
fiR
T3 S 1
0 ~
2 ~
6 ~
2 ~
3 1~
12 -F
18 1-~
12 -K
36 if
54 3-3
36 1-8
57 3-3
36 l~
108 3--K
36 3-3
(F)
T 6S3
/z
0 ~
(F)
6 0
15 ~
6 5
(F)
T 10 S 3
3 g
21 3-3 (F)
6 ~
27 -(
(F)
6 ~ (F)
II
T 3 S 1B 1L 32'33
~
~
~
~
2
20 1-2
12 "6-
35 1"2
12 I-T
T3S1B1A34
2
2
8
2
15
12
30
12
2
2
17
2
33
12
66
12
T 6 S 1B 1
2
2
8
(F)
T6S3B317
6
6 ~
21 -'C
6 g
(F)
75 3"g
36 "7-
108 3--6-
36
F Failure to satisfy necessary conditions.
this simple requirement explains clearly why certain of the elements in Tables 12.1 and 12.2 failed the single-element patch test for the relaxed boundary condition case. Thus, a successful satisfaction of the count condition requires now only the consideration of Eq. (12.15a). In the remainder of this chapter we will discuss two approaches which can successfully satisfy Eq. (12.15a). The first is the use of discrete collocation constraints in which the third of Eq. (12.3) is enforced at preselected points on the boundary and occasionally in the interior of elements. Boundary constraints are often 'shared' between two elements and thus reduce the rate at which ns increases. The other approach is to introduce bubble or enhanced modes for the rotation parameters in the interior of elements. Here, for convenience, we refer to both as a 'bubble mode' approach. The inclusion of at least as many bubble modes as shear modes will automatically satisfy Eq. (12.15a). This latter approach is similar to that used in Sec. 12.7 of Volume 1 to stabilize elements for solving the (nearly) incompressible problem and is a clear violation of 'intuition' since for the thin plate problem the rotations appear as derivatives of w. Its use in this case is justified by patch counts and performance.
Elements with discrete collocation constraints
~iiii~z~ iiiiii!i~i iiii~ili~2 ii~i~i~ii~5 iiii!~i~iiiliiie~ ii!~i~i~!!~ i~iiiiiiim~ i!iiiiliiii~ii!iiii!iiiiii~iiiiiliili!i!i!iii!i~W|~h i!i!iil!ii!iliiiii~iiii~~iiilii?ii!~iiii~i~iti~iSiiii~ii!i!liliiiiliriii~ii!itiii~ii!i!iiiiiiM i~iiiiO~!~ iiiiiiii~!ij~iiO iiii!iiiC~ iiiiiiiiitiiiiiIiiO~R ~ilili!iiiilii!i!iiiiii iiiiiiiiii~iiCO iiii!iiiiii~iiiliiii~iiii!iiii i~iiiiiii!iiiii!iii!ii!i!i~i~iiiiiii~ii~iii!iiii!ii!i!iiiiiiiiii i~iliiii~iiiiiiiiiiiiiiiiii!iiiiiil
12.5.1 General possibilities of discrete collocation constraintsquadrilaterals The possibility of using conventional interpolation to achieve satisfactory performance of mixed-type elements is limited, as is apparent from the preceding discussion. One feasible alternative is that of increasing the element order, and we have already observed that the cubic Lagrangian interpolation nearly satisfies the stability requirement and often performs well. 3'827 However, the complexity of the formulation is formidable and this direction is not often used. A different approach uses collocation constraints for the shear approximation on the element boundaries, thus limiting the number of S parameters and making the patch count more easily satisfied. This direction is indicated in the work of Hughes and Tezduyar, 35 Bathe and co-workers, 36'37 and Hinton and Huang, 38'39 as well as in generalizations by Zienkiewicz et al., 4~ and others. 41-48 The procedure bears a close relationship to the so-called DKT (discrete Kirchhoff theory) developed in Chapter 11 (see Sec. 11.18) and indeed explains why these, essentially thin plate, approximations are successful. The key to the discrete formulation is evident if we consider Fig. 12.8, where a simple bilinear element is illustrated. We observe that with a Co interpolation of q~ and w, the shear strain 0w
Ox +q~x
7X-
(12.16)
is uniquely determined at any point of the side 1-2 (such as point I, for instance) and
Sx interpolation 1
2
IV
Yt
4
3
X
z~ Sy node I> Sx node
SF interpolation
ii
Fig. 12.8 Collocation constraints on a bilinear element: independent interpolation of 5x and 5y.
397
398
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations that hence [by Eq. (12.1b)]
Sx = OCyx
(12.17)
is also uniquely determined there. Thus, if a node specifying the shear resultant distribution were placed at that point and if the constraints [or satisfaction of Eq. (12.1b)] were only imposed there, then 1. the nodal value of Sx would be shared by adjacent elements (assuming continuity of a); 2. the nodal values of S~ would be prescribed if the t~ and # values were constrained as they are in the constrained patch test. Indeed if a, the shear rigidity, were to vary between adjacent elements the values of S~ would only differ by a multiplying constant and arguments remain essentially the same. The prescription of the shear field in terms of such boundary values is simple. In the case illustrated in Fig. 12.8 we interpolate independently
Sx = NsxSx
and
Sy -- NsySy
(12.18)
using the shape functions 1
Nsx
Yl - YlII
[Y - ylII,
1
x,,-x]
Nsy -Xll
~
Yl - Y]
XIV
(12.19)
as illustrated. Such an interpolation, of course, defines Ns of Eq. (12.11). The introduction of the discrete constraint into the analysis is a little more involved. We can proceed by using different (Petrov-Galerkin) weighting functions, and in particular applying a Dirac delta weighting or point collocation to Eq. (12.1b) in the approximate form. However, it is advantageous here to return to the constrained variational principle [see Eq. (11.83)] and seek stationarity of n = l f a (s
oLq da + ~l fa s T 1c~S d~2 - fa w q df2 + Hbt = stationary (12.20)
where the first term on the fight-hand side denotes the bending and the second the transverse shear energy. In the above we again use the approximations
S
NsS
Ns=[Nsx, Nsy]
(12.21)
subject to the constraint Eq. (12.1 b)" S = c~ (Vw + qS)
(12.22)
being applied directly in a discrete manner, that is, by collocation at such points as I to I V in Fig. 12.8 and appropriate direction selection. We shall eliminate S from the computation but before proceeding with any details of the algebra it is interesting to
Elements with discrete collocation constraints
~CC= 0/0 0/0
Or.R= 9/4 139 4/3
(a)
ok; = 3/4(F) 13c = 4
o.R = 24/12 13R = 12/8
(b)
Fig. 12.9 Patch test on (a) one and (b) four elements of the type given in Fig. 12.8. (Observe that in a constrained test boundary values of S are prescribed.) I
i
zk
O and w interpolation
Sxinterpolation
and collocation nodes
Syinterpolation
and collocation nodes
I
U
Typical shape function
Fig. 12.10 The quadratic Lagrangian element with collocation constraints on boundaries and in the internal domain .38,39
observe the relation of the element of Fig. 12.8 to the patch test, noting that we still have a mixed problem requiting the count conditions to be satisfied. (This indeed is the element of references 36 and 37.) We show the counts on Fig. 12.9 and observe that although they fail in the four-element assembly the margin is small here (and for larger patches, counts are satisfactory).* The results given by this element are quite good, as will be shown in Sec. 12.9. The discrete constraints and the boundary-type interpolation can of course be used in other forms. In Fig. 12.10 we illustrate the quadratic element of Huang and Hinton. 38'39 Here two points on each side of the quadrilateral define the shears Sx and Sy but in * Reference 37 reports a mathematical study of stability for this element.
399
400
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations Table 12.3 Elements with collocation constraints: patch count. Degrees of freedom: [--1, w - 1" 0 , ~ b - 2; l~, S - 1" n, 4,,, - 1 Single-element patch Element
Four-element patch
Reference
o~c
/~c
O~R
/OR
O~C
/~C
O~R
~R
Q9D1238,39
g3
T4
24 12
y12
27 24
"9-24
72 40
40 24
Q9 D 10
3 2
T
2
24
10
27
16
72
32
(F) 0
0
]-6
21
-8-
8
1-6
15
-if
3-2
2~
Q8D8
i
o
-8-
7
T
5
8
60
24
Q5D6
2
3
2 T
12 T
6 4
15 1-2
12 -5-
36 2--0
20 1-2
Q4D436, 37
0
0
8
4
3
4
24
12
2-4
2--]
(F) Single-element patch
Six-element patch
T6D640
0
0 0
15 T
6 5
21 12
12 'if
43 24
24 23
T3D340,44
0 i
0 ~
9 ~
3 2
9 ~
6 T
45 T2
12 -C
F Failure to satisfy necessary conditions.
addition four internal parameters are introduced as shown. Now both the boundary and internal 'nodes' are again used as collocation points for imposing the constraints. The count for single-element and four-element patches is given in Table 12.3. This element only fails in a single-element patch under constrained conditions, and again numerical verification shows generally good performance. Details of numerical examples will be given later. It is clear that with discrete constraints many more altematives for design of satisfactory elements that pass the patch test are present. In Table 12.3 several quadrilaterals and triangles that satisfy the count conditions are illustrated. In the first a modification of the Hinton-Huang element with reduced intemal shear constraints is shown (second element). Here biquadratic 'bubble functions' are used in the interior shear component interpolation, as shown in Fig. 12.11. Similar improvements in the count can be achieved by using a serendipity-type interpolation, but now, of course, the distorted performance of the element may be impaired (for reasons we discussed in Volume 1, Sec. 9.7). Addition of bubble functions on all the w and q5 parameters can, as shown,
Elements with discrete collocation constraints 401
make the Bathe-Dvorkin fully satisfy the count condition. We shall pursue this further in Sec. 12.6. All quadrilateral elements can, of course, be mapped isoparametrically, remembering of course that components of shear S~ and S~ parallel to the ~, r/coordinates have to be used to ensure the preservation of the desirable constrained properties previously discussed. Such 'directional' shear interpolation is also essential when considering triangular elements, to which the next section is devoted. Before doing this, however, we shall complete the algebraic derivation of element properties.
12.5.2 Element matrices for discrete collocation constraints The starting point here will be to use the variational principle given by Eq. (12.20) with the shear variables eliminated directly. The application of the discrete constraints of Eq. (12.22) allows the 'nodal' parameters S defining the shear force distribution to be determined explicitly in terms of the and q5 parameters. This gives in general terms (12.23) in each element. For instance, for the rectangular element of Fig. 12.8 we can write S/=oL Sy
- c~
S Ill
- - oL
[/1)2-- /1)1+ ~xl + @x2] a
2
b
+
IV
-- c~
(12.24)
[l~3 -- tO4+ ~x3-~-~x4] a
Sy
2
2
[l~l -- l~4 ~Yl -t-~Y4] b
+
2
which can readily be rearranged into the form of Eq. (12.23) as
1 !a0 ab ! Qw-~-~ -b
where
b
0
0
0
0
gOl
SI l
.
Six I I
'
IV
Sy
1 0
1
0
!
1 0
Six ~ __
1!000 0 and Q ~ - ~
~g _
JPl
Ul)2
.
~1)3
'
flU4
~ _
~ 2
.
~ 3
'
~) 4
era)
402
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations
/ N&'bubble'
Fig. 12.11 A biquadratic hierarchical bubble for 5x.
Including the above discrete constraint conditions in the variational principle of Eq. (12.20) we obtain
'L
H - ~
(EN~q~) T Ds
5d ~
+ ~l f ~ [N~ (Qe4~ ~ + Q~r)]Tc~[Ns (Qe q5 ~ + Qm~)] ds2 -
(12.25)
fa wq d~ + Hbt -- minimum
This is a constrained potential energy principle from which on minimization we obtain the system of equations
Kr
Kr162
re)
(12.26)
The element contributions are Kww -- QwKs~Qw T
K~
T
=
=
Kr162 -- ./o (EN4,)TD(E,N,~) dr2 +
Q~KssQ~
(12.27)
Kss - [
Jn
with the force terms identical to those defined in Eq. (12.8d). These general expressions derived above can be used for any form of discrete constraint elements described and present no computational difficulties. In the preceding we have imposed the constraints by point collocation of nodes placed on external boundaries or indeed the interior of the element. Other integrals could be used without introducing any difficulties in the final construction of the stiffness matrix. One could, for instance, require integrals such as
fr W [Ss -
Elements with discrete collocation constraints 403
on segments of the boundary, or +
d.
_
o
in the interior of an element. All would achieve the same objective, providing elimination of the Ss parameters is still possible. The use of discrete constraints can easily be shown to be equivalent to use of substitute shear strain matrices in the irreducible formulation of Eq. (12.8a). This makes introduction of such forms easy in standard computer programs. Details of such an approach are given by Ofiate et al. 47'48
12.5.3 Relation to the discrete Kirchhoff formulation In Chapter 11, Sec. 11.18, we have discussed the so-called discrete Kirchhoff theory (DKT) formulation for beams in which the Kirchhoff constraints [i.e. Eq. (12.22) with c~ = o0] were applied in a discrete manner. The reason for the success of such discrete constraints was not obvious previously, but we believe that the formulation presented here in terms of the mixed form fully explains its basis. It is well known that the study of mixed forms frequently reveals the robustness or otherwise of irreducible approaches. In Chapter 11 of reference 4 we explained why certain elements of irreducible form perform well as the limit of incompressibility is approached and why others fail. Here an analogous situation is illustrated. It is clear that every one of the elements so far discussed has its analogue in the DKT form. Indeed, the thick plate approach we have adopted here with c~ ~ c~ is simply a penalty approach to the DKT constraints in which direct elimination of variables was used. Many opportunities for development of interesting and perhaps effective plate elements are thus available for both the thick and thin range. We shall show in the next section some triangular elements and their DKT counterparts. Perhaps the whole range of the present elements should be termed 'QnDc' and 'TnDc' (discrete Reissner-Mindlin) elements in order to ease the classification. Here 'n' is the number of displacement nodes and 'c' the number of shear constraints.
12.5.4 Collocation constraints for triangular elements Figure 12.12 illustrates a triangle in which a straightforward quadratic interpolation of q5 and w is used. In this we shall take the shear forces to be given as a complete linear field defined by six shear force values on the element boundaries in directions parallel to these. The shear 'nodes' are located at Gauss points along each edge and the constraint collocation is made at the same position. Writing the interpolation in standard area coordinates 4 we have 3
S--ZtaSa a=l
(12.28)
404
'Thick' Reissner-Mindlin plates-irreducible and mixed formulations
/~
",~ "
~,d
/~
_ ~Snodes
\s,2*~
3 (a) The parameters (0 = 12 DOF, w= 6 DOF and S = 6 DOF)
2 e2 =
n0en,
v~ors
(b) .Area coordinates and not~ion
Fig. 12.12 TheT6D6triangularplateelement. where Sa are six parameters which can be determined by writing the tangential shear at the six constraint nodes. Introducing the tangent vector to each edge of the element as
eb --- 1,eby
(12.29)
a tangential component of shear on the b-edge (for which Lb = O) is obtained from Sb = S. eb
(12.30)
Evaluation of Eq. (12.30) at the two Gauss points (defined on interval 0 to 1) 1
1
gl = 2 ~ / ~ ( ~ / ~ - 1) and g2 = 243-~-(4r3t- 1)
(12.31)
yields a set of six equations which can be used to determine the six parameters a in terms of the tangential edge shears Sbl and Sba. The final solution for the shear interpolation then becomes
3 La [ ecy,--eby] {glSbl-]-g2Sb2} S --- Za=l ~ a --ecx, ebx.] glScl + g2Sc2
(12.32)
in which a, b, c is a cyclic permutation of 1, 2, 3. This defines uniquely the shape functions of Eq. (12.11) and, on application of constraints, expresses finally the shear
Elements with rotational bubble or enhanced modes
x
r
i
I r"
(a)
137 ft
1.6
U) r
Actual . . . . . Assumed
._r 1.2 ._o r 0.8 o 0.4 e = 0 0
'
. . . . .
r -0.4
a. -0.8 (b)
~l
'|
,,1.
.
.
.
i
,
0
30
60
90 120 150 180 e (deg)
Fig. 13.10 Cooling tower: geometry and pressure load variation about circumference.
13.8.1 Cooling tower This problem of a general axisymmetric shape could be more efficiently dealt with by the axisymmetric formulations to be presented in Chapters 11 and 12. However, here this example is used as a general illustration of the accuracy attainable. The answers
444
Shells as an assembly of flat elements
lOft 15ft
CXl 4.., 0')
i~-~
r
12 at 15 ~
I
Fig. 13.11 Cooling tower of Fig. 13.10: mesh subdivisions.
against which the numerical solution is compared have been derived by Albasiny and Martin. 65 Figures 13.10 to 13.1 2 show the geometry of the mesh used and some results for a 5 inch and a 7 inch thick shell. Unsymmetric wind loading is used here.
13.8.2 Barrel vault This typical shell used in many civil engineering applications is solved using analytical methods by Scordelis and L o 66 and Scordelis. 67 The barrel is supported on rigid diaphragms and is loaded by its own weight. Figures 13.13 and 13.14 show some comparative answers, obtained by elements of type B, C and D. Elements of type C are obviously more accurate, involving more degrees of freedom, and with a mesh
Practical examples 100
5 in shell
o
9
"\
.... Finite e l e m e n t
. . . . Albasiny and Martin65
.~
N -100
-200
/L
N1 =__=._=- .
-300 -250
0
I
250
500 Ib/ft
750
1000
1250
(a)
100
7 in s h e ~ r
-100
i ~ 5 in shell
-200 ~ -300
0
~--
,, , Finite e l e m e n t . . . . Albasiny and Martin 65
0.001 0.002 0.003 0.004 0.005 0.006 0.007 ft
(b) lOO
N -100 7 in shell -200 -300
" ~ ~ ' ~ ' ~ ~"
Finite e l e m e n t
. . . . Albasiny and Martin 65 30
20
Ib ft/ft
10
5 in s h e ~
~ 0
i
10
|
0
(c)
Fig. 13.12 Cooling tower of Fig. 13.10: (a) membrane forces at ~ -- 0~ N1, tangential; N2, meridian; (b) radial displacements at ~ -- 0~ (c) moments at 8 -- 0~ MI, tangential; M2, meridian.
445
446
Shells as an assembly of flat elements z,w 9
y,v
x,u
E = 3 x 103 k/in 2 v=O
9' = 0.09 k/ft2 Support rigid diaphram U=0
e edge
\~' ~,40~ ~ ~
I
W=0
(a) _._ Analytical
I
2O
A o
4O
8 x 12 Mesh EI.B 12x18Mesh
E] 3 x 3 Mesh EI.Dc 9 8x 12 Mesh EI.D
0.1
w(ft) 0.2
0.3 (b)
40 -n ---->- r
i !
i
-0.005
v(~) -0.010 -0.015
(c) Fig. 13.13 Barrel (cylindrical) vault: flat element model results. (a) Barrel vault geometry and properties; (b) vertical displacement of centre section; (c)longitudinal displacement of support.
Practical examples 447
M~ r
M2 -1
0
10
(a)
20
$
30
40
m
~
m
l ~ ~ 10j
0
,i20
I
30
40
(b) Fig. 13.14 Barrel vault of Fig. 13.13. (a) M1, transverse; M2, longitudinal; centre-line moments; (b) M12, twisting moment at support.
of 6 • 6 elements the results are almost indistinguishable from analytical ones. This problem has become a classic on which various shell elements are compared and we shall return to it in Chapter 15. It is worthwhile remarking that only a few, secondorder, curved elements give superior results to those presented here with a fiat element approximation.
13.8.3 Folded plate structure As no analytical solution of this problem is known, comparison is made with a set of experimental results obtained by Mark and Riesa. 68
448 Shells as an assembly of flat elements
,~~
I
4.082 in
,..(bq,Y//'"'--7 ~ ~,~,/~....Plate 3
in rib
' 0 are Y-periodic, i.e. take the same values on the opposite sides of the cell of periodicity. The term scaled with the nth power of e in Eqs (18.7) and (18.8) is called a term of order n.
552 Multiscale modelling
The necessary mathematical tools are the chain rule of differentiation with respect to the micro variable and averaging over a cell of periodicity. We introduce the assumption (18.7) and (18.8) into equations of the heterogeneous problem (18.3)-(18.6b) and use the differential calculus rule (see reference 14)
df _ (Of
dxi
l Of)
~
+ -E ~
1
-- f ,i (x) -Jr -6 f ,i(y)
(18.9)
This equation also defines the notation used in the sequel for differentiation with respect to local and global independent variables. Because of Eq. (18.9) the equilibrium equations and the heat balance equation split into terms of different orders [the terms of the same power of e are equated to zero separately: e.g. Eqs (18.10a) and (18.10b) are of order l/e]. For the equilibrium equation we have (79.. tJ, J (y) (x ' y ) - 0 or~ 1 (x,y)+j~(x)=0 tj,j(x) (x ' y ) + O'ij,j(y)
(18.10a)
o1ij,j (x) (x, y) + O'ij,j 2 (y) (X, y) -- 0 We have a similar expression for the heat balance equation: 0 (x, y) = 0 qi,i(y) q0i,i(x) ( x , y ) + qi,i(y) 1 (x,y)-r(x)=0 2 (X, y) = 0 qli,i(x)(X, y) + qi,i(y) ,
o
.
From Eqs (18.5) and (18.9) it follows that the main term of depends not only on u ~ but also on u 1 e~
(18.lOb)
eij in expansions
0 1 -Jr- u(i,j)(y ) ~ eij(x) (U 0) -1" eij (y) (U 1) y) -- u(i,j)(x)
(18.82) (18.11)
The constitutive relationships (18.4) now assume the form or~
Y) - Cijkl(Y) (ekl(x)(U O) -Jr- ekl(y)(Ul)) -- O~ij(y) 0 0
o]j(x, y) -
Cijkl(Y) (ekl(x)(U 1) -q- ekl(y)(U2)) -- O~ij(y) 01
q~
00 -kk/(y) (,l(x)
y)-
q~(x, y)
-
-
--kkl (y)
01 (,l(x)
+ 01,'(y)) -JI-0 2,l(y))
(18.12a)
(18.12b)
It can be seen that the terms of order n in the asymptotic expansions for stresses (18.12a) and heat flux (18.12b) depend respectively on the displacement and temperature terms of order n and n + 1. In this way the influence of the local perturbation on the global quantifies is accounted for. This is the reason why for instance we need u 1(x, y) to define via the constitutive relationship the main term in expansion (18.8) for stresses [and u 2 (x, y) for the term of order 1, if needed, see below].
Global solution 553
i~i~i~i~iiiiiii!~!iiiiiii!i~i~iii!~iiii~iH~i~iii~i~iiiiiii~ii~iiiii~i~b~Ii~i~i~ii~ii~iii~iiiiiiiiii~i~i~i;iii~ii~i!i~;!ii;ii~i~i~iii~i~iii~iiii~Iiiiiiii!iiiiiii~i~iiiiiii!~iiiii!~i~ii~i;iiiiiiiii~i~i~ii~iiii!~iiiiiiiii Referring separately to the terms of the same powers of e leads to the following variational formulations for unknowns of successive order of the problem. Starting with the first order, using separation of variables it can be formally shown that u I (x, y) and 01 (x, y) can be represented by 14'15
U] (X, y) -- epq(x)(U 0 (X)))( pq (y) -+- Ci(x) O~(x' Y) __ oO ,p(x)(X) Op (y) + C(x)
(18.13)
We call X pq (y) and 0 p (y) the homogenization functions for displacements and temperature, respectively. The zero-order (often also referred to as first-order) component of the equation of equilibrium (18.10a) and of heat balance (18.10b) in the light of Eq. (18.13) yields the following boundary value problems for functions of homogenization: find X.pq E Vy such that: Vui ~ Vr
Pq (y) ) lfk,l(y) (y) dr2 - 0 f y Cijkl(Y) ((~ip(~jq "~ Xi,j(y)
(18.14a)
find 0 p E Vy such that: V~b ~ Vr
f y kij(Y) ((~ip AI- O,i(y)(y)) pq q~,j(y)(y)dr2 - - 0
(18.14b)
In the above equations Vr is the subset of the space of kinematically admissible functions that contains the functions with equal values on the opposite sides of the cell of periodicity Y. The tensor Xpq and the vector 0 p are functions that depend only on the geometry of the cell of periodicity and on the values of the jumps of material coefficients across Sj. Functions yi(y) and ~(y) are the usual arbitrary test functions having the meaning of Y-periodic displacements and temperature fields, respectively. They are used here to write explicitly the counterparts of the expressions (18.10a) and (18.10b), in which the prescribed differentiations again are understood in a weak (variational) sense. The solutions X pq and 0 p of the 'local' (i.e. defined for a single cell of periodicity) boundary value problems with periodic boundary condition (18.14a) and (18.14b) can be interpreted as obtained for the cell subject to a unitary average strain e pq and unitary average temperature gradient O,p(y), respectively. The true value of perturbations are obtained after by scaling X pq and 0 p with true global strains (gradient of global temperature), as prescribed by Eq. (18.13). In the asymptotic expansion for displacements and temperature given by Eq. (18.7) the dependence on x alone occurs only in the first term. The independence on y of these functions can be proved (see, for example, reference 14). The functions depending only on x define the macro behaviour of the structure and we will call these the global terms. To obtain the global behaviour of stresses and heat flux the following mean values over the cell of periodicity are defined 14
0.0j(X)_ [y[-1 [ a0j(x, y) dY dY
and
?/~
[y[-1 [ qO(x' y) dY dY
(18.15)
554 Multiscale modelling Averaging of Eqs (18.12al) and (18.12bl) results in the following, effective constitutive relationships
o'Oj(x) --- Chkl ekl (UO) -- OzhijO0
and
~0 = _ kh 00
(18.16)
where the effective material coefficients are computed according to
Ch'kl -- IYI -~ fy Cijpq(Y) ((~kp(~lq"~"X,k,l(y)(Y)) dY Pq kh = IYI-' fr
oLhj "--
kip(y)
IYI-1 fr OLij(y)
((~jp.2f_ L~,j(y) P
(18.17)
(y)) dY
dY
The macro behaviour can be defined now by averaging first-order terms in the equilibrium equations (18.10a2), flux balance equations (18.10b2) and boundary conditions (18.6a) and then substituting the averaged counterparts of stress and heat flux (18.15) [first-order perturbations vanish in averaging of Eqs (18.10a2), (18.10b2) because of periodicity]. Equation (18.4) is replaced by Eq. (18.16) and Eq. (18.6b) is no longer needed since we deal now with homogeneous uncoupled thermoelasticity. The heterogeneous structure can now be studied as a homogeneous one with effective material coefficients given by Eq. (18.17) from which the global displacements, strains and average stresses and heat fluxes can be computed. We then go back to Eq. (18.12a) to recover a local approximation of stresses. This last step corresponds to the unsmearing or localization mentioned in the introduction.
iiii iiiiiiiii iii i6i!iii
iiiii i ii iii
iiilii iiii iiiiiiii iiiiiiiiiiii!!ill ii
ii~i!~i~i~ii~i!i~i~@~!~i~i~i~!i~i~i~s~i!~i~i~!~!~i~i!~i~:~i~ii~@~i~i~!~i~!~i~!i~!~i)~i~i~ii~i~ iiiiiiiiiiiiiiii!iiii!i'~iiiii@',!i!iiii!iii~i'ii',iiiiiii iiillii',iilI l!i!ii!ii~,',i iiii',ii',',!ii ',',ii ~,i
iii iii~iiii iii~ii! iiiili
We note that the homogenization approach results in two different kinds of stress tensors. The first one is the average stress field defined by Eq. (18.16). It represents the stress tensor for the homogenized, equivalent but unreal macro body. Once the effective material coefficients are known, the stress field and the heat flux may be obtained from a standard finite element structural analysis and heat transfer program as described in the previous section. The other stress field is associated with a family of uniform states of strains epq(x) (u 0) over each cell of periodicity Y. This local stress is obtained by introducing Eq. (18.11) into Eq. (18.12a) and results in
a~
Y)=
Cijkl(Y) (r
--
Pq (U 0) Xk,l(y))epq(x)
-- O~/j(y) 0 0
(18.18)
Because of Eqs (18.10a) and (18.14a) this tensor satisfies the equations of equilibrium everywhere in Y. If needed, the stress description can be completed with a higher-order term in Eq. (18.8). This approach is presented in references 16 and 17. Finally the local approximation of heat flux is as follows: p ) (y)] 0 0,p(x) qj0 (y) = -kij(Y)[Sip + O,i(y
(18.19)
Finite element analysis applied to the local problem 555
For the finite element formulation it is convenient to introduce matrix notation for the quantities introduced above. Accordingly, the homogenization functions are ordered as defined by Eq. (18.20) [the numbers in the superscripts in Eq. (18.20) and subscripts in Eq. (18.21), refer to the reference coordinate axes 1, 2, 3]: XT(y)-- [xll(y)
X22(y) X33(y) X12(y) X23(y) X31(y)]3•
TT(y) -- [OI(y)
02(Y)
(18.20)
03(y)]1•
This is in accordance with the ordering of strains and fluxes e - - [ell q=
[ql
e22 e33 el2 q2
e23
e31]T-- {epq}6xl
q3]T'-- {qp}3xl
(18.21)
In the following a tilde again denotes a nodal value in the finite element mesh. We have the usual representations for each element: X(y) - N(y)R;
T(y) = N ( y ) l ~
(18.22)
where N contains the values of standard shape functions. It is easy to show that the variational formulation (18.14a) can be rewritten as follows: find X ~ Vr such that: u
~ Vr
r e r (v(y))D(y) (1 + EX(y)) dY = 0
(18.23)
In the above E denotes the matrix of differential operators, and D contains the material in the repetitive domain. Matrix X which contains the values of coefficients homogenization functions at the nodes of the mesh is obtained as a finite element solution of Eq. (18.23). The equation to solve is the following:
aijkt
KxX - f = 0
(18.24)
where X is Y-periodic, with zero mean value over the cell, and f = frBrD(y)dY;
Kx =
~BTD(y)BdY;
B = EN(y)
(18.25)
D contains the material coefficients aijkl. It can be shown that X in Eq. (18.23) and thus Eq. (18.24) is a solution of a boundary value problem, for which the loading consists of unitary average strains over the cell. This is seen in the form of the right-hand side of Eq. (18.24) which forms a matrix. We solve thus six equations for six functions of homogenization. The variational formulation (18.14b) can be represented in a form similar to Eq. (18.24), T being Y-periodic, with given mean zero value over the cell Kr T + f = 0
(18.26)
556
Multiscalemodelling where
Kr=frB~ko(y)BodY; B 0 = s ko contains the conductivities kij of materials in the repetitive domain. f=frB~k0(y)dY;
(18.27)
Differential operators in s are ordered suitably for the thermal problem. The periodicity conditions can be taken into account using Lagrange multipliers in the construction of a finite element code. The requirements of the zero mean value also has to be included in the program. Having computed X, I' and by consequence ll 1 and 01 one can derive the effective material coefficients, according to: O h = IY1-1 ~ O(y) (1 + BX) OY kh =
IYI-/i
k0(y) (1 + B01') d r
(18.28)
CI~h - - I Y I - 1 J I c~(y)dY With the homogenized material coefficients (18.28) any thermoelastic finite element program can be used to obtain global displacements and temperatures. For the unsmearing procedure we need the gradients of temperatures and strains in the regions of interest, see Eqs (18.18) and (18.19). Strains and temperature gradients are directly obtained from the finite element interpolations. To present the plots of stresses and heat fluxes over the single cell nodal projection can be used. To assure continuity of tangential stresses, this projection should be extended to patches of cells, for example, using SPR. 18
18.7.1 Example As an example of the procedure described above, the temperature effects on a superconducting coil are analysed. The structure has an overall shape of a large D, and is constructed by winding of a superconducting cable with a rectangular cross-section (Fig. 18.2). The whole D-shaped frame is immersed in liquid helium at a temperature of about 4K to assure superconductivity. The supports are designed to eliminate only the rigid motion of the structure. This is a typical example where asymptotic analysis can be successfully applied and the structure is clearly periodic: the macro scale is defined by a typical dimension of the coil cross-section, while the micro scale is given by the height of the section of a single cable (cell of periodicity). In the following a beam-like kinematics with deformable cross-section is used, instead of a full three-dimensional analysis. Homogenization is carried out in the cross-section plane (1,2) only, while in the axial direction (3) there is no periodic structure. 16'17 First effective elastic coefficients and six vectors of homogenization functions for unsmearing are computed. Then the thermal effective characteristics, i.e. effective conductivity and effective thermal expansion are calculated. Three scalar functions of
Finite element analysis applied to the local problem 557 Coil cross-section Yl
Cable cross-section Y2
Xl
=
21
=
[mm] Fig. 18.2 A superconducting coil cross-section and the single cell of periodicity (cable). The material properties are discontinuous, with discontinuities along regular surfaces 5j.
homogenization are further obtained for computations of the heat flux. These functions are shown in Figs 18.3-18.5. The macro analysis with the homogenized material (a standard thermoelastic finite element program) yields the global displacements and the temperature field. Unsmearing as described in the previous section then yields the local distribution of stresses in the region of interest in the coil. Stresses in each different material of a cell of periodicity can be recovered. In Fig. 18.6 the graphs of stress in the cross-section Plane homogenization functions
"
...
Z11,zee
~12 ~33
Fig. 18.3 Planefunctions of homogenization X 11, X ~2, )C22, X 33.
;;
....
558
Multiscale modelling
Y3
Homogenizationfunctions ~23
Y2~~
Y1
X,13 Y3
Fig. 18.4 Plots of the anti-plane functions of homogenization X 13,
X 23 on
four strands.
y~
~ .~
~:
Y2 2.
~,:j
81
82
83
Fig. 18.5 Three scalar functions of homogenization for temperature.
perpendicular to the fibres are shown for uniform cooling. Figure 18.7 shows the full complex state of stress in the neighbourhood of a single conductor for non-uniform cooling. Uniform cooling means here that the whole coil is immersed in liquid helium while non-uniform cooling indicates that the coil is only partially immersed in liquid helium.
18.7.2 Corrections for stresses and boundary effects A refined stress description over the cell of periodicity can be obtained by considering the second of Eq. (18.10a) (for simplicity, in the following we omit temperature effects).
Finite element analysis applied to the local problem . ~ . ~
Stress in steel
Zero level
Stress in epoxy Fig. 18.6 Graphs of stress o33 on the cross-section perpendicular to the fibres for uniform cooling (one cell of periodicity only).
This requires taking into account the second of Eq. (18.12a) and the solution for U 2 in the expansion for displacements (18.7). The rather lengthy procedure is fully described in reference 19 and has been applied to the example problem in reference 17. It also has to be extended to account for localization after the solution of the global problem. An advantage is that, over a single cell of periodicity, the equilibrium conditions are also fully satisfied in the presence of a non-uniform global state of strain. An alternative is to make the localization over a patch of neighbouring cells using the first-order homogenization described in Sec. 18.5.19 There is still another open problem: the periodicity condition used to find the local perturbation is strictly applicable only inside the body. We have hence the solution of a (thermo-)elasticity problem based on an assumed stress or displacement field which is valid nearly everywhere in the region occupied by the body under investigation, except
Stress in steel
Stress in epoxy
Zero level Fig. 18.7 Graphs of stress o33 on the cross-section perpendicular to fibres for non-uniform cooling (patch of four cells of periodicity).
559
560
Multiscale modelling on the boundary. The use of material coefficients based on the assumption of periodicity in the global solution (where the real boundary conditions are imposed) may implicitly impose some unrealistic constraints close to the boundary. This problem can be solved by some corrections which change the solution in the domain close to the boundary while away from the boundary the correction field should asymptotically decrease. 2~ Such a correction can be obtained by replacing the expansion (18.7) by
u=(x) = u~
+ e [u~(x, y) + b~(x, z)] + . . .
(18.29)
where z represents the coordinates of an additional system with the origin on the external surface, the axis z3 directed normal to the boundary surface with the other two axes oriented tangential to the surface and b is the boundary layer correction. Vector b should vanish exponentially when z3 goes to infinity. The procedure is described in detail in reference 21 and has been implemented by making use of one cell of periodicity and infinite elements with homogenized material properties around it. The use of infinite elements assures the desired exponential decay. Although the procedure is theoretically necessary near free edges, it is seldom applied in practical engineering problems.
i~ii~~i~i~ iiiiii~i~i~.~.!.~.~.~ii.~.i.~ii~i~ii~i~i~ii~i!~i!i~iii~i!i~i~i~i~iii~~i~i~i~~~i i i~i~iiiii~iii~i~i~i~i~i~!ii!ii~i~i!~ii!~i~i~i~i~!iiii~i~i~i~!~i~i~i~i~i~iii~ii~ii~i~!i~i~iiiii~~~i~!~i~ii~ ii~!~ii!i~ii!~iii~iii~i~i~ii~i~i!~i~@ii~ii~i~i~i!i~i~i~i~iiii~i~ii~!i~i~i~i~!1~iii!i~ii~i~i~ii~i~!~i~i!~ii~!ii~i~i~!i~ii~i~!~i~i~i~i~i~i~i~!~!i!i!i~i~i@~@i~i!iiiii~i~iii@ii
If applied iteratively, asymptotic theory of homogenization may also be used for nonlinear situations. Furthermore, it can obviously be used to bridge several scales. Here we deal with the case where three scales are bridged by applying in sequential manner the two-scale asymptotic analysis. The behaviour of the components is physically nonlinear. Again we refer to thermomechanical behaviour and introduce a micro, meso and macro level as shown in Fig. 18.8. At the stage of micro or meso modelling, some main features of the local structure can be extracted and used later in the macro analysis. The behaviour of the components, even if elastic-plastic, is assumed to be piecewise linear, so that the homogenization we perform is piecewise linear. Only monotonic loading and/or temperature increase (decrease) is considered, otherwise we need to store the whole history and use an incremental analysis.
X2
@mDB D
fi iS ll
X.
(a) Macro
z{
Yl (b) Meso
(c) Micro
Fig. 18.8 Example of a periodic structure with three levels: macro, meso and micro.
Asymptotic homogenization at three levels: micro, meso and macro
Because of the assumed form of material properties we deal with a sequence of problems of linear elasticity written for a non-homogeneous material domain and with coefficients that are functions of both temperature and stress level. At the top level of the hierarchy we consider an elastic body contained in the domain with a smooth boundary Of2 where on the part Of 2t of the boundary tractions are given and on the remaining part Of2, displacements are prescribed. The domain ~ is filled with repetitive cells of periodicity Y, shown in Fig. 18.8, where the material of the body is supposed to be piecewise homogeneous inside Y, as defined in Eq. (18.1). The governing equations are given by Eqs (18.3) to (18.5). For the lowest level all the formulations are formally the same with one difference: the boundary conditions are those of an infinite body. It is worth mentioning that all the macro fields at the micro level become the micro fields at the higher structural level. Effective material coefficients and mean fields obtained with the homogenization procedure at the lowest level enter as local perturbations at the next higher step. Before explaining the application of the homogenization procedure in sequential form to multi-level non-linear material behaviour we mention the solution by Terada and Kikuchi 22 who write a two-scale variational statement within the theory of homogenization. The solution of the microscopic problem at each Gauss point of the finite element mesh for the overall structure, and the deformation histories at time tn-~ must be stored until the macroscopic equilibrium state at current time tn is obtained. This procedure has not been applied to bridging of more than two scales. A three-scale asymptotic analysis is used by Fish and Yu 23 to analyse damage phenomena occurring at micro, meso, and macro scales in brittle composite materials (woven composites). These authors also retain the second-order term in the displacement expansion (18.7) and introduce a similar form for the expansion of the damage variable. We recall further that also stochastic aspects can be introduced in the homogenization procedure. 24
~!~~E~~~~i
~h~~~i
~
~i~~t~h~r~
~e~I~~l~i~ i ~r~i~'~:~
The two usual tools of homogenization of the previous section are used, i.e. volume averaging and total differentiation with respect to the global variable x that involves the local variable y. The homogenization functions are obtained similarly to Eqs (18.14a) and (18.14b). Only a factor A is introduced in Eq. (18.14a) to adapt the solution to the real strain level as explained below. find X pq ~_ Vy such that: Yui ~ Vr
fy
(A)
Cijkl(Y, ,~, O0)[(~ipt~jq "F ~i,j(y)] Pq I/k,l(y) dr2 -- O, or(A, X pq) E P
(18.30)
Material properties are assumed to depend on temperature and a set of representative temperatures is considered for the material input data where linear interpolation is used between the given values. P is the domain inside the surface of plasticity. The requirement that the stress belongs to the admissible region P [introduced in Eq. (18.30)] is verified using the classical unsmearing procedure described in the preceding section.
561
562
Multiscale modelling Table 18.2
Updating yield surface algorithm scheme i
1. 2. 3. 4. 5. 6.
Compute effective coefficients at micro level. Compute effective coefficients at meso level. Apply increment of forces and/or temperature at the macro level, solve global homogeneous problem. Compute global strain Eij" Eij -- eij (u 0) reminding that Eij -- e~j (x) [see Eq. (18.31)]. Apply Eij to meso-level cell by equivalent kinematic loading (displacement on the border). Solve the kinematic problem at the meso level for w(y), compute stress (unsmearing for meso level) and strain Eij" now Eij -- eij (w 0) and Eij -- ~ej (y). 7. Apply Eij from meso to micro level cell by equivalent kinematic loading (displacements on the border). 8. Solve the kinematic problem at the micro level for w I (z), compute stress (unsmearing for micro level). 9. Verify yielding of the material in the physically true situation at micro level. If 'yes' change mechanical parameter of the material and go to 1; else exit. ,,
,
,
,
,
It is to be noted that for 'solving the boundary value problem' mentioned in points 6 and 8 it is not always necessary to use the true finite element solution. If the cell of periodicity has not been changed before, this solution can be composed according to Eq. (18.13) (suitably rewritten).
The modification of the algorithm required by the non-linearity starts with the composite cell of periodicity with given elastic components. The uniform strain is increased step by step. Effective material coefficients are constant until the stress reaches the yield surface at some points of the cell. The yield surface in the space of stresses is different for each material component, being thus a function of position. The region where the material yield is of finite volume at the end of the step, thus it is easy to replace the material with the yielded one, with the elastic modulus equal to the hardening modulus of the elastic-plastic material and with the Poisson ratio tending to 0.5. The cell of periodicity is hence transformed into a form with one more material and the usual analysis procedure is restarted with a uniform strain, a new homogenization function and a new stress map over the cell. We identify each new region where further local yielding occurs, then redefine the cell and perform the analysis. The loop is repeated as many times as needed. In Eq. (18.30) the history of this replacement of materials at the micro level is marked by A, the level of the average stress, for which the micro yielding occurs each time. The algorithm is summarized in Table 18.2 where w(y) indicates displacement at meso level. At the end of each step we can compute also the mean stress over the cell having (generalized) homogenization functions [see Eq. (18.16)] and the effective coefficients can be computed using Eq. (18.17).
An important part of multiscale modelling is the recovery of stress and heat flux as well as strain, temperature and displacements at the level of the microstructure. This is obtained by Eqs (18.18) and (18.19) with the following procedure: first global (mean) fields are obtained from the homogeneous analysis where the material is characterized by the effective coefficients (18.17), then we go back to the original problem formulation, using homogenization functions. Thus we recover the main parts of the stress and heat flux. Because of the specified three-level hierarchical structure we are dealing with, the recovery process must be applied twice, also since material characteristics
Recovery of the micro description of the variables of the problem 563 are temperature dependent and non-linear the procedure must be applied for each representative temperature within the context of the correct stress state. We recall that the recovery process starts at the highest structural level while the homogenization begins at the lowest part of the structural hierarchy.
18.10.1 Example" the VAC strand analysis As an example of application, we consider a superconducting strand which is used to build the cable of Fig. 18.2. The structures and the three scales are shown in Figs 18.9 and 18.10, where the single filament (micro scale), groups of filaments (meso scale) and the superconducting strand (macro scale) are shown. The homogeneous effective properties will be defined for the inner part of the strand, shown on the left of Fig. 18.10. The diameter of the strand is about 0.80 mm. The application of the theory of homogenization is justified by the scale separation as clearly shown in Figs 18.9 and 18.10. As already indicated, periodic homogenization is applicable to structures obtained by a multiple translation of a representative volume element (RVE), called in this case the cell of periodicity. The considered strand shows two different levels of such a translational structure. On the meso level we have the repetitive pattern of the superconducting filament in the bronze matrix (micro scale RVE), filling the hexagonal region as illustrated in Fig. 18.11. The second translational structure is the net of the hexagonal filament groups (meso scale RVE) in the body of the single strand shown in Fig. 18.12. The homogenization splits thus into two steps, each one dealing with rather similar geometry and a comparable scale separation factor. Boundary conditions for the macro problem will be given in terms of interaction of the strand with the other strands in the cable e5 and will be of the type of Eq. (18.6). To form the superconducting alloy Nb3Sn the coil is kept for 175 hours at 923K. Afterward, to reach an operating temperature, it is cooled from 923K to 4.2K. In this example we analyse the effects of such cool-down, using the homogenization procedure to define the strain state of the strand at 4.2K which is caused by the different thermal :o ~ ~ , , ~ i ~ . ~. ~, .......... ~.~....... %
~,:~.~: .. ~.? ~:~?~:&.~:.
C~
.;:
~ ~.~ .~ :.!i~~i.:.~: ~.:"::,~-
-~L~: ~ ~io~ :~ ~.~.~~;~ 0
(18.52)
where D is the elastic constitutive tensor obtained with the elastic homogenization, k is the increment of the plastic flow, m is the flow direction and ~a tr, ~"]~(i) are the global stresses. The flow direction (which refers to an unknown plastic potential) can be calculated, at each interpolation point, in the following manner, see Fig. 18.28. Starting from an interpolation point corresponding to step i one multiplies the known strain increment E by the elastic effective tensor A, obtaining the trial stress ~a tr . Since there is plastic deformation or slip in some part of the unit cell, the strain AE actually generates the new stress ~a (i+l) , different from y]tr. The flow direction at the interpolation point for the level i + 1 can be computed as
m:cA-lI~tr
.-~-~-]~(i)]:
c IAE-A-I~(i) 1
(~8.53)
The value of k has been arbitrarily chosen to be 1/c [see local problem (18.42)], hence the quantity m not only gives information about the flow direction but also about the amount of the plastic flow. For stress points inside a patch, the flow direction is obtained again by interpolation. Once a consistency condition and a flow rule have been determined, the global constitutive law is fully defined and can be assumed as constitutive law of the homogenized material.
igi At this point a global problem can be solved. Given a periodic structure subjected to assigned external loads f and to assigned boundary conditions, the global displacements
581
582 Multiscalemodelling can be found solving the problem macroscopic constitutive law div 12 - f; macro-equilibrium global boundary conditions
(18.54)
The solution of the global problem (18.54) gives a reasonably good estimate of the displacements of the structure. Nevertheless, very often stresses are the most relevant mechanical quantities but they cannot be easily derived from global displacements. The global solution is used to evaluate the local distribution of micro stresses by solving a local problem (18.42). This is the third step of the homogenization procedure. If the stress distribution in a specific region is required, we take into consideration the integration points used in the global solution which are close to the examined region of the real structure. If the number of unit cells is high a single integration point corresponds to a group of cells, but usually one RVE corresponds to one integration point. The strains computed in the global solution are assumed as global strains of the investigated cell (or group of cells); in general the finite element computation gives a sequence of n values of strains if the load history is composed of n load steps. Such a sequence of values is taken as load history on a unit cell with periodic boundary conditions and the following local problem is solved: microscopic constitutive laws div tr - 0; micro-equilibrium Eij - Eij(t) given by the global problem
(18.55)
The solution of such a local problem gives a stress distribution which usually is a good approximation of the real one, as shown in the next example.
Numerical examples
We consider an assembly of discs shown in Fig. 18.22, under assumption of plane stress behaviour. The discs have the following mechanical properties E=128000MPa,
v=0.34,
crr=80MPa
The elastic-plastic constitutive law of the discs is of the von Mises type. The overall property of the structure will clearly be controlled by the behaviour of the discs around their contact points and by the elastic-plastic material behaviour. This character is captured by the analysis of three cases of different interface parameters. (a) Elastic-plastic material with purely frictional contact # = 0.25,
PnO --" Pto --O,
~* ~= 0
(b) Elastic-plastic material with cohesive frictional contact: #=0.25,
1 PnO0, Pto -- ~ 8 O
MPa,
~*r
(c) Elastic-plastic material with cohesive frictional contact: #-
0.10,
1 PnO0, PrO -- ---~_80MPa,
~/3
~* r 0
Global solution and stress-recovery procedure 583 ~-'11
Nt
~,'~f'~$' .....................................
(o.o)
.i ~'~;~~7~;"~ '~"ii!
~:11
E r . I,,
I
'
=",~, ~9'~I ~M l}'
~! 9
1
~,,.~.~,.I,.,.. ~..
~
~
~ (0.0)
~.,...................................... ~.,-:-~7~>,--...----f-. ~. D,li:"
~................. ~...
i'/
-+'
,
Initial
/~ ....~ yio,d~udace
~,,:
.....
~:~~ ~ ~ ~ ,7--,22
IX = 0.25
IX = 0.25
Z:11
~ 1~,~< ~ " ~.
i
Sll
! (0.0)
~r
i
~
.:,.
~,~-~~
i (o.o) .:~r
Initial
~:-. :.
~ / / /. . . . . . .
.~
i~:
...... ~:
~~/i: [J]
...... ,m -..,,,~! .. ~ : ~ . . , : .
~~l-iii/ttII
"
.:
. ,
yield surface
...~
o:~,~~..,..,, ~
.:,:!~22
.......................................
................................................. ! `7..`22
~=0.10
~
0.00
Fig. 18.29 Local numerical constitutive law for elastic-plastic bodies in contact with different material parameters: (a) purely frictional contact; (b) cohesive-frictional contact with # = 0.25; (c) cohesive-frictional contact with # = O.10; (d) purely cohesive contact.
(d) Elastic-plastic material with purely cohesive contact: # = 0.00,
PnO0,
PrO --"
1
-----~80MPa,
~/3
6* ~ 0
Pn, Pt are the limits of the normal and tangential contact forces respectively, and 6* is the gap, see Fig. 18.23. The contact algorithm used in the example is that of Zhang et al. 57 The cell of periodicity, indicated in Fig. 18.22, is discretized with four-node plane stress elements. Our interest is focused on the macro stress domain
E 11, E22 I Ell _ 0 , E22 ~ 0 } and the obtained interpolation points (yield surfaces) corresponding to the above four cases are presented in Figs 18.29 (a)-(d) in that quadrant. For the elastic case see reference 55. To verify the capability of the procedure a group of 140 discs arranged in a rectangle (10 x 14 discs) is analysed under uniform compression. For this purpose a uniform vertical displacement is assigned to the top nodes. The vertical displacements of the
584 Multiscale modelling 80 70 60 Z v
tO .D O
(D I:1:
heterog. homog.o homog.n
50 40 30
/-
20
/
f
J
/
10
o/
0.0 0.7
1.4 2.1
2.8 3.5 4.2 4.9 5.6 6.3 7.0
Top displacement (mm)
(a) 70 60
~" 50 v 40 C
o
g
30
w 20 10
0.0 0.7
1.4 2.1
2.8 3.5 4.2 4.9 5.6 6.3 7.0
Top displacement (mm)
(b)
70 60
~" 50 v
jl
= 40 o
g
30
/
w 20
o/
/
0.0 0.7 1.4 2.1
J
heterog. homog.o homog.n
2.8 3.5 4.2 4.9 5.6 6.3 7.0
Top displacement (mm) (c) Fig. 18.30 Comparisonbetween the reactions in the case of uniform top displacement. Homog.n indicates the results with the self-consistent procedure of this chapter,25 homog.othe results of reference 58.
Global solution and stress-recovery procedure VALUE OPTION:ACTUAL 8.01E@r .... ~ . , ~ tl
...... ..... ............ ,, . . . . . . . . . 9
.
.........i({~:::~c"
~,{~i"{.~
"i!.!i~:'~3i~:!i,,:.-!:~c.~. . . . . . . . .
;.;;::7~, ~.:" .:.:,..,:,: :,.~
..~,.,< 9
>>.@iS
....... ... ...... . < ~ , ~ . . . , ~ . . . .
.c
This saves the current solution data in a file that has the restart file name with an extension e x t e n d e r . For example, if the restart write file has the name 'Rprob', issuing the command SAVE
tiO
saves the data on a file named P p r o b . t i O. Alternatively, issuing the command as SAVE
References 595 saves the data on the file named P p r o b . For large problems the restart file can be quite large (especially if the elements use several history variables at each integration point) thus one should be cautious about use of too many files in these situations. To restore a file the command RESTart
is given to load the file without an extender, and the command RESTart
<extender>
to load the file with an extender.
In the discussion above we have presented a few of the ways the program FEAPpv may be used to solve non-linear finite element problems. The classes of non-linear problems which may be solved using this system is extensive and we cannot give a comprehensive summary here. The reader is encouraged to obtain a copy of the program source statements and companion documents from the publisher's website (http ://books.e lsevi er. com/companions). As noted in the introduction to this chapter the computer programs will undoubtedly contain some errors. We welcome being informed of these as well as comments and suggestions on how the programs may be improved. Although the programs available are written in Fortran it is quite easy to adapt these to permit program modules to be constructed in other languages. For example, an interface for element routines written in C has been developed by Govindjee. 12 The program system FEAPpv contains only basic commands to generate structured meshes as blocks of elements. For problems where graded meshes are needed (e.g. adaptive mesh refinements) more sophisticated mesh generation techniques are needed. There are many locations where generators may be obtained and two are given in references 13 and 14. The program GiD offers two- and three-dimensional options for fluid and structure applications.
!ii ii i !! ii!!
ii ! i i i!
ill
iii!i!i
i!
i
i
!
~ ~~~
~~~ ~~ ~~
~i~i~i~,~,~
1. W.H. Press et al., editors. Numerical Recipes in Fortran: The Art of Scientific Computing. Cambridge University Press, Cambridge, 2nd edition, 1992. 2. W.H. Press et al., editors. Numerical Recipes in Fortran 77 and 90: The Art of Scientific and Parallel Computing (Software). Cambridge University Press, Cambridge, 1997. 3. W.H. Press et al., editors. Numerical Recipes in Fortran 90: The Art of Parallel Scientific Computing, volume 2. Cambridge University Press, Cambridge, 1996. 4. G.H. Golub and C.F. Van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore MD, 3rd edition, 1996. 5. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997. 6. L. Collatz. The Numerical Treatment of Differential Equations. Springer, Berlin, 1966.
596
Computerprocedures for finite element analysis 7. H. Matthies and G. Strang. The solution of nonlinear finite element equations. International Journal for Numerical Methods in Engineering, 14:1613-1626, 1979. 8. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005. 9. J.H. Wilkinson and C. Reinsch. Linear Algebra. Handbook for Automatic Computation, volume II. Springer-Verlag, Berlin, 1971. 10. K.-J. Bathe and E.L. Wilson. Numerical Methods in Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1976. 11. K.-J. Bathe. Finite Element Procedures. Prentice-Hall, Englewood Cliffs, NJ, 1996. 12. S. Govindjee. Interface for c-language routines for feap programs. Private communication (see also at internet address: http://www.ce.berkeley.edu/~sanjay), 2000. 13. J. Shewchuk. Triangle. http://www.cs.berkeley.edu/~jrs). 14. GiD - The Personal Pre/Postprocesor. www.gidhome.com, 2004.
~i~i~i~ii~i~!~i~i~i~i~i~i~i~i~!i~i!i!~! ~i~i~i~!i~i~ii~iii~iii~ii~ii!i~i~! ii~ii~84i~~iii~!ii~i!i!i~!ii~i!ii~!i~!i!i~i~!~!ii!~i~!i~!i~i~!i~!i~!~iiii~i~i!~i~i! ~i~i!~i!~! ~i~!i!i! ~!~!~
T/I
Isoparametric finite element approximations ii~~~i~i~i~iiii~i~i~ii~ii~i~i~!i~!!!i!i~i~i~iiii!~i~ii~!!i~iii~!~!iii~!iiii!ii~i~iiiii!i!!~ii~i!~i~ii~iiiiii!iiiii~iiiii!i!ii~iiii!iiiiiiii~i~i~i!i~i!iiiiiiiiiiiii~iii!i!ii!!!ii!ii!iiiiii~i~ii!iiiiiiiii!iiiiiiii~iii~i~ii
An isoparametric formulation may be used for any problem in which the approximations are C Ocontinuous. In an isoparametric formulation a parent element is defined in terms of a set of natural coordinates. The shape functions are constructed on a parent element and used to compute the coordinates within each element using X "-- N a ( ~ ) X
(m.1)
a
where Na denotes the shape function, ~ are a set of natural coordinates and Xa are nodal coordinates. A dependent variable u is then approximated as U "~, U h =
ga(~)[l
(A.2)
a
The construction of shape functions requires the selection of an appropriate set of natural coordinates. Here we first summarize the form for quadrilateral and brick elements in two and three dimensions, respectively. We then consider triangular and tetrahedral elements. iiii',iii',i'~,ii~i~ iiiiiii i~i i~i~i:J~::~:~iiiii~ii::~Q:~:~i,'i:~,'ii~iiili~iiiii~:ii'ii'~,ii',:~iiiiii' ii'ii,~i',i~"'i,i:~':,ii~i'i~:~i~:ii~i ~i,i:!~i!':~~i':iii:, i'',~,iiliii'~iii~i~i~iiii~,i~iii:i~iii!ii!iiiiii! !~ililiiiiiiiiii"~'~i~!iiiiiii~iiiiiiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiiiii' i!i~"":~:ii! ,~iiiiiiiii' ::~iiiiiiii!!iiii ,ili:~ i!i':iiiiiii!iii:,i:,ii:.ii iiiiiiiiiiliiiiiiiiiiliiiiiiiiiii!iiiiiiiiiiii!iii' iii':i!,iiii~:i:,!iiii!iiiiiiiiiiii',ii!iiiii!ii i!iiiiiiiiii iiiiiiii
ii
The natural coordinates for a quadrilateral element are given by =(~,~/);
-l~,r/~
1
as shown in Fig. A. 1. The simplest group of elements construct the shape functions from products of one dimensional Lagrangian interpolation functions given by 1~(~)-
(~-~o)(~-~1)""" (~a--~O)(~a--~l)''"
-II bCa
(~-~a-1)(~-~a+l)""" (~a--~a--1)(~a--~a+l)
(~--~n) " " " (~a--~n)
a = 1, 2, ..- , n - 1 '
598 Appendix A
1
Fig. A.1 Natural coordinatesfor a quadrilateral.
>!
=~a~ +0 =0
On on that boundary is automatically satisfied- such a condition is known as a natural boundary condition- and (b) if the choice of 0~is restricted so as to satisfy the forced boundary conditions O~- ~ = O, we can omit the last term of Eq. (3.20) by restricting the choice of v to functions which give v - 0 on F~. The form of Eq. (3.20) is the weak form of the heat conduction statement equivalent to Eq. (3.8). It admits discontinuous conductivity coefficients k and temperature 0~ which show discontinuous first derivatives, a real possibility not easily admitted in the differential form.
If the unknown function u is approximated by the expansion (3.3), i.e., U ,~ fl - - ~
Nafla --
Nfl
I
a=l
then it is clearly impossible to satisfy both the differential equation and the boundary conditions in the general case. The integral statements (3.13) or (3.14) allow an approximation to be made if, in place of any function v, we put a finite set of approximate functions
V,~~Wb~tlb b=l
and
V:~Wb~ll b=l
b
(3.21)
Approximation to integral formulations: the weighted residuaI-Galerkin method 61 in which ~Ub the relation
are
arbitrary parameters. Inserting the approximations into Eq. (3.13) we have
and since 8fib is arbitrary we have a set of equations which is sufficient to determine the parameters fi as ~w/,A(Nfi)df2+fr,~/3(Nfi)dl-'
=0;
b = 1,2 . . . . . n
(3.22)
Performing similar steps using Eq. (3.14) gives the set f
C T(wb)79(N fi) dr2 + f r eT(qeb)'~'(N fi) dF - 0; b = 1, 2, . . . , n
(3.23)
If we note that ,A(Nfi) represents the residual or error obtained by substitution of the approximation into the differential equation [and B(Nfi), the residual of the boundary conditions], then Eq. (3.22) is a weighted integral of such residuals. The approximation may thus be called the method of weighted residuals. In its classical sense it was first described by Crandall, 1 who points out the various forms used since the end of the nineteenth century. Later a very full expos6 of the method was given by Finlayson. e Clearly, almost any set of independent functions wb could be used for the purpose of weighting and, according to the choice of function, a different name can be attached to each process. Thus the various common choices are: 1. Point collocation. 3 Wb -- 6b in (3.22), where 6b is such that for x ~ Xb; y ~ Yb, Wb -- 0 but f~ Wbd~2 = 1 (unit matrix). This procedure is equivalent to simply making the residual zero at n points within the domain and integration is 'nominal' (incidentally although Wb defined here does not satisfy all the criteria of Sec. 3.2, it is nevertheless admissible in view of its properties). Finite difference methods are particular cases of this weighting. 2. Subdomain collocation. 4 Wb -- I in subdomain ~b and zero elsewhere. This essentially makes the integral of the error zero over the specified subdomains. When used with (3.23) this is one of the many finite volume methods. 5 3. The Galerkin method (Bubnov-Galerkin). 4' 6 Wb - - Nb. Here simply the original shape (or basis) functions are used as weighting. This method, as we shall see, frequently (but by no means always) leads to symmetric matrices and for this and other reasons will be adopted in this book almost exclusively.
The name of 'weighted residuals' is clearly much older than that of the 'finite element method'. The latter uses mainly locally based (element) functions in the expansion of Eq. (3.3) but the general procedures are identical. As the process always leads to equations which, being of integral form, can be obtained by summation of contributions from various subdomains, we choose to embrace all weighted residual approximations under the name of generalized finite element method. On occasion, simultaneous use of both local and 'global' trial functions will be found to be useful. In the literature the names of Petrov and Galerkin 6 are often associated with the use of weighting functions such that Wb ~ Nb. It is important to remark that the well-known finite difference method of approximation is a particular case of collocation with locally defined
62
Generalization of the finite element concepts
Fig. 3.3 Problem description and loading for 1-d heat conduction example. basis functions and is thus a case of a Petrov-Galerkin scheme. We shall return to such unorthodox definitions in more detail in Chapter 15. To illustrate the procedure of weighted residual approximation and its relation to the finite element process let us consider some specific examples.
Example 3.5: One-dimensional equation of heat conduction. The problem here will be a one-dimensional representation of the heat conduction equation [Eq. (3.8)] with unit conductivity. (This problem could equally well represent many other physical situations, e.g., deflection of a loaded string with unit tension.) Here we have (see Fig. 3.3) d2~p A(dp) =
dx 2 + Q(x) = O (O < x < L)
(3.24a)
with Q (x) given by
Q(x) =
0
O<x = 0 throughout. Further, we shall take 4> = 0 on all boundaries for all times. An approximation for the solution is taken: M
N
~--ZZNmn(X,Y)~)mn(t) m=l n=l
gmn = cos
mrc x
L
nrc y
cos ~ ; L
(3.57)
m, n = 1, 3, 5 , . - .
with x and y measured from the centre (Fig. 3.9). The even components of the Fourier series are omitted due to the required symmetry of solution. Evaluating the coefficients (only diagonal terms exist in K), we have Kmn --
Cmn -'-
finn
ILl2 ILl2 [k (Ogmn) 2 + k (Ogmn)21 dx d y
a-L~2 a-L~2
fL/2 ;L/2 a-L~2 a-L~2
Ox
c N ~ n dx d y =
Oy
-
zr2k ~. (m 2 -k- n 2)
L2c
(3.58)
4
~[L/2[L/2 Nmn Qoe -~t dx d y a-L/2 a-L/2
=
4Qo L2
mnTr2
(_l)(m+3)/2(_l)(n+3)/2e-Ott
This leads to an ordinary differential equation with parameters
Kmnf~mn+ Cmn~d(bmn -Jr-fmn -- 0 dt
~mn"
(3.59)
with ~mn - - 0 when t = 0. The exact solution of this is easy to obtain, as is shown in Fig. 3.9 for specific values of the parameters M, N, ot and k / L 2 c . The remarkable accuracy of the approximation with M -- N -- 3 in this example should be noted. In this example we have used trigonometric functions in place of the more standard polynomials used in the finite element method. In Chapter 7 we recalculate the solution using a standard finite element method in which the solution to the time problem is computed using a finite difference method.
73
74 Generalization of the finite element concepts
Fig. 3.9 Two-dimensional transient heat development in a square prism - plot of temperature at centre.
In the previous sections we have discussed how approximate solutions can be obtained by use of an expansion of the unknown function in terms of trial or shape functions. Further, we have stated the necessary conditions that such functions have to fulfil in order that the various integrals can be evaluated over the domain. Thus if various integrals contain only the values of N and its first derivatives then N has to be Co continuous. If second derivatives are involved, C1 continuity is needed, etc. The problem which we have not yet addressed ourselves consists of the questions of just how good the approximation is and how it can be systematically improved to approach the exact answer. The first question is more difficult to answer and presumes knowledge of the exact solution (see Chapter 13). The second is more rational and can be answered if we consider some systematic way in which the number of parameters ~ in the standard expansion of Eq. (3.3),
tl = ~-~ Nafla a=l
is presumed to increase. In some examples we have assumed, in effect, a trigonometric Fourier-type series limited to a finite number of terms with a single form of trial function assumed over the whole domain. Here addition of new terms would be simply an extension of the number of terms in the series included in the analysis, and as the Fourier series is known to be able to represent any function within any accuracy desired as the number of terms increases, we can talk about convergence of the approximation to the true solution as the number of terms increases. In other examples of this chapter we have used locally based polynomial functions which are fundamental in the finite element analysis. Here we have tacitly assumed that
Convergence 75 convergence occurs as the size of elements decreases and, hence, the number of fi parameters specified at nodes increases. It is with such convergence that we need to be concerned and we have already discussed this in the context of the analysis of elastic solids in Chapter 2 (Sec. 2.6). We have now to determine (a) that, as the number of elements increases, the unknown functions can be approximated as closely as required, and (b) how the error decreases with the size, h, of the element subdivisions (h is here some typical dimension of an element). The first problem is that of completeness of the expansion and we shall here assume that all trial functions are polynomials (or at least include certain terms of a polynomial expansion). Clearly, as the approximation discussed here is to the weak, integral form typified by Eq. (3.11) or (3.14) it is necessary that every term occurring under the integral be in the limit capable of being approximated as nearly as possible and, in particular, giving a constant value over any arbitrary infinitesimal part of the domain ~2. If a derivative of order m exists in any such term, then it is obviously necessary for the local polynomial to be at least of the order m so that, in the limit, such a constant value can be obtained. We will thus state that a necessary condition for the expansion to be covergent is the criterion of completeness: that, if mth derivatives occur in the integral form, a constant value of all derivatives up to order m be attainable in the element domain when the size of any element tends to zero. This criterion is automatically ensured if the polynomials used in the shape function N are complete to mth order. This criterion is also equivalent to the one of constant strain postulated in Chapter 2 (Sec. 2.5). This, however, has to be satisfied only in the limit h --+ 0. If the actual order of a complete polynomial used in the finite element expansion is p > m, then the order of convergence can be ascertained by seeing how closely such a polynomial can follow the local Taylor expansion of the unknown u. Clearly the order of error will be simply O (h p+I) since only terms of order p can be rendered correctly. Knowledge of the order of convergence helps in ascertaining how good the approximation is if studies on several decreasing mesh sizes are conducted. Though, in Chapter 14, we shall see the asymptotic convergence rate is seldom reached if singularities occur in the problem. Once again we have re-established some of the conditions discussed in Chapter 2. We shall not discuss, at this stage, approximations which do not satisfy the postulated continuity requirements except to remark that once again, in many cases, convergence and indeed improved results can be obtained (see Chapter 9). In the above we have referred to the convergence of a given element type as its size is reduced. This is sometimes referred to as h convergence. On the other hand, it is possible to consider a subdivision into elements of a given size and to obtain convergence to the exact solution by increasing the polynomial order p of each element. This is referred to as p convergence, which is obviously assured. In general p convergence is more rapid per degree of freedom introduced. We shall discuss both types further in Chapter 14; although we have already noted in some examples how improved accuracy occurs with higher term polynomials being added at each element level.
76
Generalizationof the finite element concepts
Variational principles What are variational principles and how can they be useful in the approximation to continuum problems? It is to these questions that the following sections are addressed. First a definition: a 'variational principle' specifies a scalar quantity (functional) 17, which is defined by an integral form 17 =
F
0u ...
df~+
u, Ox'
E u,~,.., Ox
dF
(3.60)
in which u is the unknown function and F and E are specified differential operators. The solution to the continuum problem is a function u which makes 17 stationary with respect to arbitrary changes 6u. Thus, for a solution to the continuum problem, the 'variation' is 817 = 0
(3.61)
for any 8u, which defines the condition of stationarity. 15 If a 'variational principle' can be found, then means are immediately established for obtaining approximate solutions in the standard, integral form suitable for finite element analysis. Assuming a trial function expansion in the usual form [Eq. (3.3)] U ~ fl --" ~
Nail a
a=l we can insert this into Eq. (3.60) and write OH OH 017 ~l"I "-- O------~Ul+ ~u2~B2 .ql_....11_ ~Un ~ n ~. 0
(3.62)
This being true for any variations ~fi yields a set of equations 017 an _ 0~
.
= 0
(3.63)
317
from which parameters Ha are found. The equations are of an integral form necessary for the finite element approximation as the original specification of I7 was given in terms of domain and boundary integrals. The process of finding stationarity with respect to trial function parameters fi is an old one and is associated with the names of Rayleigh 16 and Ritz. 17 It has become extremely important in finite element analysis which, to many investigators, is typified as a 'variational process'.
What are 'variational principles'?
If the functional I7 is 'quadratic', i.e., if the function u and its derivatives occur in powers not exceeding 2, then Eq. (3.63) reduces to a standard linear form similar to Eq. (3.7), i.e., 017 0~
-- K~ + f = 0
(3.64)
It is easy to show that the matrix K will now always be symmetric. To do this let us consider a linearization of the vector 01-I/0~. This we can write as
zx
(0I-I) ~
=
O--~Ul ~Ul A~I
-- KT A~
(3.65)
in which KT is generally known as the tangent matrix, of significance in non-linear analysis, and A~ are small incremental changes to ~. Now it is easy to see that
02l"I
(3.66)
KTab -- O~laO~lb "- KTba
Hence KT is symmetric. For a quadratic functional we have, from Eq. (3.64), ....
A
-0--flu -- K A ~
with
K = KT
(3.67)
and hence symmetry must exist. The fact that symmetric matrices will arise whenever a variational principle exists is one of the most important merits of variational approaches for discretization. However, symmetric forms will frequently arise directly from the Galerkin process. In such cases we simply conclude that the variational principle exists but we shall not need to use it directly. Further, the discovery of symmetry from a weighted residual process leads directly to known (or previously unknown) variational principles. 18 How then do 'variational principles' arise and is it always possible to construct these for continuous problems? To answer the first part of the question we note that frequently the physical aspects of the problem can be stated directly in a variational principle form. Theorems such as minimization of total potential energy to achieve equilibrium in mechanical systems, least energy dissipation principles in viscous flow, etc., may be known to the reader and are considered by many as the basis of the formulation. We have already referred to the first of these in Sec. 2.4 of Chapter 2. Variational principles of this kind are 'natural' ones but unfortunately they do not exist for all continuum problems for which well-defined differential equations may be formulated. However, there is another category of variational principles which we may call 'contrived'. Such contrived principles can always be constructed for any differentially specified problem, either by extending the number of unknown functions u by additional variables known as Lagrange multipliers, or by procedures imposing a higher degree of continuity requirements such as in least squares problems. In subsequent sections we shall discuss, respectively, such 'natural' and 'contrived' variational principles. Before proceeding further it is worth noting that, in addition to symmetry occurring in equations derived by variational means, sometimes further motivation arises. When
77
78 Generalizationof the finite element concepts
'natural' variational principles exist the quantity FI may be of specific interest itself. If this arises a variational approach possesses the merit of easy evaluation of this functional. The reader will observe that if the functional is 'quadratic' and yields Eq. (3.64), then we can write the approximate 'functional' 1-I simply as n = I~TKfi + ~Tf
(3.68)
By simple differentiation
~l"I = ~1 (fiT)K fi + 21fiTK 8fi + 8fiTf __ 0 As K is symmetric, 3fiTKfi ~ fiTK3fi Hence FI = ~fiT (Kfi + f) = 0 which is true for all 6fi and hence Kfi+f=O when inserted into (3.68) we obtain 1 1 FI = - fiTf = _ _ fiTK 2 2
If we consider the definitions of Eqs (3.60) and (3.61) we observe that for stationarity we can write, after performing some differentiations and integrations by parts, ~n = f
~uT,A(u)d~ + f r 8uTB(u)dr - 0
(3.69)
As the above has to be true for any variations ~u, we must have ,At(u) = 0
in f2
and
B(u) = 0
on F
(3.70)
If ,At corresponds precisely to the differential equations governing the problem of interest and/3 to its boundary conditions, then the variational principle is a natural one. Equations (3.70) are known as the Euler differential equations corresponding to the variational principle requiting the stationarity of FI. It is easy to show that for any variational principle a corresponding set of Euler equations can be established. The reverse is unfortunately not true, i.e., only certain forms of differential equations are Euler equations of a variational functional. In the next section we shall consider the conditions necessary for the existence of variational principles and give a prescription for the establishing FI from a set of suitable linear differential equations. In this section we shall continue to assume that the form of the variational principle is known.
'Natural' variational principles and their relation to governing differential equations To illustrate the process let us now consider a specific example. Suppose we specify a problem by requiring the stationarity of a functional + gk
+ Q~b dr2 +
q~b dF
(3.71)
q
in which k and Q depend only on position and we assume r = ~b is satisfied on F~. We now perform the variation. 15 This can be written following the rules of differentiation
as
8I-1 =
k -3xx8 -~x + k ?y 8 -~y + a s ~
As
ar
I
dr2 +
(~/8~b)dF = 0
(3.72)
q
a
8 -~x =ff--xx(8r
(3.73)
we can integrate by parts (as in Sec. 3.3) and, since 8q~ = 0 on F4, obtain
3 ( k ~ y ) + QI dr2 +
i (
ox
&b k-~n + q
q
)
-
(3.74a)
dr=0
This is of the form of Eq. (3.69) and we immediately observe that the Euler equations are A(~b)=-~xxk
Oy
-~
k
+a=0
B(r = k a4) On + q = 0
inf2
(3.74b)
on Fq
If q~is prescribed so that q~ = ~) on F4 and 8q~ = 0 on that boundary, then the problem is precisely the one we have already discussed in Sec. 3.2 and the functional (3.71) specifies the two-dimensional heat conduction problem in an alternative way. In this case we have 'guessed' the functional but the reader will observe that the variation operation could have been carried out for any functional specified and corresponding Euler equations could have been established. Let us continue the process to obtain an approximate solution of the linear heat conduction problem. Taking, as usual,
qb~, ~ -~-~ ga~)a = N6
a
(3.75)
we substitute this approximation into the expression for the functional FI [Eq. (3.71)] and obtain
n-f
~2 (Z--~x :d f 2 + f ~ 2 !~ ( ~ - - ~ aNa~ aaNa)a - y C a ) d: r 2 (3.76)
"~ ~ O ~Na~)a d~-~"~-fF q ~a Na~)ad~ a q
79
80
Generalization of the finite element concepts
On differentiation with respect to a typical parameter ~b we have
01-1 s ( ONa )ONbd~_+_s O~)b-- k ~ ---~x ~)a OX
ONa~)ONb d~ ~a --~-ydt)a Or
(3.77)
q and a system of equations for the solution of the problem is ~
Kr
(3.78)
=0
with
Kab=Kba - s
ONaONb dr a + s Ox Ox fb -- s NbQ dS2 + J; Nb~/dF
8NaSNb dr2 Oy Oy
(3.79)
q
The reader will observe that the approximation equations are here identical with those obtained in Sec. 3.5 for the same problem using the Galerkin process. No special advantage accrues to the variational formulation here, and indeed we can predict now that Galerkin
and variationalproceduresmustgive the sameanswerfor caseswherenaturalvariational principles exist. 3.8.2 Relation of the Galerkin method to approximation via
variational principles
In the preceding example we have observed that the approximation obtained by the use of a natural variational principle and by the use of the Galerkin weighting process proved identical. That this is the case follows directly from Eq. (3.69), in which the variation was derived in terms of the original differential equations and the associated boundary conditions. If we consider the usual trial function expansion [Eq. (3.3)] u,~
fi = N ~
we can write the variation of this approximation as ~fi - N ~fi
(3.80)
and inserting the above into (3.69) yields
-
[o N
+
N
dr = 0
3.81
The above form, being true for all 6fi, requires that the expression under the integrals should be zero. The reader will immediately recognize this as simply the Galerkin form of the weighted residual statement discussed earlier [Eq. (3.22)], and identity is hereby proved. We need to underline, however, that this is only true if the Euler equations of the variational principle coincide with the governing equations of the original problem. The Galerkin process thus retains its greater range of applicability.
Establishment of natural variational principles for linear, self-adjoint, differential equations
General rules for deriving natural variational principles from non-linear differential equations are complicated and even the tests necessary to establish the existence of such variational principles are not simple. Much mathematical work has been done in this context by Vainberg, 19 Tonti, 18 Oden, 2~ 21 and others. For linear differential equations the situation is much simpler and a thorough study is available in the works of Mikhlin, 22' 23 and in this section a brief presentation of such rules is given. We shall consider here only the establishment of variational principles for a linear system of equations with forced boundary conditions, implying only variation of functions which yield 6u = 0 on their boundaries. The extension to include natural boundary conditions is simple and will be omitted. Writing a linear system of differential equations as .A(u) - L u + b = 0
(3.82)
in which E is a linear differential operator it can be shown that natural variational principles require that the operator E be such that L ~T (Z~'7) dr2 = f
7T(z2~.,) dr2 + b.t.
(3.83)
for any two function sets @ and "7. In the above, 'b.t.' stands for boundary terms which we disregard in the present context. The property required in the above operator is called that of self-adjointness or symmetry. If the operator Z2 is self-adjoint, the variational principle can be written immediately as 1-I = ~
[ 89T (Cu) + uTb] dr2 + b.t.
(3.84)
To prove the veracity of the last statement a variation needs to be considered. We thus write (omitting boundary terms)
~I'/ = f [ 8 9~uT~u -{- 1UT~ (~U) "-]-3uTb] dff2 = 0
(3.85)
Noting that for any linear operator 8(Z;u) = Z; ~u
(3.86)
and that u and 3u can be treated as any two independent functions, by identity (3.83) we can write Eq. (3.85) as ~l-I = J~ ~uT[/~U + b] dr2 - 0
(3.87)
We observe immediately that the term in the brackets, i.e., the Euler equation of the functional, is identical with the original equation postulated, and therefore the variational principle is verified.
81
82
Generalizationof the finite element concepts
The above gives a very simple test and a prescription for the establishment of natural variational principles for differential equations of the problem. Example 3.9: Helmholz problem in two dimensions. A Helmholz problem is governed by a differential equation similar to the heat conduction equation, e.g., V2~b -4- C ~ "4- Q = 0
(3.88)
with c and Q being dependent on position only. The above can be written in the general form of Eq. (3.82), with
s
Tx2 + ~ y 2 + c ;
b=Q
and
u-4~
(3.89)
Verifying that self-adjointness applies (which we leave to the reader as an exercise), we immediately have a variational principle I-I--f~ [ ~ q ~ ( ~-Yx 02q~ 02~b + c ~ )/ + q~Q ] d x d y 2 + ff~y2
(3.90)
with 4~ satisfying the forced boundary condition, i.e., 4~ - 4~ on FO. Integrating by parts of the first two terms results in b ) 2 + -~1 (O~b)2 1 c ~p2 - cp Q 1 dx d y 1-I = - f ~ [ ~ ( O ~-~x ~y - -~
(3.91)
on noting that boundary terms with prescribed ~p do not alter the principle. Example 3.10: First-order form of heat equation. This problem concerns the onedimensional heat conduction equation (Example 3.5, Sec. 3.3) written in first order form as
d~ - q - Ux
,A(u) =
dq
~+a
or, using Eq. (3.82), as s
l,
-dxx
d
.
b=
dxx' 0 1'
{0} Q
=0
and u =
{:}
Again self-adjointness of the operator can be tested and found to be satisfied. We now write the functional as
I{}T
d
Ux' dq~
{:} §
0
dx (3.92)
QI dx
Maximum, minimum, or a saddle point?
n j~
--
d21-[ t > 0
,
;o d2nl
_\
"',
~',-a
r
Fig. 3.10 Maximum, minimum and a 'saddle' point for a functional l I of one variable
The verification of the correctness of the above, by executing a variation, is left to the reader. These two examples illustrate the simplicity of application of the general expressions. The reader will observe that self-adjointness of the operator will generally exist if even orders of differentiation are present. For odd orders self-adjointness is only possible if the operator is a 'skew'-symmetric matrix such as occurs in the second example.
In discussing variational principles so far we have assumed simply that at the solution point g I-I = 0, that is the functional is stationary. It is often desirable to know whether 1-I is at a maximum, minimum, or simply at a 'saddle point'. If a maximum or a minimum is involved, then the approximation to FI will always be 'bounded', i.e., will provide approximate values of FI which are either smaller or larger than the correct ones.t The bound in itself may be of practical significance in some problems. When, in elementary calculus, we consider a stationary point of a function 1-I of one variable u, we investigate the rate of change of dFI with du and write
Ol-I d u )
d(dFl) = dk,-~u
02I'I
= -~u2 (du)2
(3.93)
The sign of the second derivative determines whether I7 is a minimum, maximum, or simply stationary (saddle point), as shown in Fig. 3.10. By analogy in the calculus of variations we shall consider changes of g I7. Noting the general form of this quantity given by Eq. (3.62) and the notion of the second derivative of Eq. (3.65) we can write, in terms of discrete parameters, g(gl"[) ~ g
~
gO -" goTg
~
-- gO T
oqUloq~lgUl
-- gOTKT glUl
(3.94)
If, in the above, g (g FI) is always negative then FI is obviously reaching a maximum, if it is always positive then 1-I is a minimum, but if the sign is indeterminate this shows only the existence of a saddle point. t Provided all integrals are exactly evaluated.
83
84
Generalizationof the finite element concepts
As ~ is an arbitrary vector this statement is equivalent to requiring the matrix KT to be negative definite for a maximum or positive definite for a minimum. The form of the matrix Kx (or in linear problems of K which is identical to it) is thus of great importance in the solution of variational problems.
Consider the problem of making a functional I-I stationary, subject to the unknown u obeying some set of additional differential relationships C(u) = 0
in f2
(3.95)
We can introduce this constraint by forming another functional l=l(u, ~k) = I-l(u) + f~ ~xTC(u)dg2
(3.96)
in which ,k is some set of functions of the independent coordinates in the domain f2 known as Lagrange multipliers. The variation of the new functional is now 81=I = 8Fl + f
)~TSC(u) d r q- f
8)~TC(u) d r = 0
(3.97)
which immediately gives C(u) = 0 and, simultaneously, an added contribution to the original 3 H involving ,,k. In a similar way, constraints can be introduced at some points or over boundaries of the domain. For instance, if we require that u obey E(u) = 0
on r'
(3.98)
we would add to the original functional the term r ,XTE(u) dF
(3.99)
with ,k now being an unknown function defined only on F. Alternatively, if the constraint C is applicable only at one or more points of the system, then the simple addition of ,xTC(u) at these points to the general functional 1-I will introduce a discrete number of constraints. It appears, therefore, possible to always introduce additional functions ,~ and modify a functional to include any prescribed constraints. In the 'discretization' process we shall now have to use trial functions to describe both u and ~k. Writing, for instance, Nafla -- Nf~
fl-- ~ a
~, = ~
Nb~b = N ~ b
(3.100)
Constrained variational principles. Lagrange multipliers we shall obtain a set of equations 01-I _
=0
0w
where
w = (~A}
(3.101)
from which both the sets of parameters fi and ,,~can be obtained. It is somewhat paradoxical that the 'constrained' problem has resulted in a larger number of unknown parameters than the original one and, indeed, has complicated the solution. We shall, nevertheless, find practical use for Lagrange multipliers in formulating some physical variational principles, and will make use of these in a more general context in Chapters 10 and 11. Before proceeding further it is of interest to investigate the form of equations resulting from the modified functional FI of Eq. (3.96). If the original functional I-I gave as its Euler equations a system ,A(u) - 0 (3.102) then we have (omitting the boundary terms) ~1=I = f~'uT.A(u) dr2 + f ~cTA dr2 + f 'ATC(u) dr2 = O
(3.103)
Substituting the trial functions (3.100) we can write for a linear set of constraints
C(u) -~ A~lU-if-C1 that
8l=I = BuT [J~ NT.,4.(fi)dff2+ J~ (ff-,1N)T~dff2] (3.104) + ~xT f a l~V(Z~lu + C1) dr2 = 0 As this has to be true for all variations 3fi and 8,~, we have a system of equations NTA(fi) dr2 + ~(~IN)T,Xdf2 = 0
f I~T(~Ifi -t" C1) dr2 =
(3.105) 0
For linear equations .,4, the first term of the first equation is precisely the ordinary, unconstrained, variational approximation Kuufi + fu
(3.106)
and inserting again the trial functions (3.100) we can write the approximated Eq. (3.105) as a linear system: =
[Ku\
'
+
fz
with KuT = ~ I~T (LIN)dr2;
fz = f I~Tc1 dr2
(3.108)
85
86 Generalizationof the finite element concepts Clearly the system of equations is symmetric but now possesses zeros on the diagonal, and therefore the variational principle H is merely stationary. Further, computational difficulties may be encountered unless the solution process allows for zero diagonal terms.
Example 3.11: Constraint enforcement using L a g r a n g e multiplier. The point about increasing the number of parameters to introduce a constraint may perhaps be best illustrated in a simple algebraic situation in which we require a stationary value of a quadratic function of two variables u 1 and u2: H = 2u 2 - 2u 1u2 "q- U2 + 18u 1 "q- 6u2
(3.109)
Ul -- /'12 = 0
(3.110)
subject to a constraint
The obvious way to proceed would be to insert directly the equality 'constraint' and obtain FI = u 2 + 24Ul
(3.111)
and write, for stationarity, OH
OUl
= 0 = 2 U l d- 2 4
Ul - - u2 = - - 1 2
(3.112)
Introducing a Lagrange multiplier )~ we can alternatively find the stationarity of (I = 2u 2 - 2 U l U 2 -~- u 2 -at- l g U l --[- 6 u 2 --[- ~,(Ul - u 2 )
(3.113)
and write three simultaneous equations
al=i = 4Ul - 2u2 + ~, -k- 18 = 0 OUl ~f-I = - - 2 U l 4- 2 u 2 -- ~, q- 6 = 0 OU2
(3.114)
=Ul--U2=0
The solution of the above system again yields the correct answer Ul - u2 - - 1 2
)~ = 6
but at considerably more effort. Unfortunately, in most continuum problems direct elimination of constraints cannot be so simply accomplished.t t In the finite element context, Szabo and Kassos 24 use such direct elimination; however, this involves considerable algebraic manipulation.
Constrained variational principles. Lagrange multipliers 87
3.11.2 Identification of Lagrange multipliers. Forced boundary conditions and modified variational principles Although the Lagrange multipliers were introduced as a mathematical concept necessary for the enforcement of certain external constraints required to satisfy the original variational principle, we shall find that in many situations they can be identified with certain physical quantities of importance to the original mathematical model. Such an identification will follow immediately from the definition of the variational principle established in Eq. (3.96) and through the first of the Euler equations in (3.105) corresponding to it. The variation l=I, written in Eq. (3.97), supplies through its third term the constraint equation. The first two terms can always be rewritten as ~ ~C(u)T)~ dr2 + f~ 6uT.gt(U)d~ = 0
(3.115a)
j~r 8E(u)Tzx dF + j~r ~uTB(u)dF = 0
(3.115b)
or
This supplies the identification of )~. In the literature of variational calculation such identification arises frequently and the reader is referred to the excellent text by Washizu e5 for numerous examples.
Example 3.12: Identification of Lagrange multiplier for boundary condition. Here we shall introduce this identification by means of the example considered in Sec. 3.8.1. As we have noted, the variational principle of Eq. (3.71) established the governing equation and the natural boundary conditions of the heat conduction problem providing the forced boundary condition E(r = r - r = 0 (3.116) was satisfied on Fr in the choice of the trial function for r The above forced boundary condition can, however, be considered as a constraint on the original problem. We can write the constrained variational principle as lYl = I7 + f
dF
)~(~b- ~) dF
(3.117)
where H is given by Eq. (3.71). Performing the variation we have
,~fI=~rI + f ~r dr" r
+f
.JFr
8X(r
~)dr =0
(3.118)
H is now given by the expression (3.74a) augmented by an integral
fr~ ar 0_r dr On which was previously disregarded (as we had assumed that 6tp = 0 on Fr the conditions of Eq. (3.74b), we now require that fr '1'(~b-~)dF+fr
'q~( k + k 0 q ~ d l - ' - 0
(3.119) In addition to
(3.120)
88
Generalizationof the finite element concepts
which must be true for all variations 3X and 3q~. The first simply reiterates the constraint (p - (p = 0
on
ro
(3.121)
The second defines X as X = -k~ On Noting that k(Ocp/On) is the negative to the flux qn on the boundary identification of the multiplier has been achieved- that is, ~. _-- qn.
(3.122) the physical
The identification of the Lagrange variable leads to the possible establishment of a modified variational principle in which ~, is replaced by the identification. We could thus write a new principle for the above example: l=I -
-n (~ fr k ~04
n -
~) dr
(3.123)
in which once again YI is given by the expression (3.71) but 4~ is not constrained to satisfy any boundary conditions. Use of such modified variational principles can be made to restore interelement continuity and appears to have been first introduced for that purpose by Kikuchi and Ando. 26 In general these present interesting new procedures for establishing useful variational principles. A further extension of such principles has been made use of by Chen and Mei 27 and Zienkiewicz et al. 28 Washizu 25 discusses many such applications in the context of structural mechanics. The reader can verify that the variational principle expressed in Eq. (3.123) leads to automatic satisfaction of all the necessary boundary conditions in the example considered. The use of modified variational principles restores the problem to the original number of unknown functions or parameters and is often computationally advantageous.
In the previous section we have seen how the process of introducing Lagrange multipliers allows constrained variational principles to be obtained at the expense of increasing the total number of unknowns. Further, we have shown that even in linear problems the algebraic equations which have to be solved are now complicated by having zero diagonal terms. In this section we shall consider alternative procedures of introducing constraints which do not possess these drawbacks.
3.12.1 Penalty functions Considering once again the problem of obtaining stationarity of FI with a set of constraint equations C(u) = 0 in domain ~, we note that the product =
+
+...
(3.124)
Constrained variational principles. Penalty function and perturbed lagrangian methods 89 where C x = [C1, C2, ...] must always be a quantity which is positive or zero. Clearly, the latter value is found when the constraints are satisfied and also clearly the variation ~ (CTC) = 0
(3.125)
as the product reaches that minimum. We can now write a new functional
fl
c T ( u ) c ( u ) d~
= n + ~ c~
(3.126)
in which ot is a 'penalty number' and then require the stationarity for the constrained solution. If FI is itself a minimum of the solution then c~ should be a positive number. The solution obtained by the stationarity of the functional 1-I will satisfy the constraints only approximately. The larger the value of c~ the better will be the constraints achieved. Further, it seems obvious that the process is best suited to cases where I-I is a minimum (or maximum) principle, but success can be obtained even with purely saddle point problems. The process is equally applicable to constraints applied on boundaries or simple discrete constraints. In this latter case integration is dropped.
3.12.2
Perturbed lagrangian ~
We consider once again the problem of obtaining stationarity of FI with a set of constraint equations C(u) = 0 in domain f2. The Lagrange multiplier form to embed the constraint is given in Eq. (3.96). Here we modify the expression by appending a quadratic term of the form ,~T~ scaled by a parameter c~. The form of the final equation is given by
~(u, ~,) = rI(u) + f~ ;~TC(u)dS2
1 ~ ,kT,,kdf2 2-~
(3.127)
We note that as the parameter c~ tends toward infinity the form approaches a Lagrange multiplier form. Accordingly, this form is called aperturbed lagrangianfunctional. Taking the variation we obtain the result
If the constraints are a linear form given by C(u) =
C0u
we can introduce the approximations (3.100) into (3.128) to obtain the set of equations
Kuu Kzu
where
1Kuz] fi -~KxxJ /,~}:
{f0}
Kuu is the coefficient array from ~l-I and Ku~ = f I~ITc0 dr2
and
Kzz = J~ I~TI~ dr2
(3.129)
90 Generalizationof the finite element concepts The second equation of (3.129) may be solved for )~ in terms of ~ and substituted into the first equation to obtain
Kuufl - [Kuu + ot KuzKx-lKxu]U = f It is now apparent that the perturbed lagrangian and penalty forms are closely related. The perturbed lagrangian uses Kux K~-~Kzu to impose the constraint whereas the penalty approach uses f
c ~ c 0 dr2
When the constraint is a simple scalar relation the two methods are identical; however, when any other form is considered the methods will yield different approximations unless the shape functions for )~ include all the terms contained in ~C(u). Example 3.13: Constraint enforcement by penalty method. To clarify ideas let us once again consider the algebraic problem of Sec. 3.11.1, in which the stationarity of a functional given by Eq. (3.109) was sought subject to a constraint. With the penalty function approach we now seek the minimum of a functional
1~I - 2u~ - 2Ul//2
+ U2 "[- 1 8 U l -~- 6u2 q- ~1 ot (Ul - - U 2 ) 2
with respect to the variation of both parameters equations
OUl we find
=
I
0,
Ul
OU2
and
U2.
(3.130)
Writing the two simultaneous
=0
] u2
3131
and note as ot is increased we approach the correct solution. In Table 3.1 the results are set out demonstrating the convergence. The reader will observe that in a problem formulated in the above manner the constraint introduces no additional unknown parameters - but neither does it decrease their original number. The process will always result in strongly positive definite matrices if the original variational principle is one of a minimum and, similarly, negative definite matrices are obtained for a maximum principle if c~ is negative. In practical applications the method of penalty functions has proved to be quite effective, 29 and indeed is often introduced intuitively. T a b l e 3.1 C o n v e r g e n c e of t w o - t e r m solution ot
1/2
1
3
5
50
u1
- 12.000
- 12.000
u2
-13.500
-13.000
500
- 12.000
- 12.000
- 12.000
- 12.000
-12.429
-12.273
-12.030
-12.003
Constrained variational principles. Penalty function and perturbed lagrangian methods 91 In the example presented next the forced boundary conditions are not introduced a priori and the problem gives, on assembly, a singular system of equations Kfi + f = 0
(3.132)
which can be obtained from the functional (providing K is symmetric) I-[ = ~1 0 T K 0 + 0Tf
Introducing a prescribed value of
U l,
(3.133)
i.e., writing Ul -- Ul -- 0
(3.134)
1 (U 1 - Ul) 2 fl _. l-I-+- ~or
(3.135)
the functional can be modified to
yielding /~11 = Kll + or
f l = fl --OtUl
(3.136)
and giving no change in any of the other matrix coefficients. Many applications of such a 'discrete' kind are discussed by Campbell. 3~ It is easy to show in another context 29'31 that the use of a high Poisson's ratio (v -+ 0.5) for the study of incompressible solids or fluids is in fact equivalent to the introduction of a penalty term to suppress any compressibility allowed by an arbitrary displacement variation. The use of the penalty function in the finite element context presents certain difficulties. First, the constrained functional of Eq. (3.126) leads to equations of the form (K1 + otK2)u + ~ = 0
(3.137)
where K1 derives from the original functional and K2 from the constraints. As o~ increases the above equation degenerates to: K2u--f/ct
--+ 0
and fi = 0 unless the matrix K2 is singular. The phenomenon where ~ ::~ 0 is known as locking and has often been encountered by researchers who failed to recognize its source. This singularity in the equations does not always arise and we shall discuss means of its introduction in Chapters 10 and 11. Second, with large but finite values ofct numerical difficulties will be encountered. Noting that discrefization errors can be of comparable magnitude to those due to not satisfying the constraint, we can make ot = constant(l/h) n ensuring a limiting convergence to the correct answer. Fried 32, 33 discusses this problem in detail. A more general discussion of the whole topic is given in reference 34 and in Chapter 11 where the relationship between Lagrange constraints and penalty forms is made clear.
92
Generalizationof the finite element concepts
A general variational principle also may be constructed if the constraints described in the previous section are simply the governing equations of the problem C(u) - A(u)
(3.138)
Obviously the same procedure can be used in the context of the penalty function approach by setting H = 0 in Eq. (3.126). We can thus write a 'variational principle' = 1 L (A 2 + A 2 + . . . ) d ~ -
1 j ~ .AT (u),A(u) d~2
(3.139)
for any set of differential equations. In the above equation the boundary conditions are assumed to be satisfied by u (forced boundary condition) and the parameter ct is dropped as it becomes a multiplier. Clearly, the above statement is a requirement that the sum of the squares of the residuals of the differential equations should be a minimum at the correct solution. This minimum is obviously zero at that point, and the process is simply the well-known least squares method of approximation. It is equally obvious that we could obtain the correct solution by minimizing any functional of the form = ~l f ~ (pl A2 + p2A 2 2 +...)dr2 -
l f~ .AT (u) p,A(u) dr2
(3.140)
in which Pl, P2 . . . . . etc., are positive valued weighting functions or constants and p is a diagonal matrix: 0 P --
P2
P3
(3.141) ".o
The above alternative form is sometimes convenient as it puts different importance on the satisfaction of individual components of the equation set and allows additional freedom in the choice of the approximate solution. Once again this weighting function could be chosen so as to ensure a constant ratio of terms contributed by various equations. A least squares method of the kind shown above is a very powerful alternative procedure for obtaining integral forms from which an approximate solution can be started, and has been used with considerable success. 35' 36 As a least squares variational principle can be written for any set of differential equations without introducing additional variables, we may well enquire what is the difference between these and the natural variational principles discussed previously. On performing a variation in a specific case the reader will find that the Euler equations which are obtained no longer give the original differential equations but give higher order derivatives of these. This introduces the possibility of spurious solutions if incorrect boundary conditions are used. Further, higher order continuity of trial functions is now generally needed. This may be a serious drawback but frequently can be by-passed by stating the original problem as a set of lower order equations.
Least squares approximations 93
We shall now consider the general form of discretized equations resulting from the least squares approximation for linear equation sets (again neglecting boundary conditions which are assumed forced). Thus, if we take ,,Zt(u) = s
+ b
(3.142)
and take the usual trial function approximation fi = Nfi
(3.143)
we can write, substituting into (3.140), lI = ~ -
1
/2
[(Z~N)fi+ b]Tp[(Z~N)fi + b] dr2
(3.144)
and obtain
= lf~
~I-I = ~
lf~ [(s163
6fiT(/2N)Tp[(/2N)fi+b]dg2+ ~
= 0 (3.145)
or, as p is symmetric, ~I~I = 6,T { [ ~ (Z~N)Tp(Z~N)dr21, + ~ (Z~N)Tpb dr2 } = 0
(3.146)
This immediately yields the approximation equation in the usual form: Kfi + f = 0
(3.147)
and the reader can observe that the matrix K is symmetric and positive definite. Example 3.14: Least squares solution for Helmholz equation. To illustrate an actual example, consider the Helmholz problem governed by Eq. (3.88) for which we have already obtained a natural variational principle [Eq. (3.91)] in which only first derivatives were involved requiting Co continuity for u. Now, if we use the operator/2 and term b defined by Eq. (3.89), we have a set of approximating equations with
gab -- f (V2Na "q-cNa)(V2Nb -k- cNb) dx dy J/.S2
(3.148)
fa = J(V2Na + cNa)Q dx dy
The reader will observe that due to the presence of second derivatives C1 continuity is now needed for the trial functions N. Example 3.15: Least squares solution for Helmholz equation in first-order form. An alternative, avoiding the requirement of C1 functions, is to write Eq. (3.88) as a first-order system. This can be written as
a(u)
=
Oqx Oqy --~-x + --~-y + cqb + Q aO ~ qx Ox a4~ Oy qY
=0
(3.149)
94
Generalization of the finite element concepts
or, introducing the vector u, (3.150)
u = [~, qx, qy]T _._ Nfl
as the unknown we can write an approximation as
0 0]
U ~, fi --
Nq 0
0 Nq
Clx
- N~
(]y
(3.151)
where N o and Nq are Co shape functions for the ~b and qx, qy variables, respectively. The least squares approximation is now given by ~l~i = 6fit f~ (Z~N)T [(Z~N)fi + b] dr2 = 0 where
0Nq cN4" s
0Nq-
Ox '
Oy
-Nq,
0
1. 8y '
(3.152a)
. {!}
(3.152b)
-Nq
O,
The reader can now perform the final steps to obtain the K and f matrices. The approximation equations in a form requiring only Co continuity are obtained, however, at the expense of additional variables. Use of such forms has been made extensively in the finite element context. 35-41
3.13.1 Galerkin least squares, stabilization It is interesting to note that the concept of penalty formulation introduced in the previous section was anticipated as early as 1943 by Courant 42 in a somewhat different manner. He used the original variational principle augmented by the differential equations of the problem employed as least squares constraints. In this manner he claimed, though never proved, that the convergence rate could be accelerated. The suggestion put forward by Courant has been used effectively by others though in a somewhat different manner. Noting that the Galerkin process is, for self-adjoint equations, equivalent to that of minimizing a functional, the least squares formulation using the original equation is simply added to the Galerkin form. Here it allows non-self-adjoint operators to be used, for instance, and this feature has been exploited with success. Consider, for instance, an equation of the form
d2O
dO
x----d 2 + ot ~
+ Q = 0
The first order term multiplying ot is a convective term and, due to its presence, no natural variational equation is available as the differential equation is non-self-adjoint. However, Galerkin methods have been successfully used in its solution providing the convection term (c~d~b/dx) remains relatively small compared to the second derivative term (the diffusion
Concluding remarks- finite difference and boundary methods 95 term). However, it is found that as the convection term increases the solution becomes highly oscillatory. Here we only consider the problem in a preliminary manner and refer the reader to references on fluid dynamics for further study (e.g., see reference 10). Suppose in a Galerkin form given by dx dx
v ot
dx
+ Q
dx = 0
(3 153)
we add a multiple of the minimization of the least squares of the total equation. The result is dx dx +
v ot
//
+Q
k,dx 2 + o t ~ -x
r
dx
)
~xZ+Ot~--X-X+ Q d x = O
(3.154)
and we see immediately that an additional diffusive term has been added which depends on the parameter r, though at the expense of having higher derivatives appearing in the integrals. If only linear elements are used and the discontinuities ignored at element interfaces, the process of adding the diffusive terms can stabilize the oscillations which would otherwise occur. The idea appears to have first been used by Hughes 43--45 and later studied by Codina. 46 This process in the view of the authors is somewhat unorthodox as discontinuity of derivatives is ignored, and alternatives to this are discussed at length in reference 10. It is interesting to note also that another application of the same Galerkin least squares process can be made to the mixed formulation with two variables u and p for incompressible problems. We shall discuss such problems in Chapter 11 of this volume and show how this process can be made applicable there. Finally, it is of interest to note that the simple procedure introduced by Courant can also be effective in the prevention of locking of other problems. The treatment for beams has been studied by Freund and Salonen 47 and it appears that quite an effective process can be reached.
This very extensive chapter presents the general possibilities of using the finite element process in almost any mathematical or mathematically modelled physical problem. The essential approximation processes have been given in as simple a form as possible, at the same time presenting a fully comprehensive picture which should allow the reader to understand much of the literature and indeed to experiment with new permutations. In the chapters that follow we shall apply to various physical problems a limited selection of the methods to which allusion has been made. In some we shall show, however, that certain extensions of the process are possible (Chapters 11 and 15) and in another (Chapter 9) how a violation of some of the rules here expounded can be accepted. The numerous approximation procedures discussed fall into several categories. To remind the reader of these, we present in Table 3.2 a comprehensive catalogue of the methods used here and in Chapter 2. The only aspect of the finite element process mentioned in
96
Generalizationof the finite element concepts
Table 3.2 Finiteelementapproximation Integral forms of continuum problems trial functions u
Direct physical model
= Y ~ NaSa a
1 Variational principles
Weighted integrals of partial differential equation governing (weak formulations)
Global physical statements (e.g. virtual work)
Meaningful physical principles Miscellaneous weight functions
Constrained lagrangian forms
--"4-1
Collocation (point or subdomain)
Penalty function forms
1 I (wb= Nb) I I Galerkin
Least square forms .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
i
this table that has not been discussed here is that of a direct physical method. In such models an 'atomic' rather than continuum concept is the starting point. While much interest exists in the possibilities offered by such models, their discussion is outside the scope of this book. In all the continuum processes discussed the first step is always the choice of suitable shape or trial functions. A few simple forms of such functions have been introduced as the need demanded and many new forms will be introduced in the next two chapters. Indeed, the reader who has mastered the essence of the present chapter will have little difficulty in applying the finite element method to any suitably defined physical problem. For further reading references 48-52 could be consulted. The methods listed do not include specifically two well-known techniques, i.e., finite difference methods and boundary solution methods (sometimes known as boundary elements). In the general sense these belong under the category of the generalized finite element method discussed here. 48 1. Boundary solution methods choose the trial functions such that the governing equation is automatically satisfied in the domain f2. Thus starting from the general approximation equation (3.22), we note that only boundary terms remain to be satisfied. We shall return to such approximations in Chapter 12. 2. Finite difference procedures can be interpreted as an approximation based on local, discontinuous, shape functions with collocation weighting applied (although usually the derivation of the approximation algorithm is based on a Taylor expansion). As Galerkin or variational approaches give, in the energy sense, the best approximation, this method has only the merit of computational simplicity and occasionally a loss of accuracy. To illustrate this process we recall the approximation carded out for the one-dimensional equation (3.24a) (viz. p. 62). We now represent a localized approximation through equally spaced nodal points by
Problems
I,..,....
i-2
i-1
9
i+ 1
i+ 2
~,,~X
Fig. 3.11 A local, discontinuous shape function by parabolic segments used to obtain a finite difference approximation 9
X--Xa)h X
' ( l - (X-Xa)2)h 2
' "~1( (X-xa)2h2
~a ~a+l
+ X--Xa)]]~..
(3.155)
where h = Xa+l -- Xa (shown in Fig. 3.11). It is now clear that adjacent parabolic approximations in this case are discontinuous between the nodes. Values of the function and its first two derivatives at a typical node i are given by
~)(Xa) = ~a 0~b] --~X X=Xa
O2~l
1 ~ =
~-~(q~a+l -- ~a-1)
(3.156)
1 x=x, = h2 (~a+~ -- 2~a 4- ~a-~)
If we insert these into the governing equation at node i, we note immediately that the approximating equation at the node becomes 1
h2 (~a-1 - 2~a + ~a+l) +
Qa - 0
(3.157)
This is identical to the result based on Taylor expansion given by Eq. (3.31). This is indeed one of the cases in which the finite difference approximation is identical to the finite element one rather than different. In Chapter 15 we shall be discussing such finite difference and point approximations in more detail. However, the reader will note the present exercise is simply given to underline the similarity of finite element and finite difference processes. Many textbooks deal exclusively with these types of approximations. References 8, 53-55 discuss finite difference approximation and references 56-59 relate to boundary methods.
3.1 Write weak forms for the following differential equations and boundary conditions. For each form state appropriate continuity conditions for approximations to the dependent
97
98
Generalization of the finite element concepts
variable u and the weighting function v. The domain for each one-dimensional differential equation is 0 < x < 1. du (a) a -~x + cu + q - O; u ( 0 ) = d(du) (b) ~--Xx a ~ xx (c) - ~
d(du) a~
d(d:u)
(d) dxx a ~ x 2
+q=0;
du
du u(0)=~, &a-~x+ku=~atx=l
+bdxx + q = 0 ; +f-0;
u(0)-g0;
u(0)-g0;
(e) - 1 7 T ( k v u ) + c b T ( 1 7 u ) + q = 0
duI
u(1)-g,2
dxx x=0
--h0 & u(1)--gl
inf2; u = ~ , o n F
The differential equations for bending of a beam are given by dV (1) ~ x + q = 0
dM (2) ~ + V = 0
dO M dw V =0 (4) -0 =0 dx EI dx GA in which V is shear force, M is moment, 0 is section rotation, w is displacement, E I is bending stiffness, GA is shear stiffness and q is load as shown in Fig. 3.12. Boundary conditions are given by (3)
(1) V = 1 2
or
w--C0
(2) M = &
or
0--0
3.2 Construct a weak form for the beam equations by multiplying (1) by 4-6 w, (2) by 4-30, (3) by 3M and (4) by 3V. Choose the correct sign for 3w and 30 to give symmetry. 3.3 Add all boundary conditions to the weak form obtained in Problem 3.2. 3.4 Construct a variational theorem which gives the weak form obtained in Problems 3.2 and 3.3 as the first variation. 3.5 For G A - c~ (no shear deformation) deduce the irreducible differential equation in terms of w. Express all boundary conditions in terms of w.
I Z
qz
ttttt V+AV
M+AM
z~
Fig. 3.12 Beam bending description.
X
Problems 99
3.6 Construct a weak form for Problem 3.5. What is the required continuity of the dependent variable needed for approximation by a finite element method? What are the natural and essential boundary conditions for the weak form? 3.7 Construct a variational theorem which has Problem 3.6 as its first variation. 3.8 For G A = c~ (no shear deformation) deduce the differential equations in terms of w and M. Express all boundary conditions in terms of these variables. 3.9 Deduce a weak form for Problem 3.8 that permits approximation using Co functions to approximate w and M. Let 2
W =
ZNaffOa and
2
ZNa]l/'Ia
M=
a=l
a--1
where Na are given by (3.28). Ensure your weak form gives a symmetric coefficient matrix for these approximations. Compute typical element matrices K and f for an element of length h with constant E l and q in the element. 3.10 For a simply supported beam of length 10 and constant cross-section E I = 3 compute the solution for a uniform load of q = 1. The boundary conditions at each end of the beam for a simple support are w = M = 0. Obtain a solution using 2, 4, and 8 elements. It is recommended that a small computer program be written using a high level language, e.g. MATLAB, 6~ to perform the numerical calculations. Compare your results to an exact solution. 3.11 Solve the one-dimensional heat equation given in Example 3.5 by enforcing the boundary conditions by the penalty formulation described in Sec. 3.12.1. How large must each penalty parameter be taken to make the boundary error less than lO-61qbmaxl? 3.12 Deduce the Euler differential equation and boundary conditions for the variational principle expressed as 1-I(u) --
E I (-~x ) - P u
dx - ug
x--b
;
u(a) = O
Classify rI as a minimum, maximum or saddle point form. 3.13 Deduce the Euler differential equation and boundary conditions for the variational principle expressed as 1-I(u) --
E A (-d--~x) 4- ku 2 - 2 q u
dx + ot [(u(a)) 2 + (u(b)) 2]
where EA and k are constant parameters and c~ is a penalty parameter. 3.14 Deduce the Euler equations and boundary conditions for the variational principle expressed as rI(u, )~a, ~.b) =
E A (~-X-x)2+ ku 2 - 2 q u
dx + 1.aU(a) + ~+u(b)
where EA, k and q are constant parameters and )~a, ~.b are Lagrange multipliers. 3.15 The transient heat equation in one dimension is given by 8x
-~x
+Q+co-7
100 Generalizationof the finite element concepts where q~ is temperature, k thermal conductivity, Q heat generation per unit length and c specific heat. Boundary conditions may be given as 4~-~
on F1 or q = -
k ~O~ =q0x
on F2
where q is the heat flux and ~, ~ are specified values. Initial conditions are given as
~(x, 0) = ~0(x). (a) Construct a weak form for the problem. (b) Using the shape functions given in Eq. (3.28) and the approximation u e = Nl(X)fil(t) + N2(x)ft2(t) 3u e = Nl(X)rfil + N2(x)rfi2
construct the semi-discrete form for a typical element of length h. (c) Consider a region of length 10, with properties k = 5, c = 1, Q = 0. Divide the region into four equal length elements and establish the set of global semi-discrete equations. (d) Consider a set of discrete times tn. Approximate time derivatives of nodal values by ddp/dt(tn) ~ ( ~ ) n - qbn-1)/At where t~n is the approximation to dp(tn) and A t = tn - t~-I and write the fully discrete equations. Write a computer program (e.g., using MATLAB) to solve the problem. Assume the initial temperature of the region is zero and boundary conditions q~(0) = 0 and 4~(10) = 1 are applied at time zero and held constant. Solve the problem using 10 steps with At = 0.01, followed by 9 steps with At = 0.1 and finally 9 steps with At = 1. Plot the finite element solution for q~ vs x at times 0.01, 0.1, 1.0 and 10.0. Replace the element matrix associated with c by a diagonal (lumped) form with c h / 2 on each diagonal (h = x~ - x~). Repeat the above solution and compare results with the consistent form for the matrix.
1. S. Crandall. Engineering Analysis. McGraw-Hill, New York, 1956. 2. B.A. Finlayson. The Method of Weighted Residuals and Variational Principles. Academic Press, New York, 1972. 3. R.A. Frazer, W.P. Jones, and S.W. Sken. Approximations to functions and to the solution of differential equations. Technical Report 1799, Aero. Research Committee Report, 1937. 4. C'B. Biezeno and R. Grammel. Technishe Dynamik. Springer-Verlag, Berlin, 1933, p. 142. 5. O.C. Zienkiewicz and E. Ofiate. Finite elements versus finite volumes. Is there a choice? In P. Wriggers and W. Wagner, editors, Nonlinear Computational Mechanics. State of the Art. Springer, Berlin, 1991. 6. B.G. Galerkin. Series solution of some problems in elastic equilibrium of rods and plates. Vestn. Inzh. Tech., 19:897-908, 1915. 7. P. Tong. Exact solution of certain problems by the finite element method. J. AIAA, 7:179-180, 1969. 8. R.V. Southwell. Relaxation Methods in Theoretical Physics. Clarendon Press, Oxford, 1st edition, 1946.
References 9. R.S. Varga. Matrix lterative Analysis. Prentice-Hall, Englewood Cliffs, N.J., 1962. 10. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 11. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 12. I.S. Sokolnikoff. The Mathematical Theory of Elasticity. McGraw-Hill, New York, 2nd edition, 1956. 13. L.V. Kantorovich and V.I. Krylov. Approximate Methods of HigherAnalysis. John Wiley & Sons (International), New York, 1964. English translation by Curtis D. Benster. 14. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 15. EB. Hildebrand. Methods of Applied Mathematics. Prentice-Hall (reprinted by Dover Publishers, 1992), 2nd edition, 1965. 16. Lord Rayleigh (J.W. Strutt). On the theory of resonance. Trans. Roy. Soc. (London),A161:77118, 1870. 17. W. Ritz. Uber eine neue Methode zur L6sung gewisser variationsproblem der mathematischen physik. J. Reine angew. Math., 135:1-61, 1908. 18. E. Tonti. Variational formulation of non-linear differential equations. Bull. Acad. Roy. Belg. Classe Sci., 55:137-165 & 263-278, 1969. 19. M.M. Vainberg. Variational Methods for the Study of Nonlinear Operators. Holden-Day Inc., San Francisco, CA, 1964. 20. J.T. Oden. A general theory of finite elements. Part I. Topological considerations. Int. J. Numer. Meth. Eng., 1:205-246, 1969. 21. J.T. Oden. A general theory of finite elements. Part II. Applications. Int. J. Numer. Meth. Eng., 1:247-254, 1969. 22. S.C. Mikhlin. Variational Methods in Mathematical Physics. Macmillan, New York, 1964. 23. S.C. Mikhlin. The Problem of the Minimum of a Quadratic Functional. Holden-Day, San Francisco, 1966. 24. B.A. Szabo and T. Kassos. Linear equation constraints in finite element approximations. Int. J. Numer. Meth. Eng., 9:563-580, 1975. 25. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd edition, 1982. 26. E Kikuchi and Y. Ando. A new variational functional for the finite element method and its application to plate and shell problems. Nucl. Eng. Des., 21(1):95-113, 1972. 27. H.S. Chen and C.C. Mei. Oscillations and water forces in an offshore harbour. Technical Report 190, Ralph M. Parsons Laboratory for Water Resources and Hydrodynamics, Massachusetts Institute of Technology, Cambridge, MA, 1974. 28. O.C. Zienkiewicz, D.W. Kelley, and P. Bettess. The coupling of the finite element and boundary solution procedures. Int. J. Numer. Meth. Eng., 11:355-375, 1977. 29. O.C. Zienkiewicz. Constrained variational principles and penalty function methods in finite element analysis. In Lecture Notes in Mathematics, No. 363, pages 207-214, Springer-Verlag, Berlin, 1974. 30. J. Campbell. A finite element system for analysis and design. Ph.D. thesis, Department of Civil Engineering, University of Wales, Swansea, 1974. 31. D.J. Naylor. Stresses in nearly incompressible materials for finite elements with application to the calculation of excess pore pressures. Int. J. Numer. Meth. Eng., 8:443-460, 1974. 32. I. Fried. Shear in c o and c 1 bending finite elements. Int. J. Solids Struct., 9:449-460, 1973. 33. I. Fried. Finite element analysis of incompressible materials by residual energy balancing. Int. J. Solids Struct., 10:993-1002, 1974. 34. O.C. Zienkiewicz and E. Hinton. Reduced integration, function smoothing and non-conformity in finite element analysis. J. Franklin Inst., 302:443-461, 1976.
101
102 Generalizationof the finite element concepts 35. EE Lynn and S.K. Arya. Finite elements formulation by the weighted discrete least squares method. Int. J. Numer. Meth. Eng., 8:71-90, 1974. 36. O.C. Zienkiewicz, D.R.J. Owen, and K.N. Lee. Least square finite element for elasto-static problems - use of reduced integration. Int. J. Numer. Meth. Eng., 8:341-358, 1974. 37. B.-N. Jiang. Optimal least-squares finite element method for elliptic problems. Comp. Meth. Appl. Mech. Eng., 102:199-212, 1993. 38. B.-N. Jiang. On the least-squares method. Comp. Meth. Appl. Mech. and Eng., 152:239-257, 1998. 39. B.-N. Jiang. Least Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Springer, New York, 1998. 40. B.-N. Jiang. The least-squares finite element method in elasticity. I. Plane stress or strain with drilling degrees of freedom. Int. J. Numer. Meth. Eng., 53:621-636, 2002. 41. B.-N. Jiang. The least-squares finite element method in elasticity. II. Bending of thin plates. Int. J. Numer. Meth. Eng., 54:1459-1475, 2002. 42. R. Courant. Variational methods for the solution of problems of equilibrium and vibration. Bull. Am. Math Soc., 49:1-61, 1943. 43. T.J.R. Hughes, L.P. Franca, and M. Balestra. A new finite element formulation for computational fluid dynamics: V. Circumventing the Babu~ka-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comp. Meth. Appl. Mech. Eng., 59:85-99, 1986. 44. T.J.R. Hughes and L.P. Franca. A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: symmetric formulation that converge for all velocity/pressure spaces. Comp. Meth. Appl. Mech. Eng., 65:85-96, 1987. 45. T.J.R. Hughes, L.P. Franca, and G.M. Hulbert. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comp. Meth. Appl. Mech. Eng., 73:173-189, 1989. 46. R. Codina, M. V~izquez, and O.C. Zienkiewicz. General algorithm for compressible and incompressible flows, Part I I I - a semi-implicit form. Int. J. Numer. Meth. Fluids, 27:13-32, 1998. 47. Jouni Freund and Eero-Matti Salonen. Sensitizing according to Courant the Timoshenko beam finite element solution. Int. J. Numer. Meth. Eng., x:129-160, 1999. 48. O.C. Zienkiewicz and K. Morgan. Finite Elements and Approximation. John Wiley & Sons, London, 1983. 49. E.B. Becker, G.E Carey, and J.T. Oden. Finite Elements: An Introduction, volume 1. PrenticeHall, Englewood Cliffs, N.J., 1981. 50. B. Szabo and I. Babu~ka. Finite Element Analysis. John Wiley & Sons, New York, 1991. 51. T.J.R. Hughes. The Finite Element Method: Linear Static and Dynamic Analysis. Dover Publications, New York, 2000. 52. C.A.T. Fletcher. Computational Galerkin Methods. Springer-Verlag, Berlin, 1984. 53. D.N. de G. Allen. Relaxation Methods. McGraw-Hill, London, 1955. 54. EB. Hildebrand. Introduction to Numerical Analysis. Dover Publishers, 2nd edition, 1987. 55. A.R. Mitchell and D. Griffiths. The Finite Difference Method in Partial Differential Equations. John Wiley & Sons, London, 1980. 56. P.K. Banerjee. The Boundary Element Methods in Engineering. McGraw-Hill, London, 1994. 57. Prem K. Kythe. An Introduction to Boundary Element Methods. CRC Press, 1995. 58. G. Beer and J.O. Watson. Programming the Boundary Element Method: An Introduction for Engineers. John Wiley & Sons, Chichester, 2001. 59. L. Gaul. Boundary Element Methods for Engineers and Scientists. Springer, Berlin, 2003. 60. MATLAB. www.mathworks.com, 2003.
In Chapters 2 and 3 the reader was shown in some detail how linear elasticity and other problems could be formulated and solved using very simple element forms. Although the detailed algebra was only concerned with shape functions which arose from triangular or rectangular shapes, it should by now be obvious that other element forms could equally well be used. Indeed, once the element and the corresponding shape functions are determined, subsequent operations follow a standard, well-defined path. It will be seen later that it is possible to program a computer to deal with wide classes of problems by specifying the shape functions only. The choice of these is, however, a matter to which intelligence has to be applied and in which the human factor remains paramount. In this chapter some rules for the generation of several families of one-, two-, and three-dimensional elements will be presented. In the problems of elasticity illustrated in Chapters 2 and 3 the displacement variable was a vector with two or three components and the shape functions were written in matrix form. They were, however, derived for each component separately and the matrix expressions in these were derived by multiplying a scalar function by an identity matrix [e.g., Eq. (2.2)]. In this chapter we shall concentrate on the scalar shape function forms, calling these simply Na. The shape functions used in the displacement formulation of elasticity problems were such that they satisfy the convergence criteria of Chapter 2: 1. The continuity of the unknown only had to occur between elements (i.e., slope continuity is not required), or, in mathematical notation, Co continuity was needed; 2. The function has to allow any arbitrary linear form to be taken so that the constant strain (constant first derivative) criterion could be observed in each element. The shape functions described in this chapter will require the satisfaction of these two criteria. They will thus be applicable to all the problems requiring Co continuity (i.e., all problems governed by first or second order differential equations). Indeed they are applicable to any situation where the functional H or 81-I (see Chapter 3) is defined by derivatives of first order only.
104 'Standard' and 'hierarchical' element shape functions The element families discussed will progressively have an increasing number of degrees of freedom. The question may well be asked as to whether any economic or other advantage is gained by increasing the complexity of an element. The answer here is not an easy one although it can be stated as a general rule that as the order of an element increases so the total number of unknowns in a problem can be reduced for a given accuracy of representation. Economic advantage requires, however, a reduction of total computation and data preparation effort, and this does not follow automatically for a reduced number of total variables. However, an overwhelming economic advantage in the case of three-dimensional analyses occurs. The same kind of advantage arises on occasion in other problems but in general the optimum element may have to be determined from case to case. In Sec. 2.6 of Chapter 2 we have shown that the order of error in the approximation to the unknown function is O(hP+l), where h is the element 'size' and p is the degree of the complete polynomial present in the expansion. Clearly, as the element shape functions increase in degree so will the order of error increase, and convergence to the exact solution becomes more rapid. While this says nothing about the magnitude of error at a particular subdivision, it is clear that we should seek element shape functions with the highest complete polynomial for a given number of degrees of freedom.
The essence of the finite element method already stated in Chapters 2 and 3 is in approximating the unknown (displacement) by an expansion given in Eqs (2.1) and (3.3). This, for a scalar variable u, can be written as
u ~ ~t - ~
Naua -- Nfi e
(4.1)
a=l
where n is the total number of functions used and fia are the unknown parameters to be determined. We have explicitly chosen to identify such variables with the values of the unknown function at element nodes, thus making ~la = 1-l(Xa)
(4.2)
The shape functions so defined will be referred to as 'standard' ones and are the basis of most finite element programs. If polynomial expansions are used and the element satisfies Criterion 1 of Chapter 2 (which specifies that rigid body displacements cause no strain), it is clear that a constant value of fia specified at all nodes must result in a constant value of fi" o
when
fia
--" /'/0-
It follows that
(4.3)
n
y~'Na = 1 a--1
(4.4)
Standard and hierarchical concepts 105 at all points of the domain. This important property is known as a partition of unity I which we will make extensive use of here and in Chapter 15. The first part of this chapter will deal with such standard shape functions. A serious drawback exists, however, with 'standard' functions, since when element refinement is made totally new shape functions have to be generated and hence all calculations repeated. It would be of advantage to avoid this difficulty by considering the expression (4.1) as a series in which the shape function Na does not depend on the number of nodes in the mesh n. This indeed is achieved with hierarchic shape functions to which the second part of this chapter is devoted. The hierarchic concept is well illustrated by the one-dimensional (elastic bar) problem of Fig. 4.1. Here for simplicity elastic properties are taken as constant (D = E) and the body force b is assumed to vary in such a manner as to produce the exact solution shown on the figure (with zero displacements at both ends). Two meshes are shown and a linear interpolation between nodal points assumed. For both standard and hierarchic forms the coarse mesh gives C
"~C
K11u I
--
(4.5)
fl
For a fine mesh two additional nodes are added and with the standard shape function the equations requiring solution are
K~
Kg
K~I
0
1{0} U2
KFJ
"-
fi3
f2
(4.6)
f3
In this form the zero matrices have been automatically inserted due to element interconnection which is here obvious, and we note that as no coefficients are the same, the new equations have to be resolved [Eq. (2.28a) shows how these coefficients are calculated and the reader is encouraged to work these out in detail]. With the 'hierarchic' form using the shape functions shown, a similar form of equation arises and an identical approximation is achieved (being simply given by a series of straight segments). The final solution is identical but the meaning of the parameters fi] is now different, as shown in Fig. 4.1. Quite generally, K F -- K~I (4.7) as an identical shape function is used for the first variable. Further, in this particular case the off-diagonal coefficients are zero and the final equations become, for the fine mesh,
[
K~I 0 0
0 Kf2 0
~ K
~;
=
f2 f3
(4.8)
The 'diagonality' feature is only true in the one-dimensional problem, but in general it will be found that the matrices obtained using hierarchic shape functions are more nearly diagonal and hence usually imply better conditioning than those with standard shape functions. Although the variables are now not subject to the obvious interpretation (as local displacement values), they can be easily transformed to those if desired. Though it is not usual
106
'Standard' and 'hierarchical' element shape functions Coarse
Fine Exact proximate
1
J
J
2
1
3
N2
N1
N3
2
1
3
N2
N1
N3
jJ
(a)
1
...""
9 ss
s
..
"',
~
ii ~
s
,2-"
,"
s ~.;r s x
~
",,"
~
"-,," s s
9
, "-~
(b) Fig. 4.1 A one-dimensional problem of stretching of a uniform elastic bar by prescribed body forces.
to use hierarchic forms in linearly interpolated elements their derivation in polynomial form is simple and very advantageous. The reader should note that with hierarchic forms it is convenient to consider the finer mesh as still using the same, coarse, elements but now adding additional refining functions. Hierarchic forms provide a link with other approximate (orthogonal) series solutions. Many problems solved in classical literature by trigonometric, Fourier series, expansion are indeed particular examples of this approach. In the next sections of this chapter we shall consider the development of shape functions for high order elements with many boundary and internal degree of freedoms. Such development will generally be made on simple geometric forms and the reader may well question the wisdom of using increased accuracy for such simple shaped domains - having already observed the advantage of generalized finite element methods in fitting arbitrary domain shapes. This concern is well founded, but in the next chapter we shall show a general method to map high order elements into quite complex shapes.
Rectangular elements-some preliminary considerations 107
Part 1. 'Standard' shape functions Two-dimensional elements
Conceptually (especially if the reader is conditioned by education to thinking in the cartesian coordinate system) the simplest element form of a two-dimensional kind is that of a rectangle with sides parallel to the x and y axes. Consider, for instance, the rectangle shown in Fig. 4.2 with nodal points numbered 1 to 8, located as shown, and at which the values of an unknown function u (here representing, for instance, one of the components of displacement) form the element parameters. How can suitable Co continuous shape functions for this element be determined? Let us first assume that u is expressed in polynomial form in x and y. To ensure interelement continuity of u along the top and bottom sides the variation must be linear. Two points at which the function is common between elements lying above or below exist, and as two values uniquely determine a linear function, its identity all along these sides is ensured with that given by adjacent elements. Use of this fact was already made in specifying linear expansions on edges for a triangle and a rectangle. Similarly, if a cubic variation along the vertical sides is assumed, continuity will be preserved there as four values determine a unique cubic polynomial. Conditions for satisfying the first criterion are now obtained. To ensure the existence of constant values of the first derivative it is necessary that all the linear polynomial terms of the expansion be retained. Finally, as eight points are to determine uniquely the variation of the function, only eight coefficients of the expansion can be retained and thus we could write
,~Y
Fig. 4.2 A rectangular element.
1
8
2
7
3
6
4
5
108 'Standard' and 'hierarchical' element shape functions U
=
Ol 1 n t- O l 2 X
-Jr"ot3y nt- Ot4Xy nt- ot5y 2 nt- Ot6Xy 2 nt- o~7y3 -k- Ot8xy 3
(4.9)
The choice can in general be made unique by retaining the lowest possible expansion terms, though in this case apparently no such choice arises.t The reader will easily verify that all the requirements have now been satisfied. Substituting coordinates of the various nodes a set of simultaneous equations will be obtained. This can be written in exactly the same manner as was done for a triangle in Eq. (2.4) as
/01/Ii Xl yl Xlyl X fi8
,
x8,
Y8,
..
.
.
.
Xlyl ]/ l/
.
x8y3J
(4.10)
or8
or simply as ~e = Cog.
(4.11)
og : c - l ~ e
(4.12)
u = P(x, y)og = P(x, y ) C - l u e
(4.13)
P(x, y) = [1, x, y, x y , y2, x y 2 , y3, xy3]
(4.14)
Formally, and we could write Eq. (4.9) as
in which Thus the shape functions for the element defined by u = Nfi e = [N1, N2 . . . . . N8] a e
(4.15)
N(x, y) - P(x, y)C -1
(4.16)
can be found as This process has, however, some considerable disadvantages. Occasionally an inverse of C may not exist 2' 3 and always considerable algebraic difficulty is experienced in obtaining an expression for the inverse in general terms suitable for all element geometries. It is therefore worthwhile to consider whether shape functions Na (x, y) can be written down directly. Before doing this some general properties of these functions have to be mentioned. Inspection of the defining relation, Eq. (4.15), reveals immediately some important characteristics. First, as this expression is valid for all components of fie, Na (Xb, Yb )
1;
-- ~ab "-
0;
a=b a~b
where ~ a b is known as the Kronecker delta. Further, the basic type of variation along boundaries defined for continuity purposes (e.g., linear in x and cubic in y in the above example) must be retained. The typical form of the shape functions for the elements considered is illustrated isometrically for two typical nodes in Fig. 4.3. It is clear that these could have been written down directly as a product of a suitable linear function in x with a t Retention of a higher order term of expansion, ignoring one of lower order, will usually lead to a poorer approximation though still retaining convergence,2 providing the linear terms are always included.
Completeness of polynomials 109
N~
/
N~
st ,,,%s
1
Fig. 4.3 Shape functions for elements of Fig. 4.2. cubic function in y. The easy solution of this example is not always as obvious but given sufficient ingenuity, a direct derivation of shape functions is always preferable. It will be convenient to use normalized coordinates in our further investigation. Such normalized coordinates are shown in Fig. 4.4 and are chosen so that their values are + 1 on the faces of the rectangle:t ~-
x -
Xc
a
r/=
Y -
b
Yc
dx
d~----
a dy dr/= m b
(4.17)
Once the shape functions are known in the normalized coordinates, translation into actual coordinates or transformation of the various expressions occurring, for instance, in the stiffness derivation is trivial for rectangular shapes. Consideration of other more convenient 'mapping' methods will be addressed in Chapter 5.
The shape function derived in the previous section was of a rather special form [viz. Eq. (4.9)]. Only a linear variation with the coordinate x was permitted, while in y a full cubic was available. The complete polynomial contained in it was thus of order 1. In general use, a convergence order corresponding to a linear variation would occur despite an increase of the total number of variables. Only in situations where the linear variation t In Chapter 5 we will show that this is convenient for purposes of numerical integration.
110 'Standard' and 'hierarchical' element shape functions ~Y
,=1
/
q= 1 ~q c
Yc
q=-I xc
/
\ ~=1 ~.
r X
Fig. 4.4 Normal coordinates for a rectangle.
in x corresponded closely to the exact solution would a higher order of convergence occur, and for this reason elements with such 'preferential' directions should be restricted to special use, e.g., in narrow beams or strips. Usually, we will seek element expansions which possess the highest order of a complete polynomial for a minimum of degrees of freedom. In this context it is useful to recall the Pascal triangle (Fig. 4.5) from which the number of terms occurring in a polynomial in two variables x, y can be readily ascertained. For instance, first-order polynomials require three terms, second order require six terms, third order require ten terms, etc.
Consider the element shown in Fig. 4.6 in which a series of nodes, external and internal, is placed on a regular grid. It is required to determine a shape function for the point indicated
Fig. 4.5 The Pascal triangle. (Cubic expansion shaded - 10 terms.)
Rectangular elements- Lagrange family (o,rn) (I, d) ce ,~u ......
(n, rn) "P
)
c)
.~
.;
.;
c
)
()
")
,i)
..')
.."
f
f
/ ' . / l l i ~ ' , ~ l l ~ - " , / l i ~ l
(0, O)
(n, O)
1
1
Fig. 4.6
A typical shape function for a lagrangian element (n = 5, m = 4, / = 1, J = 4).
by the heavy circle. Clearly the product of a fifth-order polynomial in ~ which has a value of unity at points of the second column of nodes and zero at the other nodal columns and that of a fourth-order polynomial in r/having unity on the coordinate corresponding to the top row of nodes and zero at other nodal rows satisfies all the interelement continuity conditions and gives unity at the nodal point concerned. Polynomials in one coordinate having this property are known as Lagrange polynomials and can be written down directly as
l~(~)
=
(~: -- ~:0)(~: -- ~:1)""" (~ -- ~:k-1)(~:
-- ~:k+l)"""
(~: -- ~:n)
(~k --~0)(~k - - ~ l ) ' ' ' ( ~ k --~k-1)(~k - - ~ k + l ) ' ' " (~k -- ~n)
--II i=0
ir
(4.18t giving unity at ~k and passing through zero at the remaining n points. An easy and systematic method of generating shape functions of any order now can be achieved by simple products of Lagrange polynomials in the two coordinates. 4-6 Thus, in two dimensions, if we label the node by its column and row number, I, J, we have Na ~ N i j
-
l~ (~)l7 (rl)
(4.19)
where n and m stand for the number of subdivisions in each direction. Figure 4.7 shows a few members of this unlimited family where m - n. For m -- n = 1 we obtain the simple result
111
112 'Standard' and 'hierarchical' element shape functions
0 (b)
(a) 0
0
0
0
0
0
0
(c)
0
Fig. 4.7 Three elements of the Lagrange family: (a)linear, (b) quadratic, (c) cubic.
Na
--
1 ~(1 -F ~a~)(1 -F/']aT])
(4.20)
in which ~a, Tla are the normalized coordinates at node a. Indeed, if we examine the polynomial terms present in a situation where n = m we observe in Fig. 4.8, based on the Pascal triangle, that a large number of polynomial terms is present above those needed for a complete expansion. 7 However, when mapping of shape functions is considered (viz. Chapter 5) some advantages occur for this family.
It is often more efficient to make the functions dependent on nodal values placed on the element boundary. Consider, for instance, the first three elements of Fig. 4.9. In each a progressively increasing and equal number of nodes are placed on the element boundary.
Fig. 4.8 Terms generated by a lagrangian expansion of order 3 x 3 (or m x n). Complete polynomials of order 3 (or n).
Rectangular elements-'serendipity' family 113 q=l / "--I
\ q= -I
(a) 0
0
0
0
(b) '
(c)
0
0
0
0
0
0
(d)
Fig. 4.9 Rectangles of boundary node (serendipity) family: (a)linear, (b) quadratic, (c) cubic, (d) quartic. The variation of the function on the edges to ensure continuity is linear, parabolic, and cubic in increasing element order. To achieve the shape function for the first element it is obvious that a product of linear lagrangian polynomials of the form
+ ~:)(1 + r/) 1(1 4
(4.21)
gives unity at the top right comer where ~ = r/ = 1 and zero at all the other comers. Further, a linear variation of the shape function of all sides exists and hence continuity is satisfied. Indeed this element is identical to the lagrangian one with n -- 1 and again all the shape functions may be written as one expression: 1
Na = z(1 + ~:a~:)(1 -+-?Ta/'])
As a linear combination of these shape functions yields any arbitrary linear variation of u, the second convergence criterion is satisfied. The reader can verify that the following functions satisfy all the necessary criteria for quadratic and cubic members of the family.
"Quadratic" element
Comer nodes:
i (1 + ~a~)(1 + ~a~)(~a~ "3t- ~a T] -- 1) Na = -~
(4.22a)
Mid-side nodes:
~a=0
Na = ~1 ( 1 - ~2)(1 + Fla/'])
/']a = 0
1 + ~:a~)(1Na = ~(1
(4.22b)
/72)
"Cubic" element
Comer nodes:
Na = ~2(1 + ~:a~)(1 "l- r/ar/)[9(~ :2 + 0 2)
--
10]
(4.23a)
114
'Standard' and 'hierarchical' element shape functions
Mid-side nodes: ~a -- +1
and
9(1 +
Na =
1
0a = -}-3
~a~)(1 - 02)(1 -Jr-9000)
(4.23b)
and ~a = +~1
Na -'-
and
9(1
:t::1
0a =
- ~e2)(1 + 9~'a~')(1 -I- 0a0)
(4.23c)
which all satisfy the requirement
Na (~b, 0b)
~ab --"
:
1;a=b 0; a # b
(4.23d)
The above functions were originally derived by inspection, and progression to yet higher members is difficult and requires some ingenuity. 4, 5 It was therefore appropriate to name this family 'serendipity' after the famous princes of Serendip noted for their chance discoveries (Horace Walpole, 1754). However, a quite systematic way of generating the 'serendipity' shape functions can be devised, which becomes apparent from Fig. 4.10 where the generation of a quadratic shape function is presented. 7, 8 4
1
7
5
3
2
(a) N 5 = V 2 ( 1 - ~ 2 ) ( l - q )
0.5
Step 1 1
(b) N 8 = V2(1 - ~) (1 - 1]2)
0
.
0
~
1
= (1 - ~) (1 - q)/4
I '-' 0.5
Step 2
(c)
N~ = ~ - ~ N s - Y ~ N 8 Fig. 4.10 Systematic generation of 'serendipity' shape functions.
Rectangular elements-'serendipity' family 115 As a starting point we observe that for mid-side nodes a lagrangian interpolation of a quadratic x linear type suffices to determine Na at nodes 5 to 8. N5 and N8 are shown in Fig. 4.10(a) and (b). For a comer node, such as Fig. 4.10(c), we start with a bilinear lagrangian family ~/1 and note immediately that while b~l = 1 at node 1, it is not zero at nodes 5 or 8 (step 1). Successive subtraction of 1/2 N5 (step 2) and 1/2 N8 (step 3) ensures that a zero value is obtained at these nodes. The reader can verify that the final expressions obtained coincide with those of Eqs (4.22a) and (4.22b). Indeed, it should now be obvious that for all higher order elements the mid-side and comer shape functions can be generated by an identical process. For the former a simple multiplication of mth-order and first-order lagrangian interpolations suffices. For the latter a combination of bilinear comer functions, together with appropriate fractions of mid-side shape functions to ensure zero at appropriate nodes, is necessary. It also is quite easy to generate shape functions for elements with different numbers of nodes along each side by a similar systematic algorithm. This may be very desirable if a transition between elements of different order is to be achieved, enabling a different order of accuracy in separate sections of a large problem to be studied. Figure 4.11 illustrates the necessary shape functions for a cubic/linear transition. Use of such special elements was first introduced in reference 8, but the simpler formulation used here is that of reference 7. With the mode of generating shape functions for this class of elements available it is immediately obvious that fewer degrees of freedom are now necessary for a given complete
Fig. 4.11 Shape functions for a transition 'serendipity' element, cubic/linear.
116 'Standard' and 'hierarchical' element shape functions
Fig. 4.12 Terms generated by edge shape functions in serendipity-type elements (3 x 3 and rn x m).
polynomial expansion. Figure 4.12 shows this for a cubic element where only two surplus terms arise (as compared with six surplus terms in a lagrangian of the same degree). However, when mapping to general quadrilateral shape is introduced (Chapter 5) some of these advantages are lost rendering the lagrangian form of interpolation advantageous. It is immediately evident, however, that the functions generated by nodes placed only along the edges will not generate complete polynomials beyond cubic order. For higher order ones it is necessary to supplement the expansion by internal nodes or by the use of 'nodeless' variables which contain appropriate polynomial terms. For example, in the next, quartic, member 9 of this family a central node is added [viz. Fig. 4.9(d)] so that all terms of a complete fourth-order expansion will be available. This central node adds a shape function (1 - ~2)(1 - 02) which is zero on all outer boundaries and coincides with the internal function used in the quadratic lagrangian element. Once interior nodes are added it is necessary to modify the comer and mid-side shape functions to preserve the Kronnecker delta property (4.23d).
The advantage of an arbitrary triangular shape in approximating to any boundary configuration has been amply demonstrated in earlier chapters. Its apparent superiority here over rectangular shapes needs no further discussion. However, the question of generating more elaborate higher order elements needs to be further developed. Consider a series of triangles generated on a pattern indicated in Fig. 4.13. The number of nodes in each member of the family is now such that a complete polynomial expansion, of the order needed for interelement compatibility, is ensured. This follows by comparison with the Pascal triangle of Fig. 4.5 in which we see the number of nodes coincides exactly with the number of polynomial terms required. This particular feature puts the triangle family in a special, privileged position, in which the inverse of the C matrices of Eq. (4.11) will always exist. 3 However, once again a direct generation of shape functions will be preferred- and indeed will be shown to be particularly easy. Before proceeding further it is useful to define a special set of normalized coordinates for a triangle.
.Triangularelement
family
3 3 (a)
3
1
(b)
4
8
2
(c)
1
4
7
5
2
Fig. 4.13 Triangular element family: (a)linear, (b) quadratic, (c) cubic.
4.7.1 Area coordinates ........................
.....
,.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...................................
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.......................................
. ...................................
...............
While cartesian directions parallel to the sides of a rectangle were a natural choice for that shape, in the triangle these are not convenient. A new set of coordinates, L1, L2, and L3 for a triangle 1, 2, 3 (Fig. 4.14), is defined by the following linear relation between these and the cartesian system: X -- LlXl + L2x2 + L3x3 y = L l Y l Jr- L2Y2 Jr- L3Y3 1 =
(4.24)
L1 Jr- L2 --F L3
To every set, L 1, L2, L 3 (which are not independent, but are related by the third equation), there corresponds a unique set of cartesian coordinates. At point 1, L 1 = 1 and L2 -- L3 -0, etc. A linear relation between the new and cartesian coordinates implies that contours of L1 are equally placed straight lines parallel to side 2-3 on which L1 = 0, etc.
Fig. 4.14 Area coordinates.
117
118 'Standard' and 'hierarchical' element shape functions Indeed it is easy to see that an alternative definition of the coordinate L1 of a point P is by a ratio of the area of the shaded triangle to that of the total triangle: L1 --
area P23
(4.25)
area 123
Hence the name a r e a c o o r d i n a t e s . Solving Eq. (4.24) gives Za
aa + b a x -+- Cay
--
2A
; k = 1, 2, 3
(4.26)
in which A
--
x2 Y2 1det Ei Xl yl1 x3 Y3
~
area 123
(4.27)
and al = x2Y3 - x3Y2
bl = Y2 - Y3
Cl -- x3 - x2
(4.28)
etc., with cyclic rotation of indices 1, 2, and 3. The identity of expressions with those derived in Chapter 2 [Eqs (2.6) and (2.7)] is worth noting.
4.7.2 Shape functions For the first element of the series [Fig. 4.13(a)], the shape functions are simply the area coordinates. Thus N1 -- L1 N2 -- L2 N3 = L3 (4.29) This is obvious as each individually gives unity at one node, zero at others, and varies linearly everywhere. To derive shape functions for other elements a simple recurrence relation can be derived. 3 However, it is very simple to write functions for an arbitrary triangle of order M in a manner similar to that used for the lagrangian element of Sec. 4.5. Denoting a typical node a by three numbers I, J, and K corresponding to the position of coordinates L la, L2a, and L3a we can write the shape function in terms of three lagrangian interpolations as [see Eq. (4.18)] (4.30)
Na -- l l I ( L 1 ) I J ( L 2 ) I K ( L 3 )
In the above l ], etc., are given by expression (4.18), with L1 taking the place of ~, etc. It is easy to verify that the above expression gives Na
= 1
at L1 = L I I ,
L2 -- L 2 j ,
L3 :
L3K
and zero at all other nodes. The highest term occurring in the expansion is L~ L2J L~ and as I + J + K -- M for all points the polynomial is also of order M. Expression (4.30) is valid for quite arbitrary distributions of nodes of the pattern given in Fig. 4.15 and simplifies if the spacing of the nodal lines is equal (i.e., 1/ m). The formula was first obtained by Argyris et al. 1~ and formalized in a different manner by others. 7' 11 The reader can verify the shape functions for the second- and third-order elements as given below and indeed derive ones of any higher order easily.
Line elements
.•,
O, M)
0 01/
\
Fig. 4.15 A general triangular element.
Quadratic triangle [Fig. 4.13(b)1 Comer nodes:
Na :
a=
(2La - 1)La,
1,2,3
Mid-side nodes: N4 :
4L1L2,
N5 - - 4 L 2 L 3 ,
N6 :
4L3L1
Cubic triangle [Fig. 4.13(c)1 Comer nodes:
Na :
1
-~(3La - 1 ) ( 3 L a - 2 ) L a ,
a=1,2,3
Mid-side nodes:
N 4 - 9 L I L e ( 3 L 1 - 1),
9
N5 -- ~ L 1 L 2 ( 3 L 2 -
1),
etc.
and for the internal node: N10 - - 2 7 L 1 L 2 L 3
The last shape again is a 'bubble' function giving zero contribution along boundaries and this will be found to be useful in other contexts (see the mixed forms in Chapter 11). The quadratic triangle was first derived by Veubeke 12 and used later in the context of plane stress analysis by Argyris. 13 When element matrices have to be evaluated it will follow that we are faced with integration of quantities defined in terms of area coordinates over the triangular region. It is useful to note in this context the following exact integration expression:
dx d y = (a + a! b+ a b! cc!+ 2)!2A f f A L1LbL~
(4.31)
One-dimensional elements
So far in this book the continuum was considered generally in two or three dimensions. 'One-dimensional' members, being of a kind for which exact solutions are generally available, were treated only as trivial examples in Chapter 3 and in Sec. 4.2. In many practical two- or three-dimensional problems such elements do in fact appear in conjunction with
119
120 'Standard' and 'hierarchical' element shape functions
the more usual continuum elements- and a unified treatment is desirable. In the context of elastic analysis these elements may represent lines of reinforcement (plane and threedimensional problems) or sheets of thin lining material in axisymmetric bodies. In the context of heat conduction and other field problems similar effects occur. Once the shape of such a function as displacement is chosen for an element of this kind, its properties can be determined, noting, however, that derived quantities such as strain, etc., have to be considered only in one dimension. Figure 4.16 shows such an element sandwiched between two adjacent quadratic-type elements. Clearly for continuity of the function a quadratic variation of the unknown with the one variable ~ is all that is required. Thus the shape functions are given directly by the Lagrange polynomial as defined in Eq. (4.18).
Three-dimensional elements
In a precisely analogous way to that given in previous sections equivalent lagrangian family elements of three-dimensional type can be described. Shape functions for such elements will be generated by a direct product of three Lagrange polynomials. Extending the notation of Eq. (4.19) we now have Na -~ NI j K
- - 17
(~)17 (o)l P (()
(4.32)
for n, m, and p subdivisions along each side and x -- Xc
~ = ~ ; a
~--
Y -- Yc
b
and ( =
z - - Zc
c
This element again is suggested by Zienkiewicz e t a l . 5 and elaborated upon by Argyris All the remarks about internal nodes and the properties of the formulation with mappings (to be described in the next chapter) are applicable here. The first three members of the three-dimensional Lagrange family are shown in Fig. 4.17(a). et al. 6
0
0 0
0
Fig. 4.16 A line element sandwiched between two-dimensional elements.
Rectangular prisms - 'serendipity' family
Fig. 4.17 Linear, quadratic and cubic right prisms with corresponding sheet and line elements. (Extra shading on 64-node element to show node location more clearly.)
For interelement continuity the simple rules given previously have to be modified. What is necessary to achieve such continuity is that along a whole face of an element the nodal values define a unique variation of the unknown function. It is obvious on a face that one of the l) will be unity and the remaining product defines the two-dimensional form given by (4.19), thus ensuring continuity.
The serendipity family of elements shown in Fig. 4.17(b) is precisely equivalent to that of Fig. 4.9 for the two-dimensional case. 4, 8,14 Using now three coordinates and otherwise following the terminology of Sec. 4.6 we have the following shape functions:
"Linear" element (8 nodes) Na
--
1 ~(1 + ~a~)(1 + r/at/)(1 + (a()
which is identical with the linear lagrangian element.
"Quadratic" element (20 nodes)
Comer nodes:
1 Na = ~(1 + ~a~)(1 + qaO)(1 + (a()(~a~ + qaq + (a( -- 2)
121
122
'Standard' and 'hierarchical' element shape functions
Typical mid-side node: ~a = 4-1
~a = 0
(a = +1
1 ~2 )(1 + ~Ta0)(1 + ~'a~) Na = ~ ( 1 -
"Cubic"elements (32 nodes)
Comer node:
1 (1 -+- ~a~)(1 + rlar])(1 -~- (a()[9(~ 2 + r]2 -~- ~2) _ 19] Na - -~ Typical mid-side node:
~a ---~'~1 Na - 9 ( 1 -
Oa = 4 - 1
(a - + 1
~2)(1 + 9~a~)(1 + 0a0)(1 + (a()
When (a( = (2 = 1 the above expressions reduce to those of Eqs (4.20)-(4.23c). Indeed such elements of three-dimensional type can be joined in a compatible manner to sheet or line elements of the appropriate type as shown in Fig. 4.17. Once again the procedure for generating the shape functions follows that described in Figs 4.10 and 4.11 and once again elements with varying degrees of freedom along the edges can be derived following the same steps. The equivalent of a Pascal triangle is now a tetrahedron and again we can observe the small number of surplus degrees of freedom- a situation of even greater magnitude than in two-dimensional analysis.
The tetrahedral family shown in Fig. 4.18 not surprisingly exhibits properties similar to those of the triangle family. First, once again complete polynomials in three coordinates are achieved at each stage. Second, as faces are divided in a manner identical with that of the previous triangles, the same order of polynomial in two coordinates in the plane of the face is achieved and element compatibility ensured. No surplus terms in the polynomial occur.
4.11.1 Volume coordinates Once again special coordinates are introduced defined by (Fig. 4.19): X = LlXl + L2x2 -]" L3x3 + Lnx4
y = LlYl + Ley2 + L3Y3 + L4Y4 Z = LlZl + L2Z2 + L3z3 + L4Z4
1 = L1 + Le + L3 + L4
Solving Eq. (4.33) gives Lk=
ak + bkx + ck y + dkz
6V
; k=1,2,3,4
(4.33)
Tetrahedral elements 123
Fig. 4.18 The tetrahedral family: (a)linear, (b) quadratic, (c) cubic.
with 1 1 6V = det 1 1
Xl x2 X3 x4
Yl Y2 Y3 Y4
Zl z2 Z3 z4
(4.34a)
in which, incidentally, the value V represents the volume of the tetrahedron. By expanding the other relevant determinants into their cofactors we have
al =
det
Cl = - d e t
ix2y2z21 x3 x4
x3 x4
Y3 y4
1 1
Z3
bl = - det
Z4
z3 z4
dl=
- det
[1 y2z2] 1 1
x3 x4
y3 y4
y3 y4
z3
Z4
11
(4.34b)
1 1
with the other constants defined by cyclic interchange of the subscripts in the order 1, 2, 3, 4. Again the physical nature of the coordinates can be identified as the ratio of volumes of tetrahedra based on an internal point P in the total volume, e.g., as shown in Fig. 4.19:
124 'Standard' and 'hierarchical' element shape functions volume P234 L1 -- volume 1234'
etc.
(4.35)
4.11.2 Shape functions As the volume coordinates vary linearly with the cartesian ones from unity at one node to zero at the opposite face then shape functions for the linear element [Fig. 4.18(a)] are simply Na = La a = 1, 2, 3, 4 (4.36) Formulae for shape functions of higher order tetrahedra are derived in precisely the same manner as for the triangles by establishing appropriate Lagrange-type formulae similar to Eq. (4.30). The reader may verify the following shape functions for the quadratic and cubic order cases.
"Quadratic" tetrahedron [Fig. 4.18(b)1
For comer nodes:
Na = (2Lo - 1)La
a=1,2,3,4
For mid-edge nodes: N5 - 4LIL2,
etc.
"Cubic" tetrahedron
Comer nodes:
N1-
Fig. 4.19 Volume coordinates.
1
~(3Lo - 1 ) ( 3 L o - 2)La
a = 1,2,3,4
Other simple three-dimensional elements 125 Mid-edge nodes:
N5 -- 9LIL2(3L1-
1),
etc.
Mid-face nodes: N17 -'-
27L1L2L3,
etc.
A useful integration formula may again be quoted here:
f /fvo 1LTL
L L 4
a y az =
a! b! c!+ ddl +
(a + b + c
3)!
6V
(4.37)
The possibilities of simple shapes in three dimensions are greater, for obvious reasons, than in two dimensions. A quite useful series of elements can, for instance, be based on triangular prisms (wedges) (Fig. 4.20). Here again variants of the product, Lagrange, approach or of the 'serendipity' type can be distinguished. The first element of both families, shown in Fig. 4.20(a), is identical and the shape functions are
Na --
1 =La(1 +(a() z
a=
1,2 . . . . . 6
For the 'quadratic' element illustrated in Fig. 4.20(b) the shape functions are Comer nodes
Na__l~La(2La-
1)(1 + ( a ( )
_ ~La(1 1 - ~2)
a=l,2
..... 6
Mid-edge of rectangle: N7 - L1 (1 - (2),
etc.
Mid-edge of triangles: N10 = 2LIL2(1 + (),
etc.
Such elements are not purely esoteric but have a practical application as 'fillers' in conjunction with 20-noded serendipity elements.
Part 2. Hierarchical shape functions
The general ideas of hierarchic approximation were introduced in Sec.4.2 in the context of simple, linear, elements. The idea of generating higher order hierarchic forms is again simple. We shall start from a one-dimensional expansion as this has been shown to provide a basis for the generation of two- and three-dimensional forms in previous sections. To generate a polynomial of order p along an element side we do not need to introduce nodes but can instead use parameters without any obvious physical meaning. We could use here a linear expansion specified by 'standard' functions N1 and N2 and add to this a
126 'Standard' and 'hierarchical' element shape functions
Fig. 4.20 Triangular prism elements (serendipity) family: (a)linear, (b) quadratic, (c) cubic. series of polynomials always designed so as to have zero values at the ends of the range (i.e., points 1 and 2). Thus for a quadratic approximation, we would write over the typical one-dimensional element, for instance, = Nlfil + g2u2 + N3u3
(4.38)
1 N2 = z(1 + ~) 2;
(4.39)
where 1 NI = z(1 - ~:) Z
N3 = (1 - ~2)
using in the above the normalized x coordinate [viz. Eq. (4.17)].
Hierarchic polynomials in one dimension
We note that the parameter fi3 does in fact have a meaning in this case as it is the magnitude of the departure from linearity of the approximation fi at the element centre, since N3 has been chosen here to have the value of unity at that point. In a similar manner, for a cubic element we simply have to add N4u4 tO the quadratic expansion of Eq. (4.39), where N4 is any cubic of the form (4.40)
N4 = (9/0 -+- ~IYl + ~:20t2 nI- ~:30g3
and which has zero values at ~ = 4-1 (i.e., at nodes 1 and 2). Again an infinity of choices exists, and we could select a cubic of a simple form which has a zero value at the centre of the element and for which dNn/d~ = 1 at the same point. Immediately we can write N4 = ~ ( 1
_~:2)
(4.41)
as the cubic function with the desired properties. Now the parameter U4 denotes the departure of the slope at the centre of the element from that of the linear approximation. We note that we could proceed in a similar manner and define the fourth-order hierarchical element shape function as N5 -- ~2(1 _~:2)
(4.42)
but a physical identification of the parameter associated with this now becomes more difficult (even though it is not strictly necessary). As we have already noted, the above set is not unique and many other possibilities exist. An alternative convenient form for the hierarchical functions is defined by 1 -~(~P - 1) p even
G+I (~)
//,
(4.43)
1 --7(~ p - ~) p odd p~
where p (> 2) is the degree of the introduced polynomial. 16 This yields the set of shape functions: 1
2
1
N3 = ~(~ - 1) N5 = 1 ( ~ 4 1)
N4 = ~ ( ~ 3 _ ~) 1 N 6 - 1-N(~ 5 - ~),
(4.44) etc.
We observe that all derivatives of Np+l of second or higher order have the value zero at = 0, apart from d p Np+l/d~p, which equals unity at that point, and hence, when shape functions of the form given by Eq. (4.44) are used, we can identify the parameters in the approximation as fip+l
=
d'~
~
=0
p > 2
(4.45)
This identification gives a general physical significance but is by no means necessary. In two- and three-dimensional elements a simple identification of the hierarchic parameters on interfaces will automatically ensure Co continuity of the approximation.
127
128 'Standard' and 'hierarchical' element shape functions
In deriving 'standard' finite element approximations we have shown that all shape functions for the Lagrange family could be obtained by a simple multiplication of one-dimensional ones and those for serendipity elements by a combination of such multiplications. The situation is even simpler for hierarchic elements. Here all the shape functions can be obtained by a simple multiplication process. Thus, for instance, in Fig. 4.21 we show the shape functions for a lagrangian nine-noded element and the corresponding hierarchical functions. The latter not only have simpler shapes but are more easily calculated, being simple products of linear and quadratic terms of Eq. (4.43) or (4.44). Using products of lagrangian polynomials the three functions illustrated are simply N1 -----(1 --~)(1 + r/)/4 N2 = (1 - ~ ) ( 1 - r/2)/2
(4.46)
N3 = (1 - ~2)(1 -- r] 2) The distinction between lagrangian and serendipity forms now disappears as for the latter in the present case the last shape function (N3) is simply omitted. Indeed, it is now easy to introduce interpolation for elements of the type illustrated in Fig. 4.11 in which a different expansion is used along different sides. This essential characteristic of hierarchical elements is exploited in adaptive refinement (viz. Chapter 14) where new degrees of freedom (or polynomial order increase) is made only when required by the magnitude of the error. A similar process clearly applies to the three-dimensional family of hierarchical bricktype elements.
Once again the concepts of multiplication can be introduced in terms of area or volume coordinates to define the triangle and tetrahedron family of elements. 15' 16 Starting from the linear shape functions for the comer nodes Na = Za
hierarchical functions for mid-side and interior nodes can be added. For the triangle shown in Fig. 4.14 we note that along the side 1-2, L3 is identically zero, and therefore we have (L1 + L2)1-2 = 1 (4.47) If ~, measured along side 1-2, is the usual non-dimensional local element coordinate of the type we have used in deriving hierarchical functions for one-dimensional elements, we can write 1 1 Ll11-2 -- ~(1 - ~) L211-2= ~(1 + ~) (4.48) from which it follows that we have = (L2 - L1)l-2
(4.49)
Triangle and tetrahedron family
:1 ~o3 1
:1 1
o
I
~,~%:=.L/o/"
.,~'.e
~ I x . , ".t. - ~ ,
'
-
. T r L.,< ,.,7".. L.,/.. / (a) Standard
(b) Hierarchical
Fig. 4.21 Standard and hierarchical shape functions corresponding to a lagrangian quadratic element.
This suggests that we could generate hierarchical shape functions over the triangle by generalizing the one-dimensional shape function forms produced earlier. For example, using the expressions of Eq. (4.43), we associate with the side 1-2 the polynomial of degree p (> 2) defined by Up(l-2) --
&[(L2 /]! ~.v [(L2 -
L1) p -
(L1 + L2) p]
p even
L1) p -
(L2 - L1)(L1 + L2) p-l]
p odd
(4.50)
It follows from Eq. (4.48) that these shape functions are zero at nodes 1 and 2. In addition, it can easily be shown that Np(1-2) will be zero all along the sides 3-1 and 3-2 of the triangle, and so Co continuity of the approximation fi is assured.
129
130 'Standard' and 'hierarchical' element shape functions It should be noted that in this case for p > 3 the number of hierarchical functions arising from the element sides in this manner is insufficient to define a complete polynomial of degree p, and internal hierarchical functions, which are identically zero on the boundaries, need to be introduced; for example, for p = 3 the function L1LeL3could be used, while for p - 4 the three additional functions L~L2L3,L1L~L3,LILzL~ could be adopted. In Fig. 4.22 typical, hierarchical, linear, quadratic, and cubic trial functions for a triangular element are shown. Identical procedures are obvious in the context of tetrahedra. Hierarchical functions of other forms can be found in reference 23.
We have already mentioned that hierarchic element forms give a much improved equation conditioning for steady-state (static) problems due to their form which is more nearly diagonal. In Fig. 4.23 we show the 'condition number' (which is a measure of such diagonality and is defined in standard texts on linear algebra; see Appendix A) for a single cubic element and for an assembly of four cubic elements, using standard and hierarchic forms in their
Fig. 4.22 Triangular elements and associated hierarchical shape functions of (a)linear, (b) quadratic, (c) cubic form.
Global and local finite element approximation Single element (Reduction of condition number = 10.7)
|
o
o
o
o
|
II
'11
~max/~min = 390
~max/~nin = 36
Four element assembly (Reduction of condition number = 13.2)
|
|
II
~,max/~min = 124
~max/~min- 1643
Cubic order elements ( ~ Standard shape function ( ~ Hierarchic shape function Fig. 4.23 Improvement of condition number (ratio of maximum to minimum eigenvalue of the stiffness matrix) by use of hierarchical form (isotropic elasticity, v = O.15).
formulation. The improvement of the conditioning is a distinct advantage of such forms and allows the use of iterative solution techniques to be more easily adopted. 17 Unfortunately much of this advantage disappears for transient analysis as the approximation must contain specific modes (see Chapter 16).
The very concept of hierarchic approximations (in which the shape functions are not affected by the refinement) means that it is possible to include in the expansion
U -- ~
Nabta
(4.51)
a=l
where functions N are not local in nature. Such functions may, for instance, be the exact solutions of an analytical problem which in some way resembles the problem dealt with, but do not satisfy some boundary or inhomogeneity conditions. The 'finite element', local, expansions would here be a device for correcting this solution to satisfy the real conditions,
131
132
'Standard' and 'hierarchical' element shape functions
,
(a)
/ / / /
r
(b)
/ / / / / / /
Fig. 4.24 Some possible uses of global-local approximation. (a) Rotating slotted disc. (b) Perforated beam. This use of the global-local approximation was first suggested by Mote TM in a problem where the coefficients of this function were fixed. The example involved here is that of a rotating disc with cutouts (Fig. 4.24). The global known solution is the analytical one corresponding to a disc without cutout, and finite elements are added locally to modify the solution. Other examples of such 'fixed' solutions may well be those associated with point loads, where the use of the global approximation serves to eliminate the singularity modelled badly by the discretization. In some problems the singularity itself is unknown and the appropriate function can be added with an unknown coefficient. Some aspects of this are mentioned in Chapter 15 and, for waves, in the context of fluid dynamics, in reference 19.
Internal nodes or nodeless internal parameters yield in the usual way the element properties ~I-Ie = Ke~e _+_fe a~ e
(4.52)
As ~e can be subdivided into parts which are common with other elements, ~ , and others which occur in the particular element only, ~ , we can immediately write aI-I
aI-I e
=0
Elimination of internal parameters before assembly- substructures 133 and eliminate fi~ from further consideration. Writing Eq. (4.52) in a partitioned form we have On e
O~le :
0I'I e --
1 / } /ff} _ "K~I K~2 fi~ + .K~.1 K~.2J ul l, f~
0U~
0 r[ e
N
=
{01"Ie}
Ofi--~-I
(4.53)
0
From the second set of equations given above we can write
U2 -- -- (K22) ~e
which on substitution yields
e
8I-1 e
-1
(K21u 1 -t- f~) e
~e
- e ~e : KllUl -~- fl- e
(4.54)
(4.55)
in which - e e e -1K~ Kll -- Klle - K12(K22) 1
f~ = f~ -- K~2(K~2)-lf~
(4.56)
This process of partial solution is also known in the literature as 'static condensation' .20 Assembly of the total region then follows, by considering only the element boundary variables, thus giving a saving in the equation-solving effort at the expense of a few additional manipulations carded out at the element stage. 2~ Perhaps a structural interpretation of this elimination is desirable. What in fact is involved is the separation of a part of the structure from its surroundings and determination of its solution separately for any prescribed displacements at the interconnecting boundaries. I~ *e is now simply the overall stiffness of the separated structure and f,e the equivalent set of nodal forces. If the triangulation of Fig. 4.25 is interpreted as an assembly of pin-jointed bars the reader will recognize immediately the well-known device of 'substructures' used frequently in structural engineering. Such a substructure is in fact simply a complex element from which the internal degrees of freedom have been eliminated. Immediately a new possibility for devising more elaborate, and presumably more accurate, elements is presented. Figure 4.25(a) can be interpreted as a continuum field subdivided into linear triangular elements. The substructure results in fact in one complex element shown in Fig. 4.25(b) with a number of boundary nodes. The only difference from elements derived in previous sections is the fact that the unknown u now is not approximated internally by one set of smooth shape functions but by a series of piecewise approximations. This presumably results in a slightly poorer approximation but an economic advantage may arise if the total computation time for such an assembly is saved. Substructuring is an important device in complex problems, particularly where a repetition of complicated components arises. In simple, small-scale finite element analysis, much improved use of simple triangular elements was found by the use of simple subassemblies of the triangles (or indeed tetrahedra). For instance, a quadrilateral based on four triangles from which the central node is eliminated was found to give an economic advantage over direct use of simple triangles (Fig. 4.26). This and other subassemblies based on triangles are discussed by Doherty e t al. 21 and used by Nagtegaal e t al. 22 and others.
134 'Standard' and 'hierarchical' element shape functions
I/._ i
,I (a)
I
,o
D
~3
1
i
'_....... ~._". . . . . _)'_...... ;4 ...... T
(b)
Fig. 4.25 Substructureof a complexelement.
An unlimited selection of element types has been presented here to the reader- and indeed equally unlimited alternative possibilities exist. 4' 8 What is the use of such complex elements in practice? As presented so far the triangular and tetrahedral elements are limited to situations where the real region is of a suitable shape which can be represented as an assembly of fiat facets and all other elements are limited to situations represented by an assembly of fight prisms. Such a limitation would be so severe that little practical purpose would have been served by the derivation of such shape functions unless some way could be found of distorting these elements to fit realistic curved boundaries. In fact, methods for doing this are available and will be described in the next chapter.
4.1 Develop an explicit form of the standard shape functions at nodes 1, 3 and 6 for the element shown in Fig. 4.27(a). Using a Pascal triangle in ~ and r/show the polynomials included in the element. 4.2 Develop an explicit form of the standard shape functions at nodes 2, 3 and 9 for the element shown in Fig. 4.27(b). Using a Pascal triangle in ~ and 17show the polynomials included in the element. 4.3 Develop an explicit form of the standard shape functions at nodes 1, 2 and 5 for the element shown in Fig. 4.27(c). Using a Pascal triangle in ~ and r/show the polynomials included in the element.
Fig. 4.26 A composite quadrilateral made from four simple triangles.
Problems 3
4
4
9
8
1
#
= 5
~6 8
(a)
=
2
3
~
v
@
=
5
3
7 I -
9o
1
4
v
80
(b)
=
2
=
1
=
5
(c)
_
2
Fig. 4.27 Quadrilateral element for Problems 4.1 to 4.4.
4.4 Develop an explicit expression in hierarchical form for all nodes of the element shown in Fig. 4.27(c). 4.5 Develop an explicit form of the standard shape functions at nodes 1, 2 and 5 for the element shown in Fig. 4.28(a). Using a Pascal triangle in ~ and 0 show the polynomials included in the element. 4.6 Develop an explicit form of the standard shape functions at nodes 1, 5 and 7 for the element shown in Fig. 4.28(b). Using a Pascal triangle in ~ and r/show the polynomials included in the element. 4.7 The mesh for a problem contains an 8-node quadratic serendipity rectangle adjacent to a 6-node quadratic triangle as shown in Fig. 4.29. Show that the coordinates computed from each element satisfy C Ocontinuity along the edge 3-7-11. 4.8 Determine an explicit expression for the shape function of node 1 of the linear triangular prism shown in Fig. 4.20(a). 4.9 Determine an explicit expression for the hierarchical shape function of nodes 1, 7 and 10 of the quadratic triangular prism shown in Fig. 4.20(b). 4.10 Determine an explicit expression for the shape function of nodes 1, 7, 13 and 25 of the cubic triangular prism shown in Fig. 4.20(c). 4.11 On a sketch show the location of the nodes for the quartic member of the tetrahedron family. Construct an explicit expression for the shape function of the vertex node located at (L1, L2, L3, L4) -- (1, 0, 0, 0) and the mid-edge node located at (0.25, 0.75, 0, 0). 4.12 On a sketch show the location of the nodes for the quartic member of the serendipity family. Construct an explicit expression for the shape function of the vertex node located at (~, 0, ~) = (1, 1, 1) and the mid-edge node located at (0.75, 1, 1). 4.13 On a sketch show the location of the nodes for the quartic member of the triangular prism family shown in Fig. 4.20. Construct an explicit expression for the hierarchical shape function of a vertex node, an edge node of a triangular face and an edge node of a rectangular face. 4.14 On a sketch show the location of the nodes for the quadratic member of the triangular prism family in which lagrangian interpolation is used on rectangular faces (see Fig. 4.20). Construct an explicit expression for the shape function of a vertex node, an edge node of a triangular face and an edge node of a rectangular face.
135
136
'Standard' and 'hierarchical' element shape functions
7
3 4
7o
-
=
1
5
=
=
t)'a" 6
2 1
9
(b)
v
6
2
Fig. 4.28 Quadrilateral element for Problems 4.5 and 4.6.
4 cm
9
10
I,
,qw
7q
6
q~
1 Jig 'qW
11
2
3 ~
8
4
i W
6 cm
3 cm
Fig. 4.29 Quadratic rectangle and triangle for Problem 4.7.
4.15 On a sketch show the location of the nodes for the cubic member of the triangular prism family in which lagrangian interpolation is used on rectangular faces (see Fig. 4.20). Construct an explicit expression for the shape function of a vertex node, an edge node of a triangular face, an edge node of a rectangular face, a mid-face node of a triangular face, a mid-face node of a rectangular face, and for any internal nodes.
1. W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 3rd edition, 1976. 2. P.C. Dunne. Complete polynomial displacement fields for finite element methods. Trans. Roy. Aero. Soc., 72:245, 1968. 3. B.M. Irons, J.G. Ergatoudis, and O.C. Zienkiewicz. Comments on 'complete polynomial displacement fields for finite element method' (by P.C. Dunne). Trans. Roy. Aeronaut. Soc., 72:709, 1968. 4. J.G. Ergatoudis, B.M. Irons, and O.C. Zienkiewicz. Curved, isoparametric, 'quadrilateral' elements for finite element analysis. Int. J. Solids Struct, 4:31-42, 1968.
References 137 5. O.C. Zienkiewicz, B.M. Irons, J.G. Ergatoudis, S. Ahmad, and EC. Scott. Isoparametric and associated elements families for two and three dimensional analysis. In Finite Element Methods in Stress Analysis, Chapter 13. Tapir Press, Trondheim, 1969. 6. J.H. Argyris, K.E. Buck, H.M. Hilber, G. Mareczek, and D.W. Scharpf. Some new elements for matrix displacement methods. In Proc. 2nd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-68-150, Wright Patterson Air Force Base, Ohio, Oct. 1968. 7. R.L. Taylor. On completeness of shape functions for finite element analysis. Int. J. Numer. Meth. Eng., 4:17-22, 1972. 8. O.C. Zienkiewicz, B.M. Irons, J. Campbell, and EC. Scott. Three dimensional stress analysis. In IUTAM Symposium on High Speed Computing in Elasticity, Li'ege, 1970. 9. EC. Scott. A quartic, two dimensional isoparametric element. Undergraduate Project, University of Wales, 1968. 10. J.H. Argyris, I. Fried, and D.W. Scharpf. The TET 20 and TEA 8 elements for the matrix displacement method. Aero. J., 72:618-625, 1968. 11. P. Silvester. Higher order polynomial triangular finite elements for potential problems. Int. J. Eng. Sci., 7:849-861, 1969. 12. B. Fraeijs de Veubeke. Displacement and equilibrium models in finite element method. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 9, pages 145-197. John Wiley & Sons, Chichester, 1965. 13. J.H. Argyris. Triangular elements with linearly varying strain for the matrix displacement method. J. Roy. Aero. Soc. Tech. Note, 69:711-713, 1965. 14. J.G. Ergatoudis, B.M. Irons, and O.C. Zienkiewicz. Three dimensional analysis of arch dams and their foundations. In Proc. Symp. Arch Dams, Inst. Civ. Eng., London, 1968. 15. A.G. Peano. Hierarchics of conforming finite elements for elasticity and plate bending. Comp. Math. and Applications, 2:3-4, 1976. 16. J.E de S.R. Gago. A posteri error analysis and adaptivity for the finite element method. Ph.D. thesis, Department of Civil Engineering, University of Wales, Swansea, 1982. 17. O.C. Zienkiewicz, J.E De S.R. Gago, and D.W. Kelly. The hierarchical concept in finite element analysis. Comp. Struct., 16:53-65, 1983. 18. C.D. Mote. Global-local finite element. Int. J. Numer. Meth. Eng., 3:565-574, 1971. 19. O.C. Zienkiewicz, R.L. Taylor, and E Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 20. E.L. Wilson. The static condensation algorithm. Int. J. Numer. Meth. Eng., 8:199-203, 1974. 21. W.E Doherty, E.L. Wilson, and R.L. Taylor. Stress analysis of axisymmetric solids utilizing higher-order quadrilateral finite elements. Technical Report 69-3, Structural Engineering Laboratory, Univ. of California, Berkeley, Jan. 1969. 22. J.C. Nagtegaal, D.M. Parks, and J.R. Rice. On numerical accurate finite element solutions in the fully plastic range. Comp. Meth. Appl. Mech. Eng., 4:153-177, 1974. 23. S.J. Sherwin and G.E. Karniadakis. A new triangular and tetrahedral basis for high-order (hp) finite element methods. Int. J. Numer. Meth. Eng. 38:3775-3802, 1995.
In the previous chapter we have shown how some general families of finite elements can be obtained for Co interpolations. A progressively increasing number of nodes and hence improved accuracy characterizes each new member of the family and presumably the number of such elements required to obtain an adequate solution decreases rapidly. To ensure that a small number of elements can represent a relatively complex form of the type that is liable to occur in real, rather than academic, problems, simple rectangles and triangles no longer suffice. This chapter is therefore concerned with the subject of distorting such simple forms into others of more arbitrary shape. Elements of the basic one-, two-, or three-dimensional types will be 'mapped' into distorted forms in the manner indicated in Figs 5.1 and 5.2. In these figures it is shown that the 'parent' ~, 77, (, or L1, L2, L3, L4 coordinates can be distorted to a new, curvilinear set when plotted in cartesian x, y, z space. Not only can two-dimensional elements be distorted into others in two dimensions but the mapping of these can be taken into three dimensions as indicated by the flat sheet elements of Fig. 5.2 distorting into a three-dimensional space. This principle applies generally, providing a one-to-one correspondence between cartesian and curvilinear coordinates can be established, i.e., once the mapping relations of the type
x)
fx(~, 17, ()
YI = z
fy(~, 77, () fz(~, rl, ()
fx(L1, L2, L3, L4) or
fy(L1, L2, L3, L4) fz(L1, L2, L3, L4)
(5.1)
can be established. Once such coordinate relationships are known, shape functions can be specified in local (parent) coordinates and by suitable transformations the element properties established in the global coordinate system.
Use of 'shape functions' in the establishment of coordinate transformations
44-+-
/-~,-~/*l*
L2=O
3
L2 = 0
3 L1 = 0
1
J
L1 = 0
L 2
1
Local coordinates
2 L3=O
Cartesian map
x
Fig. 5.1 Two-dimensional 'mapping' of some elements.
In what follows we shall first discuss the so-called isoparametric form of relationship (5.1) which has found a great deal of practical application. Full details of this formulation will be given, including the establishment of element matrices by numerical integration. In later sections we shall show that many other coordinate transformations also can be used effectively.
Parametric curvilinear coordinates ....5.2 Use of 'shape functions'in the establishment of coordinate transformations A most convenient method of establishing the coordinate transformations is to use the 'standard' type of Co shape functions we have already derived to represent the variation of the unknown function.
139
140 Mappedelements and numerical integration
=
=1
1]
71'
1] "---1
4
2 2
Local coordinates
Cartesian map
x
Fig. 5.2 Three-dimensional'mapping' of some elements.
If we write, for instance, for each element
' x = N lxl @ N~x2 + . . . .
,
/Xl/ N'
,
Y = N l Yl + N~Y2 + "
Z -- N~Zl -q- N~Z2 Jr- . . . .
N'
x2
= N'x
/Zl/ Y2
-- W y
z2
-
(5.2)
N'z
in which N' are standard shape functions given in terms of the local (parent) coordinates, then a relationship of the required form is immediately available 9 Further, the points with coordinates Xl, yl, Zl, etc., will lie at appropriate points of the element boundary or interior (as from the general definitions of the standard shape functions we know that these have a value of unity at the point in question and zero elsewhere). These points can establish nodes a p r i o r i . To each set of local coordinates there will correspond a set of global cartesian coordinates and in general only one such set. We shall see, however, that a non-uniqueness may arise if the nodal coordinates are placed such that a violent distortion occurs.
Use of 'shape functions' in the establishment of coordinate transformations
The concept of using such element shape functions for establishing curvilinear coordinates in the context of finite element analysis appears to have been introduced first by Taig. 1 In his first application basic linear quadrilateral relations were used. Irons generalized the idea for other elements. 2' 3 Quite independently the exercises of devising various practical methods of generating curved surfaces for purposes of engineering design led to the establishment of similar definitions by Coons 4 and Forrest, 5 and indeed today the subjects of surface definitions and analysis are drawing closer together due to this activity. In Fig. 5.3 an actual distortion of elements based on the quadratic and cubic members of the two-dimensional 'serendipity' family is shown. It is seen here that a one-to-one relationship exists between the local (~, 7) and global (x, y) coordinates. If the fixed (nodal) points are such that a violent distortion occurs then a non-uniqueness can occur in the manner indicated for two situations in Fig. 5.4. Here at internal points of the distorted element two or more local coordinates correspond to the same cartesian coordinate and in addition to some internal points being mapped outside the element. Care must be taken in practice to avoid such gross distortion. Figure 5.5 shows two examples of a two-dimensional (~, r/) element mapped into a three-dimensional (x, y, z) space. We shall often refer to the basic element in undistorted, local, coordinates as a 'parent' element. In Sec. 5.5 we shall define a quantity known as thejacobian determinant. The well-known condition for a o n e - t o - o n e mapping (such as exists in Fig. 5.3 and does not in Fig. 5.4) is that the sign of this quantity should remain unchanged at all the points of the mapped element. It can be shown that with a parametric transformation based on bilinear shape functions, the necessary condition is that no internal angle [such as ot in Fig. 5.6(a)] be equal or greater than 180~ 6 In transformations based on quadratic 'serendipity' or 'lagrangian' functions,
11
l
1
~=-1
Fig. 5.3 Plots of curvilinear coordinates for quadratic and cubic elements (reasonable distortion).
141
142
Mapped elements and numerical integration
Fig. 5.4 Unreasonableelement distortion leading to non-unique mapping and 'overspill'. Quadraticand cubic elements.
a
f-- t1x3=3 singular
2x8-3=
13 > 4 x 3 = 12 singular
(b)
6x2-3=9>2x3=6 singular
13x2-3=23,
/ Fig. 6.7 Approximation of curved surface by linear element.
In the finite element solution we divide all integrals to be sums over individual elements and approximate the weak form by
~/n 'eT[~176176 e
'uTbd~-e~fr'uTtdF=O
(6.53)
e
e
e
where ~'2e and F t denote element domains and parts of boundaries of any element where tractions are specified, respectively. The 'approximation' in this step is associated with the fact that for curved boundary surfaces the sum of element domains ~,~eis not always exactly equal to f2, nor is the sum of F t equal to Ft. This is easily observed for approximations using linear elements as shown in Fig. 6.7. We observe that the error is O(h 2) which is exactly the same as the error in displacement from shape functions using linear polynomials. Thus, the order of error in our solution is not increased by the boundary approximation.
Displacement and strain approximation
At this point we can introduce the finite element shape function expressions to define displacements. Accordingly, we have
U ,'~ ll :
{0} 0a}No { ~3
~)a
~13
t13a
-- ~
Nafla
(6.54)
a
where f i a , V a , Wa are nodal values of the displacement. Any of the three-dimensional interpolations given in Chapter 4 may be used to define the shape functions Na. Inserting
Finite element approximation 203
Fig. 6.8 8-node brick element. Local node numbering. the interpolation into Eq. (6.5) gives
FaNa
E--
Yxy Vy~ Yzx
o
ONa
0
Oy
~x
Ey Ez
o
0
~=~ a
|aNa
I ONa L Oz
aNa Oz
Oga
0
ONa
ONa
Oz
Oy
0
ONa
{OaI ~3a ~)a
-" ~
Batla
(6.55)
a
Ox
A similar expression may be written for virtual strains. Example 6.2: Strains for 8-node brick. As an example we consider the 8-node brick element shown in Fig. 6.8. The shape functions are given by
Na = g1 (1 + ~'a~')(1 + qar/)(1 + (a() for which the derivatives with respect to ~, 1/, ( are given by
204
Problems in linear elasticity
ONa
1
ONa
1
ONa
1
O~ = 8 ~a (1 + r/at/)(1 -+- (a() 0/1 = ~ /~a (1 + (a()(1 + ~a~)
For an 8-node brick element, the jacobian matrix in (5.11) may be expressed as 0b(1 +(b()(1 + ~b~)
[Xb Yb Zb]
The shape function derivatives are now obtained from (5.10) as
ONo'
ONa
~a(1 + 0a0)(1 + (a()
1
= ~ J-~
r/a(1 + (a()(1 + ~a~)
ONa
(a(1 + ~a~)(1 + /70/7)
These may be used directly to define the Ba strain matrix given in Eq. (6.55). For two-dimensional problems the finite element shape function expressions to define the displacements are given by U~fi:
(U/~ "- Z a
Na (~'la a
-:
~-~ Natla
(6.56)
a
Inserting the interpolation into Eqs (6.6) and (6.7) gives
-aNa Ox ~x
8z
za
Yxy
0
aNa Oy
0
0
aNa Oy
aNa Ox
0
{blal Ua 7t-
0 Ez 0
= ZBa~la_+_~ z
(6.57)
a
for the plane problems and
-aNa Or ~r 8O
Yrz
o
Ea
r -
0 aNa fia o
aNa
aNa
Oz
Or
-
Zo "a~
(6.58)
Finite element approximation for the axisymmetric problem, respectively. In the form for axisymmetry the radius is computed from the parametric form given by Eq. (5.2). Accordingly, for this case we have r - ~
N~ rb
(6.59)
b where rb are locations of the node points defining the N~ functions.
Stiffness and load matrices
Introducing the above approximations into the weak form (6.53) results in
e
,fT
f~ B T[O.0q_D(nbfb_~0)] d ~ - f ~ e
e
Nabds
Natdl-'] : 0
(6.60)
which after summing the element integrals and noting that ~fa is arbitrary gives the system of linear equations Kab fib + fa : 0 (6.61) where Kab = ~ / ~ e e
BTDBb dr2
fa : ~ ~ e e
(6.62)
InT(~176176
e
-
The integration over each element domain may be computed by quadrature in which
L3 f~ (.)d~'2 : fD (.)Jd[-] ~,~ ~ e l=1 L2
(.)l Jl Wl
fF (') d F -
(')l jl Wl
fD (.) jdE] ~ ~ l=1
(6.63)
where J = det J is the determinant of the jacobian transformation between the global and local volume coordinate frames, j - det j is the determinant of the jacobian transformation between the global and local surface coordinate frames, and subscript I is associated with each quadrature point with weight WI. The points and weights are taken from the tables given in Chapter 5.
Example 6.3: Quadrature for 8-node brick element. For an 8-node brick element it is sufficient to perform volume integrals using a 2 • 2 • 2 formula. Thus L3 in (6.63) is equal to 8 and the points and weights may be given ast 1 ~I Ol ~l WI
1 --C --C --C 1
2 C --C --C 1
3 --C C --C 1
4 C C --C 1
5 --C --C C 1
6 C --C C 1
7 --C C C 1
where c = 1/~/-3.
t Note that ordering is unimportant and any other permutation for I is permissible.
8 C C C 1
205
206
Problems in linear elasticity Similarly for the surface integrals a 2 • 2 formula may be used and give L2 equal 4 with points and weights ordered as l ~1 /71
Wl
1
2
3
4
--C --C 1
C --C 1
--C C 1
C C 1
in which again c = 1/~/~. These formulae always have error equal to or less than that in the approximation of domain or in the shape functions. Hence, it is never necessary to use higher order quadrature than the above. 22 A similar form holds for all the two-dimensional cases; however, the volume and surface elements are different for the plane and axisymmetric problems. For plane stress dr2 - t dx dy
and dF = t ds
(6.64)
where t is the thickness of the slab and may vary over the two-dimensional domain; for plane strain dS2 = dx dy and dF = ds (6.65) where a unit thickness is considered; and for axisymmetry t dr2 = 2zrr dr dz
and dF = 2zrr ds
(6.66)
In the above (6.67)
ds - (dx 2 -k- dy 2) 1/2
for the plane problem with a similar expression for axisymmetry. The finite element arrays may now be computed from (6.60) for quadrilateral-shaped elements of lagrangian or serendipity type; the stiffness and load matrices defined in (6.62) are computed using gaussian quadrature as Keb =
/if, 1
Z
1
BaT (~, r/) DBb (~, r/) J (~, r/) d~ dr/
BT(~I' r/l) O l b (~l, r/l) J (~l, r/l)
(6.68)
WI
l
and
f'f'
fae =
1
f_l
-
1
1
[BaT (r 17)(cro - Deo) - Na (r r/)b] J (r r/)d~ dr/
-
N~(~) t j(~) d~
(6.69)
[BT(~I' r/l)(or0 -- De0) - Na(~1, r//)b] J(~l, 77l) WI
Z l
- ~
N~ (~l) i(~l) j (~1) wt I
f Some programs omit the 2zr in the definition of d~ and dF and compute matrices for one radian of arc.
Reporting of results: displacements, strains and stresses 207
in which J=tdetJ; J = detJ;
j=t j-
[(8~)2
+
(8y)2] 1/2 ~-~
; Plane stress
[(Ox)2 8y 21 1/2 ~-~ + (~-~) ; Plane strain
J = 2zrr det J; j - 2zrr
(6.70)
[(8r)2 ( 8 z ) 2 ] 1/2 ~-~ + ~ ; Axisymmetric
for the domain and boundary. The quadrature points are denoted as ~l and/7l and weights as In (6.68) and (6.70) t, D, b, i and initial stress and strain may vary in space in an arbitrary manner and det J is computed as indicated in Sec. 5.5. The simplicity of computation using shape functions and numerical integration should be especially noted. This permits easy consideration of different types of interpolations for the shape functions and different quadrature orders for numerical integration to be assessed.
WI.
The reader will now have observed that, while the finite element representation of displacements is in a sense optimal as it is the primary variable, both the strains and the stresses are not realistic. In particular, in ordinary engineering problems both strains and stresses tend to be continuous within a single material. The answers which are obtained by the finite element calculation result in discontinuities of both strains and stresses between adjacent elements. Thus if the direct calculation of these quantities were presented the answers would be deemed unrealistic. For this reason, from the beginning of the finite element method it was sought to establish these rather important quantities in a more realistic, and possibly more accurate, way. In the very early days of finite element calculation with simple Co continuity elements an averaging of element strains and stresses, which are constant in triangular elements, was made at each node. This of course gave improved results at most points -except at those which were on the boundary. Since the simple days of averaging further attention was given to this subject and other methods were developed. The first of these methodologies was developed by Brauchli and Oden 23 in 1971 and consisted of assuming that a continuous representation of either strain or stress using the same Co functions as for displacements could be found by solving a least squares sense representation of the corresponding discontinuous (finite element) one. This method proved quite expensive but often gave results which were superior to the simple averaging - at least for some sets of problems. However, higher accuracy was not achieved despite the additional cost of solving a full set of algebraic equations. An alternative local procedure to improve results was proposed by Hinton and Campbell 24 and was once quite widely used. The methods of recovery of strain and stress have progressed much further in recent years and in Chapter 13 we discuss these fully. We find that currently an optimal procedure, which generally gives higher order accuracy and has similar cost to simple averaging, is the patch recovery method. In this the process of determining values of recovered strains or stresses assumes that:
208
Problems in linear elasticity 1. At some points of the domain or each element, the strains and stresses calculated by the direct differentiation of the shape functions are more accurate than elsewhere 9 Indeed on many occasions at such points 'superconvergence' is demonstrated which can make the accuracy at least one order higher than that of the finite element values computed from derivatives of shape functions 9 2. A continuous representation of such strains and stresses can be given by finding nodal values which in the least squares sense approximate those computed by the optimal points. Now the increased accuracy will exist over the entire domain 9 The discussion of the existence of such points at which higher order may exist is deferred to Chapter 13 but here we show how this can be easily incorporated into standard programs dealing with elasticity 9 Basically in the procedure we will assume a strain exists for an element and can be expressed by e* = ~ Nb~b (6.71) b where now e is any component of strain. A similar expression may be written for a stress component. The goal is to find appropriate values for ~b which give improved results. To do this we use a least squares method in which the strain in a patch surrounding a vertex node a on elements may be expressed in global coordinates by a polynomial expression of higher order, suitable for the number of unknown parameters in the strain expression. This polynomial expression is given by
e**--[1,
(X-Xa),
(Y--Ya),
"'']
ea o~1 C~2 = P a ( x ) C~a
(6.72)
For 3-node triangles or 4-node quadrilaterals in two dimensions and 4-node tetrahedrons or 8-node brick elements in three dimensions a linear interpolation is used. For the quadratic order elements the polynomial in Pa is also raised to quadratic order. Thus, for 6-node triangular and 8- or 9-node quadrilateral elements in two dimensions we use
Pa = [1,
(X--Xa),
(Y--Ya),
( X - - X a ) 2,
(X--Xa)(y--ya),
( y - - Y a ) 2]
The parameters in e** are determined using the least squares problem given by 1 na
1-I --- ~ Z Z [Po(X~)O~o -- 8(x~)] 2 -- min e=l l
(6.73)
where na is the number of elements attached to node a and x~ are locations where strains are computed. The minimization condition results in
(6.74)
Maoza -- fa
where
na
na
Ma "~ Z Z PT(x~)Pa(x~) e=l 1
and
fa - Z Z PaT(X~)~(X~) e=l l
(6.75)
Numerical examples 209 The values for the remaining nodes (e.g., at mid-side and boundary locations) may be computed by averaging the extrapolated values computed from (6.72). For example, from a patch, the result at node b (b r a) is given by e** (Xb) --- Pa (Xb)Ota
and averaging the result from all patches which contain node b gives the final result for ~b. An identical process may be used to compute stress values. We recommend that a patch recovery method be used to report all strain or stress values. In addition, the method serves as the basis for error assessment and methods to efficiently construct adaptive solutions to a specified accuracy as we shall present in Chapter 14.
To illustrate the application of the theory presented above we consider some example problems. Some of the problems we include can be solved by other analytic methods and thus serve to illustrate the accuracy of results obtained. Others, however, are from more practical situations where either no alternative solution method exists or the method is otherwise cumbersome to obtain thus rendering the finite element approach most useful. Here we first solve again the two-dimensional plane stress problems considered in Chapter 2 to illustrate the advantages of using 4-, 9-, and 16-node quadrilateral isoparametric elements of lagrangian type.
Example 6.4: Beam subjected to end shear. The rectangular beam considered in Sec. 2.9.1 is solved again using lagrangian rectangular elements with 4 nodes (bilinear), 9 nodes (biquadratic) and 16 nodes (bicubic). The mesh for the bilinear model initially has six elements in the depth direction and 12 along the length for a total of 72 elements and 91 nodes. This is subsequently subdivided to form meshes with 12 • 24, 24 x 48, 48 x 96 and 96 x 192 elements. All other data are as defined in Sec. 2.9.1. The analysis is repeated using 9-node biquadratic elements with an inital mesh of 3 • 6 elements, which gives the same number of nodes. Finally, the problem is solved with a mesh of 2 • 3 16-node bicubic elements which again gives a mesh with 91 nodes. Since the exact solution for displacements given in Sec. 2.9.1 contains all polynomial terms of degree 3 or less the solution with this coarse mesh is exact and no refinement is needed. In Table 6.2 we present the results for the energy obtained from each mesh and in Fig. 6.9 we show the convergence behaviour for the 4-node and 9-node element forms. Again, the expected rates of convergence are attained as indicated by the slopes of 2 and 4 in the figure. Example 6.5: Circular beam subjected to end shear. We consider next the circular beam problem described in Sec. 2.9.1. The solution to the problem is performed using isoparametric 4-node bilinear quadrilaterals, 9-node lagrangian quadrilaterals and 16-node lagrangian quadrilaterals. The geometric and material data for the problem is as given in Example 2.4. The initial mesh for all element types uses a regular subdivision of the domain that produces initial element patterns with 6 x 12 4-node elements, 3 x 6 9-node elements and 2 • 4 16-node elements. The mesh for each element form is shown in Fig. 6.10.
210
Problems in linear elasticity 10 0
9
9
i
o 4-node quads J a 9-node quads
10 -2
0 t_
>, 10-4 fLU
10 - 6
.iiiiiiiiiiiiiiiiiiill
10 -8
,_
10-1
10 0
h/h 1
Fig. 6.9 Convergencein energy error for 4-node and 9-node rectangular elements.
Table 6.2
Mesh size and energy for end loaded beam
Nodes
Elmts
4-node rectangles
91
325 1225 4753 18721 Exact
72
288 1152 4608 18432 -
Energy
3077.4986
3238.2915 3281.3465 3292.3206 3295.0790 3296.0000
9-node rectangles Elmts
18
72 288 1152 4608 -
Energy
3294.7512
3295.9174 3295.9947 3295.9997 3296.0000 3296.0000
16-node rectangles Elmts
8
Energy
3296.0000
3296.0000
Results for the energy are given in Table 6.3 and compared to the exact value computed from
Eex -- 0.02964966844238 using the geometry and properties selected. The element size is normalized to that of the coarsest mesh [shown in Fig. 6.10(a)] and the energy error computed from Table 6.3 has the expected slope for 4-node elements, for 9-node elements and for cubic elements (viz. Fig. 6.11). We now consider some practical examples for problems which have been solved using the finite element method. Some simple typical examples are given which use both tetrahedral and isoparametric brick-type elements. The isoparametric examples are all performed using Gauss quadrature to approximate the necessary integrals.
Numerical examples
(a)
(b)
(c) .
.
.
.
.
.
Fig. 6.10 End loaded circular beam' Coarsemeshfor 4-node, 9-node and 16-nodelagrangianelements. Table 6.3 Mesh size and energy for curved beam
4-node quadrilateral Nodes Elmts Energy 91 72 0.03042038175071 325 288 0.02984351371323 1225 1152 0.02969820784232 4753 4608 0.02966180825828 18721 18432 0.02965270370808 Exact - 0.02964966844238
9-node quadrilateral Elmts Energy 18 0.02970101373401 72 0.02965318188484 288 0.02964989418870 1152 0.02964968266120 4608 0.02964966933301 - 0.02964966844238
16-node quadrilateral Elmts Energy 8 0.02965327376971 32 0.02964975296446 128 0.02964966996157 512 0.02964966846707 2048 0.02964966844276 - 0.02964966844238
6.5.1 A dam subject to external and internal water pressures A buttress dam on a somewhat complex rock foundation is shown in Fig. 6.12 and analysed.25, 26 This dam (completed in 1964) is of particular interest as it is the first to which the finite element method was applied during the design stage. The heterogeneous foundation region is subject to plane strain conditions while the dam itself is considered in a state of plane stress of variable thickness. With external and gravity loading no special problems of analysis arise. When pore pressures are considered, the situation, however, requires perhaps some explanation. It is well known that in a porous material the water pressure is transmitted to the structure as a body force of magnitude
bx --
0p
Ox
by =
0p
Oy
(6.76)
and that now the external pressure need not be considered. The pore pressure p is, in fact, now a body force potential which may be determined by solving a 'field problem' as described in the next chapter. Figure 6.12 shows the element subdivision of the region and the outline of the dam. Figure 6.13(a) and (b) shows the stresses resulting from gravity (applied to the dam only) and due to water pressure assumed to be acting as an external load or, alternatively, as an internal pore pressure. Both solutions indicate large tensile regions, but the increase of stresses due to the second assumption is important.
211
212
Problems in linear elasticity 10 0
10 - 2
10 .-4
i
o [] o
9
9
4 node 9 node 16node
I
9
i
9
i
!
i
: !
>,,
t-
LU 10 .-6
10 -8 9
10-10
i
10 -1
10 0 h/h 1
Fig. 6.11 Curved beam Convergence in energy error for quadrilateral elements.
The stresses calculated here are the so-called 'effective' stresses. These represent the forces transmitted between the solid particles and are defined in terms of the total stresses o" and the pore pressures p by cr' = ,7 + m p
m T = [1, 1, 0]
(6.77)
i.e., simply by removing the hydrostatic pressure component from the total stress. 27' 28 The effective stress is of particular importance in the mechanics of porous media such as those that occur in the study of soils, rocks, or concrete. The basic assumption in deriving the body forces of Eq. (6.76) is that only the effective stress is of any importance in deforming the solid phase. This leads immediately to another possibility of formulation. 29 If we examine the equilibrium conditions of Eq. (6.8) we note that this is written in terms of total stresses. Writing the constitutive relation, Eq. (6.26a), in terms of effective stresses, i.e., ~r' -- D'(s - s0) + o'~
(6.78)
and substituting into the weak form we find that the stiffness matrix is given in terms of the matrix D' and the force terms are augmented by an additional force -
[ BTm p d ~ d~2
(6.79)
e
or, if p is interpolated by shape functions N~, the force becomes - [ BTmN ' dr2 ~e df~
(6.80)
e
This alternative form of introducing pore pressure effects allows a discontinuous interpolation of p to be used [as in Eq. (6.79) no derivatives occur] and this is now frequently used in practice.
Numerical examples 213
Here only 18 cubic serendipity elements are needed to obtain an adequate solution, arranged as shown in Fig. 6.14. It is of interest to observe that all mid-side nodes of the cubic elements may be generated within the computer program and need not be specified. Also, the problem requires the specification of body forces caused by the centrifugal effects of the rotating disk. Here, br = - p
r (.0 2
where p is the mass density of the material and co is the angular velocity.
u
i,_. Q. i,... t~
C
E
0 u
0
"0 c~
~
0
v~ if1 ~,m !,_
c ~
Lt~ ~
0 4.-J
m ~ c~ L~ Q-O 9~ Q.
~0 .m 0-0
E ~0
t ~ v~ -0 f~ Lt~ ~
c~ ~ m0
_Q
~
n t~ ~n
0
U
~ C~
o~
~
~E ~ C~
rV~ --C~
tim LL
Numerical examples 215
Fig. 6.15 Conical water tank.
6.5.3 Conical water tank In this problem cubic serendipity elements are again used as shown in Fig. 6.15. It is worth noting that single-element thickness throughout is adequate to represent the bending effects in both the thick and thin parts of the container. With simple 3-node triangular elements, several layers of elements would have been needed to give an adequate solution.
216
Problems in linear elasticity E = 10 7 Ib/in 2 v=0.2 t = 0.5 in
600 z~
400 5
M e Ib/in 200
Exact
I
15
-200
I
I
r0
~L
20
/t varies from
degrees
1to 20
Typical element Fig. 6.16 Encastr~, thin hemispherical shell. Solution with 15 and 24 cubic serendipity elements.
6.5.4 A hemispherical dome The possibilities of dealing with shells approached in the previous example are here further exploited to show how a limited number of elements can adequately solve a thin shell problem as illustrated in Fig. 6.16. This type of solution can be further improved upon from the economy viewpoint by making use of the well-known shell assumptions involving a linear variation of displacements across the thickness. Thus the number of degrees of freedom can be reduced (e.g., see reference 30).
6.5.5 Arch dam in a rigid valley This problem, perhaps a little unrealistic from the engineer's viewpoint, was the subject of a study carried out by a committee of the Institution of Civil Engineers and provided an excellent test for a convergence evaluation of three-dimensional analysis. 1~ In Fig. 6.17 two subdivisions into quadratic and two into cubic elements are shown. In Fig. 6.18 the
Problems 217
convergence of displacements in the centre-line section is shown, indicating that quite remarkable accuracy can be achieved with even one element. The comparison of stresses in Fig. 6.19 is again quite remarkable, though showing a greater 'oscillation' with coarse subdivision. The finest subdivision results can be taken as 'exact' from checks by models and alternative methods of analysis.
6.5.6 Pressure vessel problem A more ambitious problem treated with simple tetrahedra is given in reference 7. Figure 6.20 illustrates an analysis of a complex pressure vessel. Some 10000 degrees of freedom are involved in this analysis. A similar problem using higher order isoparametric elements permits a sufficiently accurate analysis for a very similar problem to be performed with only 2121 degrees of freedom (Fig. 6.21).
6.1 Use the transformation array given by
T
.__
cos0 - sin 0 0
sin0 cos 0 0
i]
with 0 -- 45 ~ to transform stress and strain components from their x, y, z components to their x t, yt, z' components. Let the material be linearly elastic with material parameters given by E and v. Show that G = E/[2(1 + v)]. 6.2 For an isotropic material expressed in E and v compute the mean stress p = (Crx + cry + crz). If the bulk modulus is given by p=Kev
where ev = ex + ey --1-e z is the volume strain, show that K = E/[3(1 - 2v)]. 6.3 The strain displacement equations for a one-dimensional problem in plane polar coordinates are given by OUr
--
{}r0 Err EO0
"--
1 OUo
Ur
r O0 ! r 1 OUr OUo (-:= uo) + r T;-r
D
The displacements are expanded in a Fourier series as Ur -- Z
un (r) COSnO and uo = Z
v" (r) sin nO
218
Problems in linear elasticity
Fig. 6.17 Arch dam in a rigid valley- various element subdivisions.
Problems 219
Level 6
120 (a) (b) (c) (d)
100
o z~ x [~i
32 El 9A El 9B El 1(96D)EI
,,, i::!.
80
r
60
,'..!....,
40
j
L
,~
y""
,///F f , ~
Level4
S
S
Level 2
Y
20
0
10
20
30
40
50
60
mm
Fig. 6.18 Archdam in a rigid valley- centre-linedisplacements.
3 ~
//Air fa!e
o 32 El z~ 9A El x 9B El Ei:i 1(96D)EI
(a) (b) (c) (d)
Water face
-80
-60 -40 -20 Compression
0
20
40
60 80 Tension
Fig. 6.19 Archdam in a rigid valley- vertical stresseson centre-line.
100 kg/cm2
220
Problems in linear elasticity
Fig. 6.20 A nuclear pressure vessel analysis using simple tetrahedral elements. 7 Geometry, subdivision, and some stress results. N.B. Not all edges are shown.
(a) Express u n ( r ) and v n ( r ) in terms of shape functions and parameter fia and Va, respectively, and determine the strain displacement matrix for each harmonic n. (b) For a linear elastic material show that the stiffness matrix for each harmonic is independent of other terms in the Fourier series 9 (Hint: Perform integrals in 0 analytically.) 6.4 Cartesian coordinates may be expressed in terms of spherical components r, 0, and ~p as
x=rcos0
sincp; y - - r
sin0 sin 4) and z = r c o s ~ p
This form permits the solution of spherically symmetric problems for which displacements depend only on r and the strain-displacement equations are expressed as 8rr
-
-
Yre~-
OUr Or '
Ou 4, Or
Ur eoo = eep4, = -7-
u4, " Ouo , YrO--" r Or
uo ; r
Y04~ = 0
(a) How many rigid body modes exist for this problem? (b) Express the displacement components Ur, uo and u, in finite element form using one-dimensional shape functions in r. (c) Determine the form of the strain-displacement matrix Ba for each shape function
Na.
Problems
Fig. 6.21 Three-dimensional analysis of a pressure vessel. (d) For a linear elastic isotropic material write the form of the stiffness matrix for the nodal pair a and b. Show that the problem decomposes into three separate problems in terms of each displacement component. (e) For linear shape functions obtain an expression for the stiffness components corresponding to the Ur displacements using a one-point quadrature formula. Check if the resulting stiffness matrix has correct rank. 6.5 For a linear elastic isotropic material the stiffness matrix may be computed by numerical integration using Eq. (6.68). Alternatively, the stiffness matrix may be computed in indicial form as indicated in Appendix B.
221
222
Problems in linear elasticity
Consider a plane strain problem which is modelled by 4-node quadrilateral elements. Assume the stiffness matrix is computed using a 2 x 2 gaussian quadrature formula. (a) Compute separately the number of additions/subtractions and multiplications necessary to evaluate the stiffness using Eq. (6.68). Count only operations involving non-zero values in B or D. (b) Repeat the above calculation using the method of Appendix B given by Eqs (B.52) and (B.54). 6.6 In the classical plane strain problem the strain normal to the plane of deformation (i.e., ez) was assumed to be zero. The problem may be 'generalized' by assuming ez is constant over the entire analysis domain. The constant strain may then be related to a resultant force Fz applied normal to the deformation plane. (a) Following the steps given in Sec. 6.3, develop the virtual work expression (weak form) for the generalized plane strain problem. (b) Write finite element approximations for all the terms in the weak form. (c) Write the expression for an element stiffness in terms of nodal parameters and the strain ez. (d) Show how the resultant force Fz is related to the constant strain ez.
Fig. 6.22 Traction loading on boundary for Problems 6.7 and 6.8.
6.7 A concentrated load, F, is applied to the edge of a two-dimensional plane strain problem which is modelled using quadratic order finite elements as shown in Fig. 6.22(a). Compute the equivalent forces acting on nodes 1, 2 and 3. 6.8 A triangular traction load is applied to the edge of a two-dimensional plane strain problem as shown in Fig. 6.22(b). (a) Compute the equivalent forces acting on nodes 1, 2 and 3 by performing the integrals exactly. (b) Use numerical integration to compute the integrals which define the equivalent forces. Use the minimum number of points that integrate the integral exactly. What is the result if one-order lower is used? 6.9 An arc of 20 for a circular boundary of radius R is approximated by the quadratic isoparametric interpolation as shown in Fig. 6.23. For this case h = R sin 0 and c - R(1 - cos0). A concentrated load, F, is applied normal to the boundary at the point labelled a (~). Let F -- 100 N, R -- 10 cm and 0 = 15 ~ For ~ - 0, 0.25, 0.50, 0.75, 1.0 determine:
Problems 223 y cL
Fig. 6.23 Concentratednormalloadon a curvedboundary.Problem6.9. ()
4
()8
()
1
L
0 7
t
3
Y
0 7
4
3
Y
b x
X
b
18
(a) Serendipityelement
2
5
5
0
J
6(
L
0 a
a
(b) Lagrangianelement
r
Fig. 6.24 Quadrilateral8- and 9-nodeelements. Problems6.10 and 6.11.
(a) The equivalent forces acting on nodes 1, 2 and 3 for the case when the normal is computed from the quadratic interpolation. (b) The equivalent nodal forces using the normal to the circular boundary. (c) The error between the two forms. Show on a sketch. 6.10 A mesh for a plane strain problem contains the quadratic order rectangular elements shown in Figs 6.24(a) and (b). The elements are subjected to a constant body force b = (0, -19 g)r where p is mass density and g is acceleration of gravity. For each element type: (a) Use standard shape functions for Na and develop a closed form expression for the nodal forces in terms of a, b and p g. (b) Use hierarchical shape functions for Na and develop a closed form expression for the nodal forces in terms of a, b and p g. 6.11 A mesh for a plane strain problem contains the quadratic order rectangular elements shown in Figs 6.24(a) and (b). The elements are subjected to a constant temperature change AT. Each element is made from an isotropic elastic material with constant properties E, v and ct. For each element type:
224 Problemsin linear elasticity
IY
qo
,~--x
L
L
r
Fig. 6.25 Uniformlyloadedcantileverbeam. Problem6.12. (a) Use standard shape functions for Na and develop a closed form expression for the nodal forces in terms of a, b and the elastic properties. (b) Use hierarchical shape functions for Na and develop a closed form expression for the nodal forces in terms of a, b and the elastic properties. 6.12 Use the program FEAPpv (or any other available program) to solve the rectangular beam problem given in Example 6.4 and verify the results shown in Table 6.2. 6.13 Use the program FEAPpv (or any other available program) to solve the curved beam problem given in Example 6.5 and verify the results shown in Table 6.3. 6.14 The uniformly loaded cantilever beam shown in Fig. 6.25 has properties L=2m;
h=0.4m;
t=0.05m
and q 0 = 1 0 0 N / m
Use FEAPpv or any other available program to perform a plane stress analysis of the problem assuming linear isotropic elastic behaviour with E = 200 GPa and v = 0.3. In your analysis: (a) Use quadratic lagrangian elements with an initial mesh of 1 element in the depth and 5 elements in the length directions. (b) Compute consistent nodal forces for the uniform loading. (c) Compute nodal forces for a parabolically distributed shear traction at the restrained end which balances the uniform loading q0. (d) Report results for the centre-line displacement in the vertical direction and the stored energy in the beam. (e) Repeat the analysis three additional times using meshes of 2 • 10, 4 x 20 and 8 • 40 elements. Tabulate the tip vertical displacement and stored energy for each solution. (f) If the energy error is given by A E = En - En-1
= Ch q
estimate C and q for your solution. Is the convergence rate as expected? Explain your answer. 6.15 A circular composite disk is restrained at its inner radius and free at the outer radius. The disk is spinning at a constant angular velocity co as shown in Fig. 6.26. The disk is manufactured by bonding a steel layer on top of an aluminium layer as shown in Fig. 6.26(b).
Problems 225
Fig. 6.26 Spinning composite disk. Problem 6.15. When spinning at an angular velocity of 50 rpm it is desired that the top surface be flat. This will be accomplished by milling the initial shape of the top to a specified level. Your task is to determine the profile for milling. To accomplish this (a) Perform an analysis for an initially flat top surface using the dimensions given in the figure (lengths given in mm). The elastic properties for steel are E = 200 GPa, v = 0.3 and p = 7.8/zg/mm3; those for aluminium are E = 70 GPa, v = 0.35 and p = 2.6/zg/mm 3 (where/z = 10-6). Be sure to use consistent units (say, mm, sec, and/zg). The inner radius of the disk is to be restrained in the radial direction (i.e., u(15, z) = 0). Axial restraint is only applied at the centre of the disk (i.e., v(15, O) - 0 ) . (b) Using the results for the vertical displacements computed in (a) reposition the top nodes to new values for which a reanalysis should give improved results. (c) Reanalyse the problem for the new coordinates. How accurate does this analysis predict the desired result? What would you do to improve your answer? 6.16 A rectangular region with a circular hole is shown in Fig. 6.27. The traction on the circular hole is zero. The region is to be used for the solution of an infinitely extending
Fig. 6.27 Rectangularregion with circular hole. Problem 6.16.
226 Problems in linear elasticity plane stress problem in which the stress at infinity is given by a uniformly distributed normal stress tr0 acting in the x direction. The stress distribution in polar coordinates for the problem is given by
o"0 --
([ 1 - (a )2] + [ 1 + 3 ( a)4 - 4 ( - )a2 ] cos 20 } r r r lif o {[1-~-(a) 2] -- [| + 3 ( a ) 4] cos20}
rr0=
go'0 1 -
ar=
1 ~ao
{
a} r
+2(-)2
r
sin 20
and the displacements by Ur
= -a~o r { [ 1 + (a )2] -t- [ 1 - (a)4 + 4( a )2 ] COS20 r r
(4,2]
o0r {[
uo=~
(a,4] cos20}
] [ a ]} l + ( a ) 4 r + 2(a)zr + v 1 + ( 4 ) 4 - 2 ( )2 sin20
In order for the region to satisfy the above solution it is necessary to: (a) Enforce symmetry conditions along the boundaries AB and DE and (b) Apply the tractions of the exact solution on the boundary BCD. Program development project: Write a program that uses numerical integration to compute the consistent nodal forces on the boundary BCD. (Hint: This may be done by adding an element to FEAPpv which computes only the nodal forces for line elements defined on the boundary BCD or by writing a MATLAB program which, given the location of nodal coordinates on BCD, computes the nodal forces.) Your program should also compute
E =t f
JB
CD
[Utxd-Vty]
dr
which is twice the stored energy in a slice of thickness t. When accurately computed (e.g., to 9 or 10 digit accuracy) this may be used as the 'exact' solution for the region. Use your program and FEAPpv (or any other available program) to solve a plane stress problem. Let the hole radius be R = 10 cm, the thickness of the slice be t = 0.1R and take E = 200 GPa and v - 0.3 for the elastic properties. The boundary BC should be placed at about 3R and the boundary CD at 2 to 3R. Assume a unit value for the stress or0. (a) Use 4-node quadrilateral elements to solve the problem on a sequence of meshes in which element sizes are reduced in half for each succeeding mesh. (b) Plot the displacement at the hole boundary and compare to the exact solution. (c) Compute the work done by your finite element program. (Note: In FEAPpvthis will be the 'energy' reported by the solver.) (d) Compute the rate of convergence for your solution and plot on a figure similar to that given in Fig. 6.9. (e) Repeat the solution using 8-node serendipity elements. (f) Repeat the solution using 9-node lagrangian elements.
References 227 Write a short report discussing your findings. 6.17 Program development project: Extend the program developed in Problem 2.17 to consider plane strain and axisymmetric geometry. 6.18 Program development project: Extend the program developed in Problem 2.17 to compute nodal forces for specified boundary tractions which are normal or tangential to the element edge. Assume tractions can vary up to quadratic order (i.e., constant, linear and parabolic distributions) and use numerical integration to compute values. Test your program for an edge with constant normal stress. Then test for linear normal and finally quadratic tangential values. Compare results with those computed by FEAPpv (or any available program). 6.19 Program development project: Extend the program developed in Problem 2.17 to compute nodal values of stress and strain. Follow the procedure given in Sec. 6.4 to project element values to nodes. Test your program using (a) the patch test of Problem 2.17 and (b) the curved beam problem shown in Fig. 2.11. 6.20 Program development project: Add a module to the program developed in Problem 2.17 to plot contours of stress and strain components for plane stress, plane strain and axisymmetric solids. Use the capability developed in Problem 6.19 to obtain nodal values and the contour routine developed in Problem 2.18. Test your program system by plotting contours of stress components for the curved beam meshes described in Problem 2.18. 6.21 Program development project: Add a 4-node quadrilateral element to the program system developed in Problem 2.17. Use shape functions and numerical integration to compute the element stiffness matrix. Also include the force vector from a constant element body force (you may need to add b to your input module). Test your program on the curved beam problems described in Problem 2.18. Compare the accuracy to that obtained using triangular elements.
1. M.J. Turner, R.W. Clough, H.C. Martin, and L.J. Topp. Stiffness and deflection analysis of complex structures. J. Aero. Sci., 23:805-823, 1956. 2. R.W. Clough. The finite element method in plane stress analysis. In Proc. 2nd ASCE Conf. on Electronic Computation, Pittsburgh, Pa., Sept. 1960. 3. R.H. Gallagher, J. Padlog, and P.P. Bijlaard. Stress analysis of heated complex shapes. ARS J., 29:700-707, 1962. 4. R.J. Melosh. Structural analysis of solids. J. Struct. Eng., ASCE, 4:205-223, Aug. 1963. 5. J.H. Argyris. Matrix analysis of three-dimensional elastic media- small and large displacements. J. AIAA, 3:45-51, Jan. 1965. 6. J.H. Argyris. Three-dimensional anisotropic and inhomogeneous media- matrix analysis for small and large displacements. Ingenieur Archiv, 34:33-55, 1965. 7. Y.R. Rashid and W. Rockenhauser. Pressure vessel analysis by finite element techniques. In Proc. Conf. Prestressed Concrete Pressure Vessels, Institute of Civil Engineering, 1968. 8. J.H. Argyris. Continua and discontinua. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, pages 11-189, Wright Patterson Air Force Base, Ohio, Oct. 1966.
228
Problems in linear elasticity 9. B.M. Irons. Engineering applications of numerical integration in stiffness methods. J. AIAA, 4:2035-2037, 1966. 10. J.G. Ergatoudis, B.M. Irons, and O.C. Zienkiewicz. Three dimensional analysis of arch dams and their foundations. In Proc. Symp. Arch Dams, Inst. Civ. Eng., London, 1968. 11. J.H. Argyris and J.C. Redshaw. Three dimensional analysis of two arch dams by a finite element method. In Proc. Symp. Arch Dams, Inst. Civ. Eng., 1968. 12. S. Fjeld. Three dimensional theory of elastics. In I. Holand and K. Bell, editors, Finite Element Methods in Stress Analysis, Trondheim, 1969. Tech. Univ. of Norway, Tapir Press. 13. A.E.H. Love. A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge, 4th edition, 1927. 14. N.I. Muskhelishvili. Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen, 3rd edition, 1953. English translation by J.R.M. Radok. 15. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 16. S.G. Lekhnitskii. Theory of Elasticity of an Anisotropic Elastic Body. Holden Day, San Francisco, 1963. (Translation from Russian by P. Fern.) 17. I.S. Sokolnikoff. The Mathematical Theory of Elasticity. McGraw-Hill, New York, 2rd edition, 1956. 18. P.G. Ciarlet. Mathematical Elasticity. Volume 1: Three-dimensional Elasticity. North-Holland, Amsterdam, 1988. 19. O.C. Zienkiewicz and EC. Scott. On the principle of repeatability and its application in analysis of turbine and pump impellers. Int. J. Numer. Meth. Eng., 9:445-452, 1972. 20. R.ES. Hearmon. An Introduction to Applied Anisotropic Elasticity. Oxford University Press, Oxford, 1961. 21. T.C.-T Ting. Anisotropic Elasticity: Theory and Applications. Oxford University Press, New York, 1996. 22. G. Strang and G.J. Fix. An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, N.J., 1973. 23. H.J. Brauchli and J.T. Oden. On the calculation bf consistent stress distributions in finite element applications. Int. J. Numer. Meth. Eng., 3:317-325, 1971. 24. E. Hinton and J. Campbell. Local and global smoothing of discontinuous finite element function using a least squares method. Int. J. Numer. Meth. Eng., 8:461-480, 1974. 25. O.C. Zienkiewicz and Y.K. Cheung. Buttress dams on complex rock foundations. Water Power, 16:193, 1964. 26. O.C. Zienkiewicz and Y.K. Cheung. Finite element procedures in the solution of plate and shell problems. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 8. John Wiley & Sons, Chichester, 1965. 27. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, B.A. Schrefler, and T. Shiomi. Computational Geomechanics: With Special Reference to Earthquake Engineering. John Wiley & Sons, Chichester, 1999. 28. K. Terzhagi. Theoretical Soil Mechanics. John Wiley & Sons, New York, 1943. 29. O.C. Zienkiewicz, C. Humpheson, and R.W. Lewis. A unified approach to soil mechanics problems, including plasticity and visco-plasticity. In Int. Symp. on Numerical Methods in Soil and Rock Mechanics, Karlsruhe, 1975. See also Chapter 4 of Finite Elements in Geomechanics (ed. G. Gudehus), pages 151-78. Wiley, 1977. 30. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005.
Field problems- heat conduction, electric and magnetic potential and fluid flow The general procedures discussed in the previous chapters can be applied to a variety of physical problems. Indeed, some such possibilities have been indicated in Chapter 3 and here more detailed attention will be given to a particular, but wide class, of such situations. Primarily we shall deal with situations governed by the general 'quasi-harmonic' equation, the particular cases of which are the well-known Laplace and Poisson equations. 1-6 The range of physical problems falling into this category is large. To list but a few frequently encountered in engineering practice we have: 9 9 9 9 9 9
Heat conduction Seepage through porous media Irrotational flow of ideal fluids Distribution of electrical (or magnetic) potential Torsion of prismatic shafts Lubrication of pad beatings, etc.
The formulation developed in this chapter is equally applicable to all, and hence only limited reference will be made to the actual physical quantities. In all the above classes of problems, the behaviour can be represented in terms of a scalar variable for which we will generally use the symbol 4~. In the applications to specific problems, however, we shall generally introduce the physical variable describing the behaviour. For instance, in discussing heat conduction applications we use the symbol T to denote the temperature. In Chapter 3 we indicated both the 'weak form' and a variational principle applicable to the Poisson and Laplace equations (see Secs 3.2 and 3.8.1). In the following sections we shall apply these approaches to a general, quasi-harmonic equation and indicate the ranges of applicability of a single, unified, approach by which one computer program can solve a large variety of physical problems. It will be observed that the Co 'shape functions' presented in Chapters 4 and 5 can be directly applied and that both isotropic and anisotropic behaviour can be treated with equal ease.
230
Field problems
In many physical situations we are concerned with the diffusion or flow of some quantity such as heat, mass, concentration, etc. In such problems, the rate of transfer per unit area (flux), q, can be written in terms of its cartesian components as
q = [qx,
qy,
qz]T
(7.1)
If the rate at which the relevant quantity is generated (or removed) per unit volume is Q, then for steady-state flow the balance or continuity requirement gives
Oqx Oqy Oqz Ox +--~-y +--~-z + Q - 0
(7.2)
Introducing the gradient operator
V=
I0Ox' OyO Oz0] T
(7.3)
VTq --I- Q = 0
(7.4)
we can write (7.2) as
Generally the rates of flow will be related to the gradient of some potential quantity q~. This may be temperature in the case of heat flow, etc. A very general linear relationship will be of the form
q =
qx i x xy xzl
qy qz
-- -
kyx,
kyy,
ky z
L~zx,
kzy,
kzzJ
= - k V~b
(7.5)
where k is a symmetric form due to energy arguments (i.e., kxy = kyx, etc.) and is variously referred to as Fourier's, Fick's, or Darcy's law depending on the physical problem. The final governing differential equation for the 'potential' q~ is obtained by substitution of Eq. (7.5) into (7.4), leading to - - V T (k Vq~) + Q = 0
(7.6)
which has to be solved in a domain f2. On the boundaries of such a domain we shall usually encounter one of the following conditions:
General quasi-harmonicequation 231 1. O n F , , (7.7a) i.e., the potential is specified (Dirichlet condition). 2. On 1-'q the normal component of flow (or flux), qn, is given as (Neumann condition) q n --" q - - H (dp -
qbo)
where H is a transfer or radiation coefficient, 4~0 is a known equilibrium value and E/is a specified value. Here q n is defined as qn
= nTq with n = [n x
, ny , n z
]T
where n is a vector of direction cosines of the normal to the boundary surface. Accordingly, we may write the second boundary condition ? / + n T (k X74~) + H (~b - q~0) = 0
(7.7b)
which holds on 1-'q.
7.2.2 Anisotropic and isotropic forms for k If we consider the general statement of Eq. (7.5) as being determined for an arbitrary set of coordinate axes x, y, z we shall find that it is always possible to determine locally another set of axes x', y', z' with respect to which the matrix k' becomes diagonal, as shown in
Fig. 7.1 Anisotropic material. Local coordinates coincide with the principal directions of stratification.
232
Field
problems
Fig. 7.1. With respect to such axes we have k'
--"
0 00]
ky,y, 0
(7.8)
kz'z'
Thus, the general form of the k has only three components which are associated with three orthogonal axes. Such materials are called anisotropic or orthotropic. The governing differential equation (7.6) for these axes can be written
---~x ~ kx,x,~ox, + ~
ky,y,~
+~Oz' kz,z,~ -- (V')T
+Q=0
(7.9)
(k' V'4~) + Q = 0
where
Ox"
Oy"
Oz'
defines the gradient operator for the 'prime' coordinate system. Alternatively, knowing k' and the orientation of the axes x', y', z' a transformation of coordinates is given by x' = Tx in which T are direction cosines defined as T -
rcos(x',x), |cos(y', x), Lcos(z', x),
cos(x',y), cos(y', y), cos(z', y),
cos(x',z)] cos(y', z) cos(z', z)
where cos(x', x) is the cosine of the angle between the x' direction and the x direction. The inverse of T is equal to its transpose; hence X = TTx /
In addition we may write the gradient with respect to the prime axes as V'(.) = T V (.) or alternatively V(.) = T T V'(.)
Using the above we obtain the expression ( v ' ) T (k'V'~b) = (V) T T T (k'TVq~)
(7.10a)
or
k=TTk'T
or k ' = T k T T
(7.10b)
Lastly for an isotropic material we can write k = kI
(7.11)
where I is an identity matrix. In two dimensions this leads to the simple form of Eq. (3.8) as discussed in Chapter 3.
Finite element solution process 233
7.2.3 Weak form and variational principle for the general
quasi-harmonic equation
Following the principles of Chapter 3, Sec. 3.2, we can obtain the weak form of Eqs (7.6) and (7.7b) by writing (using v = 8~p) f 8 4 ~ [ - V T (k Vq~) + Q] dr2 + f r 8q~ [~ + n T (k Vq~) + H (r - ~bo)] dF = o q
(7.12) for arbitrary functions &p. Integration by parts (see Appendix G) will result in the following weak statement
L [(VS~b)T(k Vq~)+ 6~b Q] J~2
/.
/.
/,
dr2 + L
to + H
~o>] dF
+ L 8q~qndF
Jl" q
=0
,]l
(7.13) Generally, the last term is omitted by requiting &p = 0 and imposing the forced (Dirichlet) boundary condition (7.7a) on 1-'q. It is also possible to express an integral form for the quasi-harmonic equation as a variational principle. The functional rI
---
j [1
~ (~7~) T
]
(k V4~) + 4~ Q dr2 +
~b0 + H (~
~2 _
~0)
]
dF
(7.14)
q
gives on minimization [subject to the constraint of Eq. (7.7a)] the original problem in Eqs (7.6) and (7.7b). The algebraic manipulations required to verify the above principle follow precisely the lines of Sec. 3.8. Clearly material properties defined by the k matrix can vary from element to element in a discontinuous manner. This is implied in both the weak and variational statements of the problem.
The finite element solution process follows the standard solution methodology and for the quasi-harmonic equation approximates the trial function using any of the Co shape function expressions given in Chapters 4 and 5. Accordingly, we use
"~ ~ -- Z
Na ~ a
-- N r
(7.15)
a
in either the weak formulation of Eq. (7.13) or the variational statement of Eq. (7.14). If, in the weak statement, we take
8cp~8~=ZWaS~a=WS(6 d
with W = N
(7.16)
234 Field problems according to the Galerkin principle, an identical form will arise with that obtained from the minimization of the variational principle. The gradient of 4~ is now given by the approximation V ~ --
Z(VNa)
a
[
~)a
ONa ONe]v
ON.
= ~a
-~X '
-~y '
OZ
(7.17)
~a = Z b a ~ a a
gradientmatrix
where now ba denotes the of shape functions. Substituting Eqs (7.15) to (7.17) into (7.13), we have a typical statement for an arbitrary 6~a giving, for each a (assuming summation convention for b),
{ IL bTkb9dK2+ Lq NaHNbdF1 ~b+fNaQd~'2+frq Na(~I-H~)o)dF} ----0 (7.18/ Evaluating the integrals for all elements leads to the set of standard discrete equations of the form
H~+f
-0
(7.19)
with
Hab=fbTakbbd~+fr NaHNbdF
q
and
fa-fNaQd~2+fr Na(~I-HcPo)dF
q
(7.20) to which prescribed values of ~ have to be imposed on boundaries 1-'~. We note that an additional 'stiffness' is contributed on boundaries for which a radiation constant H is specified. Indeed, standard operations are followed to evaluate the above arrays using quadrature. In the general three-dimensional case using Lagrange or serendipity-type 'brick' elements, use of Gauss quadrature results in
L3
nab : Z ba (~l, Ol, (l)Tkbb(~I, OI, (l) J (~l, Ol, (l) Wl /=1 L2
+ Z Na (~l, Ol) H Nb (~l, 17l)j (~l, 17l)Wl /=1
fa.
with a similar expression for Indeed in a computer program the same standard operations are followed to evaluate the fluxes using
q --- - k V4~ ~ - k Z bb ~b
(7.21)
b
The fluxes may be computed within the elements; however, it is often desirable to obtain their values at nodes. This is best accomplished by the procedure summarized in Sec. 6.4 and discussed in more detail later in Chapter 13.
Finite element solution process 235
7.3.2 Two-dimensional plane and axisymmetric problem The two-dimensional plane case is obtained by taking the gradient in the form
V= and taking the flux as q =
Ox'
(7.22)
Oy
{qx} [,xx,xy -- -
qy
kyx
kyy.
04~
(7.23)
_ff_fy
On discretization by Eqs (7.15) to (7.17) a slightly simplified form of the matrices will now be found with Da in Eq. (7.17) replaced by
ba --
Ox '
Oy
(7.24)
and the volume element by d~2 = t dx dy where t is the slab thickness. Alternatively the formulation may be specialized to cylindrical coordinates and used for the solution of axisymmetric situations by introducing the gradient 0
V=
-~r'
0] T
Oz
(7.25)
where r, z replace x, y to describe both the gradient and ba. With the flux now given by
q =
{qr} rrr 1 = -
qz
Lkzr kzzJ
(7.26)
Oqb
the discretization of Eq. (7.18) is now performed with the volume element expressed by dr2 = 2zrr dr dz and integration carried out using quadrature as described above.
Example 7.1: Plane triangular element with 3 nodes. We particularize here to the simplest triangular element (Fig. 7.2). With shape functions written in the alternative forms
aa -[- bax "Jr-Cay
Na=La
2A
in which A is defined in (4.26) and aa, ba, Ca in (4.28), we can compute the derivatives as
ONa OX
OLa OX
ba 2A
ONa Oy
OLa Oy
Ca 2A
236
Field problems
qn
r
Fig. 7.2 Division of a two-dimensional region into triangular elements.
giving the gradient matrix
1 [ba Ca]T
ba = ~-S
Since the gradient matrix is constant the element 'stiffness' matrix (ignoring the H boundary term) is given by
c1c2 C2C3 cic3] bib2 bzb3 bab3 -t- --~ kyyt [Cc~Ccl C2C2 bzb2 4A Lb3bI b3b2 b3b3 LC3C1 C3C2 C3C3 b1c2 blc3 kyx, [:;~: Clb2 c2b2 Clb31 c2b3/ kxyt rblCl /bzc I b2c2 b2c3 -t- - ~
He = kxxt rblbl |bzbl
+ - ~ Lb3cl
LC3bl c3b2 c3b3J
b3c2 b3c3
The load matrices follow a similar simple pattern and thus, for instance, due to constant Q and using a 1-point quadrature from Table 5.3 we have La : 1/3 so that
1
fe : -LaQtA = - ~ Q t A This is a very simple (almost 'obvious') result.
Example 7.2: 'Stiffness' matrix for axisymmetric triangular element with 3 nodes. The computation of the arrays for an axisymmetric problem may be performed using area coordinates as described in Sec. 4.7.1 and quadrature in Sec. 5.11. Since the integrals for
Partial discretization-transient problems 237 the 'stiffness' matrix only involve a linear function in r (from the volume element) a 1-point integration from Table 5.3 still is exact and results in
He ._
§ wlaere ~ =
{
]
bib2 bib3 kzz krr rb,bl /b2b I b2b2 b2b3 -Jr-- ~ 4-A Lb3bl b3b2 b3b3
FblCl blc2 kzr krz ib2c 1 b2c2 b2c3/ --kLb3c1 b3c2 b3c31
CLC2CC3]
[~12~11 C2C2 C2C3 LC3Cl c3c2 c3c3
I; ~11 Lc3bl
c2b2Clb2 c2b3Clb3]} 2rr ? c3b2 c3b3
(rl -k-r2 -k-r3)/3.
Example 7.3" Load matrix for axisymmetric triangular element with 3 nodes. The nodal forces from a constant source term Q are computed from
fa = fA La Q2rcrbLb dr dz
sum on b
and thus now has quadratic terms. From Table 5.3 use of a 3-point formula is adequate to obtain an exact result. For node 1 this gives fl e = 89
[1 (rl -+- r2) -f- ~(rl 1 -+- r3) 1 A g1 = l(2rl + r 2 + r3)zrQA
with results for f~ and f3e obtained by cyclic permutation. The use of a 1-point formula gives results which are of the same accuracy as that of the basic linear functions in the approximation of 4), namely, O(h 2) where h is the diameter of an element. Using this we obtain the force array fe~,~1 2zr?QA = g2 rr~QA
The above developments have assumed that the solution to the problem is independent of time. Many problems, however, require the solution to depend explicitly on time, both in the loading and in the differential equation. An example of a problem which is time dependent is a heat conduction problem in which the loading varies with time. The solution for the temperature now requires use of the differential equation given by c
OT 0t
V T (k V T) + Q = 0
(7.27)
where T is temperature (which now replaces (p), k the thermal conductivity, c the specific heat per unit volume and Q a heat source term. In addition to boundary conditions of the form given in (7.7a) and (7.7b) it is now necessary to provide the distribution of temperature at the initial time T(x, y, z, O) = To(x, y, z) (7.28)
238 Field problems Extending the method used to develop (7.13), a weak form t of the time dependent problem is given by 3T c 0----~+ (V3T)T (k V T ) + 3TQ
dr2 +
6T (71 + H ( T - To)) d r - o q
(7.29)
where we require ~ T = 0 and T = 7' on FT.
7.4.1 Finite element discretizations A finite element solution of (7.29) is constructed using an approximation of the type given in Sec. 3.5 where now we assume the separable form T(x, y, z, y) ,~ T ( x , y, z, t) - Na(x, y, z) Ta(t)
(7.30)
With this form the spatial derivatives are associated with the shape functions Na and the time derivative with the parameters 1"a. Substituting (7.30) into (7.29) yields the semi-discrete set of ordinary differential equations
dT C - : - + H'I~ + f = 0 at or for node a
(7.31)
dTb
Cab---~ + nab Tb -1I- L - - 0
where Hab and fa are given by (7.20) and Cab = f ~ NacNb dg2
In Chapters 16 and 17 we shall discuss in more detail methods of solution for large sets of equations of the form (7.31). Here, however, we consider a simple procedure in which the time dependence is given by a finite difference approximation. We will approximate the nodal temperatures at a time tn by
~r ( tn ) .~ qr. and the time derivative by
dT
~ ( T n - Tn-1) At dt t=t~ where At = tn - tn-1. An approximate solution to the semi-discrete equations at each time tn is obtained by solving the set of equation 1 c + H 1 T" = 1AtC S7
t,,-1 - f
(7.32)
If the initial condition is approximated as T(x, 0) ~ N(x) ~i'(0) with ~i'(0) = I'0 a solution for'i'l is immediately available from (7.32) by solving a set of algebraic equations. For each subsequent time step the solution process is identicalto the time independent t Note that no variational principle of the type (7.14) exists.
Numerical examples- an assessment of accuracy 239 /
2
(a)
(b)
S /
(c)
Fig. 7.3 'Regular' and 'irregular' subdivision patterns. problem except for the modified force vector and a need to use a coefficient matrix which has a term inversely proportional on the size of the time increment.
In Sec. 3.3, Example 3.6, we showed that by assembling explicitly worked-out 'stiffnesses' of triangular elements for the 'regular' mesh pattern shown in Fig. 7.3(a) the discretized equations are identical with those that are derived by well-known finite difference methods. The same result holds for the mesh pattern shown in Fig. 7.3(b). 7 For cases where all boundary conditions are given as prescribed values 4~=~
onF~
the solutions obtained by the two methods obviously will be identical, and so also will be the orders of approximation. However, if the mesh shown in Fig. 7.3(c) which is also based on a square arrangement of nodes but with 'irregular' element pattern is used a difference between the two approaches for the 'load' vector fe will be evident. The assembled equations will have the same 'stiffness' matrix as in Fig. 7.3(a) but will show 'loads' which differ by small amounts from node to node, but the sum of which is still the same as that due to the finite difference expressions. The solutions therefore differ only locally and will represent the same averages. Further advantages of the finite element process are:
240 Field problems 1. It can deal simply with non-homogeneous and anisotropic situations (particularly when the direction of anisotropy is variable). 2. The elements can be graded in shape and size to follow arbitrary boundaries and to allow for regions of rapid variation of the function sought, thus controlling the errors in a most efficient way (viz. Chapters 13 and 14). 3. Specified gradient or 'radiation' boundary conditions are introduced naturally and with a better accuracy than in standard finite difference procedures. 4. Higher order elements can be readily used to improve accuracy without complicating boundary conditions- a difficulty always arising with finite difference approximations of a higher order. 5. Finally, but of considerable importance in the computer age, standard programs may be used for assembly and solution.
7.5.1 Torsion of prismatic bars The torsion of prismatic elastic bars may be solved using a quasi-harmonic equation formulation. Here either a warping function or a stress function approach may be used. In Fig. 7.4(a) we show a rectangular bar loaded by an end torque Mt. The analysis is performed on the cross-section as shown in Fig. 7.4(b). The use of a warping function is governed by the formulation in which displacements are given as u = -yzO;
v = xzO
(7.33)
and w = ~ ( x , y ) O
where x, y are coordinates in the cross-section and z is a coordinate of the bar axis; 0 is the rate of twist and gr the warping function. The non-zero strain components resulting from these displacements are given by z
Mt I
b
-"""~
I
y r
(a) Fig. 7.4 Torsion of rectangular prismatic bar.
(b)
X
Numerical examples - an assessment of accuracy
yxz = 0
( 0--~x ~) -y
and
yyz -- O
( 0-~y ~ ) -t-- x
(7.34)
giving, for an isotropic elastic material, the stresses rxz = G yxz and ryz = G yyz
(7.35)
Inserting the stresses into the equilibrium equation gives the governing differential equation 0--~
~x
+ ~y
G~y
=0
(7.36)
and for stress-free boundary conditions (7.37)
rnz = nx rxz + ny ry z -- 0
in which nx and ny are the direction cosines for the outward normal to the boundary of the rectangular section. At least one value of the warping function must be specified to have a unique solution. The total torque acting on a cross-section is given by Mt = fA [--rxz Y + ryz X] dA =
(7.38)
/A [
G xZ+y2-Y-~x
+x
dAO=GJ~O
where G Jo is the effective torsional stiffness. A stress function formulation is deduced using the representation for stresses rxz =
0r
0y
0~
and ryz = 0--~-
(7.39)
Combining (7.36) and (7.37) with (7.39) and eliminating the warping function ~ gives the differential equation
~x ~~x +~y
+20=0
(7.40)
with 4~(s) = Constant on 1-'q
(7.41)
representing a stress-free boundary condition. The total torque acting on a cross-section is now given by Mr=
G X-~x + y
dAO=GJepO
(7.42)
where G J~ is the effective torsional stiffness. The two solutions provide a bound on the torsional stiffness with the warping function solution giving an upper bound, G J~, and the stress function a lower bound, G J~.
241
242
Field problems
42
,
~o)'
/
/
47931 (
/
5317
/
1
2
(a)
/
(b)
Fig. 7.5 Torsion of a rectangular shaft. Numbers in parenthesesshow a more accurate solution due to Southwell using a 12 x 16 mesh (values of ~/GOL=).
Example 7.4: Torsion of rectangular shaft. In Fig. 7.5 a test comparing the results obtained on an 'irregular' mesh of 3-node triangular elements with a relaxation solution of the lowest order finite difference approximation is shown. Both give results of similar accuracy, as indeed would be anticipated. In general superior accuracy is available with the finite element discretization. Furthermore, it is possible to get bounds on the torsional stiffness, as indicated above. To illustrate this latter aspect we consider a square bar which is solved using 4-node rectangular elements and a range of n x n meshes in which n is the number of spaces between nodes on each side. The results for the computed torsional stiffness values are plotted in Fig. 7.6. The improvement in the rate of convergence for higher order elements may also be illustrated by comparing the total error using 4-node and 9-node elements of lagrangian type. A very accurate solution is computed from the series solution given in reference 8 and used to compute the error in the finite element solution (see Fig. 7.7).
Example 7.5: Torsion of hollow bimetallic shaft. The pure torsion of a non-homogeneous rectangular shaft with a circular hole is illustrated in Fig. 7.8. In the finite element solution presented, the hollow section is represented by a material for which G has a value of the order of 10 -3 compared with the other materials.t The results compare well with the contours derived from an accurate finite difference solution. 9
7.5.2 Transient heat conduction Example 7.6: Transient heat conduction of a rectangular bar. In this example we consider the transient heat conduction in a long square prism with sides L x L and subjected to a rate of heat generation "~This was done to avoid difficulties due to the 'multiple connection' of the region and to permit the use of a standard program.
Numerical examples- an assessmentof accuracy 243
Fig. 7.7 Rate of convergence for square bar. 4- and 9-node lagrangian elements.
Q = Q0 e -at The problem is identical to the one considered in Sec. 3.5 where shape functions are assumed in a cosine form given by Eq. (3.57). Here, however, we use a standard finite element solution with 4-node square elements. The transient solution is performed using
244
Field problems
Fig. 7.8 Torsion of a hollow bimetallic shaft. ~p/GOL2 x 10 4.
the procedure given in Sec. 7.4.1. For the analysis we assume the following parameters: L=c=Q0=ot=l
and k =
0.75 7g 2
Using symmetry conditions, a mesh of 20 • 20 4-node elements is used to approximate one quadrant of the domain. A constant increment in time, At -- 0.01, is used to perform the solution. Results for the temperature at the centre of the prism are given in Fig. 7.9 and compared to the series solutions computed in Sec. 3.5, Fig. 3.9.
Transient heat conduction o f a rotor blade
In Fig. 7.10 we show some results for the transient temperature distributrion in a turbine rotor blade. The blade is subjected to a hot gas at 1145C ~ applied to the outer boundary in which a variable radiation constant H = ct is employed. Cooling is introduced in the internal ducts. The analysis is performed using cubic elements of serendipity type which permit the representation of the boundaries using very few elements.
7.5.3 Anisotropic seepage The next problem is concerned with the flow through highly non-homogeneous, anisotropic, and contorted strata. The basic governing equation is
Ox'
kx' x' -~x'
+ -~y' ky, y,
=0
(7.43)
in which H is the hydraulic head and kx,x, and ky,y, represent the permeability coefficients in the direction of the (inclined) principal axes. However, a special feature has to be incorporated to allow for changes of x' and y' principal directions from element to element.
Numerical examples - an assessment of accuracy 0.35
I
o M,N=I
[
[] M , N = 3
0.3 [- ........... !.... j . ~ ~ i
/
0.25
~
"~ ~
ii
n o [x denotes ~- (3 DOF)and o the 6 (2 DOF)variables].
It is of interest to note that if a higher order of interpolation is used for tr than for u the patch test is still satisfied, but in general the results will not be improved because of the principle of limitation. We do not show the similar patch test for the Co continuous No assumption but state simply that, similarly to the example of Fig. 10.3, identical interpolation of No and Nu is acceptable from the point of view of stability. However, as in Fig. 10.4, restriction of excessive continuity for stresses has to be avoided at singularities and at abrupt material property change interfaces, where only the normal and tangential tractions are continuous. The disconnection of stress variables at corner nodes can only be accomplished for all the stress variables. For this reason an alternative set of elements with continuous stress nodes at element interfaces can be introduced (see Fig. 10.6). 21 In such elements excessive continuity can easily be avoided by disconnecting only the direct stress components parallel to an interface at which material changes occur. It should be noted that even in the case when all stress components are connected at a mid-side node such elements do not ensure stress continuity along the whole interface. Indeed, the amount of such discontinuity can be useful as an error measure. However, we observe that for the linear element [Fig. 10.6(a)] the interelement stresses are continuous in the mean.
Two-field mixed formulation in elasticity
+
linear
(a)
u linear
_
T,
(~nt~'
/ T "r'x(~xx y Onn~ ~YY
(b)
/ ~tt
Fig. 10.6 Elasticity by the mixed cr-u formulation. Partially continuous ~r (continuity at nodes only). (a) ~r linear, u linear. (b) Possible transformation of interface stresses with ~rtt disconnected.
It is, of course, possible to derive elements that exhibit complete continuity of the appropriate components along interfaces and indeed this was achieved by Raviart and Thomas 22 in the case of the heat conduction problem discussed previously. Extension to the full stress problem is difficult 23 and as yet such elements have not been successfully noted.
Example 10.3: Pian-Sumihara rectangle. Today very few two-field elements based on interpolation of the full stress and displacement fields are used. One, however, deserves to be mentioned. We begin by first considering a rectangular element where interpolations may be given directly in terms of cartesian coordinates. A 4-node plane rectangular element with side lengths 2a in the x direction and 2b in the y direction, shown in Fig. 10.7, has ,~kY A
b
i i
:F-I
X
(Xo, yo)
b
.J_ "T Fig. 10.7 Geometry of rectangular ~ - u element.
367
368
Mixed formulation and constraints- complete field methods
displacement interpolation given by 4
U = ~ Na (x, y)fla a=l
The shape functions are given by
1(
Nl(X, y) = -~ 1 1(
N3(x, y) = -~ 1 +
X - - X o ) ( Y 1- - Y o ) a
x-xo)(
1+
a
1(
b
Y-Yo) b
" N2(x, y) = ~ 1+ ' 1(
" Nn(x y) -- -~ 1
'
X - - X o ) ( Y 1- - Y o ) a
x-xo)(
'
a
Id-
b
Y-Yo) b
in which x0 and Y0 are the cartesian coordinates at the element centre. The strains generated from this interpolation will be such that
8x - -
/71 d - t l 2 y ;
e y - - /73 +
/']4X;
Yxy --- 175+ 06x + O7Y
rlj are expressed in terms of ~. For isotropic linear elasticity problems these strains will lead to stresses which have a complete linear polynomial variation in each element (except for the special case when v = 0). Here the stress interpolation is restricted to each element individually and, thus, can be discontinuous between adjacent elements. The limitation principle restricts the possible choices which lead to different results from the standard displacement solution. Namely, the approximation must be less than a complete linear polynomial. To satisfy the stability condition given by Eq. (10.18) we need at least five stress parameters in each element. A viable choice for a five-term approximation is one which has the same variation in each element as the normal strains given above but only a constant shear stress. Accordingly,
where
{ax}
[i
tTy
"-"
"Cxy
0 0 1 0
0 1
Y-Yo
0
0
X -- X0
0
0
_
Otl
]
0l 2 0l 3
Ol4 015
Indeed, this approximation satisfies Eq. (10.18) and leads to excellent results for a rectangular element.
Example 10.4: Pian-Sumihara quadrilateral. We now rewrite the formulation given in Example 10.3 to permit a general quadrilateral shape to be used. The element coordinate and displacement field are given by a standard bilinear isoparametric expansion 4
x= ~ a=l
4
Na (~, r/)~a and fi = y ~ Na (~, 7)fia a=l
where now ma(~, ~) = 1(1 + ~a~)(1 +/~a~)
in which ~a and r/a are the values of the parent coordinates at node a.
Two-field mixed formulation in elasticity The problem remains to deduce an approximation for stresses for the general quadrilateral element. Here this is accomplished by first assuming stresses E on the parent element (for convenience in performing the coordinate transformation the tensor form is used, see Appendix B) in an analogous manner as the rectangle above:
~--](~: T]).__ F~]~ ]~r/] IO/1 "~'-0~4r] ' L][]r/~ ][]r/r/J -013
013 ] 13/2 + 0/5~
In the above the parent normal stresses again produce constant and bending terms while shear stress is only constant. These stresses are then transformed to cartesian space using o" = TTN(~, r/)T It remains now only to select an appropriate form for T. The transformation must 1. produce stresses in cartesian space which satisfy the patch test (i.e., can produce constant stresses and be stable); 2. be independent of the orientation of the initially chosen element coordinate system and numbering of element nodes (invariance requirement). Pian and Sumihara 24 use a constant array (to preserve constant stresses) deduced from the jacobian matrix at the centre of the element. Accordingly, with
Ox
1,O,ll ,o,11 Jo =
lJo,21
Jo,22J
=
Ox
o~
Oy Oy
~
q,o=o
the elements of the jacobian matrix at the centre are given by [see Eq. (5.11)] 1 1 J0,11 -" "~X a ~ a Jo,12 "- -~X a T]a 1 1 J0,21 -- -~Ya ~a J0,22 "- -~Ya Oa Using T = J0 gives the stresses (in matrix form)
O'y "rxy
--
I~2 (~3
"nt-
i
Jg, 12~
1
J02,21 r/
LJo, 12 Jo,21 t ]
Jo, 12 Jo,22~J
where the parameters (~i, i -- 1, 2, 3, replace the transformed quantities for the constant part of the stresses. This approximation satisfies the constant stress condition (Condition 1) and can also be shown to satisfy the invariance condition (Condition 2). The development is now complete and the arrays indicated in Eq. (10.31) may be computed. We note that the integrals are computed exactly for all quadrilateral elements (with constant D) using 2 x 2 gaussian quadrature. An alternative to the above definition for T is to use the transpose of the jacobian inverse at the centre of the element (i.e., T = JOT). This has also been suggested recently by several authors as an invariant transformation. However, as shown in Fig. 10.8, the sensitivity to element distortion is much greater for this form than the original one given by Pian and Sumihara for the above two-field approximation. The other two options (e.g., T = J~ and T = Jo 1) do not satisfy the frame invariance requirement, thus giving elements which depend on the orientation of the element with respect to the global coordinates.
369
370
Mixed formulation and constraints- complete field methods
21
\
i 0.5 ~1~, -9- i - ~
5 5 E=75, v=0
1.0 I
~- 0.5
~1 v I
0.8
.......-.-.-o 0.6
0.4 -
'i~ ,
o P-S:J [] P-S: J-inverse O Q-4
0.2 >-'0.., 0 "-0,.r ~-E3.......43 .... r .... O .......... 0 I I 0 1 2 a
~ I 3
. ........c] I 4
Fig. 10.8 Pian-Sumihara quadrilateral (P-S) compared with displacement quadrilateral (Q-4). Effect of element distortion (Exact -- 1.0).
It is, of course, possible to use an independent approximation to all the essential variables entering the elasticity problem. We can then write the three equations (10.22), (10.23), and (10.24) in their weak form as fl~r
-- t r ) d r 2 = 0
~~Or T(~II -- r dr2 = 0
(10.33)
t
where u - fi on Fu is enforced.t The variational principle equivalent to Eq. (10.33) is k n o w n by the name of H u - W a s h i z u 5 (see Problem 10.1). Introducing the approximations u ~ fi = Nufi
cr ~ & = N ~ #
and
e ~ @ - NE~
(10.34)
with corresponding 'variations' (i.e., the Galerkin form Wu = Nu, etc.) into Eq. (10.33), and writing the approximating equations in a similar fashion as we have in the previous t It is possible to include the displacement boundary conditions in Eq. (10.33) as a natural rather than imposed constraint; however, most finite element applications of the principle are in the form shown.
Three-field mixed formulations in elasticity
[A C
section yields an equation system of the following form: CT
0
o"
0
ET
u
{
--
(10.35)
f2 f3
where A = ~ NTDN~ dr2; fl = t " 2 - 0 ;
f3-
/.
E = f~ NTB dr2;
/.
C:
- f
NTNo d~2 (10.36)
LNuTbdf2+ /_ NuTidF d~l.
dl"
t
The reader will observe that in this section we have developed all the approximations directly without using a variational principle. In Problem 10.2 we suggest that the reader show the equivalence of a development from the variational principle.
10.5.2 Stability condition of three-field approximation (u-or-e) The stability condition derived in Sec. 10.3 [Eq. (10.18)] for two-field problems, which we later used in Eq. (10.32) for the simple mixed elasticity form, needs to be modified when three-field approximations of the form given in Eq. (10.35) are considered. Many other problems fall into a similar category (for instance, plate bending) and hence the conditions of stability are generally useful. The requirement now is that n~ + nu > n,r
-
(10.37)
n,r > nu
This was first stated in reference 25 and follows directly from the two-field criterion as shown below. The system of Eq. (10.35) can be 'regularized' by adding yE times the third equation to the second, with y being an arbitrary constant. We now have
c
yEE T ET
8" O
=
{ fl } t"2 + gEf3 f3
On elimination of r using the first of the above we have
[(~/EET-CTA-1C),ET, ~] (0"), -" (f2"~-~/Ef3-CTA-lfl/f3 From the two-field requirement [Eq. (10.18)] it follows that we require no > nu
for the equation system to have a solution. To establish the second condition we rearrange Eq. (10.35) as
(10.38)
371
372
Mixed formulation and constraints - complete field methods
This again can be regularized by adding multiples ?'C and y E T of the third of the above equations to the first and second respectively obtaining YETCT,
YETE E
C T,
fi ~
--
f3 + vETf2 f2
By partitioning as above it is evident that we require (10.39)
n~ + nu > no
We shall not discuss in detail any of the possible approximations to the 6-or-u formulation or their corresponding patch tests as the arguments are similar to those of two-field problems. In some practical applications of the three-field form the approximation of the second and third equations in (10.33) is used directly to eliminate all but the displacement terms. This leads to a special form of the displacement method which has been called a B (B-bar) form.26, 27 In the 171form the shape function derivatives are replaced by approximations resulting from the mixed form. We shall illustrate this concept with an example of a nearly incompressible material in Sec. 11.4.
10.5.3 The U--O'--Sen form. Enhanced strain formulation In the previous two sections the general form and stability conditions of the three-field formulation for elasticity problems are given in Eqs (10.32) and (10.37). Here we consider a special case of this form from which several useful elements may be deduced. In the special form considered the strain approximation is split into two parts: one the usual displacement-gradient term and, second, an added or enhanced strain part. Accordingly, we write : S U "4- ~en ~ --- ~ ( S U ) + ~ e n (10.40) Substitution into Eq. (10.33) yields the weak forms as
~
~ (SU) T
f
(D ( S u +
E:en) -- or)
dg2 = 0
3~eT (D (SU + ~en) -- or) dg2 -- 0 (10.41) / ~ ~iorT~en d ~ = 0
/, (Su)Toraft.-/,uTbd~-/r ,uT{dF= 0 t
where, as before, u = fi is enforced on Fu. We can directly discretize Eq. (10.41) by taking the following approximations u ~ fi - Nufi
or ,~ & = N~rr
~en ~'~ ~en :
Nen~en
(10.42)
with corresponding expressions for variations. Substituting the approximations into Eq. (10.41) yields the discrete equation system
[AoT T'1}oo i]{'en} "0 {
(10.43)
Three-field mixed formulations in elasticity where A =
NenD Nen d~2;
LT
C -- -
NenN~rdr2;
K--/fBTDBdff2;
G :
/T
NenD B d ~
f,-~NTbdff2+fr NTtdF
fl = t " 2 = 0 ;
(10.44)
t
In this form there is only one zero diagonal term and the stability condition reduces to the single condition nu +nen _> n~r (10.45) Further, the use of the strains deduced from the displacement interpolation leads to a matrix which is identical to that from the irreducible form and we have thus included this in Eq. (1 0.44) as K.
Example 10.5: Simo-Rifai quadrilateral. An enhanced strain formulation for application to problems in plain elasticity was introduced by Simo and Rifai. 28 The element has 4 nodes and employs isoparametric interpolation for the displacement field. The derivatives of the shape functions yield a form
~Na aNa
=
ax,a(Yb) -4;-bx,a(Yb)~ "[-Cx,a(Yb)rl) J(~, o)
ay,a(Xb) -J- by,a(Xb)~ + Cy,a(Xb)r]
where aa, ba and C a depend on the nodal coordinates, and the jacobian determinant for the 4-node quadrilateral is given byt d e t J = J(~, O) = Jo + J ~ + Jorl The enhanced strains are first assumed in the parent coordinate frame and transformed to the cartesian frame using a transformation similar to that used in developing the PianSumihara quadrilateral in Example 10.4. Due to the presence of the jacobian determinant in the strains computed from the displacements (as well as the requirement to later pass the patch test for constant stress states) the enhanced strains are computed from ~en --
I
J (~, O)
TTE(~, r/)T
where E-
F
LE.~
Er
E,,TJ
In matrix form this may be written as
•y
---
Yxy en
J(~i O)
T2 2TilT12
T2
T12T22
]{} , |
2T21T222 TilT22 + T12T21J
Eoo
2Er176
t In general, the determinantof the jacobian for the two-dimensionalLagrangefamilyof elementswill not contain the term with the product of the highest order polynomial, e.g., ~r/for the 4-node element, ~2172for the 9-node element, etc.
373
374
Mixed formulation and constraints-complete field methods The parent strains (strains with components in the parent element frame) are assumed as
Eo0
=
0
2E~0
0
0
772 773
~ 77 /74
0
The above is motivated by the fact that the derivatives of the shape functions with respect to parent coordinates yields
ONa = ~
ONo = oo
a~ + b~ 0
a,7 + b,7~
and these may be combined to form strains in the usual manner, but in the parent frame. Thus, by design, the above enhanced strains are specified to generate complete polynomials in the parent coordinates for each strain component. References 29 and 30 discuss the relationship between the design of assumed stress elements using the two-field form and the selection of enhanced strain modes so as to produce the same result.
Remarks
1. The above enhanced strains are defined so that the C array is identically zero for constant assumed stresses in each element. 2. Parent normal strains have linearly independent terms added. However, the assumed parent shear strains are linearly dependent. Due to this linear dependence the final sheafing strain will usually be nearly constant in each element. Accordingly, to be more explicit, normal strains are enhanced while sheafing strain is de-enhanced. Since the C array vanishes, the equation set to be solved becomes fl [GAT GI /Ee~n/~If3/ and in this form no additional count conditions are apparently needed. The solution may be accomplished partly at the element level by eliminating the equation associated with the enhanced strain parameters. Accordingly, K*fi = f ~ where K* = K - GTA-1G
and
f~ -- t"3 - GTA-lfl
The sensitivity of the enhanced strain element to geometric distortion is evaluated using the problem shown in Fig. 10.9. The transformation from the parent to the global frame is assessed using T = J0 and T = Jo T. These are the only options which maintain frame invafiance for the element. As observed in Fig. 10.9 the results are now better using the inverse transpose. Since the stress and strain are conjugates in an energy sense, this result could be anticipated from the equivalence relationship
1L tr Te dr2 -
E = ~
-2
~2TEd [7
Complementary forms with direct constraint
where E is energy and D denotes the domain of the element in the parent coordinate system (i.e., the bi-unit square for a quadrilateral element). The performance of the enhanced element is compared to the Pian-Sumihara element for a shear loading on the mesh shown in Fig. 10.10. In Fig. 10.11 the convergence results for various order meshes are shown for linear elastic, plane strain conditions with: (a) E = 70 and v = 1/3 and (b) for E = 70 and v = 0.499995. The results shown in Fig. 10.11 clearly show the strong dependence of the displacement formulation on Poisson's ratio - namely the tendency for the element to lock for values which approach the incompressibility limit of v = 1/2. On the other hand, the performance of both the enhanced strain and the Pian-Sumihara element are nearly insensitive to the value of Poisson's ratio selected, with somewhat better performance of the enhanced element on coarse meshing.
In the introduction to this chapter we defined the irreducible and mixed forms and indicated that on occasion it is possible to obtain more than one 'irreducible' form. To illustrate this in the problem of heat transfer given by Eqs (10.2) and (10.3) we introduced a penalty function ot in Eq. (10.6) and derived a corresponding single governing equation (10.7) given in terms of q. This penalty function here has no obvious physical meaning and served simply as a device to obtain a close enough approximation to the satisfaction of the continuity of flow equations. On occasion it is possible to solve the problem as an irreducible one assuming a priori that the choice of the variable satisfies one of the equations. We call such forms directly constrained and obviously the choice of the shape function becomes difficult. We shall consider two examples.
1.0 0.8 0 v
0.6 i
0.4 'X b"
0.2
~-~>... " ~ , 0
I 1
o S-R:J [] S-R: J-inverse ~ Q-4
" " C , . . . . ~, . . . . . . . . . . I 2 a
C, I 3
I 4
Fig. 10.9 Simo-Rifai enhanced strain quadrilateral (S-R) compared with displacement quadrilateral (Q-4). Effect of element distortion (Exact - 1.0).
375
376
Mixed formulation and constraints- complete field methods
"l
48
L~
I
J~
16
44
!' t t t |
,,.__1
F
F/8 1=/4 F/4 FI4 FI8
,
(b) Mesh and nodal loads
(a) Geometry Fig. 10.10 Mesh with 4 x 4 elements for shear load.
35 30
f
35-
o
30-
25 . ~ , , , , , , , , , , - - - - - ~ O
25
w,"
.-- 20 ~" 15 ~, v
o
0
~
I 10
I I I I 20 30 40 50 n-elements/side (v = 1/3)
20
o o S-R o.-- - .----o P-S ............o Q-4
=" 15
0 ............~ Q-4
10 0
o S-R
D - - - - - - - o P-S
10
I 60
0
~ . - 0 .......~ ................0 ....................................
0
I
I
I
I
I
I
10 20 30 40 50 60 n- elements/side (v = 0.499995)
Fig. 10.11 Convergence behaviour for: (a) v = 1/3; (b) v = 0.499995.
The complementary heat transfer problem
In this we assume a priori that the choice of q is such that it satisfies Eq. (10.3) and the natural boundary conditions V T q -- - Q in f2
and
qn = q Tn : On o n ~ q
(10.46)
where n is the unit normal to the boundary. Thus we only have to satisfy the constitutive relation (10.2), i.e., k - l q + V~p = 0 in f2
with
~p = ~ on FO
(10.47)
A weak statement of the above is f
,~qT(k-aq + V ~ ) dr2 - f r 'qn(q~ - ~) dF - 0 r
(10.48)
Complementary forms with direct constraint 377
in which 3 q n - - 3qTn represents the variation of normal flux on the boundary. Use of Green's theorem transforms the above into ~SqTk-lqdf2-f
VTBq~pdf2+fr 8 q n @ d F + f r ' q n ~ p d F - - 0 r
(10.49)
q
If we further assume that ~7 T*q ~ 0 in if2 and 3qn = 0 on Fq, i.e., that the weighting functions are simply the variations of q, the equation reduces to
'qn~dF--O
L ,qTk-lqdf2 + f r
s
(10.50)
This is in fact the variation of a complementary flux principle
5q k
q da +
qn~ dr'
(10.51)
Numerical solutions can obviously be started from either of the above equations but the difficulty is the choice of the trial function satisfying the constraints. We shall return to this problem in Sec. 10.6.2.
The complementary elastic energy principle
In the elasticity problem specified in Sec. 10.4 we can proceed similarly, assuming stress fields which satisfy the equilibrium conditions both on the boundary 1-'tand in the domain f2. Thus in an analogous manner to that of the previous example we impose on the permissible stress field the constraints which we assume to be satisfied by the approximation identically, i.e., (10.52) ,sT0. + b -- 0 in f2 and t - t on I" t Thus only the constitutive relations and displacement boundary conditions remain to be satisfied, i.e., D-10. - ,Su - 0 in f2 and u - ~ on Fu (10.53) The weak statement of the above can be written as /_ 8o'T(D-lor -- Su)dr2 + f
8 t T ( u - fi)dF = 0
(10.54)
all' u
d~2
which on integration by Green's theorem gives f ,~rTD-IordQ + f ( S T , o - ) T u d Q - - f r , t T ~ d F - f r u
,tTudF = 0
(10.55)
t
Again assuming that the test functions are complete variations satisfying the homogeneous equilibrium equation, i.e., 8T30" =
0 in f2
and
3t -- 0 on Ft
(10.56)
we have as the weak statement L 30.TD-10. aft2 - ~
,tTfi dI" = 0
(10.57)
u
The corresponding complementary energy variational principle is FI = ~1 ~ o.TD-1 0. dr2 -
tT~ dF
(10.58)
u
Once again in practical use the difficulties connected with the choice of the approximating function arise but on occasion a direct choice is possible. 31
378
Mixedformulation and constraints-complete field methods
10.6.2 Solution using auxiliary functions Both the complementary forms can be solved using auxiliary functions to ensure the satisfaction of the constraints.
Example 10.6: Heat transfer solution by potential function. In the lem it is easy to verify that the homogeneous equation
heat transfer prob-
VTq -- -~x Oy = 0 Oqx + OqY
(10.59)
is automatically satisfied by defining a function ~p such that qx =
00
00
qy "-
Oy
(10.60)
Ox
Thus we define and
q= gTr+q0
6q = s
(10.61)
where q0 is any flux chosen so that VTq0 = - Q
(10.62)
and 12 =
0y
0x
(10.63)
the formulations of Eqs (10.50) and (10.51) can be used without any constraints and, for instance, the stationarity
I-[ =
falg(Z~0+ q0)Tk- 1(gO + q0) df2 - j~F (01~) -~s
~>dF
(10.64)
will suffice to so formulate the problem (here s is the tangential direction to the boundary). The above form will require shape functions for ~ satisfying Co continuity.
Example 10.7: Elasticity solution by Airy stress function. In the elasticity problem a two-dimensional form can be obtained by the use of the so-called Airy stress function ~.32 Now the equilibrium equations 0ax
~ T o" "Jr-
b-
+~oy +bx -fix OTxy -+---~y +by
/
= 0
(10.65)
are identically solved by choosing (10.66)
a = C~+~ro where 12 "--
2
02
Oy 2'
OX 2'
02 1 OX Oy
(10.67)
Concluding remarks - mixed formulation or a test of element 'robustness'
and tr0 is an arbitrary stress chosen so that ,STtr0 + b = 0
(10.68)
Again the substitution of (10.66) into the weak statement (10.57) or the complementary variational problem (10.58) will yield a direct formulation to which no additional constraints need be applied. However, use of the above forms does lead to further complexity in multiply connected regions where further conditions are needed. The reader will note that in Chapter 7 we encountered this in a similar problem in torsion and suggested a very simple procedure of avoidance (see Sec. 7.5). The use of this stress function formulation in the two-dimensional context was first made by de Veubeke and Zienkiewicz 33 and Elias, 34 but the reader should note that now with second order operators present, C1 continuity of shape functions is needed in a similar manner to the problems which we have to consider in plate bending (see reference 6). Incidentally, analogies with plate bending go further here and indeed it can be shown that some of these can be usefully employed for other problems. 35
The mixed form of finite element formulation outlined in this chapter opens a new range of possibilities, many with potentially higher accuracy and robustness than those offered by irreducible forms. However, an additional advantage arises even in situations where, by the principle of limitation, the irreducible and mixed forms yield identical results. Here the study of the behaviour of the mixed form can frequently reveal weaknesses or lack of 'robustness' in the irreducible form which otherwise would be difficult to determine. The mixed approximation, if properly understood, expands the potential of the finite element method and presents almost limitless possibilities of detailed improvement. Some of these will be discussed further in the next two chapters, and others in references 6 and 36.
10.1 Show that the stationarity of the variational principle given by
t
where u - fi on Fu is equivalent to Eq. (10.33). 10.2 Using the variational principle of Problem 10.1 with the approximations (10.34) show that the stationarity condition gives (10.35) and (10.36). 10.3 Show that the variational principle given by stationarity of
lien --- ~ 1 (SU+r
(SU-q-r
d~+fo'T~end~
-- ~ uTbd" - frt uT'dF with u = fi enforced on Fu is equivalent to Eq. (10.41).
379
380
Mixed formulation and constraints- complete field methods 10.4 For the rectangular element shown in Fig. 10.7 develop the expressions for T]i for the Pian-Sumihara element described in Sec. 10.4.3. For an isotropic elastic material and a plane stress problem compute the expressions for the stresses which result from the strains (these are those of the displacement model described in Chap. 6). How do these differ from those assumed for the mixed element? 10.5 For the enhanced strain formulation described in Sec. 10.5.3 use the constant stress patch test for a plane strain problem to show that ~en • O. Show that a necessary condition to satisfy this requirement is
f
Nend~-O e
10.6 Generalize the Simo-Rifai quadrilateral given as Example 10.5 in Sec. 10.5.3 for a three-dimensional solid modelled by 8-node hexahedral elements. 10.7 Generalize the Simo-Rifai quadrilateral given as Example 10.5 in Sec. 10.5.3 for an axisymmetric geometry. 10.8 A plane stress problem has the geometry shown in Fig. 10.11 and is loaded by a uniformly distributed shear traction (i.e., ty = const.). Use FEAPpv to solve the problem using a series of 3-node triangular meshes. The first mesh should be as shown with each quadrilateral divided into two triangles. Consider two values for the elastic properties: (a) E = 70, v = 1/3 and (b) E = 70, v = 0.499995. Let the thickness of the slab be one unit. Next, perform the solution using 4-node quadrilaterals based on (a) the displacement solution described in Chap. 6; (b) the Simo-Rifai enhanced element described in Sec. 10.5.3. Plot the displacement convergence for the top and bottom points at the loaded end. Plot contours for displacement and principal stresses. Repeat the calculations assuming plane strain conditions. Briefly discuss your findings.
1. S.N. Atluri, R.H. Gallagher, and O.C. Zienkiewicz, editors. Hybrid and Mixed Finite Element Methods. John Wiley & Sons, New York, 1983. 2. O.C. Zienkiewicz, R.L. Taylor, and J.A.W. Baynham. Mixed and irreducible formulations in finite element analysis. In S.N. Atluri, R.H. Gallagher, and O.C. Zienkiewicz, editors, Hybrid and Mixed Finite Element Methods, pages 405-431. John Wiley & Sons, 1983. 3. I. Babu~ka and J.E. Osborn. Generalized finite element methods and their relations to mixed problems. SlAM J. Num. Anal., 30:510-536, 1983. 4. R.L. Taylor and O.C. Zienkiewicz. Complementary energy with penalty function in finite element analysis. In R. Glowinski, E.Y. Rodin, and O.C. Zienkiewicz, editors, Energy Methods in Finite Element Analysis, Chapter 8. John Wiley & Sons, Chichester, 1979. 5. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd edition, 1982. 6. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 7. L.R. Herrmann. Finite element bending analysis of plates. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, AFFDL-TR-66-80, pages 577-602, Wright-Patterson Air Force Base, Ohio, 1965.
References 381 8. K. Hellan. Analysis of elastic plates in flexure by a simplified finite element method. Technical Report Civ. Eng. Series 46, Acta Polytechnica Scandinavia, Trondheim, 1967. 9. R.S. Dunham and K.S. Pister. A finite element application of the Hellinger-Reissner variational theorem. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-66-80, Wright Patterson Air Force Base, Ohio, Oct. 1966. 10. R.L. Taylor and O.C. Zienkiewicz. Mixed finite element solution of fluid flow problems. In R.H. Gallagher, G.E Carey, J.T. Oden, and O.C. Zienkiewicz, editors, Finite Elements in Fluids, volume 1, Chapter 4, pages 1-20. John Wiley & Sons, 1982. 11. B. Fraeijs de Veubeke. Displacement and equilibrium models in finite element method. In O.C. Zienkiewicz and G.S. Holister, editors, Stress Analysis, Chapter 9, pages 145-197. John Wiley & Sons, Chichester, 1965. 12. I. Babu~ka. Error bounds for finite element methods. Numer. Math., 16:322-333, 1971. 13. I. Babu~ka. The finite element method with lagrangian multipliers. Numer. Math., 20:179-192, 1973. 14. F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. Rev. Fran9aise d'Automatique Inform. Rech. Opdr., Ser. Rouge Anal. Numgr., 8(R-2):129-151, 1974. 15. O.C. Zienkiewicz, S. Qu, R.L. Taylor, and S. Nakazawa. The patch test for mixed formulations. Int. J. Numer. Meth. Eng., 23:1873-1883, 1986. 16. J.T. Oden and N. Kikuchi. Finite element methods for constrained problems in elasticity. Int. J. Numer. Meth. Eng., 18:701-725, 1982. 17. E. Hellinger. Die allgemeine Aussetze der Mechanik der Kontinua. In E Klein and C. Muller, editors, Encyclopedia der Mathematishen Wissnschafien, volume 4. Tebner, Leipzig, 1914. 18. E. Reissner. On a variational theorem in elasticity. J. Math. Phys., 29(2):90-95, 1950. 19. L.R. Herrmann. Finite element bending analysis of plates. J. Eng. Mech., ASCE, 94(EM5): 13-25, 1968. 20. L.R. Herrmann and D.M. Campbell. Finite element analysis for thin shells. J. AIAA, 6:18421847, 1968. 21. O.C. Zienkiewicz and D. Lefebvre. Mixed methods for FEM and the patch test. Some recent developments. In E Murat and O. Pirenneau, editors, Analyse Mathematique of Application. Gauthier Villars, Paris, 1988. 22. P.A. Raviart and J.M. Thomas. A mixed finite element method for second order elliptic problems. In Lect. Notes in Math., no. 606, pages 292-315. Springer-Verlag, Berlin, 1977. 23. D.N. Arnold, E Brezzi, and J. Douglas. PEERS, a new mixed finite element for plane elasticity. Japan J. Appl. Math., 1:347-367, 1984. 24. T.H.H. Pian and K. Sumihara. Rational approach for assumed stress finite elements. Int. J. Numer. Meth. Eng., 20:1685-1695, 1985. 25. O.C. Zienkiewicz and D. Lefebvre. Three field mixed approximation and the plate bending problem. Comm. Appl. Num. Meth., 3:301-309, 1987. 26. T.J.R. Hughes. Generalization of selective integration procedures to anisotropic and non-linear media. Int. J. Numer. Meth. Eng., 15:1413-1418, 1980. 27. J.C. Simo, R.L. Taylor, and K.S. Pister. Variational and projection methods for the volume constraint in finite deformation plasticity. Comp. Meth. Appl. Mech. Eng., 51:177-208, 1985. 28. J.C. Simo and M.S. Rifai. A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Meth. Eng., 29:1595-1638, 1990. 29. U. Andelfinger and E. Ramm. EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. Int. J. Numer. Meth. Eng., 36:1311-1337, 1993. 30. M. Bischoff, E. Ramm, and D. Braess. A class of equivalent enhanced assumed strain and hybrid stress finite elements. Comput. Mech., 22:443-449, 1999.
382
Mixed formulation and constraints-complete field methods 31. C. Loubignac, G. Cantin, and C. Touzot. Continuous stress fields in finite element analysis. J. AIAA, 15:1645-1647, 1978. 32. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition, 1969. 33. B. Fraeijs de Veubeke and O.C. Zienkiewicz. Strain energy bounds in finite element analysis by slab analogy. J. Strain Anal., 2:265-271, 1967. 34. Z.M. Elias. Duality in finite element methods. Proc. Am. Soc. Civ. Eng., 94(EM4):931-946, 1968. 35. R.V. Southwell. On the analogues relating flexure and displacement of flat plates. Quart. J. Mech. Appl. Math., 3:257-270, 1950. 36. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005.
Incompressible problems, mixed
methods and other procedures of solution
We have noted earlier that the standard displacement formulation of elastic problems fails when Poisson's ratio v becomes 0.5 or when the material becomes incompressible. Indeed, problems arise even when the material is nearly incompressible with v > 0.4 and the simple linear approximation with triangular elements gives highly oscillatory results in such cases. The application of a mixed formulation for such problems can avoid the difficulties and is of great practical interest as nearly incompressible behaviour is encountered in a variety of real engineering problems ranging from soil mechanics to aerospace engineering. Identical problems also arise when the flow of incompressible fluids is encountered. In this chapter we shall discuss fully the mixed approaches to incompressible problems, generally using a two-field manner where displacement (or fluid velocity) u and the pressure p are the variables. Such formulation will allow us to deal with full incompressibility as well as near incompressibility as it occurs. However, what we will find is that the interpolations used will be very much limited by the stability conditions of the mixed patch test. For this reason much interest has been focused on the development of so-called stabilized procedures in which the violation of the mixed patch test (or Babugka-Brezzi conditions) is artificially compensated. A part of this chapter will be devoted to such stabilized methods.
The main problem in the application of a 'standard' displacement formulation to incompressible or nearly incompressible problems lies in the determination of the mean stress or pressure which is related to the volumetric part of the strain (for isotropic materials). For this reason it is convenient to separate this from the total stress field and treat it as an independent variable. Using the 'vector' notation of stress, the mean stress or pressure is given by
p - - 5 (O'x + Cry + o-z)
~mTo 9
(11.1)
384 Incompressibleproblems, mixed methods and other procedures of solution where m for the general three-dimensional state of stress is given by
m--J1,
1,
1,
0,
0,
0] T
For isotropic behaviour the 'pressure' is related to the volumetric strain, ev, by the bulk modulus of the material, K. Thus, e v - - e x -at- e y + E z ---
mTe = P
(11.2)
For an incompressible material K = ~ (v = 0.5) and the volumetric strain is simply zero. The deviatoric strain e a is defined by 1 ~mev -- (I - l m m T ) e
E d --E-
=
Ide
(11.3)
where Id is a deviatoric projection matrix which also proves useful in problems with more general constitutive relations. 1 In isotropic elasticity the deviatoric strain is related to the deviatoric stress by the shear modulus G as tr d = Ido" - - 2 G I o e d
= 2G (Io - ~ m m l T) e
(11.4)
where the diagonal matrix -2 I0=
1
2
1 1 1
-
is introduced because of the vector notation. A deviatoric form for the elastic moduli of an isotropic material is written as Dd = 2G (I0 - l m m x)
(11.5)
for convenience in writing subsequent equations. The above relationships are but an alternate way of determining the stress-strain relations shown in Chapters 2 and 6, with the material parameters related through G= K=
E 2 (1 + v) E
(11.6)
3(1 - 2 v )
and indeed Eqs (11.4) and (11.2) can be used to define the standard D matrix in an alternative manner.
In the mixed form considered next we shall use as variables the displacement u and the pressure p.
Two-field incompressible elasticity (u-p form) Now the equilibrium equation (10.22) is rewritten using (11.4), treating p as an independent variable, as f
6eTDae dr2 + s 6eTmp d r 2 - s 6uTb d r 2 - f r 6uVidP = 0
(11.7)
t
and in addition we shall impose a weak form of Eq. (11.2), i.e., f
Sp
mTe-- P ] d~2 = 0
(11.8)
with e = Su. Independent approximation of u and p as and
u~fi=Nufi
p ~ ~ = Np[~
(11.9)
immediately gives the mixed approximation in the form EA
_v {o}= {ff,}
n p (11.12) and is necessary for prevention of locking (or instability) with the pressure acting now as the constraint variable of the lagrangian multiplier enforcing zero volumetric strain. In the form of a patch test this condition is most critical and we show in Figs 11.1 and 11.2 a series of such patch tests on elements with Co continuous interpolation of u and either discontinuous or continuous interpolation of p. For each we have included all combinations of constant, linear and quadratic functions. In the test we prescribe all the displacements on the boundaries of the patch and one pressure variable as it is well known that in fully incompressible situations pressure will be indeterminate by a constant for the problem with all boundary displacements prescribed.t *Alternatively, it is possible to omit all boundary conditions on pressure if one displacement with a component normal to the boundary is allowed to exist.
385
386
Incompressible problems, mixed methods and other procedures of solution The single-element test is very stringent and eliminates most continuous pressure approximations whose performance is known to be acceptable in many situations. For this reason we attach more importance to the assembly test and it would appear that the following elements could be permissible according to the criteria of Eq. (11.12) (indeed all pass the B-B condition fully):
Triangles: T6/1; T10/3; T6/C3 Quadrilaterals" Q9/3; Q8/C4; Q9/C4 We note, however, that in practical applications quite adequate answers have been reported with Q4/1, Q8/3 and Q9/4 quadrilaterals, although severe oscillations of p may occur. If full robustness is sought the choice of the elements is limited. 4 It is unfortunate that in the present 'acceptable' list, the linear triangle and quadrilateral are missing. This appreciably restricts the use of these simplest elements. A possible and indeed effective procedure here is not to apply the pressure constraint at the level of a single element but on an assembly. This was done by Herrmann in his original presentation 2 where four elements were chosen for such a constraint as shown in Fig. 11.3(a). This composite 'element' passes the single-element (and multiple-element) patch tests but apparently so do several others fitting into this category. In Fig. 11.3(b) we show how a single triangle can be internally subdivided into three parts by the introduction of a central node. This coupled with constant pressure on the assembly allows the necessary count condition to be satisfied and a standard element procedure applies to the original triangle treating the central node as an internal variable. Indeed, the same effect could be achieved by the introduction of any other internal element function which gives zero value on the main triangle perimeter. Such a bubble function can simply be written in terms of the area coordinates (see Chapter 4) as
IT 3/1 I
IT 6/1 I
IT 6/3 I
IT 10/31
nu=O np=O
=0 =0 (pass)
=0 =2 (fail)
=2 =2 (pass)
(pass)
IQ
I
I Q 8/4 1 9
A
I Q 8/31 9
A
LQ 9/41 9
9 zx
nu=O np=O (a)
(pass)
A
o
=0 =3 (fail)
zx
zx
=0 =2 (fail)
zx
zx w
=2 =3 (fail)
I Q 9/3 I 9
A
o zx
=2 =2 (pass)
Fig. 11.1 Incompressible elasticity u-p formulation. Discontinuous pressure approximation. (a) Single-element patch tests, u variable (o restrained, o free) 2 DOF, p variable (A restrained,/x free) 1 DOE
Two-field incompressible elasticity (u-p form)
I T3/11
[T6/1 I
nu= 2 np=5
= 7 x 2 = 14 =5 (pass)
(fail)
IQ4/' I
w
[
w
w
~
IQ8/4 J
= 19x2=38 =17 (pass)
[ Q 9/3 ]
-9x2-18 -11
(fail)
(pass)
IQ9/41
IQ8/31 m,
14
=15
(fail)
d==
=7x2 =17 (fail)
= 5 x 2 = 10
nu= 2 np=3
IT ~o/31
IT 6/3 I
m,
9
9
A
"A )A
(b)
n u = 5 X 2 = 10 np=11 (fail)
w
= 9 x 2 = 18 =15 (pass)
,w
Fig. 11.1 (Cont.) Incompressible elasticity u-p formulation. Discontinuous pressure approximation. (b) Multiple-element patch tests.
L 1L2L3. However, as we have stated before, the degree of freedom count is a necessary but not sufficient condition for stability and a direct rank test is always required. In particular it can be verified by algebra that the conditions stated in Sec. 10.3 are not fulfilled for this triple subdivision of a linear triangle (or the case with the bubble function) and thus C~ = 0 for some non-zero values of indicating instability.
387
388 Incompressible problems, mixed methods and other procedures of solution
9 o u variable (restrained, free) 2 DOF A ~ p variable (restrained, free) 1 DOF I T3/03 I
I T6/06 1
i T6/03 I
nu=O np=2 (fail)
=0 =5 (fail)
=0 =2 (fail)
I Q4/04 I
I Q8/08 I
I Q8/04 I
I Q9/04.,.1
o
(a)
nu=O np=3 (fail)
=0 =7 (fail)
=0 =3 (fail)
=2 =3 (fail)
T3/c3 !
I To/co I
I T6/c3 I
nu=2 np=6 (fail)
=7x2= 14 =18 (fail)
=7x2= 14 =6 (pass)
I Q4/04 I
I Q 8/04 I
Z
I Q 9/04 ..I A
(b)
nu=2 np=8 (fail)
=5x2= 10 =8 (pass)
=9x2= 18 =8 (pass)
Fig. 11.2 Incompressibleelasticity u-p formulation. Continuous (Co) pressure approximation. (a) Singleelement patch tests. (b) Multiple-element patch tests.
Two-field incompressible elasticity (u-p form) 389
(a)
..i-
nu= 2
(b)
w
nu = 2
np=O
w
np=0 (Bubble function)
(c)
w
nu=2
w
np=2
OR (Bubble function) w
w
OR ubble function)+,
(d)
nu=2
~
np=2
Fig. 11.3 Somesimplecombinationsof lineartrianglesand quadrilateralsthat passthe necessarypatchtest counts. Combinations(a), (c),and (d) are successfulbut (b)is still singularand not usable.
390
Incompressibleproblems, mixed methods and other procedures of solution
~r
~
~Load Only vertical movement ,-o possiblefor ," ~ no volume ,," change
Triangle 1
9
I~~/2 / Z,/ / / ,/ / / ,/,/ / /~ t
a
s
t
s
t
s
t
t
s
)-~}~ Only horizontal movement possible for no volume change Triangle 2
(np
Fig. 11.4 Locking (zero displacements) of a simple assembly of linear triangles for which incompressibility is - n u - 24). fully required In Fig. 11.3(c) we show, however, that the same concept can be used with good effect for Co continuous p.5 Similar internal subdivision into quadrilaterals or the introduction of bubble functions in quadratic triangles can be used, as shown in Fig. 11.3(d), with success. The performance of all the elements mentioned above has been extensively discussed 6-11 but detailed comparative assessment of merit is difficult. As we have observed, it is essential to have nu > n p but if near equality is only obtained in a large problem no meaningful answers will result for u as we observe, for example, in Fig. 11.4 in which linear triangles for u are used with the element constant p. Here the only permissible answer is of course u -- 0 as the triangles have to preserve constant volumes. The ratio nu / np which occurs as the field of elements is enlarged gives some indication of the relative performance, and we show this in Fig. 11.5. This approximates to the behaviour of a very large element assembly, but of course for any practical problem such a ratio will depend on the boundary conditions imposed. We see that for the discontinuous pressure approximation this ratio for 'good' elements is 2-3 while for Co continuous pressure it is 6-8. All the elements shown in Fig. 11.5 perform very well, though two (Q4/1 and Q9/4) can on occasion lock when most boundary conditions are on u. Example 11.1: Simple triangle with bubble- MINI element. In Fig. 11.3(c) we indicate that the simple triangle with Co linear interpolation and an added bubble for the displacements u together with continuous Co linear interpolation for the pressure p satisfied the count test part of the mixed patch test and, verifying the consistency condition, can be used with success. 5 Here we consider this element further to develop some understanding about its performance at the incompressible limit. The displacement field with the bubble is written in hierarchical form as U ~, !1 -- ~
N a !1a Jr" Nbub tlbub a
(11.13)
Two-field incompressible elasticity (u-p form) (a)
T6/1
3,=4 2 "Y" / / / / / / / / ~ / / / / / / / / / ~
".m2"//////////////////~/////
"/////
(
A y=3
T10/3
Q9/3
"/~'/////~/////'/~'/////~'/////
~ t
y = 2.66
"/~////////7~////////2"Y'/////
A Q9/4
y= 2
T6B1/3
ZX
O ZX
ZX
0 t ZX
O
ZX
y=2
A
~Y////////~'/////////~'/////
~Y'///////,/~'////////2"~'/////
(b) T6/3C
h,
z~t:////////7~////////~L2,'////
T3B1/3C
y= 8
Q9/4C
y=8
,..,
~///////7~////////z/lYt2z////
y=6 zz~,'////////////////z-z~v////
Fig. 11.5 The freedom index or infinite patch ratio for various u - p elements for incompressible elasticity (y = nu/np). (a) Discontinuous pressure. (b) Continuous pressure. B-bubble, C-continuous.
where here Nbub -- L 1 L 2 L 3
(11.14)
Ua are nodal parameters of displacement and Llbub are parameters of the hierarchical bubble function. The pressures are similarly given by p ~/~ = ~ a
N~Pa
(11.15)
391
392
Incompressibleproblems, mixed methods and other procedures of solution
where Pa are nodal parameters of the pressure. In the above the shape functions are given by (e.g., see Eqs (4.26) and (4.29)) 1
Na -- La = 2A (aa + bax -!-Cay)
(11.16)
where ba = Yb -- Yc;
aa = XbYc -- XcYb;
Ca =
Xc -- Yb
b, c are cyclic permutations of a and 1 X1 Yl] 2A = det 1 X2 Y2 = al + a2 + a3 1 X3 Y3 The derivatives of the shape functions are thus given by ONa
ba
Ox
2A
ONa __ C__5_a 2A Oy
and
Similarly the derivatives of the bubble are given by ONbub 8x 8Nb~b Oy
1
2A 1
2A
(blL2L3 + b2L3L1 W b3L1L2) (ClL2L3 + c2L3L1 + c3L1L2)
r0a0]
0]
The strains may be expressed in terms of the above and the nodal parameters ast E -=. Za
"~
a
Ca f i a + ba
LCa
2A
Ca flb~b LCa ba
(11.17)
where again b, c are cyclic permutations of a. Substituting the above strains into Eq. (11.11) and evaluating the integrals give All A
A12
A13
A32
A33
0 0 0
0
0
Abb
A21 A22 A23
__.
A~I
(11.18)
where G [(4babb + 3CaCb) [(3baCb -- 2Cabb)
(3Cabb -- 2baCb)] (3babb + 4CaCb)J
Aab -- ~
G Abubbub = 2160A
[ (4bTb + 3cTc) b Tc
bTc ] (3bTb + 4cTc)J
and
b = [bl,
b2,
b3] T
and
c = [Cl,
C2,
C3] T
t At this point it is also possible to consider the term added to the derivatives to be enhancedmodes and delete the bubble mode from displacement terms.
Three-field nearly incompressible elasticity (u-p-ev form) 393 Note in the above that all terms except Abubbub are standard displacement stiffnesses for the deviatoric part. Similarly,
C --
Cll C21 C31
C12 C22 C32
C13 ] C23 / C33 |
(11.19)
Cbubl Cbub2 Cbub3J
'Ebb]
where
Cab = g r
120
In all the above arrays a and b have values from 1 to 3 and b u b denotes the bubble mode. We note that the bubble mode is decoupled from the other entries in the A array - it is precisely for this reason that the discontinuous constant pressure case shown in Fig. 11.3(b) cannot be improved by the addition of the internal parameters associated with fibub. Also, the parameters fibub are defined separately for each element. Consequently, we may perform a partial solution at the element levell2 to obtain the set of equations in the form Eq. (11.10) where now
rAil A12 A131 rCll C12 C13] A = |A21 A22 A23|; C= |C21 C22 C23|; ]A31 A32 A33J LC31 C32 C33J with Vob =
and
ba Ca]F11
2--A ~
L~'=I r=2
3 A2 [(3bTb-F- 4cTc) -bXc
"1" = l O G c L
rV l V= []/21
/bb/
V12 V13] V22 V23 V31 V32 V33
J
(11.20a)
r
--bTc ] (4bXb+ 3eTc )
(ll.20b)
in which c -- 12 (bTb)2 + 25 (bTb)(cTc) + 12 (cTc) 2 -- (bTc) 2 The reader may recognize the V array given above as that for the two-dimensional, steady heat equation with conductivity k = 7- and discretized by linear triangular elements. The direct reduction of the bubble matrix Abubbub as given above leads to a full matrix 7-. Some numerical experiments including the above formulation are presented in Sec. 11.7.
!iii ii
iiiiiiiiiiiiiiiiii!iii
A direct approximation of the three-field form leads to an important method in finite element solution procedures for nearly incompressible materials which has sometimes been called the B-bar method. The methodology can be illustrated for the nearly incompressible isotropic problem. For this problem the method often reduces to the same two-field form previously discussed. However, for more general anisotropic or inelastic materials and in
394 Incompressibleproblems, mixed methods and other procedures of solution finite deformation problems the method has distinct advantages as are discussed in reference 1. The usual irreducible form (displacement method) has been shown to 'lock' for the nearly incompressible problem. As shown in Sec. 11.3, the use of a two-field mixed method can avoid this locking phenomenon when properly implemented (e.g., using the Q9/3 two-field form). Below we present an alternative which leads to an efficient and accurate implementation in many situations. For the development shown we shall assume that the material is isotropic linear elastic but it may be extended easily to include anisotropic materials. Assuming an independent approximation to ev and p we can formulate the problem by use of Eq. (11.7) and the weak statement of relation (11.2) written as f6p[m TSu-ev] d~--0
(11.21)
f 3e,, [Keo - p] dr2 = 0
(11.22)
and If we approximate the u and p fields by Eq. (11.9) and eo ~ ~v = Noffo
(11.23)
we obtain a mixed approximation in the form of Sec. (10.5.3) but now only for p and eo
[
A C CT 0 0 .... E T
0 -E H
]{} {} fit P ~v
:
fl f2 f3
(11.24)
where A, C, fl, f2 are given by Eq. (11.11) and H = / ~ N~KNv dg2;
E = f N~Npdf2;
1'3=0
(11.25)
For completeness we give the variational theorem whose first variation gives Eqs (11.7), (11.21) and (11.22). First we define the strain deduced from the standard displacement approximation as Cu = S u ,~ Bfi (11.26) The variational theorem is then given as
rI=~l f~ (euTDdeu + evKev)d~2+
f
p (mTcu - e v ) d g 2 (11.27)
- ~ u T b d f 2 - ~r uTidF
Example 11.2: An enhanced strain triangle. In Example 11.1 we presented a two-field formulation using continuous u and p approximations together with an added hierarchical bubble mode to the displacements. For more general applications this form is not the most convenient. For example, if transient problems are considered the accelerations will also involve the bubble mode and affect the inertial terms. We will also find in the Sec. 11.7 that use of the above bubble is not fully effective in eliminating pressure oscillations in
Three-field nearly incompressible elasticity (u-p-~v form) 395 solutions. An alternative form is discussed in which we use a three-field approximation involving u, p and ev discussed above, together with an enhanced strain formulation as discussed in Sec. 10.5.3. The enhanced strains are added to those computed from displacements as (11.28)
-- ~u -~- ~e
in which 8e represents a set of enhanced strain terms. The internal strain energy is represented by = 1 (~TDa ~ + evKev) (11.29)
W(~, ev)
Using the above notation a Hu-Washizu-type variational theorem for the deviatoric-spherical split may be written as I-lnw = ~
[W(~, eo) + p (mT~ -- ev) + crT (eu -- ~)] dr2 + I-Iex t
(11.30)
where FIex t represents the terms associated with body and traction forces. After substitution for the mixed enhanced strain the last term simplifies to
/s2 crT (eu -- ~) d~2 = -- ~ o'T ee d~2
(11.31)
Taking variations with respect to u, p, ev, ~e and cr the principle yields the weak form ~Flnw = / ~ 8uTB T [Da~ + mp] dr2 + ~I"[ex t t/~G
q-~'~T[Dd~-q-mp--o'] d~2"q-~'o'T~ed~'-O Equal order interpolation with shape functions N are used to approximate u, p and to as u~fi-Nfi
p ~ p = N~
(11.33)
e~ ~ ~ = N ~ However, only approximations for u and p are Co continuous between elements. The approximation for ev may be discontinuous between elements. The stress ~r in each element is assumed constant. Thus, only the approximation for Se remains to be constructed in such a way that the third equation in (10.41) is satisfied. For the present we shall assume that this approximation may be represented by
Se ~ ge = Be&e
(11.34)
so that the terms involving cr and its variation in Eq. (11.32) are zero and thus do not appear in the final discrete equations. With the above approximations, Eq. (11.32) may be evaluated as
Aue Cu AeuAeeCeO
Auu Cux o
0
Cex
0
-E
0
-E x
H
fi
He P
~v
fl
=
0
t"2
f3
(11.35)
396 Incompressibleproblems, mixed methods and other proceduresof solution where Auu = A, Cu = C, fi, E and H are as defined in Eqs (11.11), (11.25) and /,
Aue = _/o BDdBe d~ -- AeT Aee - " / ~ BeDdne d~2
(1 1.36)
Ce = J~ BemN d~ Since the approximations for eo and ~e are discontinuous between elements we can again perform a partial solution for ev and &e using the second and fourth row of (11.35). After eliminating these variables from the first and third equation we again, as in Example 11.1, obtain a form identical to Eq. (11.10). As an example we consider again the 3-noded triangular element with linear approximations for N in terms of area coordinates Li. We will construct enhanced strain terms from the derivatives of an assumed function. Here we consider three enhanced functions given by
N~ --/~Li + L j L k
(11.37)
in which i, j, k is a cyclic permutation and/3 is a parameter to be determined. Note that this form only involves quadratic terms and thus gives linear strains which are fully consistent with the linear interpolations for p and ev. The derivatives of the enhanced function are given by DN~
1
cgx = 2A [/3bi 4- Ljbk + Lkbj] Oy
(11.38)
= 1. [gci + Z.jc + L cj] 2A
where
bi -- yj - Yk
and
Ci --" Xk --Xj
and A is the area of a triangular element. For constant p the requirement imposed by Eq. 11.21 gives fl = 1/3. The derivatives are inserted in the usual strain-displacement matrix
"aN~
o
o
aN~
Ox
B~=
ONie . Oy
Oy ONe
(11.39)
Ox.
While the use of added enhanced modes leads to increased cost (over use of a simple bubble mode, as in Example 11.1) in eliminating the ~o and Ore parameters in Eq. (11.35) the results obtained are improved considerably, as indicated in the numerical results presented in Sec. 11.7. Furthermore, this form leads to improved consistency between the pressure and strain.
Three-field nearly incompressible elasticity (u-p-ev form)
11.4.1 The B-bar method for nearly incompressible problems The second of (11.24) has the solution ~v - - E - 1 c T f i
"- W f i
(11.40)
In the above we assume that E may be inverted, which implies that Nv and N p have the same number of terms. Furthermore, the approximations for the volumetric strain and pressure are constructed for each element individually and are not continuous across element boundaries. Thus, the solution of Eq. (11.40) may be performed for each individual element. In practice No is normally assumed identical to N p so that E is symmetric positive definite. The solution of the third equation of (11.24) yields the pressure parameters in terms of the volumetric strain parameters and is given by = E-THffv
(11.41)
Substitution of (11.40) and (11.41) into the first of (11.24) gives a solution that is in terms of displacements only. Accordingly, Aft = fl (11.42) where for isotropy /.
A =/,.,
BTDd B dr2 + WTHW
(11.43)
= A + WTHW The solution of (11.42) yields the nodal parameters for the displacements. Use of (11.40) and (11.41) then gives the approximations for the volumetric strain and pressure. The result given by (11.43) may be further modified to obtain a form that is similar to the standard displacement method. Accordingly, we write = L BTD[I dr2
(1 1.44)
where the strain-displacement matrix is now 1 171= IdB + ~mNvW
(11.45)
For isotropy the modulus matrix is D = Dd + K m m T
(11.46)
We note that the above form is identical to a standard displacement model except that B is replaced by 1). The method has been discussed more extensively in references 13, 14 and 15. The equivalence of (11.43) and (11.44) can be verified by simple matrix multiplication. Extension to treat general small strain formulations can be simply performed by replacing the isotropic D matrix by an appropriate form for the general material model. The formulation shown above has been implemented into an element included as part of the program available on the web site. The elegance of the method is more fully utilized when
397
398 Incompressibleproblems, mixed methods and other procedures of solution considering non-linear problems, such as plasticity and finite deformation elasticity (see reference 1). We note that elimination starting with the third equation of (11.24) could be accomplished leading to a u - p two-field form using K as a penalty number. This is convenient for the case where p is continuous but ev remains discontinuous - as already discussed in Example 11.2. Such an elimination, however, points out that precisely the same stability criteria operate here as in the two-field approximation discussed earlier.
In Chapter 5 we mentioned the lowest order numerical integration rules that still preserve the required convergence order for various elements, but at the same time pointed out the possibility of a singularity in the resulting element matrices. In Chapter 9 we again referred to such low order integration rules, introducing the name 'reduced integration' for those that did not evaluate the stiffness exactly for simple elements and pointed out some dangers of its indiscriminate use due to resulting instability. Nevertheless, such reduced integration and selective integration (where the low order integration is only applied to certain parts of the matrix) has proved its worth in practice, often yielding much more accurate results than the use of more precise integration rules. This was particularly noticeable in nearly incompressible elasticity (or Stokes fluid flow which is similar) 16-18 and in problems of plate and shell flexure dealt with as a case of a degenerate solid 19' 20 (see reference 1 for more information on plate and shell problems). The success of these procedures derived initially by heuristic arguments proved quite spectacular- though some consider it somewhat verging on immorality to obtain improved results while doing less work! Obviously fuller justification of such processes is required. 21 The main reason for success is associated with the fact that it provides the necessary singularity of the constraint part of the matrix [viz. Eqs (10.19)-(10.21)] which avoids locking. Such singularity can be deduced from a count of integration points, 22, 23 but it is simpler to show that there is a complete equivalence between reduced (or selective) integration procedures and the mixed formulation already discussed in Sec. 11.3. This equivalence was first shown by Malkus and Hughes 24 and later in a general context by Zienkiewicz and Nakazawa. 25 We shall demonstrate this equivalence on the basis of the nearly incompressible elasticity problem for which the mixed weak Galerkin integral statement is given by Eqs (11.7) and (11.8). It should be noted, however, that equivalence holds only for the discontinuous pressure approximation. The corresponding irreducible form can be written by satisfying the second of Eq. (11.8) exactly, implying p = Km Te
and substituting above into (11.7) as
(11.47)
Reduced and selective integration and its equivalence to penalized mixed problems 399 ~ 8r
(Io -- ~l m T m ) e d S 2 + f ~ e T m K m T e d ~ (11.48) --~~uTbd~-
~r ~uTt, d r = 0 t
On substituting u ,~ fi = Nu~
and
e ,~ ~ = SNu~ = B~
(11.49)
we have (A +/i,) ~ = fl
(11.50)
where A and fl are exactly as given in Eq. (11.11) and /i = f~ BTmKmTB dr2
(11.51)
The solution of Eq. (11.50) for fi allows the pressures to be determined at all points by Eq. (11.47). In particular, if we have used an integration scheme for evaluating (11.51) which samples at points (~k) we can write
Npa(~k)tga
P(~k) = KmTe(~k) = KmTB(~k)fi = ~
(11.52)
a
Now if we turn our attention to the penalized mixed form of Eqs (11.7)-(11.11) we note that the second of Eq. (11.10) is explicitly Np mTBfi-- ~ N p ~
d~ = 0
(11.53)
If a numerical integration is applied to the above sampling at the pressure nodes located at coordinate (~l), previously defined in Eq. ( 11.52), we can write for each scalar component of Np Npa(~l)
(
1
mTB(~'I) ~1 -- -~-
p(~l)P
l
)
Wl : 0
(11.54)
in which the summation is over all integration points (~l) and Wl are the appropriate weights and jacobian determinants. Now as Npa(~l) = ~al
if ~l is located at the pressure node a and zero at other pressure nodes, Eq. (11.54) reduces simply to the requirement that at all pressure nodes mTB(~/) ~ = ~I N p(~l)P
(11.55)
This is precisely the same condition as that given by Eq. (11.52) and the equivalence of the procedures is proved, providing the integrating scheme usedfor evaluating A gives an
identical integral of the mixed form of Eq. (11.53).
This is true in many cases and for these the reduced integration-mixed equivalence is exact. In all other cases this equivalence exists for a mixed problem in which an inexact rule of integration has been used in evaluating equations such as (11.53).
400 Incompressibleproblems, mixed methods and other proceduresof solution For curved isoparametric elements the equivalence is in fact inexact, and slightly different results will be obtained using reduced integration and mixed forms. This is illustrated in examples given in reference 26. We can conclude without detailed proof that this type of equivalence is quite general and that with any problem of a similar type the application of numerical quadrature at n p points in evaluating the matrix ,~ within each element is equivalent to a mixed problem in which the variable p is interpolated element by element using as p-nodal values the same integrating points. The equivalence is only complete for the selective integration process, i.e., application of reduced numerical quadrature only to the matrix/~, and ensures that this matrix is singular, i.e., no locking occurs if we have satisfied the previously stated conditions (nu > n p). The full use of reduced integration on the remainder of the matrix determining ~, i.e., A, is only permissible if that remains non-singular- the case which we have discussed previously for the Q8/4 element. It can therefore be concluded that all the elements with discontinuous interpolation of p which we have verified as applicable to the mixed problem (viz. Fig. 11.1, for instance) can be implemented for nearly incompressible situations by a penalized irreducible form using corresponding selective integration.t In Fig. 11.6 we show an example which clearly indicates the improvement of displacements achieved by such reduced integration as the compressibility modulus K increases (or the Poisson ratio tends to 0.5). We note also in this example the dramatically improved performance of such points for stress sampling. For problems in which the p (constraint) variable is continuously interpolated (Co) the arguments given above fail as quantities such as rote are not interelement continuous in the irreducible form. A very interesting corollary of the equivalence just proved for (nearly) incompressible behaviour is observed if we note the rapid increase of order of integrating formulae with the number of quadrature points (viz. Chapter 5). For high order elements the number of quadrature points equivalent to the p constraint permissible for stability rapidly reaches that required for exact integration and hence their performance in nearly incompressible situations is excellent, even if exact integration is used. This was observed on many occasions 27-29 and Sloan and Randolf 3~ have shown good performance with the quintic triangle. Unfortunately such high order elements pose other difficulties and are seldom used in practice. A final remark concerns the use of 'reduced' integration in particular and of penalized, mixed, methods in general. As we have pointed out in Sec. 10.3.1 it is possible in such forms to obtain sensible results for the primary variable (u in the present example) even though the general stability conditions are violated, providing some of the constraint equations are linearly dependent. Now of course the constraint variable (p in the present example) is not determinate in the limit. This situation occurs with some elements that are occasionally used for the solution of incompressible problems but which do not pass our mixed patch test, such as Q8/4 and Q9/4 of Fig. 11.1. If we take the latter number to correspond to the integrating points these will yield acceptable u fields, though not p. t The Q9/3 element would involve three-point quadrature which is somewhatunnatural for quadrilaterals. It is therefore better to simplyuse the mixedformhere - and, indeed, in any problemwhichhas non-linearbehaviour between p and u (see reference 1).
~0 14row c~ in
0 r~
~0
CIJ
~m
~z o~
Ls~
O
im
0
1in
in C~
Ill
q~m Vm
~D
olm WR
E
im
O
C13 Im Q.
C~ tm Im CIJ
CIJ 1ram
Fig. 11.6 Sphere under internal pressure. Effect of numerical integration rules on results with different Poission ratios.
402
Incompressibleproblems, mixed methods and other proceduresof solution
Fig. 11.7 Steady-state, low Reynolds number flow through an orifice. Note that pressure variation for element Q8/4 is so large it cannot be plotted. Solution with u/p elements Q8/3, Q8/4, Q9/3, Q9/4.
Figure 11.7 illustrates the point on an application involving slow viscous flow through an orifice - a problem that obeys identical equations to those of incompressible elasticity. Here elements Q8/4, Q8/3, Q9/4 and Q9/3 are compared although only the last completely satisfies the stability requirements of the mixed patch test. All elements are found to give a reasonable velocity (u) field but pressures are acceptable only for the last one, with element Q8/4 failing to give results which can be plotted. 4
Reduced and selective integration and its equivalence to penalized mixed problems 403
Fig. 11.8 A quadrilateral with intersecting diagonals forming an assembly of four T311 elements. This allows displacements to be determined for nearly incompressible behaviour but does not yield pressure results. It is of passing interest to note that a similar situation develops if four triangles of the T3/1 type are assembled to form a quadrilateral in the manner of Fig. 11.8. Although the original element locks, as we have previously demonstrated, a linear dependence of the constraint equation allows the assembly to be used quite effectively in many incompressible situations, as shown in reference 31. Example 11.3: A weak patch test - selective integration.
In order to illustrate the performance of an element which only satisfies a weak patch test we consider an axisymmetric linear elastic problem modelled by 4-noded isoparametric elements. The material is assumed isotropic and the finite element stiffness and reaction force matrices are computed using a selective integration method where terms associated with the bulk modulus are evaluated by a single-point Gauss quadrature, whereas all other terms are computed using a 2 • 2 (standard) gaussian quadrature. It may be readily verified that the stiffness matrix is of proper rank and thus stability of solutions is not an issue. On the other hand, consistency must still be evaluated. In order to assess the performance of a selective reduced quadrature formulation we consider the patch of elements shown in Fig. 11.9. The patch is not as generally shaped as desirable and is only used to illustrate performance of an element that satisfies a weak patch test. The polynomial solution considered is u-2r
(11.56)
v=O
and material constants E = 1 and v -- 0 are used in the analysis. The resulting stress field is given by t7r
--"
O"0
"--
2
(11.57)
with other components identically zero. The exact solution for the nodal quantities of the mesh shown in Fig. 11.9 are summarized in Table 11.1. Patch tests have been performed for this problem using the selective reduced integration scheme described above and values of h of 0.8, 0.4, 0.2, 0.1, and 0.05. The result for the radial displacement at nodes 2 and 5 (reported to six digits) is given in Table 11.2. All other quantities (displacements, strains, and stresses) have a similar performance with convergence rates of at least O (h) or more. Based on this assessment we conclude the element passes a weak patch test. A similar result will be found for elements which are not rectangular and thus the element produces convergent results.
404
Incompressible problems, mixed methods and other procedures of solution tl
h
~b "I ~'
,..._I ~I
r=l
Fig. 11.9 Patchfor selective,reducedquadrature on axisymmetric4-noded elements. Table 11.1 Exact solution for patch Displacement
Force
Node a
Radius ra
tla
~)a
Era
1,4 2,5 3, 6
1 -h 1 1+ h
2(1 - h ) 2 2(1 + h)
0 0 0
-(1 - h ) h 0 (1 + h)h
Fza
Table 11.2 Radial displacement at nodes 2 and 5 h 0.8 0.4 0.2 0.1 0.05
2.01114 2.00049 2.00003 2.00000
2.00000
In the general remarks on the algebraic solution of mixed problems characterized by equations of the type [viz. Eq. (10.14)] A
fl
we have remarked on the difficulties posed by the zero diagonal and the increased number of unknowns (nx + ny) a s compared with the irreducible form (nx o r ny). A general iterative form of solution is possible, however, which substantially reduces the cost. 32 In this we solve successively y(k+l)
_
y(k) +
pr(k)
(11.59)
where r (k~ is the residual of the second equation computed as r (k~ = CTx (k~ - t"2
(11.60)
A simple iterative solution process for mixed problems: Uzawa method and follow with solution of the first equation, i.e., x(k+l) - -
A-1 (fl - CY(k+l))
(11.61)
In the above p is a 'convergence accelerator matrix' and is chosen to be efficient and simple to use. The algorithm is similar to that described initially by Uzawa 33 and has been widely applied in an optimization context. 28, 34--38 Its relative simplicity can best be grasped when a particular example is considered.
11.6.2 Iterative solution for incompressible elasticity In this case we start from Eq. (11.10) now written with V - 0, i.e., complete incompressibility is assumed. The various matrices are defined in (11.11), resulting in the form
fi
IC/]kT 0el {~) -- { l~'0 lj
(11.62)
Now, however, for three-dimensional problems the matrix A is singular (as volumetric changes are not restrained) and it is necessary to augment it to make it non-singular. We can do this in the manner described in Sec. 10.3.1, or equivalently by the addition of a fictitious compressibility matrix, thus replacing A by /~ = A + f~ B T()~GmmT)B dr2
(11.63)
If the second matrix uses an integration consistent with the number of discontinuous pressure parameters assumed, then this is precisely equivalent to writing /~ = A + )~GCC T
(11.64)
and is simpler to evaluate. Clearly this addition does not change the equation system. The iteration of the algorithm (11.59)-(11.61) is now conveniently taken with the 'convergence accelerator' being simply defined as p = ~.GI
(11.65)
We now have the iterative system given as
~(k+l) _ ~(k) ..1_XGr(k) where
r (k) --
cTfi (k)
(11.66)
(11.67)
the residual of the incompressible constraint, and fi(k+l) = ~ - l ( f 1 _ C~(k+l))
(11,68)
In this A can be interpreted as the stiffness matrix of a compressible material with bulk modulus K = ~.G and the process may be interpreted as the successive addition of volumetric 'initial' strains designed to reduce the volumetric strain to zero. Indeed this simple
405
406
Incompressible problems, mixed methods and other procedures of solution
approach led to the first realization of this algorithm. 39-41 Alternatively the process can be visualized as an amendment of the original equation (11.62) by subtracting the term p/(~.G) from each side of the second to give (this is often called an augmented lagrangian form)32,37, 38 [Ac"r
C} ] {1~ } =1 { _ f~ ~ XG E-G
and adopting the iteration [c/~T
C1 i ) { ; } ( k + l ) ( kG
fl~ - ~ G p(k) }
(11.69)
(11.70)
With this, on elimination, a sequence similar to Eqs (11.66)-(11.68) will be obtained provided A is defined by Eq. (11.64). Starting the iteration from 8(o) = 0
and
~(o) = 0
in Fig. 11.10 we show the convergence of the maximum div u computed at any of the integrating points used. We note that this convergence becomes quite rapid for large values of ~. = (103-104). 12.0 A
u=1t10.01 100
"" ,'~
O(h2)
0 (h2)
I
A
I
O(h2)
0 (h 4)
0 (h2)
I. I
0 (h2)
:> 0 (h 3)
O(h 4) O(h 4) O(h 4)
G•
0 (h 3) O(h4 ) O(h4 )
Fig. 13.6 Optimalsuperconvergentsamplingand minimumintegrationpointsfor some Co elements. In Fig. 13.7 representing an analysis of a cantilever beam by four rectangular quadratic serendipity elements we see how well the stresses sampled at superconvergent points behave compared to the overall stress pattern computed in each element. The extension of the idea of superconvergent points from one-dimensional elements to two-dimensional rectangles is fairly obvious. However, the full order of superconvergence is lost when isoparametric distortion of elements occurs. We have shown, however, that results at the pth order Gauss-Legendre points still remain excellent and we suggest that superconvergent properties of the integration points continue to be used for sampling. In all of the above discussion we have assumed that the weighting matrix A is diagonal. If a diagonal structure does not exist the existence of superconvergent points is questionable. However, excellent results are still available through the sampling points defined as above. Finally, we refer readers to references 3-8 for surveys on the superconvergence phenomenon and its detailed analyses.
In the previous section we have shown that sampling of the gradients and stresses at certain points within an element is optimal and higher order accuracy can be achieved. However, we would also like to have similarly accurate quantities elsewhere within each element for general analysis purposes, and in particular we need such highly accurate displacements,
466
Errors, recovery processes and error estimates
40 i !
30
! | ! !
20
i
! i
I
i s
."
Exact average 9 shear stressl ',
'
; ~ Nodal values extrapolated E I" from Gauss points
10
i
-10
i
i
". ,'
'
I
l
values
9
Land 0.24 per unit 9
)
9
2
9
40
o
o
-I T
2 Gauss points
Fig. 13.7 Cantilever beam with four quadratic (Q8) elements. Stress sampling at cubic order (2 x 2) Gauss points with extrapolation to nodes. gradients and stresses when energy norm or other norms representing the particular quantity of interest have to be evaluated in error estimates. We have already shown how with some elements very large errors exist beyond the superconvergent point and attempts have been made from the earliest days to obtain a complete picture of stresses which is more accurate overall. Here attempts are generally made to recover the nodal values of stresses and gradients from those sampled internally and then to assume that throughout the element the recovered stresses tr* are obtained by interpolation in the same manner as the displacements o-* = N.8-*
(13.21)
We have already suggested a process used almost from the beginning of finite element calculations for triangular elements, where elements are sampled at the centroid (assuming linear shape functions have been used) and then the stresses are averaged at nodes. We have referred to such recovery in Chapter 6. However, this is not the best for triangles and for higher order elements such averaging is inadequate. Here other procedures were necessary, for instance Hinton and Campbell 9 suggested a method in which stresses at all nodes were calculated by extrapolating the Gauss point values. A method of a similar kind was suggested by Brauchli and Oden 1~ who used the stresses in the manner given by Eq. (13.21) and assumed that these stresses should represent in a least squares sense the actual finite element stresses. This is therefore an L2 projection. Although this has a
Superconvergent patch recovery- SPR 467
Fig. 13.8 Interior superconvergent patches for quadrilateral elements (linear, quadratic, and cubic) and triangles (linear and quadratic).
similarity with the ideas contained in the Herrmann theorem it reverses the order of least squares application and has not proved to be always stable and accurate, especially for even order elements. In the following presentation we will show that highly improved results can be obtained by direct polynomial 'smoothing' of the optimal values. Here the first method of importance is called superconvergent patch recovery. 11-13
We have noted above that the stresses sampled at certain points in an element possess a superconvergent property (i.e., converge at a rate comparable to that of displacement) and have errors of order O (h p+ 1). A fairly obvious procedure for utilizing such sampled values seems to the authors to be that of involving a smoothing of such values by a polynomial of order p within a patch of elements for which the number of sampling points can be taken as greater than the number of parameters in the polynomial. In Fig. 13.8 we show several such patches each assembled around an interior vertex (comer) node. The first four represent rectangular elements where the superconvergent points are well defined. The last two give patches of triangles where the 'optimal' sampling points used are not quite superconvergent. If we accept the superconvergence of t~ at certain points k in each element then it is a simple matter (which also turns out computationally much less expensive than the L2 projection) to compute tr* which is superconvergent at all points within the element. The procedure is illustrated for two dimensions in Fig. 13.8, where we shall consider interior patches (assembling all elements at interior nodes) as shown. At each superconvergent point the values of & are accurate to order p + 1 (not p as is true elsewhere). However, we can easily obtain an approximation tY* given by a polynomial of
468
Errors,recovery processes and error estimates
degree p, with identical order to those occurring in the shape function for displacement, which has superconvergent accuracy everywhere when this polynomial is made to fit the superconvergent points in a least squares manner. Thus we proceed for each component ~i of t~"as follows: writing the recovered solution as (13.22a)
c7/* = p(x, y ) a i in which p(x,y)=[1, ai --
2, al,
~,
a2,
..., "" ,
~P]
(13.22b)
am
with Yc - x - Xc, ~ - y - Yc where Xc, yc are the coordinates of the interior vertex node describing the patch. For each element patch we minimize a least squares functional with n sampling points, n
l"I -- -~ Z [ffi(Xk, k=l
Yk)
-
pkai] 2
(13.23)
where Pk : p(xk, Yk) [(Xk, Yk) correspond to the coordinates of the sampling superconvergent point k)] obtaining immediately the coefficient ai as ai -" A - 1bi (13.24)
where
n
A - ~ k=l
P~Pk
and
bi --
pgT ,cri , (Xk,
Yl,)
(13.25)
k=l
The availability of ~* allows superconvergent values of t~* to be determined at all nodes. For example, each component of the recovered solution at node a in the element patch is obtained by (O'?)a -- O'?(Xa, Ya) - - p ( X a , y a ) a i (13.26) It should be noted that on external boundaries or indeed on interfaces where stresses are discontinuous the nodal values should be calculated from interior patches and evaluated in the manner shown in Fig. 13.9. As some nodes belong to more than one patch, average values of #* are best obtained. The superconvergence of tr* throughout each element is established by Eq. (13.21). In Fig. 13.10 we show in a one-dimensional example how the superconvergent patch recovery reproduces e x a c t l y the stress (gradient) solutions of order p + 1 for linear or quadratic elements. Following the arguments of Chapter 9 on the patch test it is evident that superconvergent recovery is now achieved at all points. Indeed, the same figure shows why averaging (or L2 projection) is inferior (particularly on boundaries). Figure 13.11 shows experimentally determined convergence rates for a one-dimensional problem (stress distribution in a bar of length L = 1; 0 < x < 1 and prescribed body forces). A uniform subdivision is used here to form the elements, and the convergence rates for the stress error at x = 0.5 are shown using the direct stress approximation 8, the L2 recovery a L and cr* obtained by the SPR procedure using elements from order p = 1
Superconvergent patch recovery- SPR 469
Fig. 13.9 Recoveryof boundary or interface gradients.
to p = 6. It is immediately evident that ~r* is superconvergent with a rate of convergence being at least one order higher than that of 8. However, as anticipated, the L2 recovery gives much poorer answers, showing superconvergence only for odd values of p and almost no improvement for even values of p, while or* shows a two-order increase of convergence rate for even order elements (tests on higher order polynomials are reported in reference 14). This ultraconvergence has been verified mathematically. 15-17 Although it is not observed when elements of varying size are used, the important tests shown in Figs 13.12 and 13.13 indicate how well the recovery process works for problems in two dimensions. In the first of these, Fig. 13.12, a field problem is solved in two dimensions using a very irregular mesh for which the existence of superconvergent points is only inferred heuristically. The very small error in Crx*is compared with the error of 8x and the improvement is obvious. Here Crx = 8u/Ox where u is the field variable. In the second, i.e., Fig. 13.13, a problem of stress analysis, for which an exact solution is known, is solved using three different recovery methods. Once again the recovered solution cr* (SPR) shows much improved values compared with crL. It is clear that the SPR process should be included in all codes if simply to present improved stress values, to which we have already alluded in Chapters 6 and 7. The SPR procedure which we have just outlined has proved to be a very powerful tool leading to superconvergent results on regular meshes and much improved results (nearly superconvergent) on irregular meshes. It has been shown numerically that it produces superconvergent recovery even for triangular elements which do not have superconvergent points within the element. Recent mathematical proofs confirm these capabilities of SPR. 16-21 It is also found, for linear elements on irregular meshes, that SPR produces superconvergence of order O(h 1+~) with ot greater than zero. 22 The SPR procedure, introduced by Zienkiewicz and Zhu in 1992,11-13 is recommended as the best recovery procedure which is simple to use. However, the procedure has been modified by various investigators. 23-27 Some of the modifications have been shown to produce improved results in certain instances but with additional computational costs. One such modification appends satisfaction of discrete equilibrium equations and/or boundary conditions to the functional where the least squares fit is performed. While the satisfaction of known boundary tractions can on occasion
470
Errors, recovery processes and error estimates
S
l
(~
Exact (~ and (~* (5 h
"13 c-
Interior patch
()
O
9,' r t ' t
9
) rX
(a)
9Superconvergent values [] Nodal SPR values
l~
Quadratic exact solution
~ B o u n d a r y r-X
(b)
Fig. 13.10 Recovery of exact o- of degree p by linear elements (/9 - 1) and quadratic elements (p - 2).
be useful most of the additional constraints introduced have affected the superconvergent properties adversely and in general the modified versions of SPR such as those by Wiberg et al 23, 24 and by Blacker and Belytschko 25 have not proved to be effective. Example 13.1: SPR stress projection for rectangular element patch. As an example we consider the SPR projection for a stress component ai on the patch of rectangular elements shown in Fig. 13.14. The elements are 4-node rectangles in which shape functions are given by bilinear interpolations. Thus, the optimal sampling points are given by the points at the centre of each element. The recovered solution is given by a linear polynomial expressed as a* -- [1,
(x
-- Xl),
(Y - Yl)]
{01} a2 63
Superconvergent patch recovery- SPR 471
Fig. 13.12 Poisson equation in two dimensions solved using arbitrary-shaped quadratic quadrilaterals.
For this patch of elements, (13.23) is given by 1
4
H= ~ ~ k=l
[~i (xk, Yk) - Pka] 2
472
Errors,recovery processes and error estimates
Fig. 13.13 Plane stress analysis of stresses around a circular hole in a uniaxial field.
where [1 -a/2
Pk=
-b/2] [1 a/2 -b/2] b/2] [1 a/2 [1 -a/2 b/Z]
fork=l fork=2 fork=3 fork-4
Superconvergent patch recovery- SPR 473 8
(
(
() 1
2L.
(
3 a
~..~
6
5
4
a
Fig. 13.14 Patch of rectangular elements for SPR projection. Optimal points to sample stresses indicated by D.
Evaluating the minimum for FI and performing the sum gives the equations
[ioo]
Aa=b where
a2 0
A
and
b=
(F 1 ] -a/2
L-b/2J
(7il ~-
Ill
a/2
t~i2-~-
L-b/2J
0 b2
E1]
a/2 t~i3-4;- - a / 2 t~i4 b/2J L b/2J
The solution for the parameters is given by a l "-
1
~[
/
t~il + ~ i 2
+~i3
+~i4]
1 a2 - ~-s [-d'il + 6"/2 + ~i3 - 6"/4] 1
a3 -- =-7 [--tTil -- t7i2 -~- t7i3 -~- t~i4] 2/9
Inserting the parameters into the equation for the recovered stress gives , [~ O'i --'
(X -- X1)
2a
y_ym]{ 2b
--~i 1 "~- tTi 2 "nt- tTi 3 -- t7i4 - - ~ i l -- t7i2 nt- ~i3 "1-"~i4
)
474
Errors, recovery processes and error estimates
We note that the above yields SPR values at an internal node of a regular mesh which are the same as that obtained by averaging. Unfortunately, this is not the case when the mesh is irregular or boundary nodes are considered, as the reader can easily establish, where SPR will retain high accuracy but averaging will not.
13.4.2 SPR for displacements and stresses The superconvergent patch recovery can be extended to produce superconvergent displacements. The procedure for the displacements is quite simple if we assume the superconvergent points to be at nodes of the patch. However, as we have already observed it is always necessary to have more data than the number of coefficients in the particular polynomial to be able to execute a least squares minimization. Here of course we occasionally need a patch which extends further than before, particularly since the displacements will be given by a polynomial one order higher than that used for the shape functions. In Fig. 13.8, however, we show for most assemblies that an identical patch to that used for stresses will suffice. Larger element patches have also been suggested in reference 28 but it does not seem anything is gained. The recovered solution u* has on occasion been used in dynamic problems (e.g., Wiberg28, 29), since in this class of problems the displacements themselves are often important. We also find such recovery useful in problems of fluid dynamics. When both recovered displacements and stresses are desired, it is advantageous to compute the recovered stresses directly using the derivatives of the recovered displacements. The advantage of computing recovered stresses directly from displacements means that we have now obtained fully superconvergent results for all element types. Indeed, a recent study by Zhang and Naga, 3~ for field problems, has found that SPR using nodal field variable sampling produces better recovered gradients in certain instances. For example, although both SPR using gradient sampling and SPR using field variable sampling achieve ultraconvergence in the recovered gradient at vertex nodes of quadratic triangles, ultraconvergence of the recovered gradient at the mid-edge nodes can only be obtained by SPR using field variable sampling. A similar procedure to that studied in reference 30 has been used by Wiberg and Hager 31 in eigenfrequency computations. Thus, field variable recovery should probably always be used for triangular and tetrahedral elements, as well as for other element types when both superconvergent displacements and stresses or strains are required. The SPR recovery technique described in this section takes advantage of the superconvergence property of the finite element solutions and/or the availability of optimal sampling points. A recovery method which does not need such information has been devised and will be discussed in the next section.
Although SPR has proved to work well generally and much research has been devoted to its mathematical analyses, the reason behind its capability of producing an accurate recovered solution even when superconvergent points do not in fact exist remains an open question. We have therefore sought to determine viable recovery alternatives. One of these, known
Recovery by equilibration of patches- REP 475 by the acronym REP (recovery by equilibrium of patches), will be described next. This procedure was first presented in reference 32 and later improved in reference 33. To some extent the motivation is similar to that of Ladev~ze et al. 34' 35 who sought to establish (for somewhat different reasons) a fully equilibrating stress field which can replace that of the finite element approximation. However, we believe that the process presented here and in reference 33 is simpler although equilibrium is satisfied in an approximate manner. The starting point for REP is the governing equilibrium equation S TO" -~- b
= 0
(13.27)
In a finite element approximation this becomes fa BT~df2-fa p
NTbdfZ-fr p
NTtdF-0
(13.28)
p
where & are the stresses from the finite element solution. In the above f2p is the domain of a patch and the last term comes from the tractions on the boundary of the patch domain Fp. These can, of course, represent the whole problem, a patch of a few elements or a single element. As is well known the stresses dr which result from the finite element analysis will in general be discontinuous and we shall seek to replace them in every element patch by a recovered system which is smooth and continuous. To achieve the recovery we proceed in an analogous way to that used in the SPR procedure, first approximating the stress in each patch by a polynomial of appropriate order ~*, second using this approximation to obtain nodal values of #* andfinally interpolating these values by standard shape functions. The stress cr is taken as a vector of appropriate components, which for convenience we write as: crl if2 cr = . (13.29) Crn The above notation is general with, for instance, Crl -- Crx,or2 = Cry and Cr3 - - "Cxy describing a two-dimensional plane elastic analysis. We shall write each component of the above as a polynomial expansion of the form: c~* -- [1,
~:,
~,
...] ai :-- p(x, y)ai
(13.30)
where p is a vector of polynomials, ai is a set of unknown coefficients for the ith component of stress and 2, ~ are as described for (13.22b). For equilibrium we shall always attempt to ensure that the smoothed stress #* satisfies in a least squares sense the same patch equilibrium conditions as the finite element solution. Accordingly, ~
B T & d ~ 2 - ~ ~ BT&*d~2 p
(13.31)
p
where #* = Pa =
p
a2
0
a3
(13.32)
476
Errors,recovery processes and error estimates
written here again for the case of three stress components. Obvious modifications are made for more or fewer components. It has been found in practice that the constraints provided by Eq. ( 13.31) are not sufficient to always produce non-singular least squares minimization. Accordingly, the equilibrium constraints are split into an alternative form in which each component of stress is subjected to equilibrium requirements. This may be achieved by expressing the stress as
0"* =
i
~" = Z
li o"i ~
i
O"i (13.33)
lit~i -- Z t ~ i i i
in which
11--[1,
0,
0] T',
12 -- [0,
1,
0] T etc.
(13.34)
The equations are now obtained by imposing the set of constraints
3~~ BTO'id~'~ 3f~ BTO'*df2 = 3f~ B T l i p d ~ a i P
P
(13.35)
P
The imposition of the approximate equation (13.35) allows each set of coefficients ai to be solved independently reducing considerably the solution cost and here repeating a procedure used with success in SPR. A least squares minimization of Eq. (13.35) is expressed as
n=~1 (H/a/ where
Hi=fnBTlipdf2
- f;)T (H/a/- f;) and
f/P = / ~
P
BTr"i dr2
(13.36)
(13.37)
P
The minimization condition results in
ai = [HTI-Ii] -1 HTf;
(13.38)
Nodal values &* are obtained from Eq. (13.30) and the final recovered solution is given by Eq. (13.21). The REP procedure follows precisely the details of SPR near boundaries and gives overall an approximation which does not require knowledge of any superconvergent points. The accuracy of both processes is comparable.
One of the most important applications of the recovery methods is its use in the computation of error estimators. With the recovered solutions available, we can now evaluate errors simply by replacing the exact values of quantities such as u, or, etc., which are in general unknown, in Eqs (13.1) and (13.2), by the recovered values which are more accurate
aposteriori
Error estimates by recovery 477
than the direct finite element solution. We write the error estimators in various norms such
as
Ilell ~ I1~11 = I l u * - ~ l l (13.39)
IlellL= ~ II~IIL= - - I l u * - ~ I I L = Ile~ IlL= ~ IIG IlL= = liar* - &ILL=
For example, an error estimator of the energy norm for elasticity problems has the form 1
,,e,, = [f~ (o'*--o')TD-1 (o'*--~') d~"2]~
(13.40)
Similarly, estimates of the RMS error in displacement and stress can be obtained through Eqs (13.9)-(13.10). Error estimators formulated by replacing the exact solution with the recovered solution are sometimes called recovery-based error estimators. This type of error estimator was first introduced by Zienkiewicz and Zhu. 36 The accuracy or the quality of the error estimators is measured by the effectivity index O, which is defined as I1~11 0 = (13.41) Ilell A theorem presented by Zienkiewicz and Zhu 12 shows that for all estimators based on recovery we can establish the following bounds for the effectivity index: 1
Ile*ll
2 is necessary for the full
General single-step algorithms for first- and second-order equations 603 A
I
(Zn 2 At2/2
Un + ilnAt= On
Un
r
At
tn
tn+ l
Fig. 17.9 A second-order time approximation.
dynamic equation and p > 1 is necessary for the first-order equation. Indeed the lowest approximation, that is p = 1, is the basis of the algorithm derived in the previous section. The recurrence algorithm will now be obtained by inserting u, fi and ii [obtained by differentiating Eq. (17.34)] into Eq. (17.1) and satisfying the weighted residual equation with a single weighting function W(r). This gives (p _1 p) f0At W(r) IM ( On + r'l~l'n -'1-""" + 2)-------------~rP-2ot
1
+ C (fin -I- z'fin + " " + ~
( p - 1)!
+ K
z'P--10tp
)
( Iln + Z'l]ln-l-""" + -p1-! r
(17.36) =
0
as the basic equation for determining anP. Without explicitly specifying the weighting function used we can, as in Sec. 17.2.1, generalize its effects by writing
Ok = f~ Wr~ dr A t k fOt W dr = fot Wf dr f~t W d r
k=0, 1,...,p (17.37)
where we note 00 is always unity. Equation (17.36) can now be written more compactly as
Ac~p + M~n+l -t- C~n+l "-I-KUn+l + P -- 0
(17.38)
604
The time dimension - discrete approximation in time
where A
--
Op-2Atp-2
(p - 2)!
M+
Op-1 A t p-1 C -]- O p A t p K
( p - 1)!
p!
x-~P-10qAt q q
Un+l -- ~ q=0 9
Un+l
q!
Un
(17.39)
_ V TMp-10q-1 A t q-1 q
/---' q=l
( q - 1)*
Un
p-10q-zAt q-2 q fi~+l=~ ( q - 2 ) v u,~ q=2 " As an+l, ~n+l and Un+l can be computed directly from the initial values we can solve Eq. (17.38) to obtain O~p = - A
(17.40)
-1 [MUn+l + Can+l .qt..KUn+l + f]
It is important to observe that an+l, ~n+l and Un+l here represent some mean predicted values of Un+l, Lln+l and fin+l in the interval and satisfy the governing Eq. (17.1) in a weighted sense if C~nPis chosen as zero. The procedure is now complete as knowledge of the vector C~nPpermits the evaluation of p-1 Un+l to Un+l from the expansion originally used in Eq. (17.34) by putting r = At. This gives Un+l ---Un -~- Atfl~ + - . . + I]ln+1 --Ill n --[- Atii~ + . . .
+
At p
p i9 a p = fi"+l + A t p-1
D
.
At p
p.
i ap A t p-1
+ 1)--------5'~n (p= fin+l ( p -
D
1)! C~n
(17.41)
p-1 p-1 Un+l -- Un + Atc~nP
In the above fi, fi, etc., are again quantities that can be written down a priori (before solving for c~). These represent predicted values at the end of the interval with C~nP= 0. To summarize, the general algorithm necessitates the choice of values for 01 to Op and requires (a) computation of an+l, ~n-k-1 and fin-k-1 using the definitions of Eqs (17.39); (b) computation of a ff by solution of Eq. (17.40); p-1 (c) computation of Un+l to Un+l by Eqs (17.41). After completion of stage (c) a new time step can be started. In first-order problems the computation of ~ can obviously be omitted 9 If matrices C and M are diagonal the solution of Eq. (17.40) is trivial providing we choose Op = 0 (17.42)
General single-step algorithms for first- and second-order equations 605 With this choice the algorithms are explicit but, as we shall find later, only sometimes
conditionally stable. When Op > O, implicit algorithms of various kinds will be available and some of these will be found to be unconditionally stable. Indeed, it is such algorithms that are of great practical use. Important special cases of the general algorithm are the SS 11 and SS22 forms given below. Example 17.1: The S S l l algorithm. If we consider the first-order equation (that is j = 1) it is evident that only the value of Un is necessarily specified as the initial value for any computation. For this reason the choice of a linear expansion in the time interval is natural (p = 1) and the SS 11 algorithm is for that reason most widely used. Now the approximation of Eq. (17.34) is simply U --- U n -~- "CO:
(Og1
--
a = r)
(17.43)
and the approximation to the average satisfaction of Eq. (17.2) is simply Cc~ + K(fin+l + 0 A t a ) + f = 0 with
Uln+ 1 --" Un.
(17.44)
Solution of Eq. (17.44) determines c~ as -- - - ( C + OAtK) -1 (f
+ Nun)
(17.45)
and finally U n + l " - Un +
At c~
(17.46)
The reader will verify that this process is identical to that developed in Eqs (17.7)-(17.13) and hence will not be further discussed except perhaps for noting the more elegant computation form above. Example 17.2: The SS22 algorithm. With Eq. (17.1) we considered a second-order system (j = 2) in which the necessary initial conditions require the specification of two quantities, Un and fin- The simplest and most natural choice here is to specify the minimum value of p, that is p -- 2, as this does not require computation of additional derivatives at the start. This algorithm, SS22, is thus basic for dynamic equations and we present it here in full. From Eq. (17.34) the approximation is a quadratic 1 2r u = Un + rfin + ~r
~
(c~2 = c~ ~ ii)
(17.47)
The approximate form of the 'weighted' dynamic equation is now Ms +
C(Un+l
-~- 0 1 A t c ~ )
+ K(Un+l + 102At200 + f -- 0
(17.48)
with predicted 'mean' values fin+ l ~- On -+- O1Atfln O n + l - - tin
(17.49)
606 The time dimension- discrete approximation in time
After evaluation of c~ from Eq. (17.40), the values of Un+1 are found by Eqs (17.41) which become simply 1
Un+l "-- Un -~- Atfin + ~ At2ot
(17.50)
I~ln+l = fin -+- Atc~
This completes the algorithm which is of much practical value in the solution of dynamics problems. In many respects the previous example resembles the Newmark algorithm 44 which we shall discuss in the next section and which is widely used in practice with the forms an+ 1 "- U n -+- Atfi, + ( 89- / ~ ) a t 2 0 n I]ln+ 1 --" II n -~" (1
-
+/~At20n+l
+ y Atiin+l
y)Atfin
Indeed, the stability properties of the SS22 algorithm turn out to be identical with the Newmark algorithm if 01 --" y;
02 "-- 2/3;
1 01 _> 02 _> ~
(17.51)
for unconditional stability. In the above y and/3 are conventionally used Newmark parameters. For 02 = 0 the algorithm is 'explicit' (assuming both M and C to be diagonal) and can be made conditionally stable if 01 >__ 1/2. The algorithm is clearly applicable to first-order equations described as SS21 and we shall find that the stability conditions are identical. In this case, however, it is necessary to identify an initial condition for fi0 and I~lo -- - C -1 ( K u o -+- fo)
is one possibility.
17.3.3 TruncatedTaylor series collocation algorithm GNpj In the derivation using collocation, we consider the satisfaction of the governing equation (17.1) only at the end points of the interval At [which results from the weighting function shown in Fig. 17.3(c)] and write
Miin+1 +
Cl~ln+ 1 -[-
Ku~+1 "~- fn+ 1 :
0
(17.52)
with appropriate approximations for the values of Un+l, Iln+l and an+l- It will be shown that again as in Sec. 17.2.2 a non-self-starting process is obtained, which in most cases, however, gives an algorithm similar to the SSpj one we have derived. The classical Newmark method 44 will be recognized as a particular case together with its derivation process in a form presented generally in existing texts. 45 Because of this similarity we shall term the new algorithm generalized Newmark (GNpj).
General single-step algorithms for first- and second-order equations 607 If we consider a truncated Taylor series expansion similar to Eq. (17.17a) for the function u and its derivatives, we can write Un+l ---Un -+- Atfin -F""-Il[ln+l = Il n "~- A t fin + " "
AtPp AtP(p P) p V un -']-'/~PT.I n+l -- Un
Atp-I P AtP-1 (~1 P ) d- ( p - 1)! Un +/3p-1 ( p _ 1)----~ n + l - Un
(17.53)
p-1 p-1 (~ p) Un+l "-- Un -~- At ~_1n --/~1 At n+l -- Un
In Eqs ( 17.53) we have effectively allowed for a polynomial of degree p (i.e., by including terms up to A t p) plus a Taylor series remainder term in each of the expansions for the function and its derivatives with a parameter flj, j -- 1, 2 . . . . . p, which can be chosen to give good approximation properties to the algorithm. Insertion of the first three expressions of (17.53) into Eq. (17.52) gives a single equation p p-1 from which Un+l can be found. When this is determined, Unq-1 to un+l can be evaluated using Eqs (17.53). Satisfying Eq. (17.52) is almost a 'collocation' which could be obtained by inserting the expressions (17.53) into a weighted residual form (17.36) with W = 8 (t,,+l) (the Dirac delta function). However, the expansion does not correspond to a unique function u. In detail we can write the first three expansions of Eqs (17.53) as At p p Un+l = fin+l +/3p ~ U.+I 9 A t p-1 p Iln+l : Iln+l -~-tip--1 ( p _ 1)! Un+l
(17.54)
A t p-2 p Un+l : Iln+l -~-/~p-2 (p _ 2)! Un+l ..
where At p p Iln+ 1 "-" Un --~ Atiln + . . . + (1 - ~ p ) 7 . 1 U n .t-
AtP-1 P ( p - 1)! A t p-2 p Iln+l = fin -]- A t ll'n -+-"" + (1 - J ~ p - 2 ) ~ Un I~n+l
illn + Atiin +
.
+ (1
-
.
.
t/p-l) .
.
U
n
(17.55)
..
(p - 2)!
P Inserting the above into Eq. (17.52) and solving for an+ 1 gives P 1-- - A -1 [Man+l-+-Cl~n+l-+-Ki)n+l an+
where A "- " - 2 A t p - E M (p - 2)!
+
+ fn+l]
(17.56)
Atp-1
C-q- flpAt p K (p - 1)! p!
We note immediately that the above expression is formally identical to that of the SSpj algorithm, Eq. (17.40), if we make the substitutions
608 The time dimension- discrete approximation in time ~p = Op
/~p-1 --- Op-1
/~p-2 = Op-2
(17.57)
However, Un+l, U~+l, etc., in the generalized Newmark, GNpj, are not identical to Un+l, fin+l, etc., in the SSpj algorithms. In the SSpj algorithm these represent predicted mean values in the interval At while in the GNpj algorithms they represent predicted values at tn+l.
The computation procedure for the GN algorithms is very similar to that for the SS algorithms, starting now with known values of Un to Un. As before we have the given initial conditions and we can often arrange to use the differential equation and its derivatives to generate higher derivatives for u at t - 0. However, the GN algorithm requires use of u0 in the computation of the next time step. An important member of this family is the GN22 algorithm.
The Newmark algorithm (GN22)
We have already mentioned the classical Newmark algorithm as it is one of the most popular for dynamic analysis. It is indeed a special case of the general algorithm of the preceding section in which a quadratic (p = 2) expansion is used, this being the minimum required for second-order problems. We describe here the details in view of its widespread use. The expansion of Eq. (17.53) for p = 2 gives 1 1 Un+l -- Un "[- Atiln + 5(1 -/~2)At2iin -[- ~/~2At2iin+l -- Un+l "-{-1/~2At2Un+l
(17.58) lin+l --/In + (1 - ill)Atlln +/31Atfln+l -- I]n+l +/31Atlln+l and this together with the dynamic equation (17.52),
Miin+l + Clin+l --1-KUn+l + fn+l -- 0
(17.59)
allows the three unknowns Un+l, Iln+ 1 and fin+l to be determined. We now proceed as we have already indicated and solve first for iin+l by substituting (17.58) into (17.59). This yields as the first step
iin+l ~-- - A - 1 {fn+l -'[- C~n+l --[-Kiln+l}
(17.60)
1 A = M +/31AtC + ~/32AteK
(17.61)
where After this step the values of Un+l and lln+ 1 can be found using Eqs (17.58). As in the general case,/32 = 0 produces an explicit algorithm whose solution is very simple if M and C are assumed diagonal. It is of interest to remark that the accuracy can be slightly improved and yet the advantages of the explicit form preserved for SS/GN algorithms by a simple iterative process within each time increment. In this, for the GN algorithm, we predict Un+ l i , Un+l" i and Un+l..i using expressions (17.54) with (~ln+l) i - (~ln+l) i-1 and setting for i - 1
(Un+l) 0 __ 0
Stability of general algorithms 609 This is followed by rewriting the governing equation (17.52) as
M [/1 Un+l @ (p -- 2)!
IOn+llil
+ CIlin-+l "+- Kuin-+l -+- f,,+l = 0
(17.62)
( )
p i and solving for On+1 This predictor-corrector iteration has been successfully used for various algorithms, though of course the stability conditions remain unaltered from those of a simple explicit scheme. 46 For implicit schemes we note that in the general case, Eqs (17.58) have scalar coefficients while Eq. (17.59) has matrix coefficients. Thus, for the implicit case some users prefer a slightly more complicated procedure than indicated above in which the first unknown determined is Un+l. This may be achieved by expressing Eqs (17.58) in terms of the U~+l to obtain 2 Un+l -- Un+l -+" /32At 2 Un+l (17.63) 2/31 Iln+l = Un+l ~ Unq-1 /~2At where .~ Un+l --
2 2 ~1n /32At2 an -- /~2A--~
an+l----/~2A------~Un-+-
1 -/32 .. TUn
1----~-2jan-+-
1-~-~2
(17.64) Atiin
These are now substituted into Eq. (17.59) to give the result
Un+l : _A-1 (fn+l + C~n+l -'~ M~n+l)
(17.65)
where now
2/31 2 M+ C+K A = 32At-------~ 32At which again on using Eqs (17.63) and (17.64) gives fi and ii. The inversion is here identical to within a scalar multiplier and, thus, precludes use of the explicit form where/32 is zero.
Consistency of the general algorithms of SS and GN type is self-evident and assured by their formulation. In a similar manner to that used in Sec. 17.2.5 we can conclude from this that the local truncation error is O (At p+I) as the expansion contains all terms up to r p for SS and At p for GN algorithms. However, the total truncation error after n steps is only O (At P) for the first-order equation system and O (At p-l) for the second-order one. Details of accuracy discussions and reasons for this can be found in reference 6. The question of stability is paramount and in this section we shall discuss it in detail for the SS type of algorithms. The establishment of similar conditions for the GN algorithms follows precisely the same pattern and is left as an exercise to the reader. It is, however, important to remark here that it can be Shown that
610 The time dimension- discrete approximation in time
(a) the SS and GN algorithms are generally similar in performance; (b) their stability conditions are identical when Op - ~p. The proof of the last statement requires some elaborate algebra and is given in reference 6. The determination of stability requirements follows precisely the pattern outlined in Sec. 17.2.5. However, for practical reasons we shall (a) avoid writing explicitly the amplification matrix A; (b) immediately consider the scalar equation system implying modal decomposition and no forcing, i.e., mii + cit + ku = 0
(17.66)
Equations (17.37), (17.40) and (17.41) written in scalar terms define the recurrence algorithms. For the homogeneous case the general solution can be written down as Un+l ~ ~Un k n + l = /ZUn
(17.67) p-1 p-1 Un+ 1 --" /Z Un
and substitution of the above into the equations governing the recurrence can be written quite generally as SX,, = 0
(17.68)
where Un
Atitn X n ~-
(17.69)
p-1 A t p-1 Un A t p Pn
The matrix S is given below in a compact form which can be verified by the reader:
S __
bo
bl
1-/~
1
0
0 0
b2
"'"
bp-1
bp
2-~ "'"
(Pll)!
/]!
1 -/z
1
...
(p - 2)!
0 0
0 0
-....
1 1-/x
1
1
1
(p-
1)!
1
25 1
(17.70)
Stability of general algorithms where bo = OoAt2k bl -- OoAtc q-O1At2k
Oq-2 Oq-lAt OqAt 2 bq -- (q _ 2)!m + (q _ 1)-----------~c + q! k,
q - 2, 3 . . . . . p
and00 = 1. For non-trivial solutions for the vector Xn to exist it is necessary for detS = 0
(17.71)
This provides a characteristic polynomial of order p for/z which yields the eigenvalues of the amplification matrix 9 For stability it is sufficient and necessary that the moduli of all eigenvalues [see Eq. (17.27)] satisfy I~1 _< 1 (17.72) We remark that in the case of repeated roots the equality sign does not apply 9 The reader will have noticed that the direct derivation of the determinant of S is much simpler than writing down matrix A and finding the eigenvalues. The results are, of course, identical 9 The calculation of stability limits, even with the scalar (modal) equation system, is nontrivial. For this reason in what follows we shall only do it for p = 2 and p = 3. However, two general procedures will be introduced here. The first of these is the so-called z transformation. In this we use a change of variables in the polynomial putting l+z /z = (17.73) 1-z where z as well as/x are in general complex numbers. It is easy to show that the requirement of Eq. (17.72) is identical to that demanding the realpart ofz to be negative (see Fig. 17.10). The second procedure introduced is the well-known Routh-Hurwitz condition 47-49 which states that for a polynomial with
CoZ n -+" C lZ n - 1 "Jl- " ' " "~- C n - l Z nt- Cn = 0
[cc3c]
co > 0
(17.74)
the real part of all roots will be negative if, for Cl > 0, det cl
c3 > 0 ;
CO
det
C2
c2
ca
Cl
C3
>0
(17.75)
and generally
det
-Cl
C3
C5
C7
Co
C2
C4
C6
"'" "'"
0
Cl
c3
c5
...
0
0
C2
C4
"'"
.0
0
0
"'"
Cn_ 2
>0
(17.76)
Cn
With these tools in hand we can discuss in detail the stability of specific algorithms 9
611
612
The time dimension-discrete approximation in time
The recurrence relations for the algorithm given in Eqs (17.48) and (17.50) can be written after inserting (17.77) Un+l = ]J~Un; ~ln+l = ]J~{gn a n d f = 0 as
mot + c (Un -~- O1 mtot) + k (U n -~- 01Atitn + 102 At2ot) = 0 --~Un -~- Un + Ati~n + ~1 At2ot = 0 --lZi~n + ion + Atot = 0
Changing the variable according to Eq. (17.73) results in the characteristic polynomial coz 2 + ClZ + c2 = 0
(17.78)
with co = 4m + (401 - 2)Atc + 2(02 -- 0 1 ) A t 2 k (17.79)
cl = 2 A t c + (201 -- 1)AtZk c2 = A t Z k
The Routh-Hurwitz requirement for stability is simply that C0>0
el _>0
det IClco c201 > 0
or simply co > 0
Cl >__0
c2 > 0
(17.80)
These inequalities give for unconditional stability the condition that 02 >__01 >__
1
(17.81)
Stability of general algorithms This condition is also generally valid when m = 0, i.e., for the SS21 algorithm (the firstorder equation) though now 02 = 01 must be excluded. It is possible to satisfy the inequalities (17.80) only at some values of At yielding conditional stability. For the explicit process 02 = 0 with SS22/SS21 algorithms the inequalities (17.80) demand that 2m + (201 - 1 ) A t c - 01At2k > 0 2c + (201 - 1)Atk >_ 0
(17.82)
The second one is satisfied whenever
01 ~ ~1
(17.83)
and for 01 --- 1/2 the first supplies the requirement that At 2 < 4m - k
(17.84)
The last condition does not permit an explicit scheme for SS21, i.e., when m = 0. Here, however, if we take 01 > 1/2 we have from the first equation of Eq. (17.82) 201-1
At
01 -at- 1 02 > 1 _ ~ -- ~ 30102 - - 302 + 01 >__ 03
03 > 3 - g
(17.86)
For first-order problems (m = 0), i.e., SS31, the first requirements are as in dynamics but the last one becomes 3 0 1 0 2 - - 302 -k-01 >__ 03 - -
[601(01 -- 1) + 112 9 ( 2 0 1 - 1)
(17.87)
With 03 = 0, i.e., an explicit scheme when c = 0, At 2 < -
12(201 - 1) m -602- 1 k
(17.88)
and when m = 0, 02--01
C
At < - 602- 1 k
(17.89)
SS42/41. For this (and indeed higher orders) unconditional stability in dynamics problems m 7~ 0 does not exist. This is a consequence of a theorem by Dahlquist. 5~ The SS41 scheme can have unconditional stability but the general expressions for this are cumbersome. We quote one example that is unconditionally stable: 01 - - ~5
02 - - ~
03 - - T 25
04-
24
This set of values corresponds to a backward difference four-step algorithm of Gear. 51 It is of general interest to remark that certain members of the SS (or GN) families of algorithms are similar in performance and identical in the stability (and hence recurrence) properties to others published in the large literature on the subject. Each algorithm claims particular advantages and properties. In Tables 17.2-17.4 we show some members of this family.a1, 50-56 Clearly many more algorithms that are applicable are present in the general f o r / I l d 1 ae.
We remark here that identity of stability and recurrence always occurs with multistep algorithms, which we shall discuss briefly in the next section. Table 17.2
SS21 equivalents
Algorithms
Theta values
Zlama141
O1 -- 5, 02 = 2
Gear 51 Liniger 52 Liniger 52
01 ~-~ 3, 02 = 2 01 = 1.0848, 02 = 1 01 = 1.2184, 02 = 1.292
Multistep recurrence algorithms Table 17.3
SS31 equivalents
Algorithms
Theta values
Gear 51 Liniger 52 Liniger 52
01 = 2 , 0 2 = ~ , 0 3 = 6 01 : - 1.84, 02 = 3.07, 03 = 4.5 01 = 0.8, 02 = 1.03, 03 = 1.29
Table 17.4
SS32 equivalents
Algorithms
Theta values
Houbolt 53 Wilson {~)54 B o s s a k - N e w m a r k 55 (mfi + ku = 0, YB = 89- orB) B o s s a k - N e w m a r k 55 (mii + cil + ku = O, YB ~--- ~ -- OlB,
H i l b e r - H u g h e s - T a y l o r s6 (mfi + ku = O, •
=
89 -
2,02 = ~ , 0 3 = 6 01 = O, 02 = 0 2, 03 -- O 3 (| = 1.4 c o m m o n l y used) 01 = 1 - CrB
01 - -
,~,,)
o2 = 2 _ ,~8 + 2/~B 03 01 02 03
= = = =
6/3B 1 --O~B 1--2uB 1 -- 3C~B
01=1 02 = 2 + 2/3H -- 20t 2 03 = 6/3H(1 + C~H)
Multistep methods
In the previous sections we have been concerned with recurrence algorithms valid within a single time step and relating the values of Un+l, tin+l, fin+l to Un, fin, ii~, etc. It is possible to derive, using very similar procedures to those previously introduced, multistep algorithms in which we relate Un+l to the values u~, U~-l, Un-e, etc., without explicitly introducing the derivatives. Much classical work on stability and accuracy has been introduced on such multistep algorithms and hence they deserve mention here. We shall show in this section that a series of such algorithms may be simply derived using the weighted residual process. For constant time increments At, it can be shown that this set possesses identical stability and accuracy properties to the SSpj procedures.
17.5.2 The approximation procedure for a general multistep
algorithm
As in Sec. 17.3.2 we shall approximate the function u of the second-order equation M i i + Cli + K u + f = 0
(17.90)
615
616 The time dimension- discrete approximation in time
by a polynomial expansion of the order p, now containing a single unknown Un_t_1. This polynomial assumes knowledge of the value of Un, Un-1. . . . . U,-p+l at appropriate times tn, tn-1 . . . . . tn-p+l (Fig. 17.11). We can write this polynomial as 1
U(t)-- Z Nj(t)Un+j j=l-p
(17.91)
where Lagrange interpolation in time is given by (see Chapter 4) 1
Nj(t) = I I t - tn+k k=l-p tn+j -- tn+k k#j
(17.92)
Substituting this approximation into Eq. (17.91) gives 1
1
]Vj(t)lln+j and ii= Z
!1-- Z j=l-p
(17.93)
~[j(t)Un+j
j=l-p
where Nj and Nj denote the time derivatives of the shape functions. Insertion of u, li and ti into the weighted residual equation form yields ltn+l a tn
1 W(t) Z [(]VjM + NjC + NjK) Un+j + Njfn+j] d t = 0 j=l-p
(17.94)
with the forcing functions interpolated similarly from its nodal values. Using (17.92) and the definition for Ok given by (17.37) leads to a recurrence relation which may be used to compute u,,+a.
i
Un-p+l n-p+1
n-2 i~!:e::ig.na~!l~c~~t ~ .p~f .' [ 9
.>,(
At -F n
n-1
_
At
n+l
~ ~ ' ~~ I
11
...............
Approxi domaimnate Fig. 17.11 Multistep polynomial approximation.
Un+l
,,
Multistep recurrence algorithms Example 17.3: Two-point interpolation: p = 1. Evaluating Eq. (17.92) for the two points we obtain N1
--
No where At =
tn+ 1 -
tn
and
t -
tn
tn+l
-
tn
tn+l -
t
1
~(t At
-
r
tn) =
1
t n + l - - tn
z = t -
=
At
- t) = 1
= ~(tn+l At
(17.95a) At
Here the derivative is computed directly as
tn.
dN1 dt
dN0 dt
1 At
(17.95b)
Second derivatives are obviously zero, hence this form may only be used for first-order equations as 1 ~C(un+
1 - a n ) -~- K[(1 - O ) u n + 0Un+l] -~- f = 0 At which is, obviously, identical to the SS 11 result given previously.
(17.95c)
Example 17.4: Three-point interpolation: p = 2. Evaluating Eq. (17.92) for the three points gives N1
(t -- tn-1)(t
:
-- tn)
(tn+l - t n - 1 ) ( t n + l (t -
No =
tn-1)(t
-
(t -- tn)(t (tn-1
tn)
tn+l)
-- tn+l)
(tn - - t n - 1 ) ( t n
N_I =
-
(17.96a)
-- tn+l)
-- tn)(tn-1
tn+l)
--
The derivatives follow immediately from Eqs (17.92) and (17.93) as dN1
(t -
dt
dN0 dt
t n ) -t- ( t -
tn-1)
(tn+l -- t n - 1 ) ( t n + l
=
dN_l
(t (tn -
tn+l)
+ (t -
tn-1)(tn
(t - tn+l)
dt
-- tn)
--
tn-1)
tn+l)
+ (t -
(17.96b)
tn)
-- tn+l)
(tn--1 - - t n ) ( t n - - 1
This is the lowest order which can be used for second-order equations and has second derivatives d2N1
2
dt 2 d2No dt 2
(tn+l - - t n - 1 ) ( t n + l
=
2 (tn -
tn-1)(tn
(tn-1
-- tn)(tn-1
d2N_l dt 2
-- tn)
--
tn+l)
2 --
tn+l)
(17.96c)
617
618 The time dimension- discrete approximation in time
The recurrence relation for the two-step method with At constant is given by
(~-~M 1 + I
1 1 -Jr-~l(02+01)K]Un+l + ~-~(01 -~- ~)C
2 - ~sM-
2_
_
~ - 7 0 1 C -~- (1 - 0 2 ) K
Un'+"
(17.96d)
[ A 1t 2 M + -A-~(01 1 -- ~1 ) C + ~1(02 - 01)K] Un-1 + f = 0 where f is the effect of the integrated force resultant and Ok is computed using (17.92), but now has different values for stability than given for Ok in the SS22 form. The above form is identical to the form originally derived by Newmark 44 (however, the conventional parameters are usually/3 and 3/) and also corresponds to the SS22 and GN22 forms when parameters are related by: Y -- 01 + ~1 = 01 = fll
and
1 = ~1 132 /~ = ~1(02 ~-01) m_ ~02
The explicit form of this algorithm with 2/3 = 02 = 02 =/32 = 0 and V = 01 + 1/2 = 01 --/31 = 1/2 is frequently used as an alternative to the single-step explicit form. It is then known as the central difference approximation obtained by direct differencing. The reader can easily verify that the simplest finite difference approximation of Eq. (17.1) in fact corresponds to the above with 02 - 0 and 01 = 0. Higher order multistep forms follow the general pattern given above for the two- and three-point forms and need not be discussed more here. In general there are no added advantages using the multistep form and, quite generally, we recommend use of the onestep forms SSpj and GNpj given above.
In Secs 17.2.5 and 17.3.3 we have considered the exact solution of the approximate recurrence algorithm given in the form etc.
Un+ 1 - - ].ZUn,
(17.97)
for the modally decomposed, single degree of freedom systems typical of Eqs (17.4) and (17.5). The evaluation of/z was important to ensure that its modulus does not exceed unity so that stability is preserved. However, analytical solution of the linear homogeneous differential equations is also easy to obtain in the form fi -- fie x/ or
Un+ 1
Un e~.At
=
(17.98)
and comparison of/z with such a solution is always instructive to provide information on the performance of algorithms in the particular range of eigenvalues. In Fig. 17.5 we plotted the exact solution e -~~ and compared it with the values of/z for various 0 algorithms approximating the first-order equation, noting that here ~, -- -o9 -and is real.
k C
Time discontinuous Galerkin approximation 619
Immediately we see there that the performance error is very different for various values of At and obviously deteriorates at large values. Such values in a real multivariable problem correspond of course to the 'high frequency' responses which are often less important, and for smooth solutions we favour algorithms where/z tends to values much less than unity for such problems. However, response through the whole time range is important and attempts to choose an optimal value of 0 for various time ranges has been performed by Liniger. 52 Table 17.1 of Sec. 17.2.6 illustrates how an algorithm with 0 = 2/3 and a higher truncation error than that of 0 = 1/2 can perform better in a multidimensional system because of such properties. Similar analysis can be applied to the second-order equation. Here, to simplify matters, we consider 0nly the homogeneous undamped equation in the form
mii + ku = 0
(17.99)
in which the value of ~. is purely imaginary and corresponds to a simple oscillator. By examining/z we can find not only the amplitude ratio (which for high accuracy should be unity) but also the phase error. In Fig. 17.12(a) we show both the variation of the modulus of/z (which is called the spectral radius) and in Fig. 17.12(b) that of the relative period for the SS22/GN22 schemes, which of course are also applicable to the two-step equivalent. The results are plotted against At
2zr k where T = ~ ; o92=T co m In Fig. 17.13(a) and (b) similar curves are given for the SS23 and GN23 schemes frequently used in practice and discussed previously. Here as in the first-order problem we often wish to suppress (or damp out) the response to frequencies in which At / T is large (say greater than 0.1) in multidegree of freedom systems, as such a response will invariably be inaccurate. At the same time below this limit it is desirable to have amplitude ratios as close to unity as possible. It is clear that the stability limit with 01 - 02 = 1/2 giving unit response everywhere is often undesirable (unless physical damping is sufficient to damp high frequency modes) and that some algorithmic damping is necessary in these cases. The various schemes shown in Figs 17.12 and 17.13 can be judged accordingly and provide the reason for a search for an optimum algorithm. We have remarked frequently that although schemes can be identical with regard to stability their performances may differ slightly. In Fig. 17.14 we illustrate the application of SS22 and GN22 to a single degree of freedom system showing results and errors in each scheme.
A time discontinuous Galerkin formulation may be deduced from the finite element in the time approximation procedure considered in this chapter. This is achieved by assuming the weight function w and solution variables u are approximated within each time interval At as +
u-
u n + Au(t)
W =
W n
+
-~-
Aw(t)
tn < t < t~-+l t n < t < tn+ 1
(17.100)
where the time tn is the limit from times smaller than tn and tn+ is the limit from times larger than tn and, thus, admit a discontinuity in the approximation to occur at each discrete
620 The time d i m e n s i o n - discrete approximation 1.0
in time
~':-:4,-,~, ,,
02/2 = 13-- 0.25, 01 = 7 = 0.5
0.902/2 = 13= 0.5, 01 = 7 = 0.6
:=L m
02/2 = 13= 0.3025, 01 = 7 = 0.6
0.8-
0.7 10-2
I
I 11
1 0 -1
(a) Spectral radius I~tl
At/T
I 0
I 102
103
0.5 e e
01=7=0.6,02/2=13=0.5;
a .... n~ "1 - t ......
;
a /o_ 2 " - - 13 = 0 . 3 0 2 5
0.4
0.3~1... 0.2 -
0
0
,~
0.1
025
I
I
0.2 0.3 0.4 At/T (b) Relative period elongation, T = exact period, T= numerical period Fig. 17.12 SS22, GN22 (Newmark) or their two-step equivalent.
time location. The functions Au and Aw are defined to be zero at tn and continuous up to the time tn-+l where again a discontinuity can occur during the next time interval. The discrete form of the governing equations may be deduced starting from the time dependent partial differential equations where standard finite elements in space are combined with the time discontinuous Galerkin approximation and defining a weak form in a spacetime slab. Alternatively, we may begin with the semi-discrete form as done previously in this chapter for other finite element in time methods. In this second form, for the first-order case, we write
Time discontinuous Galerkin approximation 1.1
1.0
0.9
_~ 0.8
Wilson 54 e= 1.4
A 0.7
0.6
0.5 10-2
I 10-1
I 1
I 10
I 102
103
At/T
(a) Spectral radius 0.5 H 0.4 -
0.3 F... 0.2
Hilber et al. 56 0.1
/
/ /
0 S 0
_Wilson 54 01=1"4
0.1
(b) Relative period elongation
0.2
At/T
I 0.3
0.4
Fig. 17.13 SS23,GN23 or their two-step equivalent (see Problem 17.10 for description of o~).
621
622
The time dimension - discrete approximation in time
Fig. 17.14 Comparison of the 5S22 and GN22 (Newmark) algorithms: a single DOF dynamic equation with periodic forcing term, 01 -/31 - 1/2, 02 --/32 - 0.
I =
f
t;+l w x (Cfi + K u + f) d r = 0
(17.101)
Jt~-
Due to the discontinuity at tn it is necessary to split the integral into I =
J t~-
w T (Cfi + Ku + f) dr +
J t+
w T (Cfi + K u + f) dr = 0
(17.102)
which gives
] = (Wn+)T[C(un+ - an)]-[-(Wn+)
T
[ tn-bl (Cfi + K u + f ) dr Jt +
+
(17.103)
(Aw) T (Cli + K u + f) d r = 0 J t~+
in which now all integrals involve approximations to functions which are continuous. To apply the above process to a second-order equation it is necessary first to reduce the equation to a pair of first-order equations. This may be achieved by defining the momenta p = Mfi
(17.104)
Time discontinuous Galerkin approximation 623 and then writing the pair Mli-p
=0
(17.105)
+ Cfi+ Ku+ f = 0
The time discrete process may now be applied by introducing two weighting functions as described in reference 39. E x a m p l e 17.5: Solution of a scalar equation. To illustrate the process we consider the
simple first-order scalar equation cft + k u + f = 0
(17.106)
We consider the specific approximations u ( t ) = u ,+ + r AUn+ 1
(17.107)
+
w(t) = w n + rAw,+ 1
- 1 -where Au2+ 1 = Un+
+ , etc., and
U n
t -- tn 72
---
tn+l - tn
t -- t, ---
At
defines the time interval 0 < r < At. This approximation gives the integral form I
=
w
+ [c(u +
n
-
Un)] +
w
+At
cZXUn+I + ~(U+n +
n
+
+ At
foI +
Awn-+lr
[1
- - ~ c A u 2 + 1 + k (u + + rAU~+l) + f
~AUn+l) + I a~
1
dr = 0 (17.108)
Evaluation of the integrals gives the pair of equations
[~c + ~t,t) l~At 1 1 89 { ~n+ lkAt ~c+
-
C~n { 0 }
(17.109a)
where
Thus, with linear approximation of the variables the time discontinuous Galerkin method - 1. gives two equations to be solved for the two unknowns u,+ and u,+ To illustrate the performance of the above scheme we compare the amplification matrix for the discontinuous Galerkin and standard Galerkin method in Fig. 17.15. In addition we use the method to solve the example described in Fig. 17.4 and present the results in Fig. 17.16. It is possible to also perform the solution with c o n s t a n t approximation. Based on the above this is achieved by setting Au~-+l and Aw,+ 1 to zero yielding the single equation (17.110) (c + k A t ) u , + + A t f = cunand now since the approximation is constant over the entire time the u n+ also defines exactly the Un+ 1 value. This form will now be recognized as identical to the b a c k w a r d difference implicit scheme defined in Fig. 17.4 for 0 = 1.
624
The time dimension- discrete approximation in time
Fig. 17.15 The amplification A for standard and discontinuous Galerkin schemes. Finally, we compare the error in the amplification matrix for different step sizes. The error defined by
E =]~(At)-Aex(At) ^
where the A is the amplification for the approximate form and Aex -- e x p ( - A t ) , the exact value. In Fig. 17.17(a) we present the values for the single-step algorithms and in Fig. 17.17(b) those for the discontinuous Galerkin and two-step quadratic continuous Galerkin solution. We note that the 0 = 1/2 (Crank-Nicolson), discontinuous Galerkin and p = 2 continuous Galerkin solutions are all second-order accurate (slope zero for small At -- 0) while other values have finite slope and hence are only first-order accurate. It is also evident that the error at larger steps for the p -- 2 continuous Galerkin is more accurate than the discontinuous Galerkin. Thus, for the same computational effort the use of the continuous form is more appropriate in this class of problems. For this reason we will not pursue use of the discontinuous Galerkin time integration procedure further here.
The derivation and examples presented in this chapter cover, we believe, the necessary toolkit for efficient solution of many transient problems governed by Eqs (17.1) and (17.2). In the next chapter we shall elaborate further on the application of the procedures discussed here and show that they can be extended to solve coupled problems which frequently arise in practice and where simultaneous solution by time stepping is often needed. Finally, as we have indicated in Eq. (17.3), many problems have coefficient matrices or other variations which render the problem non-linear. This topic is addressed further for structural and solid mechanics problems in reference 57 and we note also that the issue of stability after many time steps is more involved than the procedures introduced here to investigate local stability.
Concluding
121
I--~xact
11%,,J,.
~ eI -=-o2 / 3D1G1
................................................................. ~ ~ ~
0.8 I ~ , , ,
:~..............
......
~
::...............
i ...............
::...............
121
I"
9
1
I--~xact I I
"-'.,.--.i ........................................................................ ~,,
0.6
9
E o 0.6
........................
.
-0.2
0
0.5
1
1.5 ttime
' 2.5
2
-0.2
3
i ,oG_.o. 311
...... i ................................................
0.8
::.............
.
.
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0.5
.
.
.
.
.
1
.
.
.
.
1 ..................................................................
Exact DG
o
I i
I--.
1 ,
:
:
::...............
::..............
~ .............
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
i 2
1.5 ttime
.................................................................
i
1
:
1.2
1.2
remarks
i
i
.
.
.
.
.
.
.
.
.
2.5
3
___' Exact o
DG
- - - 0=2/31
I
Ij....
0.8 E 0.6
0.6
'
iliiiiI ii
n
.~ 0.4 0.2
'0.4 0.2 ___
0
-0.2
0
.........................................................................................
0
2
3
4
..................................
-0.2
5
0
!. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
ttime
3
ttime
4
Fig. 17.16 Comparison of standard and discontinuous Galerkin schemes on a first-order initial value problem.
0.25 0.2 0.15
.
.
.
.
0.1
~ i ..................... .......... i ........... '
.
--;---.--..-. ..............
.......... i:.......... ~ ......... ~........... - i.........i :.. o=O ..... i .......... .......... ~
.
0.05
.
0
~-0.05 ILl -0.1 -0.15
.o--
..........
*. -. :: '. : - ....
.... ! .........
~
: ..........
" :
~
:
..........
......
......
! ...........
-= : :
, ......................
-~:
:
"~.
i .........
.....
0
i"-: i + :, '0.5
i i i 1
t
/ t ...................
o.o8 ~ o.o, 0.02
'~'~
:: "= :....
z .....
"
: ......
.::........... i i 1.5
::....... i :, 2
: .......
" .
.
.
9
.
.
i. :
~ ' . ~ .
.
.
.
:: i .
.........
"e..
.
:
,~
! ..........
:: "o.
: . . . . . . . . . .
.
:
i i
~m~'"
i
"
.,
0
,~=-0.02 ~
-0.04
1
-0.06
::. . . . . . . . . . i ; 3.5 4
0.08
. i
:-.......... !"a. : i -.L 2.5 3"-.
"....--o. ..........
k z~#c step size
(a)
! ..........
"e
::
: ...........
,= .....
:
...........
::
: ...........
P ""
:
e ~ . i
i": . ! ""m, . .!........... . ! "'n . .......... i "'.. ....... i ...........
-0.2 -0.25
/
-..~Z..~
, , ,-o-FE-p=2 - m. DG
0.08
"",,
Fig. 17.17 Error in amplification matrix for single steps.
-0.1
. 0
1
.
!iiiii,i.i,,iiill
. 2
.
"
3
4 t time
(b)
i'~" " 5
,~" D 6
7
8
625
626
The time dimension- discrete approximation in time
17.1 Verify the recurrence relation given in Eq. (17.19) using a least squares minimization process in which (17.7) is substituted into (17.2). 17.2 Determine the stability characteristics for a scalar form of the least squares recurrence relation given in Eq. (17.19). Plot the behaviour of the amplification matrix vs At. 17.3 Houbolt's method was originally developed as a multi-step method for the equation of motion written as
Miin+1 -+- C l h n +
1 "at-
Kun+1 3r- f n +
1 --" 0
with updates given by Ihn+l "-- ~
1 1
Un+l -- ~ ~
( l l U n + l -- 18Un + 9Un-1 -- 2Un-2) (2Un+l -- 5Un + 4Un-I -- Un-2)
(a) Following the approach given in Sec. 17.2.5 determine the amplification matrix in the form Xn+I=AXn
with X n + l = [Un+l
Un
Un_I]T
(b) For the undamped and unloaded case (i.e., c - f = 0) determine and plot the spectral radius I/~l and period elongation A T / T vs the time increment A t / T as shown in Fig. 17.13. T is the period of the undamped equation. 17.4 Consider the scalar first-order equation: cit + ku + f -
0
Construct the discrete form for transient solution using the SS 11 algorithm described in Sec. 17.3. For the data c = k = 1 and f = sin 2 t obtain, by hand, the solution for the first five steps using a time step of At = 0.1. Write a MATLAB program to solve the problem for 0 < t < 2. 17.5 Consider the scalar second-order equation: m i;t + cfa + ku + f = 0
Construct the discrete form for transient solution using the SS22 algorithm described in Sec. 17.3. (a) For the data m = k = 1, c = 0 and f = sin 2 t obtain, by hand, the solution for the first five steps using a time step of At -- 0.05. Write a MATLAB program to solve the problem for 0 < t < 2. (b) Repeat (a) for c = 0.05. 17.6 Consider the scalar second-order equation: mi2 + cfa + ku + f = 0
Construct the discrete form for transient solution using the GN22 algorithm described in Sec. 17.3.
Problems 627 (a) For the data m = k = 1, c = 0 and f = sin 2 t obtain, by hand, the solution for the first five steps using a time step of At = 0.05. Write a MATLAB program to solve the problem for 0 < t < 2. (b) Repeat (a) for c = 0.05. 17.7 For the general second-order equation
mfi + cit + ku + f = 0 develop the discrete form for the SS32 algorithm. 17.8 For the general second-order equation
mii +cit + k u + f = 0 develop the discrete form for the GN32 algorithm. 17.9 The general second-order equation may be split into the pair of first-order equations given by M~, + Cv + Ku + f = 0
u-v
=0
(a) Develop the discrete form of the equations using the SS 11 algorithm. (b) For the scalar form of the equations determine the amplification matrix and the stability characteristics of the method. (c) For the data m = k = 1, c = 0 and f = sin 2 t obtain, by hand, the solution for the first five steps using a time step of At -- 0.05. (d) Write a MATLAB program to solve the problem for 0 _< t _< 2. 17.10 The Hilber-Hughes-Taylor (HHT) algorithm 56 is given b y t Miin +~ + Cdn+~ + Kun +~ -+- fn+~ -- 0
where tn+~ -- (1 - an)tn + antn+l, Un+ot :
(1 - an)Un + anUn+l
fln+a :- (1 -- all)fin -t- aHtln+l fin +or :
fin+ 1
and fn+~ is the force at t,,+~. The algorithm is completed using the GN22 relations (17.58) with 3 1 /31 = 2 a n and /~2 "-- ~ ( 2 - a n ) 2 (a) For the scalar form of the equations determine the amplification matrix and the stability characteristics of the method. (If necessary, use MATLAB to determine the roots of the stability equation.) (b) For the data m -- k - 1, c = 0 and f -- sin 2 t obtain, by hand, the solution for the first five steps using a time step of At -- 0.05. (c) Write a MATLAB program to solve the problem for 0 < t < 2. t In the original publication a H = 1 + a and a had negative values. The definition used here is more consistent with other usage in this chapter and an is always positive.
628 The time dimension- discrete approximation in time
17.11 Using the Routh-Hurwitz criterion described in Sec. 17.4 perform the stability analysis for the first-order problem described by SS 11. 17.12 Using the Routh-Hurwitz criterion described in Sec. 17.4 perform the stability analysis for the second-order problem described by SS22. 17.13 Using the Routh-Hurwitz criterion, described in Sec. 17.4 perform the stability analysis for the second-order problem described by GN22. 17.14 Use FEAPpv to solve the rectangular beam problem described in Problem 16.13. Compare the solution with that computed by modal analysis. (Note: The comparison with the full modal solution gives the error between the discrete integration and an exact integration of the semi-discrete system.) 17.15 Use FEAPpv to solve the curved beam problem described in Problem 16.11. Compare the solution with that computed by modal analysis. 17.16 Program development project: Extend the program system started in Problem 2.17 to perform time integration using the single-step algorithms described in Sec. 17.3. Your implementation should include: (a) SS 11 to integrate a first-order system such as encountered for thermal analysis. (b) SS22 to integrate a second-order system for transient analysis of solids. (c) GN22 to integrate a second-order system for transient analysis of solids. Test your program by integrating a single degree of freedom problem for which you have a hand calculation for verification use. Solve the rectangular beam problem described in Problem 16.13. Compare the solution with that computed by modal analysis. (Note: The comparison with the full modal solution gives the error between the discrete integration and the exact integration of the semi-discrete system.)
1. R.D. Richtmyer and K.W. Morton. Difference Methods for Initial Value Problems. Wiley (Interscience), New York, 1967. 2. T.D. Lambert. Computational Methods in Ordinary Differential Equations. John Wiley & Sons, Chichester, 1973. 3. P. Henrici. Discrete Variable Methods in Ordinary Differential Equations. John Wiley & Sons, New York, 1962. 4. EB. Hildebrand. Finite Difference Equations and Simulations. Prentice-Hall, Englewood Cliffs, N.J., 1968. 5. G.W. Gear. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, N.J., 1971. 6. W.L. Wood. Practical Time Stepping Schemes. Clarendon Press, Oxford, 1990. 7. O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 8. J.T. Oden. A general theory of finite elements. Part II. Applications. Int. J. Numer. Meth. Eng., 1:247-254, 1969. 9. I. Fried. Finite element analysis of time-dependent phenomena. J. AIAA, 7:1170-1173, 1969. 10. J.H. Argyris and D.W. Scharpf. Finite elements in time and space. Nucl. Eng. and Design, 10:456--469, 1969.
References 629 11. O.C. Zienkiewicz and C.J. Parekh. Transient field problems - two and three dimensional analysis by isoparametric finite elements. Int. J. Numer. Meth. Eng., 2:61-71, 1970. 12. O.C. Zienkiewicz. The Finite Element Method in Engineering Science. McGraw-Hill, London, 2nd edition, 1971. 13. O.C. Zienkiewicz and R.W. Lewis. An analysis of various time stepping schemes for initial value problems. Earth. Eng. Struct. Dyn., 1:407-408, 1973. 14. W.L. Wood and R.W. Lewis. A comparison of time marching schemes for the transient heat conduction equation. Int. J. Numer. Meth. Eng., 9:679-689, 1975. 15. O.C. Zienkiewicz. A new look at the Newmark, Houbolt and other time stepping formulas. A weighted residual approach. Earth. Eng. Struct. Dyn., 5:413-4 18, 1977. 16. W.L. Wood. On the Zienkiewicz four-time-level scheme for numerical integration of vibration problems. Int. J. Numer. Meth. Eng., 11:1519-1528, 1977. 17. O.C. Zienkiewicz, W.L. Wood, and R.L. Taylor. An alternative single-step algorithm for dynamic problems. Earth. Eng. Struct. Dyn., 8:31-40, 1980. 18. W.L. Wood. A further look at Newmark, Houbolt, etc. time-stepping formulae. Int. J. Numer. Meth. Eng., 20:1009-1017, 1984. 19. O.C. Zienkiewicz, W.L. Wood, N.W. Hine, and R.L. Taylor. A unified set of single-step algorithms. Part 1: General formulation and applications. Int. J. Numer. Meth. Eng., 20:1529-1552, 1984. 20. W.L. Wood. A unified set of single-step algorithms. Part 2: Theory. Int. J. Numer. Meth. Eng., 20:2302-2309, 1984. 21. M. Katona and O.C. Zienkiewicz. A unified set of single-step algorithms. Part 3: The beta-m method, a generalization of the Newmark scheme. Int. J. Numer. Meth. Eng., 21:1345-1359, 1985. 22. E. Varoglu and N.D.L. Finn. A finite element method for the diffusion convection equations with concurrent coefficients. Adv. Water Res., 1:337-341, 1973. 23. C. Johnson, U. N~ivert, and J. Pitk~anta. Finite element methods for linear hyperbolic problems. Comp. Meth. Appl. Mech. Eng., 45:285-312, 1984. 24. T.J.R. Hughes, L.P. Franca, and G.M. Hulbert. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comp. Meth. Appl. Mech. Eng., 73:173-189, 1989. 25. T.J.R. Hughes and G.M. Hulbert. Space-time finite element methods in elastodynamics: formulation and error estimates. Comp. Meth. Appl. Mech. Eng., 66:339-363, 1988. 26. G.M. Hulbert and T.J.R. Hughes. Space-time finite element methods for second-order hyperbolic equations. Comp. Meth. Appl. Mech. Eng., 84:327-348, 1990. 27. G.M. Hulbert. Time finite element methods for structural dynamics. Int. J. Numer. Meth. Eng., 33:307-331, 1992. 28. B.M. Irons and C. Treharne. A bound theorem for eigenvalues and its practical application. In Proc. 3rd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-71-160, pages 245-254, Wright-Patterson Air Force Base, Ohio, 1972. 29. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3rd edition, 1982. 30. M.E. Gurtin. Variational principles for linear initial-value problems. Q. Appl. Math., 22: 252-256, 1964. 31. M.E. Gurtin. Variational principles for linear elastodynamics. Arch. Rat. Mech. Anal., 16:34-50, 1969. 32. E.L. Wilson and R.E. Nickell. Application of finite element method to heat conduction analysis. Nucl. Eng. Design, 4:1-11, 1966. 33. J. Crank and P. Nicolson. A practical method for numerical integration of solutions of partial differential equations of heat conduction type. Proc. Camb. Phil. Soc., 43:50, 1947.
630 The time dimension- discrete approximation in time 34. R.L. Taylor and O.C. Zienkiewicz. A note on the 'order of approximation'. Int. J. Solids Struct., 21:793-798, 1985. 35. W.H. Reed and T.R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973. 36. P. Lesaint and P.-A. Raviart. On a finite element method for solving the neutron transport equation. In C. de Boor, editor, Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, New York, 1974. 37. C. Johnson. Numerical Solutions of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge, 1987. 38. K. Eriksson and C. Johnson. Adaptive finite element methods for parabolic problems I: A linear model problem. SIAM J. Numer. Anal., 28:43-77, 1991. 39. X.D. Li and N.-E. Wiberg. Structural dynamic analysis by a time-discontinuous Galerkin finite element method. Int. J. Numer. Meth. Eng., 39:2131-2152, 1996. 40. M. Salvadori and M. Baron. Numerical Methods in Engineering. Prentice-Hall, New York, 1952. 41. M. Zlamal. Finite element methods in heat conduction problems. In J. Whiteman, editor, The Mathematics of Finite Elements and Applications, pages 85-104. Academic Press, London, 1977. 42. W. Liniger. Optimisation of a numerical integration method for stiff systems of ordinary differential equations. Technical Report RC2198, IBM Research, 1968. 43. J.M. Bettencourt, O.C. Zienkiewicz, and G. Cantin. Consistent use of finite elements in time and the performance of various recurrence schemes for heat diffusion equation. Int. J. Numer. Meth. Eng., 17:931-938, 1981. 44. N. Newmark. A method of computation for structural dynamics. J. Eng. Mech., ASCE, 85:67-94, 1959. 45. T. Belytschko and T.J.R. Hughes, editors. Computational Methods for Transient Analysis. North-Holland, Amsterdam, 1983. 46. I. Miranda, R.M. Ferencz, and T.J.R. Hughes. An improved implicit-explicit time integration method for structural dynamics. Earth. Eng. Struct. Dyn., 18:643-655, 1989. 47. E.J. Routh. A Treatise on the Stability of a Given State or Motion. Macmillan, London, 1977. 48. A. Hurwitz. Uber die Bedingungen, unter welchen eine Gleichung nur Wtirzeln mit negativen reellen teilen besitzt. Math. Ann., 46:273-284, 1895. 49. ER. Gantmacher. The Theory of Matrices. Chelsea, New York, 1959. 50. G.G. Dahlquist. A special stability problem for linear multistep methods. BIT, 3:27-43, 1963. 51. C.W. Gear. The automatic integration of stiff ordinary differential equations. In A.J.H. Morrell, editor, Information Processing 68. North-Holland, Dordrecht, 1969. 52. W. Liniger. Global accuracy and A-stability of one and two step integration formulae for stiff ordinary differential equations. In Proc. Conf. on Numerical Solution of Differential Equations, Dundee University, 1969. 53. J.C. Houbolt. A recurrence matrix solution for dynamic response of elastic aircraft. J. Aero. Sci., 17:540-550, 1950. 54. K.-J. Bathe and E.L. Wilson. Stability and accuracy analysis of direct integration methods. Earth. Eng. Struct. Dyn., 1:283-291, 1973. 55. W. Wood, M. Bossak, and O.C. Zienkiewicz. An alpha modification of Newmark's method. Int. J. Numer. Meth. Eng., 15:1562-1566, 1980. 56. H. Hilber, T.J.R. Hughes, and R.L. Taylor. Improved numerical dissipation for the time integration algorithms in structural dynamics. Earth. Eng. Struct. Dyn., 5:283-292, 1977. 57. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005.
Frequently two or more physical systems interact with each other, with the independent solution of any one system being impossible without simultaneous solution of the others. Such systems are known as coupled and of course such coupling may be weak or strong depending on the degree of interaction. An obvious 'coupled' problem is that of dynamic fluid-structure interaction. Here neither the fluid nor the structural system can be solved independently of the other due to the unknown interface forces. A definition of coupled systems may be generalized to include a wide range of problems and their numerical discretization as: 1
Coupled systems and formulations are those applicable to multiple domains and dependent variables which usually (but not always) describe different physical phenomena and in which (a) neither domain can be solved while separated from the other; (b) neither set of dependent variables can be explicitly eliminated at the differential equation level. The reader may well contrast this with definitions of mixed and irreducible formulations introduced in Chapter 3 and discussed fully in Chapter 10 and find some similarities. Clearly 'mixed' and 'coupled' formulations are analogous, with the main difference being that in the former elimination of some dependent variables is possible at the governing differential equation level. In the coupled system a full analytical solution or inversion of a (discretized) single system is necessary before such elimination is possible. Indeed, a further distinction can be made. In coupled systems the solution of any single system is a well-posed problem and is possible when the variables corresponding to the other system are prescribed. This is not always the case in mixed formulations. It is convenient to classify coupled systems into two categories:
Class L This class contains problems in which coupling occurs on domain interfaces via the boundary conditions imposed there. Generally the domains describe different physical situations but it is possible to consider coupling between domains that are physically similar in which different discretization processes have been used.
632 Coupledsystems Class II. This class contains problems in which the various domains overlap (totally or
partially). Here the coupling occurs through the governing differential equations describing different physical phenomena. Typical of the first category are the problems of fluid-structure interaction illustrated in Fig. 18.1(a) where physically different problems interact and also structure-structure interactions of Fig. 18.1(b) where the interface simply divides arbitrarily chosen regions in which different numerical discretizations are used. The need for the use of different discretizations may arise from different causes. Here for instance:
Fig. 18.1 Class I problems with coupling via interfaces (shown as thick line).
Coupled problems-definition and classification 633 1. Different finite element meshes may be advantageous to describe the subdomains. 2. Different procedures such as the combination of boundary method and finite elements in respective regions may be computationally desirable. 3. Domains may simply be divided by the choice of different time-stepping procedures, e.g., of an implicit and explicit kind. In the second category, typical problems are illustrated in Fig. 18.2. One of these is that of metal extrusion where the plastic flow is strongly coupled with the temperature field while at the same time the latter is influenced by the heat generated in the plastic flow. This problem is included to illustrate a form of coupling that commonly occurs in analyses of solids. The other problem shown in Fig. 18.2 is that of soil dynamics (earthquake response of a dam) in which the seepage flow and pressures interact with the dynamic behaviour of the soil 'skeleton'. We observe that, in the examples illustrated, motion invariably occurs. Indeed, the vast majority of coupled problems involve such transient behaviour and for this reason the present chapter will only consider this area. It will thus follow and expand the analysis techniques presented in Chapters 16 and 17. As the problems encountered in coupled analysis of various kinds are similar, we shall focus the presentation on three examples: 1. fluid-structure interaction (confined to small amplitudes); 2. soil-fluid interaction;
Fig. 18.2 Class II problems with coupling in overlapping domains.
634
Coupledsystems 3. implicit-explicit dynamic analysis of a structure where the separation involves the process of temporal discretization. In these problems all the typical features of coupled analysis will be found and extension to others will normally follow similar lines. As a final remark, it is worthwhile mentioning that problems such as linear thermal stress analysis to which we have referred frequently in this volume are not coupled in the terms defined here. In this the stress analysis problem requires a knowledge of the temperature field but the temperature problem can be solved independently of the stress field.t Thus the problem decouples in one direction. Many examples of truly coupled problems will be found in available books. 3-5
The problem of fluid-structure interaction is a wide one and covers many forms of fluid which we do not discuss in this book. The consideration of problems in which the fluid is in substantial motion is considered in standard texts on fluid dynamics (e.g., see reference 6) and, thus, we exclude at this stage such problems as flutter where movement of an aerofoil influences the flow pattern and forces around it leading to possible instability. For the same reason we also exclude here the 'singing wire' problem in which the shedding of vortices reacts with the motion of the wire. However, in a very considerable range of problems the fluid displacement remains small while interaction is substantial. In this category fall the first two examples of Fig. 18.1 in which the structural motions influence and react with the generation of pressures in a reservoir or a container. A number of symposia have been entirely devoted to this class of problems which is of considerable engineering interest, and here fortunately considerable simplifications are possible in the description of the fluid phase. References 7-22 give some typical studies. In such problems it is possible to write the linearized dynamic equations of fluid behaviour about the hydrostatic state as 0(pv) at
Ov po~ = - Vp + b at
(18.1)
where v is the fluid velocity, p is the fluid density (with P0 the density in the hydrostatic state), p the pressure and b is a constant body force of gravity. In postulating the above we have assumed 1. that the density P0 varies by a small amount only so may be considered constant; 2. that velocities are small enough for convective effects to be omitted; 3. that viscous effects by which deviatoric stresses are introduced can be neglected in the fluid. t In a general settingthe temperaturefield does dependuponthe strainrate. However,these terms are not included in the form presented in this volumeand in many instances produce insignificantchanges to the solution.2
Fluid-structure interaction (Class I problems) The reader can in fact note that with the preceding assumption Eq. (18.1) is a special form of a more general relation (described in reference 6). The linearized continuity equation based on the same assumption is P0 div v - p0VTv =
Op Ot
(18.2)
and noting that
Op Po Op ~ Ot K Ot
(18.3)
where K is the bulk modulus of the fluid, we can write VT v =
1 Op K Ot
(18.4)
Elimination of v between (18.1) and (18.4) gives the well-known scalar wave equation governing the pressure p: 1 02p ~72p-- c 2 0 t 2 (18.5) where
c=( K P0
(18.6)
denotes the speed of sound in the fluid. The equations described above are the basis of acoustic problems.
18.2.2 Boundary conditions for the fluid. Coupling and radiation In Fig. 18.3 we focus on the Class I problem illustrated in Fig. 18.1 (a) and on the boundary conditions possible for the fluid part described by the governing equation (18.5). As we know well, either normal gradients or values of p now need to be specified.
Fig. 18.3 Boundary conditions for the fluid component of the fluid-structure interaction.
635
636
Coupledsystems
Interface with solid
On the boundaries (D and ~ in Fig. 18.3 the normal velocities (or their time derivatives) are prescribed. Considering the pressure gradient in the normal direction to the face n we can thus write, by Eq. (18.1), Op On
= - P o O n = - p0nTv
(18.7)
where n is the direction cosine vector for an outward pointing normal to the fluid region and 0,, is prescribed. Thus, for instance, on boundary (!)coupling with the motion of the structure described by displacement u occurs. Here we put U n --- iL n "--
nTii
(18.8)
while on boundary ~ where only horizontal motion exists we have Vz - 0
(18.9)
Coupling with the structure motion occurs only via boundary ~ .
Free surface
On the free surface (boundary Q in Fig. 18.3) the simplest assumption is that p =0
(18.10)
However, this does not allow for any possibility of surface gravity waves. These can be approximated by assuming the actual surface to be at an elevation 0 relative to the mean surface. Now p = pogrl (18.11) where g is the acceleration due to gravity. From Eq. (18.1) we have, on noting Vz = O0/Ot and assuming P0 to be constant, 0277 Op = (18.12) Po Ot 2
OZ
and on elimination of r/, using Eq. (18.11), we have a specified normal gradient condition Op
az
=
10Zp
1 = --~0 g 0t 2 g
(18.13)
This allows for gravity waves to be approximately incorporated in the analysis and is known as the linearized surface wave condition.
Radiation boundary
Boundary (~)physically terminates an infinite domain and some approximation to account for the effect of such a termination is necessary. The main dynamic effect is simply that the wave solution of the governing equation (18.5) must here be composed of outgoing waves only as no input from the infinite domain exists. If we consider only variations in x (the horizontal direction) we know that the general solution of Eq. (18.5) can be written as p = F ( x - ct) + G(x + ct)
(18.14)
Fluid-structure interaction (Class I problems)
where c is the wave velocity given by Eq. (18.6) and the two waves F and G travel in positive and negative directions of x, respectively. The absence of the incoming wave G means that on boundary (~) we have only p -- F ( x - ct)
(18.15)
Op = Op = F' On Ox
(18.16)
Thus and
Op Ot
=
(18.17)
-cF'
where F' denotes the derivative of F with respect to (x - ct). We can therefore eliminate the unknown function F' and write Op
1
(18.18) On c which is a condition very similar to that of Eq. (18.13). This boundary condition was first presented in reference 7 for radiating boundaries and has an analogy with a damping element placed there. More accurate forms are possible to represent far field radiation conditions. For example, use of so-called perfectly matched layers (PML) is reported in references 23, 24. --
=
--~b
18.2.3 Weak form for coupled systems A weak form for each part of the coupled system may be written as described in Chapter 3. Accordingly, for the fluid we can write the differential equation as 8I-lf --
6p ~5~0-
p
d~ = 0
(18.19)
f
which after integration by parts and substitution of the boundary conditions described above yields fa
f
[~P ~-5 1 )~ + (V~P)T(VP) ] dr2+ Jfr ~pp0nTiidF+ f r 8 p -1~ d I ' + 1 3 g
= 0 f r ~p-/~dF 1 4 C (18.20)
where ~ f is the fluid domain and Fi the integral over boundary part O. Similarly for the solid the weak form after integration by parts is given by guT[ps i] +/Z/It + s T I ) S u - b] dr2 - ~ ~uTtdF = 0
(18.21)
,/1 t
where for pressure defined positive in compression the surface traction is defined as i = -pns
= pn
(18.22)
since the outward normal to the solid is ns = - n . The traction integral in Eq. (18.21) is now expressed as f r ,uTidF = f r ,uTnpdF t
t
(18.23)
637
638
Coupledsystems (1) In complex physical situations, the interaction between compressibility and internal gravity waves (interaction between acoustic modes and sloshing modes) leads to a modified scalar wave equation. Equation (18.5) should then be replaced by a more complex equation: in a stratified medium for instance, the irrotationality condition for the fluid is not totally verified (the fluid is irrotational in a plane perpendicular to the stratification axis). 16 (2) The variational formulation defined by Eq. (18.20) is valid in the static case provided the following constraints conditions are added f~i p d~ + po c2 fs~2y nTu dl-'= 0 for a compressible fluid filling a cavity, frl nTu dF + fr2 P/Pog dF = 0, for an incompressible liquid with a free surface contained inside a reservoir. The static behaviour is important for the modal response of coupled systems when modal truncation needs static corrections in order to accelerate the convergence of the method. This static behaviour is also of prime importance for the construction of reduced matrix models when using dynamic substructuring methods for fluid structure interaction problems. 17' 18
18.2.4 The discrete coupled system We shall now consider the coupled problem discretized in the standard (displacement) manner with the displacement vector approximated as u ~ fi = NuR
(18.24)
and the fluid similarly approximated by p ~ ~b = Np~
(18.25)
where ~ and ~ are the nodal parameters of each field and Nu and N p are appropriate shape functions. The discrete structural problem thus becomes Mfi + C~ + K~ - Q~ + f = 0
(18.26)
where the coupling term arises due to the pressures (tractions) specified on the boundary as f r NuTtdl-' - f r NuTnNpdl-'P = QP t
(18.27)
t
The terms of the other matrices are already well known to the reader as mass, damping, stiffness and force. Standard Galerkin discretization applied to the weak form of the fluid equation (18.20) leads to (including the possibility of a source term, q) S~ + Cb + H~ + p0QTfi + q = 0
(18.28)
where T%N d~-+Npc2 P
S=
T1NpdF C-- f r Npc 4
I - I - f (~TNp)TVNp dr2 and Q is identical to that of Eq. (18.27).
3 NpgNpdl7 (18.29)
Fluid-structure interaction (Class I problems)
18.2.5 Free vibrations If we consider free vibrations and omit all force and damping terms (noting that in the fluid component the damping is strictly that due to radiation energy loss) we can write the two equations (18.26) and (18.28) as a set:
M
[o
0
{~
J = 0
(18.30)
and attempt to proceed to establish the eigenvalues corresponding to natural frequencies. However, we note immediately that the system is not symmetric (nor positive definite) and that standard eigenvalue computation methods are not directly applicable. Physically it is, however, clear that the eigenvalues are real and that free vibration modes exist. The above problem is similar to that arising in vibration of rotating solids and special solution methods are available, though costly. 25 It is possible by various manipulations to arrive at a symmetric form and reduce the problem to a standard eigenvalue one. 14-22' 25-28 A simple method proposed by Ohayon proceeds to achieve the symmetrization objective by putting ~ - lie it~ p : p e ic~ and rewriting Eq. (18.30) as Kfl - Q~ - co2M~ = 0
(18.31)
H~ - co2S~ - co2p0QTl~ -- 0 and an additional variable t) such that = co2q)
(18.32)
After some manipulation and substitution we can write the new system as
" ~ I:~ ~ ~
1p0S
0
-co 2
QT
0
•p0
0
1 ST
_1 n
l~
p0
= 0
(18.33)
which is a symmetric generalized eigenproblem. Further, the variable q can now be eliminated by static condensation and the final system becomes symmetric and now contains only the basic variables. The system (18.32), with static corrections, may lead to convenient reduced matrix models through appropriate dynamic substructuring methods. 19 An alternative that has frequently been used is to introduce a new symmetrizing variable at the governing equation level, but this is clearly not necessary. 14' 15 As an example of a simple problem in the present category we show an analysis of a three-dimensional flexible wall vibrating with a fluid encased in a 'rigid' container 29 (Fig. 18.4).
18.2.6 Forced vibrations and transient step-by-step algorithms The reader can easily verify that the steady-state, linear response to periodic input can be readily computed in the complex frequency domain by the procedures described in Chapter 16. Here no difficulties arise due to the non-symmetric nature of equations and standard procedures can be applied. Chopra and coworkers have, for instance, done many
639
640
Coupled
systems
":2~
."""
,,
"
9
"
9
O
iiii
.~:k i
o
9
.
_[':2
0
TI o -"
..""
()
o
..( L..
9s. - 9
.e"
o
.(b.
..El"'"
.-"
( "-.
() ":()
Mode2 Frequency:43.6 Hz
9149149
9
[]
D 9149149149 D
9
Frequency:9"8 Hz
. ~ r 9149 ,
".~'.. ....O'"" "-....-" ]
"''''''()
Mode 1
9
.-'""" 9 9S
'
iiii
..Q,,
(a)
(:
9149149149
':)
.0.
()
Mode3
.-'""
---iii ..o-----
Fig. 18.4 Bodyof fluidwith a freesurfaceoscillatingwith a wall. Circlesshowpressureamplitudeandsquares indicateoppositesigns.Three-dimensionalapproachusingparabolicelements.
Fluid-structure interaction (Class I problems) studies of dam/reservoir interaction using such methods. 3~ 31 However, such methods are not generally economical for very large problems and fail in non-linear response studies. Here time-stepping procedures are required in the manner discussed in the previous chapter. However, simple application of methods developed there leads to an unsymmetric problem for the combined system (with fi and ~ as variables) due to the form of the matrices appearing in (18.30) and a modified approach is desirable. 32 In this, each of the equations (18.26) and (18.28) is first discretized in time separately using the general approaches of Chapter 17. Thus in the time interval At we can approximate 0 using, say, the general SS22 procedure as follows. First we write r2 UI = Un + an z" + O ~ 2
(18.34)
with a similar expression for p, T2 = Pn + 0n "g + ~ ~ -
(18.35)
where r = t - tn. Insertion of the above into Eqs (18.26) and (18.28) and weighting with two separate weighting f u n c t i o n s results in two relations in which cz and/3 are the unknowns. These are Mo~ + C(Un+l + 01Alex) -+- K(Un+l -+- 102A/2(x) -- Q(Pn+l + 102At2/~) + fn+l -- 0
(18.36a)
and 8/3 + QTcz + H (Pn+l + ~102At2/3) + qn+l -" 0
(18.36b)
where 1Uln+l = an + 01Attin ~n+l -- [In
(18.37)
Pn+l = Pn + 01Atpn
are the predictors for the n + 1 time step. In the above the parameters 0 i and 0i are similar to those of Eq. (18.49) and can be chosen by the user. It is interesting to note that the equation system can be put in symmetric form as
/. + where the second equation has been multiplied by - 1 , the unknown/3 has been replaced by /~ -- 102At2/3 (18.39) and the forces are given by F1 -- --fn+l -- C~n+l - KUn+l -+- Qpn+l
(18.40)
F2 - qn+l + H0n+l It is not necessary to go into detail about the computation steps as these follow the usual patterns of determining cz and/3 and then evaluation of the problem variables, that is Un+l,
641
642
Coupledsystems Pn+l, tln+l and Pn+l at tn+l before proceeding with the next time step. Non-linearity of structural behaviour can also be accommodated (e.g., see reference 33). It is, however, important to consider the stability of the linear system which will, of course, depend on the choice of Oi and Oi. Here we find, by using procedures described in Chapter 17, that unconditional stability is obtained when 1
02 >__O1 02 ~ O1
O1 >_ ~ O1 ~
--
--
(18.41)
1
It is instructive to note that precisely the same result would be obtained if GN22 approximations were used in Eqs (18.34) and (18.35). The derivation of such stability conditions is straightforward and follows precisely the lines of Sec.17.4 of the previous chapter. However, the algebra is sometimes tedious. Nevertheless, to allow the reader to repeat such calculations for any case encountered we shall outline the calculations for the present example.
Stability of the fluid-structure time-stepping scheme 32
For stability evaluations it is always advisable to consider the modally decomposed system with scalar variables. We thus rewrite Eqs (18.36a) and (18.36b) omitting the forcing terms and putting Oi = Oi as mot + C([l n
-+- 01Atot) + k(un + O1AtUn -k- 102 At2ot)
-- q ( p . + 01Atpn + 102At2/3) = 0
and
1 s/3 + q a + h(pn -~- 01Atp + ~02At2/~)
-- 0
(18.42a)
(18.42b)
To complete the recurrence relations we have 1
Un+ 1 ~" Un -+- A t { t n Jr ~ AtZot
/~n+l = i/n -~- Atot Pn+l = Pn + A t p n -t-
1 At2/3
(18.42c)
ign+l = Pn + At/3
The exact solution of the above system will always be of the form Un+l ~ ll~Un
/~/n+l --" /Z/~n
(18.43)
Pn+l -'-/zp, /)n+l = /ZPn and immediately we put /z=
l+z
1-z knowing that for stability we require the real part of z to be negative. Eliminating all n + 1 values from Eqs (18.42c) and (18.43) leads to 2z
~n -- - - ~ Ign
4Z2 ot = (1 - z) At 2 Un
2z
19n = - ~ pn
/~
4Z2 (1 - z) At 2 Pn
(18.44)
Fluid-structure interaction (Class I problems) Inserting (18.44) into the system (18.42a) and (18.42b) gives (allZ 2 -+- bllZ --1-k)un + (al2z 2 q- bl2z - q)Pn = 0 4qZ2Un -]- (a22z 2 + b22z -+- h ' ) p n - 0
(18.45)
where all
--
4 m ' - 2(1
-
a12 =
2q(01 - 02)
a22 - -
4s
-
201)c t-
2k(01
-02)
2(01 - O 2 ) h '
(18.46)
bll = 2 c ' - k(1 - 201) b12 = (1 -
201)q
b22 - - - ( 1
-
201)h'
m At 2
ct =
in which mt =
c At
ht _ A t 2 h
For non-trivial solutions to exist the determinant of the coefficient matrix Eq. (18.45) has to be zero. This determinant provides the characteristic equation for z which, in the present case, is a polynomial of fourth order of the form aoz 4 -Jr a l z 3 + a2z 2 + a3z -t- a4 -- 0
Thus use of the Routh-Hurwitz conditions given in Sec. 17.4 ensures stability requirements are satisfied, i.e., that the roots of z have negative real parts. For the present case the requirements are the following a0 > 0
and
ai > 0,
i = 1,2,3,4
The inequality a l i a 2 2 - - 8 q 2 ( 0 1 - - 02) > 0
(18.47)
is satisfied for m', c', k, s, h' > 0 if 1
O1 ~> ~
02 ~ O1
The inequality al = all [ - h ' ( 1 - 201)] + [ 2 c ' - k(1 - 201)] a22 - 4qb12 > 0
(18.48)
is also satisfied if 1
O1 >__ ~
02 >__O1
The inequalities all h p q-bllb22 -+ a22k -+-4q 2 > 0 a3 -- bll h t + b:z2k > 0 a2
=
(18.49)
are satisfied if (18.47) and (18.48) are satisfied. The inequality a4 -- k h ' > 0
(18.50)
643
644
Coupledsystems is automatically satisfied. Finally the two inequalities a l a 2 -- aoa3 >__ 0
(18.51)
a l a 2 a 3 -- a o a 2 -- a 4 a 2 > 0
are also satisfied if (18.47) and (18.48) are satisfied. If all the equalities hold then m ' s > 0 has to be satisfied. In case then 02 > 01 must be enforced.
m's
= 0 and c' = 0
18.2.7 Special case of incompressible fluids If the fluid is incompressible as well as being inviscid, its behaviour is described by a simple laplacian equation VZp = 0
(18.52)
obtained by putting c = cx~ in Eq. (18.5). In the absence of surface wave effects and of non-zero prescribed pressures the discrete equation (18.28) becomes simply I-I~ = _QT~
(18.53)
as wave radiation disappears. It is now simple to obtain = --H-1QTfi
(18.54)
and substitution of the above into the structure equation (18.26) results in (M + QH-1QT)~ + C~ + K~ + f = 0
(18.55)
This is now a standard structural system in which the mass matrix has been augmented by an a d d e d m a s s m a t r i x as Mu = QH-1QT
(18.56)
and its solution follows the standard procedures of previous chapters. We have to remark that: 1. In general the complete inverse of H is not required as pressures at interface nodes only are needed. 2. In general the question of when compressibility effects can be ignored is a difficult one and will depend much on the frequencies that have to be considered in the analysis. For instance, in the analysis of the reservoir-dam interaction much debate on the subject has been recorded. 34 Here the fundamental compressible period may be of order H / c where H is a typical dimension (such as height of the dam). If this period is of the same order as that of, say, earthquake forcing motion then, of course, compressibility must be taken into account. If it is much shorter then its neglect can be justified.
Soil-pore fluid interaction (Class II problems)
18.2.8 Cavitation effects in fluids In fluids such as water the linear behaviour under volumetric strain ceases when pressures fall below a certain threshold. This is the vapour pressure limit. When this is reached cavities or distributed bubbles form and the pressure remains almost constant. To follow such behaviour a non-linear constitutive law has to be introduced. Although this book is primarily devoted to linear problems we here indicate some of the steps which are necessary to extend analyses to account for non-linear behaviour. A convenient variable useful in cavitation analysis was defined by Newton 35 s = div(pu)
(18.57)
-- V T (pu)
where u is the fluid displacement. The non-linearity now is such that p = - K divu P = Pa -
=
c2s,
if s < (Pa -- P v ) / c 2 if s > (Pa -- P v ) / c 2
Pv,
(18.58)
Here Pa is the atmospheric pressure (at which u = 0 is assumed), pv is the vapour pressure and c is the sound velocity in the fluid. Clearly monitoring strains is a difficult problem in the formulation using the velocity and pressure variables [Eq. (18.1) and (18.5)]. Here it is convenient to introduce a displacement potential ~ such that pu = - V ~ (18.59) From the momentum equation (18.1) we see that =
=
-re
and thus ~) = p
(18.60)
The continuity equation (18.2) now gives s-
1
1 ..
p d i v u - - - V 2 1 / / " = ~ - - ~ p - - ~--~lp
(18.61)
in the linear case [with an appropriate change according to conditions (18.58) during cavitation]. Details of boundary conditions, discretization and coupling are fully described in reference 36 and follow the standard methodology previously given. Figure 18.5, taken from that reference, illustrates the results of a non-linear analysis showing the development of cavity zones in a reservoir.
It is well known that the behaviour of soils (and indeed other geomaterials) is strongly influenced by the pressures of the fluid present in the pores of the material. Indeed, the
645
646
Coupledsystems
Fig. 18.5 The Bhakra dam-reservoir system36 Interaction during the first second of earthquake motion showing the development of cavitation. concept of effective stress is here of paramount importance. Thus if ~r describes the total stress (positive in tension) acting on the total area of the soil and the pores, and p is the pressure of the fluid (positive in compression) in the pores (generally of water), the effective stress is defined as cr' = ~r + m p
(18.62)
Here m T = [1, 1, 1, 0, 0, 0] if we use the notation in Chapter 11. Now it is well known that it is only the stress o" which is responsible for the deformations (or failure) of the solid skeleton of the soil (excluding here a very small volumetric grain compression which has to be included in some cases). Assuming for the development given here that the soil can be represented by a linear elastic model we have o - ' = De
(18.63)
Soil-pore fluid interaction (Class II problems) Immediately the total discrete equilibrium equations for the soil-fluid mixture can be written in exactly the same form as is done for all problems of solid mechanics: Mfi + Cfi + f~ BTtr dr2 + f = 0
(18.64)
where fi are the displacement discrefization parameters, i.e., u ~
N~
fi -
(18.65)
B is the strain-displacement matrix and M, C, f have the usual meaning of mass, damping and force matrices, respectively. Now, however, the term involving the stress must be split as f BTtr dr2 = ~ BTtr' dr2 - ~ BTmp dr2
(18.66)
to allow the direct relationship between effective stresses and strains (and hence displacements) to be incorporated. For a linear elastic soil skeleton we immediately have Mfi + Cd + Kfi - Q~ + f = 0
(18.67)
where K is the standard stiffness matrix written as j ( BTtr' dr2 = ( f
BTDBdf2) fi = Kfi
(18.68)
and Q couples the field of pressures in the equilibrium equations assuming these are discretized as p ~ ~b = Np~ (18.69) Thus Q - f~ BTmNp dr2
(18.70)
In the above discretization conventionally the same element shapes are used for the fi and variables, though not necessarily identical interpolations. With the dynamic equations coupled to the pressure field an additional equation is clearly needed from which the pressure field can be derived. This is provided by the transient seepage equation of the form 1
- V T ( k V p ) + ~ b + ev = 0
(18.71)
where Q is related to the compressibility of the fluid, k is the permeability and ev is the volumetric strain in the soil skeleton, which on discretization of displacements is given by ev = m Te = mTBfi
(18.72)
The equation of seepage can now be discretized in the standard Galerkin manner as QTfi + S~ + n ~ + q = 0
(18.73)
where Q is precisely that of Eq. (18.70), and S = f~ N pT~1 N p dr2
H = f~ ( V N p ) T k V N p dr2
(18.74)
647
648
Coupledsystems with q containing the forcing and boundary terms. The derivation of coupled flow-soil equations was first introduced by Biot 37 but the present formulation is elaborated upon in references 32, 34--45 where various approximations, as well as the effect of various non-linear constitutive relations, are discussed. We shall not comment in detail on any of the boundary conditions as these are of standard type and are well documented in previous chapters.
18.3.2 The format of the coupled equations The solution of coupled equations often involves non-linear behaviour, as noted previously in the cavitation problem. However, it is instructive to consider the linear version of Eqs (18.67) and (18.73). This can be written as
[o :1
C
:]
[:
d
{o} {q} fi
f
(18.75)
Once again, like in the fluid-structure interaction problem, overall asymmetry occurs despite the inherent symmetry of the M, C, K, S and H matrices. As the free vibration problem is of no great interest here, we shall not discuss its symmetrization. In the transient solution algorithm we shall proceed in a similar manner to that described in Sec. 18.2.6 and again symmetry will be observed.
18.3.3 Transient step-by-step algorithm Time-stepping procedures can be derived in a manner analogous to that presented in Sec. 18.2.6. Here we choose to use the GNpj algorithm of lowest order to approximate each variable. Thus for fi we shall use GN22, writing Un+l -m- Un +
1
Atfin + ~ Ataiin
1
+ ~/~2At2AUn+l
P + 21/~2 At2 Afin+l Un+l
(18.76)
I]ln+l = tlln + A t tl n - t - / 3 1 A t A O n + l .p Un+ 1 + / 3 1 A t A O n + l
For the variables p that occur in first-order form we shall use GN11, as Pn+l - - P n + A t p n - + - O A t A p n + l p = Pn+ 1 -+- 0 At Apn +1
(18.77)
In the above u.+ p 1, etc., denote values that can be 'predicted' from known parameters at time tn and AiJn+l = Un+l -- iin APn+I = 0n+l -- Pn (18.78) are the unknowns. To complete the recurrence algorithm it is necessary to insert the above into the coupled governing equations [(18.64) and (18.73)] written at time tn+l. Thus we require the following equalities
Soil-pore fluid interaction (Class II problems) MUn+l at- Ciin+l at-
B O'n+ 1 - Qpn+l + fn+l --" 0
(18.79)
QTlin+l @ Spn+l + Hpn+l + qn+l : 0 in which o"+1 is evaluated using the constitutive equation (18.63) in incremental form and knowledge of trnf as I ! O'n4_1 -- or n +
DA6n+I
I = or n -'~ DBAu,,+I
(18.80)
In general the above system may be non-linear and indeed on many occasions the H matrix itself may be dependent on the values of u due to permeability variations with strain. It is of interest to look at the linear form as the non-linear system usually solves a similar form iteratively. 33 Here insertion of Eqs (18.76), (18.77) and (18.80) into (18.79) results in the equation system [ (M d-/31AtC lfl2At2K)_QT d-
+ ~-~S)] (AU. }nPn+l 'n+I } _ {F1 F2
_ (-Q1H
(18.81)
where F1 and F2 are vectors that can be evaluated from loads and solution values at Symmetry in the above is obtained by multiplying Eq. (18.36b) by - 1 and defining
tn.
(18.82)
Apn+I -- /~1AtA0n+l
The solution of Eq. (18.81) and the use of Eqs (18.76) and (18.77) complete the recurrence relation. The stability of the linear scheme can be found by following identical procedures to those used in Sec. 18.2.6 and the result is that stability is unconditional when 27
/32 >_ /~1
/~1 >_ ~1
0 >_ ~1
(18.83)
18.3.4 Special cases and robustness requirements Frequently the compressibility of the fluid phase, which forms the matrix S, is such that S~0 compared with other terms. Further, the permeability k may on occasion also be very small (as, say, in clays) and H~0 leading to so-called 'undrained' behaviour. Now the coefficient matrix in (18.81) becomes of the lagrangian constrained form (see Chapter 10), i.e., _QT
-Q01 { A_i_i A n +Pln}+ l
{F:}
and is solvable only if nu > n p
where
nu
and n p denote the number of fi and ~ parameters, respectively.
(18.84)
649
650 Coupledsystems
OU
Dp
0
Fig. 18.6 'Robust' interpolations for the coupled soil-fluid problem. The problem is indeed identical to that encountered in incompressible behaviour and the interpolations used for the u and p variables have to satisfy identical criteria. As Co interpolation for both variables is necessary for the general case, suitable element forms are shown in Fig. 18.6 and can be used with confidence. Alternatively, equal order interpolation may be used for u and p in conjunction with stabilized forms discussed in Sec.11.7. The formulation can of course be used for steady-state solutions but it must be remarked that in such cases an uncoupling occurs as the seepage equation can be solved independently. Finally, it is worth remarking that the formulation also solves the well-known soil consolidation problem where the phenomena are so slow that the dynamic term Mfi tends to 0. However, no special modifications are necessary and the algorithm form is again applicable.
18.3.5 Examples - soil liquefaction As we have already mentioned, the most interesting applications of the coupled soil-fluid behaviour is when non-linear soil properties are taken into account. In particular, it is a well-known fact that repeated straining of a granular, soil-like material in the absence of the pore fluid results in a decrease of volume (densification) due to particle rearrangement. Constitutive equations which include this effect are available; 33 however, here we only represent a typical result which they can achieve when used in a coupled soil-fluid solution. When a pore fluid is present, densification will (via the coupling terms) tend to increase the fluid pressures and hence reduce the soil strength. This, as is well known, decreases with the compressive mean effective stress. It is not surprising therefore that under dynamic action the soil frequently loses all of its strength (i.e., liquefies) and behaves almost like a fluid, leading occasionally to catastrophic failures of structural foundations in earthquakes. The reproduction of such phenomena with computational models is not easy as a complete constitutive behaviour description for soils is imperfect. However, much effort devoted to the subject has produced good results 38-45
Soil-pore fluid interaction (Class II problems)
Fig. 18.7 Soil-pressure water interaction. Computation and centrifuge model results compared on a problem of a dyke foundation subject to a simulated earthquake.
651
652 Coupledsystems
Fig. 18.7 (Cont.) and a reasonable confidence in predictions achieved by comparison with experimental studies exists. One such study is illustrated in Fig. 18.7 where a comparison with tests carried out in a centrifuge is made. 44, 45 In particular the close correlation between computed pressure and displacement with experiments should be noted.
18.3.6 Biomechanics, oil recovery and other applications The interaction between a porous medium and interstitial fluid is not confined to soils. The same equations describe, for instance, the biomechanics problem of bone-fluid interaction in vivo. Applications in this field have been documented. 46, 47 On occasion two (or more) fluids are present in the pores and here similar equations can again be written 46' 47 to describe the interaction. Problems of ground settlement in oil fields
Partitioned single-phase systems -implicit-explicit partitions (Class I problems) 653 due to oil extraction, or flow of water/oil mixtures in oil recovery are good examples of application of techniques described here.
In Fig. 18.1 (b), describing problems coupled by an interface, we have already indicated the possibility of a structure being partitioned into substructures and linked along an interface only. Here the substructures will in general be of a similar kind but may differ in the manner (or simply size) of discretization used in each or even in the transient recurrence algorithms employed. In Chapter 12 we have described special kinds of mixed formulations allowing the linking of domains in which, say, boundary-type approximations are used in one and standard finite elements in the other. We shall not return to this phase and will simply assume that the total system can be described using such procedures by a single set of equations in time. Here we shall only consider a first-order problem (but a similar approach can be extended to the second-order dynamic system): Cd + Kfi + f -- 0
(18.85)
which can be partitioned into two (or more) components, writing
Cll C12] ~1 C21 C22J / ~ 2 ) "l- IK121 K12]
(18.86)
Now for various reasons it may be desirable to use in each partition a different time-step algorithm. Here we shall assume the same structure of the algorithm (SS 11) and the same time step (At) but simply a different parameter 0 in each. Proceeding thus as in the other coupled analyses we write
UI.1= Uln "+""~'0~1 1HI2~ U2n -I'-T'O~2
(18.87)
Inserting the above into each of the partitions and using different weight functions, we obtain
CllO~l -+- C120~2 -+- Kll(Uln + 0AtCzl) + K12(U2n -+- 0Ato~2) + f l = 0 C210~1 + C220~2 + K21 (Uln -+- 0At~l) -+- K22(U2n -+-0Atcz2) + f2 = 0
(18.88)
This system may be solved in the usual manner for O~ 1 and O~2 and recurrence relations obtained even if 0 and 0 differ. The remaining details of the time-step calculations follow the obvious pattern but the question of coupling stability must be addressed. Details of such stability evaluation in this case are given elsewhere 48 but the result is interesting. 1. Unconditional stability of the whole system occurs if 0>~ 1 2. Conditional stability requires that
0 >_1~
654
Coupledsystems At _< Atcrit where the Atcrit condition is that pertaining to each partitioned system considered without its coupling terms.
Indeed, similar results will be obtained for the second-order systems Mfi + Cfi + Kfi + f - 0
(18.89)
partitioned in a similar manner with SS22 or GN22 used in each. The reader may well ask why different schemes should be used in each partition of the domain. The answer in the case of implicit-implicit schemes may be simply the desire to introduce different degrees of algorithmic damping. However, much more important is the use of implicit-explicit partitions. As we have shown in both 'thermal' and dynamictype problems the critical time step is inversely proportional to h e and h (the element size), respectively. Clearly if a single explicit scheme were to be used with very small elements (or very large material property differences) occurring in one partition, this time step may become too short for economy to be preserved in its use. In such cases it may be advantageous to use an explicit scheme (with 0 - 0 in first-order problems, 02 = 0 in dynamics) for a part of the domain with larger elements while maintaining unconditional stability with the same time step in the partition in which elements are small or otherwise very 'stiff'. For this reason such implicit-explicit partitions are frequently used in practice. Indeed, with a lumped representation of matrices C or M such schemes are in effect staggered as the explicit part can be advanced independently of the implicit part and immediately provides the boundary values for the implicit partition. We shall return to such staggered solutions in the next section. The use of explicit-implicit partitions was first recorded in 1978. 49-51 In the first reference the process is given in an identical manner as presented here; in the second, a different algorithm is given based on an element split (instead of the implied nodal split above) as described next.
partition
Implicit-explicit solution - element
We again consider the first-order problem given in Eq. (18.85) and split as
Ci~I -JI--CEUE ']- Klfll + Kefie + f = 0
(18.90)
where the subscript I denotes an implicit partition and subscript E an explicit one. An iteration process may be used in which one or more iterations per time step are used. The recurrence relation for u at iteration j is written using GN11 as i.t(j) n+l
with
. n( +j -l 1) + -- U
0
9( j ) AtUn+l
a(0)
n+ 1 -- Un + (1 -- O) Ati~n
(18.91) (18.92)
Using the iteration process an approximation for the implicit-explicit split is now taken as ~ Hi
. (J) -- Un+ 1
~ BE
(j-l) -- Un+ 1
~I
=
~E
=
"U(nJ +) 1
thus yielding the system of equations at iteration j as
Staggered solution processes 655 (C +
OAtKI)fi~J)+l + F (j)
(18.93)
= 0
where F (j) contains the loading terms which depend on known values at tn and possibly previous iterate values (j - 1). The above algorithm has stability properties which depend on the choice of 0. For a linear system with 0 >_ 0.5 the implicit part is unconditionally stable and stability depends onthe Atcr,t of the explicit elements. 5~ 51 Performing only one iteration in each time step is normally used; however, improved accuracy in the explicit partition can occur if additional iterations are used, although the cost of each time step is obviously increased.
We have observed in the previous section that in the nodal-based implicit-explicit partitioning of time stepping it was possible to proceed in a staggered fashion, achieving a complete solution of the explicit scheme independently of the implicit one and then using the results to progress with the implicit partition. It is tempting to examine the possibility of such staggered procedures generally even if each uses an independent algorithm. In such procedures the first equation would be solved with some assumed (predicted) values for thevariable of the other. Once the solution for the first system was obtained its values could be substituted in the second system, again allowing its independent treatment. If such procedures can be made stable and reasonably accurate many possibilities are immediately open, for instance: 1. Completely different methodologies could be used in each part of the coupled system. 2. Independently developed codes dealing efficiently with single systems could be combined. 3. Parallel computation with its inherent advantages could be used. 4. Finally, in systems of the same physics, efficient iterative solvers could easily be developed. The problems of such staggered solutions have been frequently discussed 36' 52-55 and on occasion unconditional stability could not be achieved without substantial modification. In the following we shall indicate some options available.
18.5.2 Staggered process of solution in single-phase systems We shall look at this possibility first, having already mentioned it as a special form arising naturally in the implicit-explicit processes of Sec~l8.4. We return here to consider the problem of Eq. (18.85) and the partitioning given in Eq. (18.86). Further, for simplicity we shall assume a diagonal form of the C matrix, i.e., that the problem is posed as
1 c 2]
{:}
1894
656 Coupledsystems As we have already remarked, the use of 0 = 0 in the first equation and 0 > 0.5 in the second [see Eq. (18.88)] allowed the explicit part to be solved independently of the implicit. Now, however, we shall use the same 0 in both equations but in the first of the approximations, analogous to Eq. (18.88), we shall insert a predicted value for the second variable: UI2 -" U~ "-- UZn
(18.95)
This is similar to the treatment of the explicit part in the element split of the implicit-explicit scheme and gives in place of Eq. (18.88) (18.96)
CllOtl -~- Kll(Uln -~-0AtOtl) = - f l - K121LI2n
allowing direct solution for c~1. Following this step, the second equation can be solved for ot2 with the previous value of c~1 inserted, i.e., (18.97)
C220t2 -+- K22(u2 n -+-0Atot2) = - f 2 - K21 (Uln -+-0AtOtl)
This scheme is unconditionally stable if 0 > 0.5, i.e., the total system is stable provided each stagger is unconditionally stable. A similar condition holds for linear second-order dynamic problems. Obviously, however, some accuracy will be lost as the approximation of Eq. (18.96) is that of the predicted value of u2. The approximation is consistent and hence convergence will occur as At ~ 0. The advantage of using the staggered process in the above is clear as the equation solving, even though not explicit, is now confined to the magnitude of each partition and computational economy occurs. Further, it is obvious that precisely the same procedures can be used for any number of partitions and that again the same stability conditions will apply. Define the arrays "ell
C22 ~
C
__.
(18.98a)
Cii Ckk
"Kll
K21
0
~
1
4
9
1
4
0
9
K22
+
.
K
m
K12
o
0
..
.
Klk
.
~
~
Kii .
Kkl
""
Kk,k-1
0
Kkk.
Kk- l,k 0
.
o
.
0
"-'KL + K u (18.98b) and consider the partition
Staggered solution processes
0.8
I
m
0.6 0.4 0.2
m
Implicit = 1 At/Atcrit = 4
y/
~..'"
'
~s S
###"
-
J
~
iExplicit'-sPlit
,""
At/Atcrit = 4
_-......"~_ _ . I.. -'~176176176I
0
20
40
T-, 1
I
I
I
I
60
80
100
120
t/Atcrit
~
140
I
# To=0
I
critical time step for standard explicit form Tc = temperature on centre-line
Atcrit =
Fig. 18.8 Accuracyof an explicit-split procedure compared with a standard implicit process for heat conduction of a bar.
C~ + KL~ + Ku~ + f = 0
(18.99)
Introducing now the approximation Ui --Uin
Jr- 150t i
(18.100)
and using Eq. (18.95) gives the discrete form (C + 0AtKL) c~ + Kuun 4- t' - 0
(18.101)
where f contains the load and effects from u,,. In approximating the first equation set it is necessary to use predicted values for u2, u3, . . . , uk, writing in place of Eq. (18.96),
CllO~l -q- Kll(Uln -+-0AtOll) -+- K12u2n "q- K13u3n "~-""" "~-fl -" 0
(18.102)
and continue similarly to (18.97), with the predicted values now continually being replaced by better approximations as the solution progresses. The partitioning of Eq. (18.98a) can be continued until only a single equation set is obtained. Then at each step the equation that requires solving for Ot i is of the form ( C i i -3t- O A t K i i )
ot i - - Fi
(18.103)
where Fi contains the effects of the load and all the previously computed u i . For partitions where each submatrix is a scalar Eq. (18.103) is a scalar equation and computation is thus fully explicit and yet preserves unconditional stability for 0 > 0.5. This type of partitioning and the derivation of an unconditionally stable explicit scheme was first proposed by Zienkiewicz et al. 56 An alternative and somewhat more limited scheme of a similar kind was given by Trujillo. 57
657
658 Coupledsystems 9
lw
9 9
=A~, ~ 5,,
~,
lw
i
9
9
3,, 4
,
Z,
~
8.,
9
lw
.~,, 9
9r
9
~w
9F
2. ~, 4,,~. ~,
9
~,~
9
9
; 9,;,10,, 11(, 12,
13,,; 14,, , 15,,) 1~)
13, 14 15, 1~ ',27.
~~"
""
~"
:
Fig. 18.9 Partitionscorrespondingto the well-known ADI (alternating direction implicit) finite difference scheme. Clearly the error in the approximation in the time step decreases as the solution sweeps through the partitions and hence it is advisable to alter the sweep directions during the computation. For instance, in Fig. 18.8 we show quite reasonable accuracy for a onedimensional heat-conduction problem in which the explicit-split process was used with alternating direction of sweeps. Of course the accuracy is much inferior to that exhibited by a standard implicit scheme with the same time step, though the process could be used quite effectively as an iteration to obtain steady-state solutions. Here many other options are also possible. It is, for instance, of interest to consider the system given in Eqs (18.98a), (18.98b) and (18.99) as originating from a simple finite difference approximation to, say, a heatconduction equation on the rectangular mesh of Fig. 18.9. Here it is well known that the so-called alternating direction implicit (ADI) scheme 58 presents an efficient solution for both transient and steady-state problems. It is fairly obvious that the scheme simply represents the procedure just outlined with partitions representing lines of nodes such as (1, 5, 9, 13), (2, 6, 10, 14), etc., ofFig. 18.9 alternating with partitions (1, 2, 3, 4), (5, 6, 7, 8), etc. Obviously the bigger the partition, the more accurate the scheme becomes, though of course at the expense of computational costs. The concept of the staggered partition clearly allows easy adoption of such procedures in the finite element context. Here irregular partitions arbitrarily chosen could be made but so far applications have only been recorded in regular mesh subdivisions. 58 The field of possibilities is obviously large. Use in parallel computation is obvious for such procedures. A further possibility which has many advantages is to use hierarchical variables based on, say, linear, quadratic and higher expansions and to consider each set of these variables as a partition. 59 Such procedures are particularly efficient in iteration if coupled with suitable preconditioning 60 and form a basis of multigridprocedures. 61-63
18.5.3 Staggered schemes in fluid-structure systems and
stabilization processes
The application of staggered solution methods in coupled problems representing different phenomena is more obvious, though, as it turns out, more difficult.
Staggered solution processes 659 For instance, let us consider the linear discrete fluid-structure equations with Po = 1 and damping omitted, written as [see Eqs (18.26) and 18.28)]
sl {;}+
(18.104)
where we have omitted the tilde superscript for simplicity. For illustration purposes we shall use the GN22 type of approximation for both variables and write using Eq. (18.76) p 1 Un_l_1 -~ Un+ 1 -~- ~/32AtZAfin+l .p I]ln+l = Un+l "]"/~1 At Aiin+l p 1 P n + l = P n + l -+- ~ / ~ 2 A t 2 A l J n + l 1J-n+l -" P"np+ l + / ~ 1
(18.105)
At A P n + l
which together with Eq. (18.104) written at t -- tn+l completes the system of equations requiring simultaneous solution for A n n + 1 and AlJn+l. Now a staggered solution of a fairly obvious kind would be to write the first set of equations (18.104) corresponding to the structural behaviour with a predicted (approximate) p value of Pn+l "- Pn+l, as this would allow an independent solution for Aiin+l writing Miin+l -+- KUn+l
-
-
P - f + QPn+I
(18.106)
This would then be followed by the solution of the fluid problem for AlJn+ 1 writing Sii~+l + Hp~+I = - q - QTiin+I
(18.107)
This scheme turns out, however, to be only conditionally stable, 48 even if/3/and/3i are chosen so that unconditional stability of a simultaneous solution is achieved. (The stability limit is indeed the same as if a fully explicit scheme were chosen for the f l u i d phase.) Various stabilization schemes can be used here. 27' 48 One of these is given below. In this Eq. (18.106) is augmented to MUn+l -I-- (K -~- QS-1QT)un+I = - f -t- QpnP+I -I- Q s -1QTun+lp
(18.108)
before solving for Aiin+l. It turns out that this scheme is now unconditionally stable provided the usual conditions
/~2 >__/~1
~1 >__
1
are satisfied. Such stabilization involves the inverse of S but again it should be noted that this needs to be obtained only for the coupling nodes on the interface. Another stable scheme involves a similar inversion of H and is useful as incompressible behaviour is automatically given. Similar stabilization processes have been applied with success to the soil-fluid system.64, 65
660
Coupled systems
iii ig77:7 Sq!iiiiiiii !{iiSiiiii!:!ii!{ili;iii;!:i i!!!iiiiii iiiiiiii iiiiii !i :;==:::i : i :;i:~71: : ~:17i77i{77i 7!!!: 7!17 :iT::iiii! i777i7~;ili!~i!i~i77i1i1iii~7i:!ii!i!iiTi=~:Ti ii~;'=i!:~iT=iii!~::i:i77i1T1ii!i:i~i~i !i~:iii!i:T ~i~illT ~ii~!~:~::i~~!i ~i!:i~~i ::=ii77 !i=i717 ~~ili:iii7i!7= ~=i?i!i?i~!i:i!iili;iliii; !i:i,:7i~!iiil ;~i;i:7~=:i;7~~7i!!i7:?i;:i~!ii;ii i!~~:?1T1i7!7i?!!~i i:~i!!?; i!!7i ii;i; i!iiii7:i:Tiii~!i!!7~i !iiiTiiTii!;ilii i!i)i~!iTiiii!il i7ii7ii7~i??ii7iiiii ii~Ti 7i!iiii! iii~7i!~iiii;:i!iT;i~!!17~~77:?!?i ~i??i~?iT?i{!i i?ii?iii;7iiiii!!~i :iTiiiiT7!ii!:!i~if:!i;~iiiilii; :,ii:7iii! ~i;i;!;ii?i;1;;i7ii;!?7~:71;?=:~:?i:i:!::~I.ii:!~!?; T h e r a n g e o f p r o b l e m s w h i c h m a y b e c o n s i d e r e d as c o u p l e d is v e r y large a n d f o r m s studies w h i c h are n o w o f t e n r e f e r r e d to as ' m u l t i - p h y s i c s ' p r o b l e m s . T h e r a n g e o f p o s s i b l e algor i t h m s to s o l v e s u c h p r o b l e m s has b e e n s u m m a r i z e d above; h o w e v e r , n e w m e t h o d s often are p r o p o s e d (e.g., see r e f e r e n c e 66). A n o t h e r class of p r o b l e m s w h i c h m a y be c o n s i d e r e d as c o u p l e d c o n s i d e r s ' m u l t i - s c a l e ' effects. T h e s e a t t e m p t to b r i d g e the b e h a v i o u r o f m a t e r i a l s from, for e x a m p l e , a m i c r o to a m a c r o scale. T h i s topic is v e r y p o p u l a r t o d a y a n d is d i s c u s s e d f u r t h e r in r e f e r e n c e 33.
1. O.C. Zienkiewicz. Coupled problems and their numerical solution. In R.W. Lewis, R Bettess, and E. Hinton, editors, Numerical Methods in Coupled Systems, Chapter 1, pages 35-68. John Wiley & Sons, Chichester, 1984. 2. B.A. Boley and J.H. Weiner. Theory of Thermal Stresses. Dover Publications, Mineola, New York, 1997. 3. R.W. Lewis, E Bettess, and E. Hinton, editors. Numerical Methods in Coupled Systems. John Wiley & Sons, Chichester, 1984. 4. R.W. Lewis, E. Hinton, E Bettess, and B.A. Schrefler, editors. Numerical Methods in Coupled Systems. John Wiley & Sons, Chichester, 1987. 5. J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of Interdisciplinary Applied Mathematics. Springer-Verlag, Berlin, 1998. 6. O.C. Zienkiewicz, R.L. Taylor, and E Nithiarasu. The Finite Element Method for Fluid Dynamics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 7. O.C. Zienkiewicz and R.E. Newton. Coupled vibration of a structure submerged in a compressible fluid. In Proc. Int. Symp. on Finite Element Techniques, pages 359-371, Stuttgart, 1969. 8. R Bettess and O.C. Zienkiewicz. Diffraction and refraction of surface waves using finite and infinite elements. Int. J. Numer. Meth. Eng., 11:1271-1290, 1977. 9. O.C. Zienkiewicz, D.W. Kelly, and R Bettess. The Sommerfield (radiation) condition on infinite domains and its modelling in numerical procedures. In Proc. IRIA 3rd Int. Symp. on Computing Methods in Applied Science and Engineering, Versailles, Dec. 1977. 10. O.C. Zienkiewicz, R Bettess, and D.W. Kelly. The finite element method for determining fluid loadings on rigid structures. Two- and three-dimensional formulations. In O.C. Zienkiewicz, R.W. Lewis, and K.G. Stagg, editors, Numerical Methods in Offshore Engineering, pages 141183. John Wiley & Sons, Chichester, 1978. 11. O.C. Zienkiewicz and R Bettess. Dynamic fluid-structure interaction. Numerical modelling of the coupled problem. In O.C. Zienkiewicz, R.W. Lewis, and K.G. Stagg, editors, Numerical Methods in Offshore Engineering, pages 185-193. John Wiley & Sons, Chichester, 1978. 12. O.C. Zienkiewicz and R Bettess. Fluid-structure dynamic interaction and wave forces. An introduction to numerical treatment. Int. J. Numer. Meth. Eng., 13:1-16, 1978. 13. O.C. Zienkiewicz and E Bettess. Fluid-structure dynamic interaction and some 'unified' approximation processes. In Proc. 5th Int. Symp. on Unification of Finite Elements, Finite Differences and Calculus of Variations. University of Connecticut, May 1980. 14. R. Ohayon. Symmetric variational formulations for harmonic vibration problems coupling primal and dual v a r i a b l e s - applications to fluid-structure coupled systems. La Rechereche Aerospatiale, 3:69-77, 1979.
% :7
References 661 15. R. Ohayon. True symmetric formulation of free vibrations for fluid-structure interaction in bounded media. In R.W. Lewis, P. Bettess, and E. Hinton, editors, Numerical Methods in Coupled Systems. John Wiley & Sons, Chichester, 1984. 16. R. Ohayon. Fluid-structure interaction. In Proc. of the ECCM'99 Conference IACM/ECCM'99, Munich, Germany, Sept. 1999 (on CD-ROM). 17. H. Morand and R. Ohayon. Fluid-Structure Interaction. John Wiley & Sons, London, 1995. 18. M.P. Paidoussis and P.P. Friedmann, editors. 4th International Symposium on Fluid-Structure Interactions, Aeroelasticity, Flow-Induced Vibration and Noise, vol. 1, 2, 3, volume AD-vo. 52-3, Dallas, Texas, Nov. 1997. ASME/Winter Annual Meeting. 19. T. Kvamsdal et al., editors. Computational Methods for Fluid-Structure Interaction. Tapir Publishers, Trondheim, 1997. 20. R. Ohayon and C.A. Felippa (editors). Computational Methods for Fluid-Structure Interaction and Coupled Problems. Comp. Meth. Appl. Mech. Eng., 190:2977-3292 (Special issue). 21. M. Geradin, G. Roberts, and J. Huck. Eigenvalue analysis and transient response of fluid structure interaction problems. Eng. Comput., 1:152-160, 1984. 22. G. Sandberg and P. Gorensson. A symmetric finite element formation of acoustic fluid-structure interaction analysis. J. Sound Vibr., 123:507-515, 1988. 23. U. Basu and A.K. Chopra. Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation. Comp. Meth. Appl. Mech. Eng., 192:1337-1375, 2003. 24. U. Basu and A.K. Chopra. Perfectly matched layers for transient elastodynamics of unbounded domains. Int. J. Numer. Meth. Eng., 59:1039-1074, 2004. 25. K.K. Gupta. On a numerical solution of the supersonic panel flutter eigenproblem. Int. J. Numer. Meth. Eng., 10:637-645, 1976. 26. B.M. Irons. The role of part inversion in fluid-structure problems with mixed variables. J. AIAA, 7:568, 1970. 27. W.J.T. Daniel. Modal methods in finite element fluid-structure eigenvalue problems. Int. J. Numer. Meth. Eng., 15:1161-1175, 1980. 28. C.A. Felippa. Symmetrization of coupled eigenproblems by eigenvector augmentation. Comm. Appl. Numer. Meth., 4:561-563, 1988. 29. J. Holbeche. Ph.D. thesis, Department of Civil Engineering, University of Wales, Swansea, 1971. 30. A.K. Chopra and S. Gupta. Hydrodynamic and foundation interaction effects in earthquake response of a concrete gravity dam. J. Struct. Div. Am. Soc. Civ. Eng., 578:1399-1412, 1981. 31. J.E Hall and A.K. Chopra. Hydrodynamic effects in the dynamic response of concrete gravity dams. Earth. Eng. Struct. Dyn., 10:333-395, 1982. 32. O.C. Zienkiewicz and R.L. Taylor. Coupled problems- a simple time-stepping procedure. Comm. Appl. Numer. Meth., 1:233-239, 1985. 33. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 34. O.C. Zienkiewicz, R.W. Clough, and H.B. Seed. Earthquake analysis procedures for concrete and earth dams - state of the art. Technical Report Bulletin 32, Int. Commission on Large Dams, Paris, 1986. 35. R.E. Newton. Finite element study of shock induced cavitation. In ASCE Spring Convention, Portland, Oregon, 1980. 36. O.C. Zienkiewicz, D.K. Paul, and E. Hinton. Cavitation in fluid-structure response (with particular reference to dams under earthquake loading). Earth. Eng. Struct. Dyn., 11:463-481, 1983. 37. M.A. Biot. Theory of propagation of elastic waves in a fluid saturated porous medium, Part I: Low frequency range; Part II: High frequency range. J. Acoust. Soc. Am., 28:168-191, 1956.
662
Coupledsystems 38. O.C. Zienkiewicz, C.T. Chang, and E. Hinton. Non-linear seismic responses and liquefaction. Int. J. Numer. Anal. Meth. Geomech., 2:381-404, 1978. 39. O.C. Zienkiewicz and T. Shiomi. Dynamic behaviour of saturated porous media, the generalized Biot formulation and its numerical solution. Int. J. Numer. Anal. Meth. Geomech., 8:71-96, 1984. 40. O.C. Zienkiewicz, K.H. Leung, and M. Pastor. Simple model for transient soil loading in earthquake analysis: Part I - Basic model and its application. Int. J. Numer. Anal. Meth. Geomech., 9:453-476, 1985. 41. O.C. Zienkiewicz, K.H. Leung, and M. Pastor. Simple model for transient soil loading in earthquake analysis: Part I I - non-associative models for sands. Int. J. Numer. Anal. Meth. Geomech., 9:477-498, 1985. 42. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, and T. Shiomi. Computational approach to soil dynamics. In A.S. Czamak, editor, Soil Dynamics and Liquefaction, volume Developments in Geotechnical Engineering 42. Elsevier, Amsterdam, 1987. 43. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, D.K. Paul, and T. Shiomi. Static and dynamic behaviour of soils: a rational approach to quantitative solutions, I. Proc. R. Soc. London, 429:285-309, 1990. 44. O.C. Zienkiewicz, Y.M. Xie, B.A. Schrefler, A. Ledesma, and N. Bi~ani~. Static and dynamic behaviour of soils: a rational approach to quantitative solutions, II. Proc. R. Soc. London, 429:311-321, 1990. 45. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, B.A. Schrefler, and T. Shiomi. Computational Geomechanics: With Special Reference to Earthquake Engineering. John Wiley & Sons, Chichester, 1999. 46. B.R. Simon, J. S-S. Wu, M.W. Carlton, L.E. Kazarian, E/P. France, J.H. Evans, and O.C. Zienkiewicz. Poroelastic dynamic structural models of rhesus spinal motion segments. Spine, 10(6):494-507, 1985. 47. B.R. Simon, J. S-S. Wu, and O.C. Zienkiewicz. Higher order mixed and Hermitian finite element procedures for dynamic analysis of saturated porous media. Int. J. Numer. Meth. Eng., 10:483499, 1986. 48. O.C. Zienkiewicz and A.H.C. Chan. Coupled problems and their numerical solution. In I.S. Doltsinis, editor, Advances in Computational Non-linear Mechanics, Chapter 3, pages 109-176. Springer-Vedag, Berlin, 1988. 49. T. Belytschko and R. Mullen. Stability of explicit-implicit time domain solution. Int. J. Numer. Meth. Eng., 12:1575-1586, 1978. 50. T.J.R. Hughes and W.K. Liu. Implicit-explicit finite elements in transient analyses. Part I and Part II. J. Appl. Mech., 45:371-378, 1978. 51. T. Belytschko and T.J.R. Hughes, editors. Computational Methods for Transient Analysis. NorthHolland, Amsterdam, 1983. 52. C.A. Felippa and K.C. Park. Staggered transient analysis procedures for coupled mechanical systems: formulation. Comp. Meth. Appl. Mech. Eng., 24:61-111, 1980. 53. K.C. Park. Partitioned transient analysis procedures for coupled field problems: stability analysis. J. Appl. Mech., 47:370-376, 1980. 54. K.C. Park and C.A. Felippa. Partitioned transient analysis procedures for coupled field problems: accuracy analysis. J. Appl. Mech., 47:919-926, 1980. 55. O.C. Zienkiewicz, E. Hinton, K.H. Leung, and R.L. Taylor. Staggered time marching schemes in dynamic soil analysis and selective explicit extrapolation algorithms. In R. Shaw et al., editors, Proc. Conf. on Innovative Numerical Analysis for the Engineering Sciences, University of Virginia Press, 1980. 56. O.C. Zienkiewicz, C.T. Chang, and P. Bettess. Drained, undrained, consolidating dynamic behaviour assumptions in soils. Geotechnique, 30:385-395, 1980.
References 663 57. D.M. Trujillo. An unconditionally stable explicit scheme of structural dynamics. Int. J. Numer. Meth. Eng., 11:1579-1592, 1977. 58. L.J. Hayes. Implementation of finite element alternating-direction methods on non-rectangular regions. Int. J. Numer. Meth. Eng., 16:35-49, 1980. 59. A.W. Craig and O.C. Zienkiewicz. A multigrid algorithm using a hierarchical finite element basis. In D.J. Pedolon and H. Holstein, editors, Multigrid Methods in Integral and Differential Equations, pages 310-312. Clarendon Press, Oxford, 1985. 60. I. Babu~ka, A.W. Craig, J. Mandel, and J. Pitk~anta. Efficient preconditioning for the p-inversion finite element method in two dimensions. SlAM J. Num. Anal., 28:624-661,1991. 61. R. Lthner and K. Morgan. An unstructured multigrid method for elliptic problems. Int. J. Numer. Meth. Eng., 24:101-115, 1987. 62. M. Adams. Heuristics for automatic construction of coarse grids in multigrid solvers for finite element matrices. Technical Report UCB//CSD-98-994, University of California, Berkeley, 1998. 63. M. Adams. Parallel multigrid algorithms for unstructured 3D large deformation elasticity and plasticity finite element problems. Technical Report UCB//CSD-99-1036, University of California, Berkeley, 1999. 64. K.C. Park. Stabilization of partitioned solution procedures for pore fluid-soil interaction analysis. Int. J. Numer. Meth. Eng., 19:1669-1673, 1983. 65. O.C. Zienkiewicz, D.K. Paul, and A.H.C. Chan. Unconditionally stable staggered solution procedures for soil-pore fluid interaction problems. Int. J. Numer. Meth. Eng., 26:1039-1055, 1988. 66. J.Y. Kim, N.R. Aluru, and D.A. Tortorelli. Improved multi-level Newton solvers for fully coupled multi-physics problems. Int. J. Numer. Meth. Eng., 58:463-480, 2003.
Computer procedures for finite element analysis
A companion program to this book is available which can carry out analyses for most of the theory presented in previous chapters. In particular the computer program discussed here may be used to solve any one-, two-, or three-dimensional linear steady-state or transient problem. The program also has capabilities to perform non-linear analysis for the type of problems discussed in reference 1. Source listings and a user manual may be obtained at no charge from the author's intemet web site (http://www.ce.berkeley.edu/~rlt) or the publisher's internet web site (http:// books.elsevier.com/companions). The program is written mostly in Fortran with some routines in C (see author's web site for more information on using C for user modules). Any errors reported by readers will be corrected so that up-to-date versions are available. The version available for download is called FEAPpv which is an acronym for Finite Element Analysis Program-personal version. It is intended mainly for use in learning finite element programming methodologies and in solving small to moderate size problems on single processor computers. A simple management scheme is employed to permit efficient use of main memory with limited need to read and write information to disk. Finite element programs can be separated into three basic parts: 1. Data input module and pre-processor 2. Solution module 3. Results module and post-processor.
FEAPpv is mainly a solution module but provides simple data input and pre-processor capabilitites which permit generation of meshes using the multiblock schemes of Zienkiewicz and Phillips 2 and Gordon and Hall. 3 Alternatively the data may be input from neutral files written by other pre-processing systems (e.g., GiD4). Data input for the program consists of specification (or generation) of: (1) the coordinates for each node; (2) the element form and the nodal connection list for each element;
Pre-processing module: mesh creation 665 (3) boundary conditions and loads to be applied; and (4) material property data. The user manual describes the format for specifying the data to be used by FEAPpv.
19.2.1 Element library As part of the input data it is necessary to describe the element formulation to be used in forming the 'stiffness' matrix and 'load' vector of each problem. This may be provided either by user written modules (see below) or using the element library provided with the program. Currently, the element library in FEAPpv includes:
1. Solid elements for two-dimensional linear elasticity. Forms are provided for the irreducible formulation described in Chapters 2 and 6; the three-field mixed form described in Sec. 11.3 and the enhanced strain form described in Sec. 10.5.3. The elements permit consideration of elastic models which are isotropic or orthotropic as described in Chapter 6. (a) For the irreducible form the element shape may range from a 3-node triangle to a 9-node lagrangian quadrilateral. (b) For the three-field mixed form the element shape may be a 4-node, 8-node or 9-node quadrilateral form. (c) For the enhanced strain model the element is restricted to a 4-node quadrilateral form. 2. Solid elements for three-dimensional linear elasticity. Only the irreducible form for a 4-node tetrahedron or an 8-node brick may be used. The 8-node brick may be degenerated into other forms by giving the same node number to nodes used to perform the degenerate shape (see Sec.5.8). The elastic material model may be isotropic or orthotropic as described in Chapter 6. 3. Frame (rod) elements for two- and three-dimensional elasticity. Conventional structural elements are provided to perform analysis of elastic two- and three-dimensional frame structures. While these forms have not been discussed in this text, except as suggested problems for solution, they are useful for use in general analysis. The theory is contained in standard references for structural analysis and also in reference 1. 4. Truss elements for two- and three-dimensional elasticity. Similar to frame elements, the FEAPpv system includes conventional truss elements which may be used to analyse plane and space truss structures. 5. Plate element for linear elasticity. A plate bending element for use in the analysis of plates which include the primary effects of transverse shear (so-called Reissner-Mindlin theory 1) is provided. The element form may be either a 3-node triangle or a 4-node quadrilateral. The theory is described in references 1, 5, 6. 6. Shell element for three dimensions with linear elasticity. A 4-node quadrilateral element form for use in modelling general shell forms is provided. The element includes membrane and bending effects only and, thus, may be used only for analysis of 'thin' shells. The theory for the element is given in reference 7. The element form should be a 4-node quadrilateral.
666 Computerprocedures for finite element analysis
7. Membrane element for linear elasticity. A general elastic membrane form is provided which is the same as the shell element but without the bending terms. The element form should be a 4-node quadrilateral. 8. Thermal elements for two- and three-dimensional Fourier heat conduction. The theory described in Chapter 7 for transient heat conduction is provided in elements which solve two- and three-dimensional problems. The Fourier model may be isotropic or orthotropic. 9. User developed elements. Users may develop and add element modules for any problem which can be formed by the finite element approach described in this book. Details for writing modules will be found in the Programers Manual available at the web sites.
The main part of FEAPpv is a solution module which permits users to analyse a large range of problems formulated by the finite element method. Specific solution methods are prepared by the user using a unique command language, which is a sequence of statements which describe each algorithm. The current version of FEAPpv permits both 'batch' and 'interactive' problem solution. The commands provided permit specification of problems with either symmetric or unsymmetric 'stiffness' matrices, selection of direct or iterative solution of the linear algebraic equation system, selection of different transient solution algorithms, and output of solution results in either a text or graphics format. Commands which permit solution of a symmetric generalized linear eigenproblem (see Chapter 16) using a 'subspace' method 8, 9 are also available as well as a feature to compute the eigenvalues and vectors for an element stiffness. While the main thrust of this book is the solution of linear problems, the system FEAPpv is capable of solving both linear and non-linear problems. The use of special 'loop' commands permits the construction of algorithms which require iteration or time stepping. In addition features to solve problems in which load following is needed are provided in the form of 'arc-length'-type methods. 1~ The solution of problems for which it is not possible to deduce an accurate 'stiffness' matrix may be attempted using a quasi-Newton method based on the BFGS method. TM 14 The user manual available at the web site provides examples for several algorithms as well as a list of all available commands.
As noted above the FEAPpv system contains capabilities to report results as text data written to an output file or in graphical form which may be displayed on the screen or written to files for processing by other systems. Files are written in PostScript format (in an encapsulated form which may be used by many programs -e.g., TeX or LaTeX). The general features of graphical post-processing are limited to displaying two-dimensional objects. More complex forms require an interface to a separate pre-/post-processing system (e.g., GiD.4). The two-dimensional capabilities in FEAPpv include display of the mesh including node and element numbers, boundary conditions and loads. Contour plots for each degree of freedom of the solution system may be displayed as well as contours of
User modules 667
element values such as stress or strain components. The user manual provides a list of all commands for constructing graphical outputs. The available version for graphics is limited to X-window applications and compilers compatible with the current HP Fortran 95 compiler for Windows-based systems. 15
A key ingredient of the F E A P p v system is the ability of a user to add their own modules to extend the capabilities of the program to other classes of problems, material models, or solution strategies. Some user developed modules are available at the authors' web site given above and include element modules for other problem forms, an interface to other linear equation solvers, etc. Experienced programmers should be able to easily adapt these routines to include additional features. Programming additions to the system may be performed following descriptions in the Programmer Manual available at the web sites.
1. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford, 6th edition, 2005. 2. O.C. Zienkiewicz and D.V. Phillips. An automatic mesh generation scheme for plane and curved surfaces by isoparametric coordinates. Int. J. Numer. Meth. Eng., 3:519-528, 1971. 3. W.J. Gordon and C.A. Hall. Transfinite element methods - blending-function interpolation over arbitrary curved element domains. Numer. Math., 21:109-129, 1973. 4. GiD - The Personal Pre/Postprocessor. www.gidhome.com, 2004. 5. E Auricchio and R.L. Taylor. A shear deformable plate element with an exact thin limit. Comp. Meth. AppL Mech. Eng., 118:393-412, 1994. 6. E Auricchio and R.L. Taylor. A triangular thick plate finite element with an exact thin limit. Finite Elements in Analysis and Design, 19:57-68, 1995. 7. R.L. Taylor. Finite element analysis of linear shell problems. In J.R. Whiteman, editor, The Mathematics of Finite Elements and Applications VI, pages 191-203. Academic Press, London, 1988. 8. J.H. Wilkinson and C. Reinsch. Linear Algebra. Handbook for Automatic Computation, volume II. Springer-Verlag, Berlin, 1971. 9. K.-J. Bathe. Finite Element Procedures. Prentice Hall, Englewood Cliffs, N.J., 1996. 10. E. Riks. An incremental approach to the solution of snapping and buckling problems. Int. J. Solids Struct., 15:529-551, 1979. 11. K. Schweizerhof. Nitchlineare Berechnung von Tragwerken unter verformungsabhangiger belastung mitfiniten Elementen. Doctoral dissertation, U. Stuttgart, Stuttgart, Germany, 1982. 12. J.C. Simo, P. Wriggers, K.H. Schweizerhof, and R.L Taylor. Finite deformation post-buckling analysis involving inelasticity and contact constraints. Int. J. Numer. Meth. Eng., 23:779-800, 1986. 13. J.E. Dennis and J. More. Quasi-Newton methods- motivation and theory. SlAM Rev., 19:46-89, 1977. 14. H. Matthies and G. Strang. The solution of nonlinear finite element equations. Int. J. Numer. Meth. Eng., 14:1613-1626, 1979. 15. HP Fortran home page. http://h18009.wwl.hp.com/fortran, 2004.
Matrix algebra
The mystique surrounding matrix algebra is perhaps due to the texts on the subject requiring a student to 'swallow too much' at one time. It will be found that in order to follow the present text and carry out the necessary computation only a limited knowledge of a few basic definitions is required.
The linear relationship between a set of variables x and b allXl -~- a12x2 -~- a13x3 q- a14x4 :- bl a21x1 d- a22x2 q-- a23x3 -]- a24x4 -- b2
(A.1)
a31xl 9 a32x2 9 a33x3 9 a34x4 "- b3 can be written, in a short-hand way, as [A] {x} = {b}
(A.2)
Ax - b
(A.3)
or where all
A -- [A] -
a12
a13
a14]
a21 a22 a23 a24 a31
a32
a33
a34
x1 X ~ {X} --
X2 X3 X4
b -- {b} -
b2 b3
(A.4)
The above notation contains within it the definition of both a matrix and the process of multiplication of two matrices. Matrices are defined as 'arrays of number' of the
Matrix addition or subtraction
type shown in Eq. (A.4). The particular form listing a single column of numbers is often referred to as a vector or column matrix, whereas a matrix with multiple columns and rows is called a rectangular matrix. The multiplication of a matrix by a column vector is defined by the equivalence of the left and fight sides of Eqs (A.1) and (A.2). The use of bold characters to define both vectors and matrices will be followed throughout the t e x t - generally lower case letters denoting vectors and capital letters matrices. If another relationship, using the same a constants, but a different set of x and b, exists and is written as l
I
l
!
!
l
!
l
I
l
t
/
!
I
/
allX 1 -~- al2x 2 -~- al3x 3 + al4x 4 -- b 1 a21x 1 -[- a22x 2 -t- a23x 3 -t- a24x 4 "- b 2
(A.5)
a31x 1 nt- a32x 2 + a33x 3 -[- a34x 4 --- b 3
then we could write [A] [ X ] = [ B ]
xa Xl
in which
or A X = B
(A.6)
I
X--[X]=
x2,
x2,
X3, X4,
x3, X4
B--[B]=
b2, b3,
b; b~
(A.7)
implying both the statements (A. 1) and (A.5) arranged simultaneously as
[
allXl+''',
allX 1 + " "
a21xl + . . . ,
azlx 1 + . . .
a31xl + ' '
]
I
",
/
= B-
[B] -
a31x 1 + " "
I
bl,
b~l
b2,
b; b~
b3,
]
(A.8)
It is seen, incidentally, that matrices can be equal only if each of the individual terms is equal. The multiplication of full matrices is defined above, and it is obvious that it has a meaning only if the number of columns in A is equal to the number of rows in X for a relation of the type (A.6). One property that distinguishes matrix multiplication is that, in general, AX # XA i.e., multiplication of matrices is not commutative as in ordinary algebra.
If relations of the form from (A. 1) and (A.5) are added then we have a l l (Xl + X~l) -+- alE(X2 + x;) + a13(x3 nt- x ; ) + a14(x4 + x 4) = bl -t- b'~ a21 (Xl --[- Xtl) 71--a22(x2 + x ; ) -[- a23(x3 nt- x ; ) -~- a24(x4 -~- x ; ) -- b2 +
b;
a31 (Xl + Xtl) nt- a32(x2 --[- x ; ) -[- a33(x3 -q- x ; ) -[-- a34(x4 -[- x ; ) -- b3 nt- b;
which will also follow from Ax + Ax' = b + b'
(A.9)
669
670
Matrix algebra
9
if we define the addition of matrices by a simple addition of the individual terms of the array. Clearly this can be done only if the size of the matrices is identical, i.e., for example,
[al1 a121 [bll b12] Jail+b11a12+b121 a21 a31
a22 a32
--1--
b21 b31
b22 b32
-
a21 + b21 a31 -+- b31
a22 -+- b22 a32 q-- b32
or
A+B:C
(A.10)
implies that every term of C is equal to the sum of the appropriate terms of A and B. Subtraction obviously follows similar rules.
This is simply a definition for reordering the terms in an array in the following manner: all a21
a12 a22
a13 a23
--
Iall a21] a12 a13
a22 a23
(A.11)
and will be indicated by the symbol T as shown. Its use is not immediately obvious but will be indicated later and can be treated here as a simple prescribed operation.
If in the relationship (A.3) the matrix A is 'square', i.e., it represents the coefficients of simultaneous equations of type (A. 1) equal in number to the number of unknowns x, then in general it is possible to solve for the unknowns in terms of the known coefficients b. This solution can be written as x = A-lb
(A.12)
in which the matrix A -1 is known as the 'inverse' of the square matrix A. Clearly A -1 is also square and of the same size as A. We could obtain (A.12) by multiplying both sides of (A.3) by A -1 and hence A-1A = I = AA -1
(A.13)
where I is an 'identity' matrix having zero on all off-diagonal positions and unity on each of the diagonal positions. If the equations are 'singular' and have no solution then clearly an inverse does not exist.
A s u m of products
In problems of mechanics we often encounter a number of quantifies such as force that can be listed as a matrix 'vector':
fl f2
f=
(A. 14)
fn These, in turn, are often associated with the same number of displacements given by another vector, say, Ul U2 .
a --
(A. 15)
Un
It is known that the work is represented as a sum of products of force and displacement n W
=
k=l
Clearly the transpose becomes useful here as we can write, by the rule of matrix multiplication, b/1 W
--
[fl
f2...
U2 .
f.]
--
fTu -- uTf
(A.16)
bl n
Use of this fact is made frequently in this book.
An operation that sometimes occurs is that of taking the transpose of a matrix product. It can be left to the reader to prove from previous definitions that (A B) T = BTA T
(A.17)
In structural problems symmetric matrices are often encountered. If a term of a matrix A is defined as a i j , then for a symmetric matrix aij -- aji
or A = A T
A symmetric matrix must be square. It can be shown that the inverse of a symmetric matrix is also symmetric A -1 _
(A-l)
T = A -T
671
672
Matrix algebra
It is easy to verify that a matrix product AB in which, for example,
A
all a21
a12 a13 a14 a15 a22 a23 a24 a25
a31
a32 a33 a34 a35 - bll
b12
-
b22 b31 b32
b21
B
b41 b42 . b51 b52
_
could be obtained by dividing each matrix into submatrices, indicated by the lines, and applying the rules of matrix multiplication first to each of such submatrices as if it were a scalar number and then carrying out further multiplication in the usual way. Thus, if we write I All A12 I [ B1 ] A -- A21 A22 BB2 then AB -
A11B1 A12B2 ] A21B1 A22B2
can be verified as representing the complete product by further multiplication. The essential feature of partitioning is that the size of subdivisions has to be such as to make the products of the type AllB1 meaningful, i.e., the number of columns in All must be equal to the number of rows in B1, etc. If the above definition holds, then all further operations can be conducted on partitioned matrices, treating each partition as if it were a scalar. It should be noted that any matrix can be multiplied by a scalar (number). Here, obviously, the requirements of equality of appropriate rows and columns no longer apply. If a symmetric matrix is divided into an equal number of submatfices Aq in rows and columns then Aij - AjT
An eigenvalue of a symmetric matrix A of size n • n is a scalar ,~i which allows the solution of ( A - ,~i I)qSi = 0 and det I A - ,~i I I= 0 (A.18) where r
is called the eigenvector.
The eigenvalue problem 673 There are, of course, n such eigenvalues ~i to each of which corresponds an eigenvector q5i. Such vectors can be shown to be orthonormal and we write
lfori--j 0fori # j The full set of eigenvalues and eigenvectors can be written as
A-
I
A1 0
"..
0 1
q' = [q~l,
A~
Using these the matrix A may be written in its spectral form by noting from the orthonormality conditions on the eigenvectors that
~I~-1 ---CI~T then from Aq, = q,A it follows immediately that A
=
~A~
T
(A.19)
The condition number tc (which is related to equation solution round-off) is defined as X=
I Amax I I Amin I
(A.20)
Tensor-indicial notation in the approximation of elasticity
problems
The matrix type of notation used in this volume for description of tensor quantities such as stresses and strains is compact and we believe easy to understand. However, in a computer program each quantity often will still have to be identified by appropriate indices and the conciseness of matrix notation does not always carry over to the programming steps. Further, many readers are accustomed to the use of indicial-tensor notation which is a standard tool in the study of solid mechanics. For this reason we summarize here the formulation of the finite element arrays in an indicial form. Some advantages of such reformulation from the matrix setting become apparent when evaluation of stiffness arrays for isotropic materials is considered. Here some multiplication operations previously necessary become redundant and the element module programs can be written more economically. When finite deformation problems in solid mechanics have to be considered the use of indicial notation is almost essential to form many of the arrays needed for the residual and tangent terms. This appendix adds little new to the discretization ideas - it merely repeats in a different language the results already presented.
A point P in three-dimensional space may be represented in terms of its cartesian coordinates xi, i - 1, 2, 3. The limits that i can take define its range. To define these components we must first establish an oriented orthogonal set of coordinate directions as shown in Fig. B. 1. The distance from the origin of the coordinate axes to the point define a position vector x. If along each of the coordinate axes we define the set of unit orthonormal base vectors, ii, i = 1, 2, 3 which have the property l
[i 9 ij - ~o -
~ 1 for i - j Ofori # j
(
(B.1)
Indicial notation: summation convention
x2
x2(P)
>xl Fig. B.10rthogonal axes and a point: Cartesian coordinates.
where ( ) 9( ) denotes the vector dot product. The components of the position vector are constructed from the vector dot product Xi
:
ii 9 X;
i -- 1, 2, 3
(B.2)
From this construction it is easy to observe that the vector x may be represented as 3 X -- ~
X i ii
(B.3)
i--1
In dealing with vectors, and later tensors, the form x is called the intrinsic notation of the coordinates and xi ii the indicialform.t An intrinsic form is a physical entity which is independent of the coordinate system selected, whereas an indicial form depends on a particular coordinate system. To simplify notation we adopt the common convention that any index which is repeated in any given term implies a summation over the range of the index. Thus, our short-hand notation for Eq. (B.3) is X - - X i ii - - X1 il d- X2 i2 "if- X3 i3
(B.4)
For two-dimensional problems, unless otherwise stated, it will be understood that the range of the index is two. Similarly, we can define the components of the displacement vector u as u =
ui ii
(B.5)
Note that the components (u 1, u2, u3) replace the components (u, v, w) used throughout most of this volume. To avoid confusion with nodal quantities to which we previously also attached subscripts we shall simply change their position to a superscript. Thus, ~ has the same meaning as ~a used previously, etc. t Often in an indicial form of equations the base vectors are omitted from the final equation.
675
676 Tensor-indicial notation in the approximation of elasticity problems
In indicial notation the derivative of any quantity with respect to a coordinate component xi is written compactly as a = ( ),i (B.6) Oxi Thus we can write the gradient of the displacement vector as Oui OXj ~" Ui'j; i, j = 1, 2, 3
(B.7)
In a cartesian coordinate system the base vectors do not change their magnitude or direction along any coordinate direction. Accordingly their derivatives with respect to any coordinate is zero as indicated in Eq. B.8 Oii OXj
(B.8)
= ii,j -- 0
Thus, in cartesian coordinates the derivative of the intrinsic displacement u is given by U,j -- gi,ji i at- Uiii,j -- Ui,jii
(B.9)
The collection of all the derivatives defines the displacement gradient which we write in intrinsic notation as ~Tu -- Ui,j ii | ij (B. 10) The symbol | denotes the tensor product between two base vectors and since only two vectors are involved the gradient of the displacement is called second rank. The notation used to define a tensor product follows that used in reference 1. Any second rank intrinsic quantity can be split into symmetric and a skew symmetric (anti-symmetric) parts as
1
A = ~1 [A + AT] + ~ [A - AT] = A (~) + A(a'
(B.11)
where A and its transpose have cartesian components
A-Aijii|
AT-Ayiii|
(B.12)
The symmetric part of the displacement gradient defines the (small) strain~
= Vu,S _ ! [Vu + (Vu T] 2 -- ! [Ui,j "[" Uj,i] ii (~ __
2
Eij
ii | ij
- - Eji
ij
(B.13)
ii | ij
t Note that this definition is slightly different from that occurring in Chapters 2 to 6. Now the shearing strain is given by eij -- 1/2 Yij when i # j.
Coordinate transformation 677
and the skew symmetric part gives the (small) rotation w -
Vu
1 [Vu-(Vu)
~'~ -
__ 1__ [Ui, j __ U j ' i ]
2
O)ij ii @ i: = - o ) j i
:
(B.14)
ii (~)ij ii @ i j
The strain expression is analogous to Eq. (2.13). The components/3ij and wij may be represented by a matrix as /3ij
--
/321 /331
/322 /332
/323 /333
wij
=
c021 0 0923 0931 0932 0
/312 /322 /323 /313 /323 /333
:
--
-0912 0 -0913 -o923
(B.15)
0923 0
(B.16)
Consider now the representation of the intrinsic coordinates in a system which has different orientation than that given in Fig. B.1. We represent the components in the new system by I *I x - x i, l i, (B. 17) Using Eq. (B.2) we can relate the components in the prime system to those in the original system as (B.18) X i,' - - I"'i 9X = I" i. i j x j - - A i , j x j where " A i , j --- li,
9ij = cos(x~,, x j )
(B 19)
define the direction cosines of the coordinate in a manner similar to that of Eq. (6.18). Equation (B.18) defines how the cartesian coordinate components transform from one coordinate frame to another. Recall that summation convention implies X i,'
-- Ai'IX1 --[- Ai,2x2 q- Ai,3x3
i' -- 1, 2, 3
(B.20)
In Eq. (B. 18) i' is called a f r e e i n d e x whereas j is called a d u m m y i n d e x since it may be replaced by any other unique index without changing the meaning of the term (note that the notation used does not permit an index to appear more than twice in any term). Summation convention will be employed throughout the remainder of this discussion and the reader should ensure that the concept is fully understood before proceeding. Some examples will be given occasionally to illustrate its use. Using the notion of the direction cosines, Eq. (B.18) may be used to transform any vector with three components. Thus, transformation of the components of the displacement vector is given by !
u i, :
Ai,juj
i', j = 1, 2, 3
(B.21)
678
Tensor-indicial notation in the approximation of elasticity problems Indeed we can also use the above to express the transformation for the base vectors since i I, -- ( i l , - i j ) ij -- A i , j i j (B.22) Similarly, by interchanging the role of the base vectors we obtain
(ij
ij --"
. l"') i,
I" i, -- Ai,jl "'i,
B23)
which indicates that the inverse of the direction cosine coefficient array is the same as its transpose. The strain transformation follows from the intrinsic form written as !
9
9
e = ei,j,1 i, Q lj, -- eklik Q il
(B.24)
Substitution of the base vectors from Eq. (B.23) into Eq. (B.24) gives e -- A i , k e k l A j , l i i ,
Q ij,
(B.25)
Comparing Eq. (B.25) with Eq. (B.24) the components of the strain transform according to the relation E~,j, - Ai,kEkll~j,l (B.26) Variables that transform according to Eq. (B.21) are called first rank cartesian tensors whereas quantities that transform according to Eq. (B.26) are called second rank cartesian tensors. The use of indicial notation in the context of cartesian coordinates will lead naturally to each mechanics variable being defined in terms of a cartesian tensor of an appropriate rank. Stress may be written in terms of its components crij which may be written in a matrix form similar to Eq. (B. 15) O'ij --
O'11 O'12 O'13 ] O'21 0"22 0"23 ; i, j = 1, 2, 3 O"31 0"32 0"33
J
(B.27)
In intrinsic form stress is given by tr -- ffijii |
ij
(B.28)
and, using similar logic as used for strain, can be shown to transform as a second rank cartesian tensor. The symmetry of the components of stress may be established by summing moments (angular momentum balance) about each of the coordinate axes to obtain O'ij : O'ji (B.29)
Introducing a body force vector b-
biii
(B.30)
Elastic constitutive equations 679
we can write the static equilibrium equations (linear momentum balance) for a differential element as div tr + b - (0"ji,j -3t- 0 (B.31)
bi)i/
where the repeated index again implies summation over the range of the index, i.e., 3
0"ji,j ~ ~_~ 0"ji,j -- 0"1i,1 -JI- 0"2i,2 + 0"3i,3 j=l
Note that the free index i must appear in each term for the equation to be meaningful. As a further example of the summation convention consider an internal energy term W - 0"ij8ij
(B.32)
This expression implies a double summation; hence summing first on i gives W = 0"1j81j '}- 0"2j82j + 0"3j83j
and then summing on j gives finally W - 0"11811 -~0"12812 +0"13813 "-['- 0"21 821 + 0"22 822 + 0"23 E23 ~- 0-31 831 "+" 0-32 832 + 0"33 833
We may use symmetry conditions on 0"ij and Eij to reduce the nine terms to six terms. Accordingly, W - 0"11811 + 0"22822 -+-0"33833 "[- 2(0"12812 -+" 0"23823 + 0"31831) -- 0"11811 -'l'- 0"22822 -~" 0"33E33 -+- 0"12}/12 + 0"23Y23 -1- 0"31}/31
(B.33)
Following a similar expansion we can also show the result 0-ijO)ij ~ 0
(B.34)
For an elastic material the most general linear relationship we can write for components of the stress-strain characterization is 0
0-ij~ Dijkl (Ekl ~ 8kl) 9 (70
(B.35)
Equation (B.35) is the equivalent of Eq. (2.16) but now written in indicial notation. We note that the elastic moduli which appear in Eq. (B.35)are components of the fourth rank tensor D : Dijklii ~ ij Q ik | it (B.36) The elastic moduli possess the following symmetry conditions Dijkl -- Djikl -- Dijlk --" Dklij
(B.37)
680
Tensor-indicial notation in the approximation of elasticity problems the latter arising from the existence of an internal energy density in the form z
1
W ( ~ ) -- -~EijOijkiEkl "+" Eij [(7.0 __ OijkiEkO]
(B.38)
which yields the stress from OW
(B.39)
O'ij = OEiJ
By writing the constitutive equation with respect to x~, and using properties of the base vectors we can deduce the transformation equation for moduli as D~,j,k,l, -- A i , m A j , n A k , pAl,qDmnpq
(B.40)
A common notation for the intrinsic form of Eq. (B.35) is (B.41) in which : denotes the double summation (contraction) between the elastic moduli and the strains. The elastic moduli for an isotropic elastic material may be written in indicial form as Oijkl "- /~ij ~kl -Jr- ].L((~ik(~jl -Jr- (~il~jk ) (B.42) where A, /z are the Lam6 constants. An isotropic linear elastic material is always characterized by two independent elastic constants. Instead of the Lam6 constants we can use Young's modulus, E, and Poisson's ratio, v, to characterize the material. The Lam6 constants may be deduced from E
/z = 2(1 + v)
and
A=
vE
(1 + v ) ( 1 - 2v)
(B.43)
If we now introduce the finite element displacement approximation given by Eq. (2.1), using indicial notation we may write for a single element U i ~ ~li : Na~-I a i -- 1,2,3; a -- 1,2 . . . . ,n
(B.44)
where n is the total number of nodes on an element. The strain approximation in each element is given by the definition of Eq. (B. 13) as /
,, -- -~1 [Na, jUi ~a .3t_ Na , i~lj ] Eij
(B.45)
The internal virtual work for an element is given as
(~ul - f~ 8r erdfl-f~ e
3eqaq d~ e
(B.46)
Finite element approximation 681 Using Eqs (B.45) and (B.46) and noting symmetries in Dijkl w e may write the internal virtual work for a linear elastic material as
8U I - ~
Na,jDijklNb,l d~2 uk (B.47)
e
P
+ ~ft a /.. Na,j(ff 0 - DijklSkO) d a e
which replaces in indicial notation the matrix form presented in Chapters 2 to 6. In describing a stiffness coefficient two subscripts have been used previously and the submatrix Kab implied 2 x 2 or 3 x 3 entries for the ab nodal pair, depending on whether two- or three-dimension displacement components were involved. Now the scalar components K~ b i , j 1,2,3; a , b - 1,2 . . . . . n (B.48) define completely the appropriate stiffness coefficient with ij indicating the relative submatrix position (in this case for a three-dimensional displacement). Note that for a symmetric matrix we have previously required that
Kab
-- K L
(B.49)
In indicial notation the same symmetry is implied if
K~jb -- K jb'~
(B.50)
The stiffness tensor is now defined from Eq. (B.47) as
Kiakb -- ~_ Na,j Dijkl Nb,l dr2
(B.51)
,.IS2 e
When the elastic properties are constant over the element we may separate the integration from the material constants by defining
Wij b ' - f~ Na,iNb,jd~
(B.52)
e
and then perform the summations with the material moduli as
Ki? - w;b nijkl
(B.53)
In the case of isotropy a particularly simple result is obtained
giakb -- )~W~kb -3I- ~b[W:b "3I- r W; b ]
(B.54)
which allows the construction of the stiffness to be carried out using fewer arithmetic operations as compared with the use of matrix form. 3 Using indicial notation the final equilibrium equations of the system are written as
gab~b ikuk + fi a - - 0
i--1,2,3
(B.55)
and in this scalar form every coefficient is simply identified. The reader can, as a simple exercise, complete the derivation of the force terms due to the initial strain e ~ stress cr~ body force bi and external traction ti. Indicial notation is at times useful in clarifying individual terms, and this introduction should be helpful as a key to reading some of the current literature.
682
Tensor-indicial notation in the approximation of elasticity problems Table B.1 Mapping between matrix and tensor indices for second rank symmetric tensors Form
Index number
Matrix
1
2
3
4
5
6
Tensor
11 xx
22 yy
33 zz
12 & 21 xy & yx
23 & 32 yz & zy
31 & 13 zx & xz
The matrix form used throughout most of this volume can be deduced from the indicial form by a simple transformation between the indices. The relationship between the indices of the second rank tensors and their corresponding matrix form can be performed by an inspection of the ordering in the matrix for stress and its representation shown in Eq. (B.27). In the matrix form the stress was given in Chapter 6 as O'=
If ill
0 " 2 2 033
O"12 0 " 2 3 O"31 ]T
(B.56)
This form includes use of symmetry of stress components. The mapping of the indices follows that shown in Table B. 1. Table B. 1 may also be used to perform the map of the material moduli by noting that the components in the energy are associated with the index pairs from the stress and the strain. Accordingly, the moduli transform as Dllll -+ Dll;
D2233-+ D23; D1231~ D46; etc.
(B.57)
The symmetry of the stress and strain is imbedded in Table B.1 and existence of an energy function yields symmetry of the modulus matrix, i.e., Dab -- Dba.
1. E Chadwick. Continuum Mechanics. John Wiley & Sons, New York, 1976. 2. I.S. Sokolnikoff. The Mathematical Theory of Elasticity. McGraw-Hill, New York, 2nd edition, 1956. 3. A.K. Gupta and B. Mohraz. A method of computing numerically integrated stiffness matrices. Int. J. Numer. Meth. Eng., 5:83-89, 1972.
Solution of simultaneous linear algebraic equations A finite element problem leads to a large set of simultaneous linear algebraic equations whose solution provides the nodal and element parameters in the formulation. For example, in the analysis of linear steady-state problems the direct assembly of the element coefficient matrices and load vectors leads to a set of linear algebraic equations. In this section methods to solve the simultaneous algebraic equations are summarized. We consider both direct methods where an a priori calculation of the number of numerical operations can be made, and indirect or iterative methods where no such estimate can be made.
Consider first the general problem of direct solution of a set of algebraic equations given by Ka=f (C.1) where K is a square coefficient matrix, ~ is a vector of unknown parameters and f is a vector of known values. The reader can associate these with the quantities described previously" namely, the stiffness matrix, the nodal unknowns, and the specified forces or residuals. In the discussion to follow it is assumed that the coefficient matrix has properties such that row and/or column interchanges are unnecessary to achieve an accurate solution. This is true in cases where K is symmetric positive (or negative) definite.t Pivoting may or may not be required with unsymmetric, or indefinite, conditions which can occur when the finite element formulation is based on some weighted residual methods. In these cases some checks or modifications may be necessary to ensure that the equations can be solved accurately. 1-3 For the moment consider that the coefficient matrix can be written as the product of a lower triangular matrix with unit diagonals and an upper triangular matrix. Accordingly, K = LU
(C.2)
t For mixed methods which lead to forms of the type given in Eq. (10.14) the solution is given in terms of a positive definite part for (1 followed by a negative definite part for t~.
684 Solutionof simultaneous linear algebraic equations where
I
1 L21
L -
01
.
LL~I and
Ull i
U -
Ln2
... 9 0i .. -.
"'"
U12 ''U22 '".' ' 0
(C.3)
..i
Uln U2n .
(C.4)
Un n
This form is called a triangular decomposition of K. The solution to the equations can now be obtained by solving the pair of equations Ly = f
(C.5)
Ufi = y
(C.6)
and where y is introduced to facilitate the separation, e.g., see references 1-5 for additional details. The reader can easily observe that the solution to these equations is trivial. In terms of the individual equations the solution is given by
Y l - fl
i-1
Yi -- f i - Z
j=l
Lij yj
(C.7)
i -2,3 ..... n
and
1(
Yn bl n - - -
~li ---
Unn
~ii
Yi-
j=i+l
Uijuj
)
i-n-l,n-2
..... 1
(c.8)
Equation (C.7) is commonly called forward elimination while Eq. (C.8) is called back substitution. The problem remains to construct the triangular decomposition of the coefficient matrix. This step is accomplished using variations on gaussian elimination. In practice, the operations necessary for the triangular decomposition are performed directly in the coefficient array; however, to make the steps clear the basic steps are shown in Fig. C. 1 using separate arrays. The decomposition is performed in the same way as that used in the subprogram DATRI contained in the FEAPpv program; thus, the reader can easily grasp the details of the subprograms included once the steps in Fig. C. 1 are mastered. Additional details on this step may be found in references 3-5. In DATRI a Crout form of gaussian elimination is used to successively reduce the original coefficient array to upper triangular form. The lower portion of the array is
Direct solution Active zone
Step 1. Active zone. First row and column to principal diagonal. rK21
Reduced zone i,/-- Active zone K22 K23
K31 K32 K3a
L21 = K21/Ull
L22 = 1
0
U22 = K22- L21 U12
Step 2. Active zone. Second row and column to principal diagonal. Use first row of K to eliminate L21 Ull. The active zone uses only values of K from the active zone and values of L and U which have already been computed in steps 1 and 2. ~/-- Reduced zone ~/-- Active zone
K31
K13
1
0
0
Ull
K23
L21 1
0
0
K32 K33
L31 L32 L33 = 1 0
U12 U13-- K13 U22 U23 = K23- L21U13 0
U33 = K33- L31U13- L32U23
L31 = K31/U11 L32 = (K32-L31U12)/U22
Step 3. Active zone. Third row and column to principal diagonal. Use first row to
eliminate L31 Ul1" use second row of reduced terms to eliminate L32 U22 (reduced coefficient K32). Reduce column 3 to reflect eliminations below diagonal.
Fig. C.1 Triangular decomposition of K. used to store L - I as shown in Fig. C.1. With this form, the unit diagonals for L are not stored. Based on the organization of Fig. C. 1 it is convenient to consider the coefficient array to be divided into three parts: part one being the region that is fully reduced; part two the region that is currently being reduced (called the active zone); and part three the region that contains the original unreduced coefficients. These regions are shown in Fig. C.2 where the jth column above the diagonal and the jth row to the left of the diagonal constitute the active zone. The algorithm for the triangular decomposition of an n • n square matrix can be deduced from Fig. C. 1 and Fig. C.3 as follows: Ull = K l l ;
Lll-
(C.9)
1
For each active zone j from 2 to n,
Kjl Ljl : ~ ;
(C.10)
Ulj -- Klj
U11 i-1
1
Lji : Uii (Kji - ~
LjmUmi)
m=l
(C.11)
i-1
Uij -- Kij - ~ m--1
Limgmj
i = 2, 3 .....
j -
1
685
686
Solution of simultaneous linear algebraic equations jth column active zone
U
K~E "x
Reduced zone
jth row active zone - - ~
Kjl, KE2
9
9
9
~'jj
",,,
Unreduced zone
Fig. C.2 Reduced,active and unreduced parts.
U1i
Ulj
uj_~,j
Ui_~ Lil Li2
9
9
9
Li, i-I
"_j
Lj~ Lj~
9
9
9
Lj,,_, k
%
%
% %
%.
%
Fig. C.3 Terms used to construct Uij and Lji.
and finally Ljj -- 1 j-1
Ujj
-- Kjj - y ~ L jmUmj m=l
(C.12)
Direct solution Table C.1 Example: triangular decomposition of 3 x 3 matrix
I'
K
Step 1.
I' I'
[ 1
U23
0.5
--
0.5 0.25
3
1 0.5
3
1
L31 = 1 = 0.25, U13 -- 1, L32 -2 - 0.5 x 1 = 1.5, L33
0.5 0.25
Step 4.
1
L21 -- 2 __ 0.5, U12 -- 2, U22 -- 1, U22 = 4 - 0.5 x 2 = 3
2
Step 3.
] ][421]
L l l -- 1, U l l -- 4
4 2
Step 2.
] [4
L
1 0.5
-'-
2 - 0.25 x 2 3
1.5 3
1.5 -----0.5 3
1, U33 = 4 - 0.25 x 1 - 0.5 x 1.5 = 3
3 1
1.5 3
=
4 2
Check
The ordering of the reduction process and the terms used are shown in Fig. C.3. The results from Fig. C.1 and Eqs (C.9)-(C.12) can be verified using the matrix given in the example shown in Table C. 1. Once the triangular decomposition of the coefficient matrix is computed, several solutions for different fight-hand sides f can be computed using Eqs (C.7) and (C.8). This process is often called a resolution since it is not necessary to recompute the L and U arrays. For large size coefficient matrices the triangular decomposition step is very costly while a resolution is relatively cheap; consequently, a resolution capability is necessary in any finite element solution system using a direct method. The above discussion considered the general case of equation solving (without row or column interchanges). In coefficient matrices resulting from a finite element formulation some special properties are usually present. Often the coefficient matrix is symmetric (Kij "-- K j i ) and it is easy to verify in this case that Uij -
L j i Uii
(no sum)
(C.13)
For this problem class it is not necessary to store the entire coefficient matrix. It is sufficient to store only the coefficients above (or below) the principal diagonal and the diagonal coefficients. Equation (C.13) may be used to construct the missing part. This reduces by almost half the required storage for the coefficient array as well as the computational effort to compute the triangular decomposition. The required storage can be further reduced by storing only those rows and columns which lie within the region of non-zero entries of the coefficient array. Problems formulated by the finite element method and the Galerkin process normally have a
687
688 Solutionof simultaneous linear algebraic equations symmetric profile which further simplifies the storage form. Storing the upper and lower parts in separate arrays and the diagonal entries of U in a third array is used in DATRI. Figure C.4 shows a typical profile matrix and the storage order adopted for the upper array AU, the lower array AL and the diagonal array AD. An integer array JD is used to locate the start and end of entries in each column. With this scheme it is necessary to store and compute only withiJa the non-zero profile of the equations. This form of storage does not severely penalize the presence of a few large columns/rows and is also an easy form to program a resolution process (e.g., see subprogram DASOL in FEAPpv and reference 4). The routines included in FEAPpv are restricted to problems for which the coefficient matrix can fit within the space allocated in the main storage array. In two-dimensional formulations, problems with several thousand degrees of freedom can be solved on today's personal computers. In three-dimensional cases, however, problems are restricted to a few thousand equations. To solve larger size problems there are several options. The first is to retain only part of the coefficient matrix in the main array with the rest saved on backing store (e.g., hard disk). This can be quite easily achieved but the size of problem is not greatly increased due to the very large solve times required and the rapid growth in the size of the profile-stored coefficient matrix in three-dimensional problems. A second option is to use sparse solution schemes. These lead to significant program complexity over the procedure discussed above but can lead to significant savings in storage demands and compute time- especially for problems in three dimensions. Nevertheless, capacity in terms of storage and compute time is again rapidly encountered and alternatives are needed.
One of the main problems in direct solutions is that terms within the coefficient matrix which are zero from a finite element formulation become non-zero during the triangular decomposition step. While sparse methods are better at limiting this fill than profile methods they still lead to a very large increase in the number of non-zero terms in the factored coefficient matrix. To be more specific consider the case of a three-dimensional linear elastic problem solved using 8-node isoparametric hexahedron elements. In a regular mesh each interior node is associated with 26 other nodes, thus, the equation of such a node has 81 non-zero coefficients - three for each of the 27 associated nodes. On the other hand, for a rectangular block of elements with n nodes on each of the sides the typical column height is approximately proportional to n 2 and the number of equations t o n 3. In Table C.2 we show the size and approximate number of non-zero terms in K from an element finite formulation for linear elasticity (i.e., with three degrees of freedom per node). The table also indicates the size growth with column height and storage requirements for a direct solution based on a profile solution method. From the table it can be observed that the demands for a direct solution are growing very rapidly (storage is approximately proportional to n 5) while at the same time the demands for storing the non-zero terms in the stiffness matrix grow proportional to the number of equations (i.e., proportional to n 3 for the block).
Iterative solution ~A
\
Half band width ~
Half band width ._
Profile
v
Kll K12 K13 K14
Kll
i
___/_:
K22 K23 K24 K33 K34 K35 K44 K45 K46
iK44 K45 K46 , . . . .
K55 K56
Symmetric' K55 K56 |
!
. . . . . . . . .
! . . . .
1
i
K77 K78
K77 K78
. . . . . i
i /
K66 K67 K68
K66 K67 K68
i i . . . .
i
i
i
,
i
K88
K88' i
Banded storage array ADi
AUi
ALi
JDi
K11 K22 K33
K12
I(21
0
K13 K23
K31 %2
3
K14 K24 K34
K41 K42 K43
K35 K45
Ks3 Ks4
10
K46 K56
K64 K6~
11
K67
K7~
12
/(18
K81
18
K7~
K8;
K44 Kss K66 K77 K88 Diagonals
_•
9
Profile
-- !~:~;iK~ K~4i
i K,,iK,~ K4~i
i _ _
r _ _ .
I
i
Symmetric iK55:K56:
iK~8I K,8 I
K581
!K881 Fig. C.4 Profile storage for coefficient matrix.
Storage of arrays
1 6 8 10 11 18
689
690
Solutionof simultaneous linear algebraic equations Table C.2 Partial list of solutions commands
Non-zeros in K
Profile storage data
Side nodes
Number of equations
Words (x 10 -6)
Mbytes
Col. Ht.
Words (x 10 -6)
Mbytes
5 10 20 40 80
375 3000 24000 192000 1536000
0.02 0.12 0.96 7.68 61.44
0.12 0.96 7.68 61.44 491.52
90 330 1260 4920 18440
0.03 0.99 30.24 944.64 28323.84
0.27 7.92 241.82 7557.12 226584.72
Iterative solution methods use the terms in the stiffness matrix directly and thus for large problems have the potential to be very efficient for large three-dimensional problems. On the other hand, iterative methods require the resolution of a set of equations until the residual of the linear equations, given by r (i) - -
f
- K f i (i)
(C.
14)
becomes less than a specified tolerance. In order to be effective the number of iterations i to achieve a solution must be quite small - generally no larger than a few hundred. Otherwise, excessive solution costs will result. At the time of writing this book the subject of iterative solution for general finite element problems remains a topic of intense research. There are some impressive results available for the case where K is symmetric positive (or negative) definite; however, those for other classes (e.g., unsymmetric or indefinite forms) are generally not efficient enough for reliable use in the solution of general problems. For the symmetric positive definite case methods based on a preconditioned conjugate gradient method have been particularly effective. 6-8 The convergence of the method depends on the condition number of the matrix K - the larger the condition number, the slower the convergence (see reference 3 for more discussion). The condition number for a finite element problem with a symmetric positive definite stiffness matrix K is defined as A, tc = - (C.15) )~1 where ,km and )kn a r e the smallest and largest eigenvalue from the solution of the eigenproblem (viz. Chapter 16) KcI, = ~ A
(C.16)
in which A is a diagonal matrix containing the individual eigenvalues/~i and the columns of 9 are the eigenvectors r associated with each of the eigenvalues. Usually, the condition number for an elasticity problem modelled by the finite element method is too large to achieve rapid convergence and a preconditionedconjugate gradient (PCG) is used. 6 A symmetric form of preconditioned system is written as Kpz --
pKpTz - Pf
(C.17)
where Prz - fi
(C.18)
References 691
Now the convergence of the PCG algorithm depends on the condition number of Kp. The problem remains to construct a preconditioner which adequately reduces the condition number. In FEAPpv the diagonal of K is used; however, more efficient schemes incorporating also multigrid methods are discussed in references 7 and 8.
1. A. Ralston. A First Course in Numerical Analysis. McGraw-Hill, New York, 1965. 2. J.H. Wilkinson and C. Reinsch. LinearAlgebra. HandbookforAutomatic Computation, volume II. Springer-Verlag, Berlin, 1971. 3. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997. 4. R.L. Taylor. Solution of linear equations by a profile solver. Eng. Comput., 2:344-350, 1985. 5. G. Strang. Linear Algebra and its Application. Academic Press, New York, 1976. 6. R.M. Ferencz. Element-by-element preconditioning techniques for large-scale, vectorized finite element analysis in nonlinear solid and structural mechanics. Ph.D. thesis, Department of Mechanical Engineering, Stanford University, Stanford, California, 1989. 7. M. Adams. A parallel maximal independent set algorithm. In Proc. 5th Copper Mountain Conference on Iterative Methods, 1998. 8. M. Adams. Parallel multigrid solver algorithms and implementations for 3D unstructured finite element problems. In Supercomputing '99: High Performance Networking and Computing, volume http://www.sc99.org/proceedings, Portland, Oregon, Nov. 1999.
Some integration formulae for a triangle Let a triangle be defined in the xy plane by three points (Xl, yj), (x2, Y2), (x3, Y3) with the origin of the coordinates taken at the centroid (or baricentre), i.e., Xl "~-X2 -~-X3
3
--
Yl "+- Y2 nt- Y3
3
=0
Then integrating over the triangle area we obtain:
1 dx dy - -~
xdxdy = f
1
Xl
Yl
1
X2
Y2
1
x 3
Y3
--
A - area of triangle
y d x d y =O
A x 2dxdy = -~ (x~ +x~ +x~) A y2 dx dy - -i2 (y~ 4- y22 + y32) f
A (Xlyl q- x2Y2 + x3Y3) x y dx dy - -i2
Some integration formulae for a tetrahedron Let a tetrahedron be defined in the x y z coordinate system by four points (Xl, yl, z 1), (x2, y2, z2), (x3, Y3, z3) (x4, Y4, z4) with the origin of the coordinates taken at the centroid, i.e., Xl "q- X2 @ X3 "~- X4
Yl -+- Y2 -+- Y3 q- Y4 ~"
4
Zl + Z2 "q" Z3 @ Z4 "-'-
4
4
~"
0
Then integrating over the tetrahedron volume 1 dx dy dz - -~
1
X1
Yl
Zl
1
x2
Y2
z2
1
X3
Y3
Z3
1
X4
Y4
Z4
- V - tetrahedron volume
Provided the order of numbering the nodes is as indicated on Fig. 4.18(a) then also"
/x xdydz=/ y x y z=/zdx ydz=O x 2 dx dy dz = ~ (x~ + x~ + x~ + x24) y2 dx dy dz = ~
f
z 2 dx dy dz --
(y~ + y~ + y~ + y~)
v (z~+ z~+ z~+ zl)
x y dx dy dz = 2-0 (XlYl nt- x2Y2 -t- x3Y3 -t- X424) y z dx dy dz - - ~ (ylzl + Y2Z2 + y3z3 + 2424) z x dx dy dz = ~ (ZlXl + ZzX2 + z3x3 + Z4X4)
Some vector algebra
Some knowledge and understanding of basic vector algebra is needed in dealing with complexities of elements oriented in space as occur in beams, shells, etc. Some of the operations are summarized here. Vectors (in the geometric sense) can be described by their components along the directions of the x, y, z axes. Thus, the vector V01 shown in Fig. E 1 can be written as Vol
(El)
xli q- Ylj q- zlk
-
in which i, j, k are unit vectors in the direction of the x, y, z axes. Alternatively, the same vector could be written as
Vol
:
{xl}
(F.2)
Yl Zl
(now a 'vector' in the matrix sense) in which the components are distinguished by positions in the column.
Addition and subtraction is defined by addition and subtraction of components. Thus, for example, V02 -- V01 -- (x2 -
xl)i +
(Y2 - -
Yl)j +
(Z2 - -
zl)k
(E3)
The same result is achieved by the definitions of matrix algebra; thus
V02 -
V0I
-'- V 2 1 =
I X2~X1 I Y2 - - Yl Z2 - - Z l
(F.4)
'Scalar' products z
2
.'"""
Z2
, i
V01
i
'
Y .
.
.
.
.
.
.
.
.
x2 .
.
.
X
.
.
.
.
.
.
.
.
.
........
r
""
~'x'=~i
!
"""
~'X
Fig. R1 Vector addition.
A scalar product of two vectors is defined as 3 A . B - B. A = Z
akbk
(ES)
k=l
If A
=
axi + ayj + azk
B
=
bxi + byj -+- bzk
(E6)
then A 9B = axbx + ayby + azbz
Using the matrix notation A =
{ax} Bay az
bx
by }
(E7)
(E8)
bz
the scalar product becomes A . B = ATB = BTA
(E9)
The length of the vector V21 is given, purely geometrically, as 121 -- r
-- Xl) 2 "[" (Y2 -- Yl) 2 + (Z2 -- Zl) 2
(El0)
695
696
Some vector algebra
Fig. R2 Vector multiplication (cross product). or in terms of matrix algebra as 12,- 4V21 " V21-
(Ell)
4VT1V21
Direction cosines of a vector are simply, from the definition of the projected component of lengths, given as (Fig. F. 1) cos Otx :
X2 -- X1 121
=
V21 9i
(F. 12)
121
The scalar product may also be written as (Fig. E2) A . B = B . A = lalb
(F.13)
COS y
where y is the angle between the two vectors A and B and respectively.
la
and
lb a r e
their lengths,
Another product of vectors is defined as a vector oriented normally to the plane given by two vectors and equal in magnitude to the product of the length of the two vectors multiplied by the sine of the angle between them. Further, the direction of the normal vector follows the fight-hand rule as shown in Fig. F.2 in which A • B - C
(El4)
is shown. Thus, from the fight-hand rule, we have A x B - -B x A
(F.15)
Elements of area and volume
It is worth noting that the magnitude (or length) of C is equal to the area of the parallelogram shown in Fig. E2. Using the definition of Eq. (E6) and noting that ixi
--
jxj-kxk-0
i•
=
k,j•
(F.16)
we have AxB
i ax bx
j ay by
k az bz
-
det
=
(aybz - azby)i + (azbx - axbz)j + (axby - a y b x ) k
In matrix algebra this does not find a simple counterpart but we can use the above to define the vector C t
{ ayz-azy}
C - A x B -
(F. 17)
azbx - axbz axby - aybx
The vector product will be found particularly useful when the problem of erecting a normal direction to a surface is considered.
If~ and r/are some curvilinear coordinates, then the following vectors in two-dimensional plane ax ax d~ --
0-~
d~
Oy
drl =
~
Oy
d77
(F. 18)
defined from the relationship between the cartesian and curvilinear coordinates, are vectors directed tangentially to the ~ and 0 equal constant contours, respectively. As the length or the vector resulting from a cross product of d~ x dr/is equal to the area of the elementary parallelogram we can write
d(area) -- det
0x
0x
0-~
~
Oy
Oy
d~ dr/
by Eq. (F.17). t If we rewrite A as a skewsymmetricmatrix A,=
az
0
ay
ax
aox]
then an altemativerepresentationof the vectorproductin matrixform is C = .~B.
(E19)
697
698 Somevector algebra
Similarly, if we have three curvilinear coordinates ~, r/, ( in the cartesian space, the 'triple' or box product defines a differential volume " Ox
d(vol) = (d~ x dr/). d~ = det
Ox
Ox-
Oy
Oy
Oy
Oz
Oz
Oz
d~ dr/d(
(F.20)
this follows simply from the geometry. The bracketed product, by definition, forms a vector whose length is equal to the parallelogram area with sides tangent to two of the coordinates. The second scalar multiplication by a length and the cosine of the angle between that length and the normal to the parallelogram establishes a differential volume element. The above equations serve in changing the variables in surface and volume integrals.
Integration by parts in two or three
dimensions (Green's theorem)
Consider the integration by parts of the following two-dimensional expression / 4~ 0x dx dy
(G.1)
~2
Integrating first with respect to x and using the well-known relation for integration by parts in one dimension
u dv - -
/xl
v du + (UV)x=x~ - (UV)x=xz,
(G.2)
we have, using the symbols of Fig. G. 1,
ff
iOx gOdxdy--
ff
-~x O~ 0 dx dy + fyyl'~ [(~b gr)x=xR - (qb gr)x=xL] dy (G.3)
If now we consider a direct segment of the boundary dl-" on the fight-hand boundary, we note that dy = nx dl-" (G.4) where nx is the direction cosine between the outward normal and the x direction. Similarly on the left-hand section we have
dy - - nx dI"
(G.5)
The final term of Eq. (G.3) can thus be expressed as the integral taken around an anticlockwise direction of the complete closed boundary:
/ dp~/nx dI"
(G.6)
r If several closed contours are encountered this integration has to be taken around each such contour. The general expression in all cases is
Ox ~2
-~x ~ dx dy + ~2
dp~nx dr" F
(6.7)
700
Integration by parts in two or three dimensions (Green's theorem)
'
YT
f~
F
~x Fig. G.1 Definitions for integrations in two dimensions.
Similarly, if differentiation in the y direction arises we can write f/
Oy~ d x d y - - f f O
(G.8)
-~y Ocp ~ dx dy + f dp~ny dF g2
F
n y is the direction cosine between the outward normal and the y axis. In three dimensions by identical procedure we can write
where
f/j dp O~p ~ gr dx dy dz + / 6q/ny Oy dx dy dz - - JjJ~-~y ~2
~
dl-"
(G.9)
F
where dl-' becomes the element of the surface area and the last integral is taken over the whole surface.
Solutions exact at nodes
The finite element solution of ordinary differential equations may be made exact at the interelement nodes by a proper choice of the weighting function in the weak (Galerkin) form. To be more specific, let us consider the set of ordinary differential equations given by A(u) + f(x) = 0 (H.1) where u is the set of dependent variables which are functions of the single independent variable 'X' and f is a vector of specified load functions. The weak form of this set of differential equations is given by (H.2)
fxl RV T [A(u) + f] dx = 0
The weak form may be integrated by parts to remove all the derivatives from u and place them on v. The result of this step may be expressed as ~xl=R [uTA*(v)-+-vTf] dx -4- [B* (v)] T B(u)
I
XR
- 0
(H.3)
XL
where A*(v) is the adjoint differential equation and B* (v) and B(u) are terms on the boundary resulting from integration by parts. If we can find the general integral to the homogeneous adjoint differential equation A* (v) = 0
(H.4)
then the weak form of the problem reduces to XR
fxl ~RvTf dx + [B* (v)] T B(u)
= 0
(H.5)
XL
The first term is merely an expression to generate equivalent forces from the solution to the adjoint equation and the last term is used to construct the residual equation for the problem. If the differential equation is linear these lead to a residual which depends linearly on the values of u at the ends XL and X R. If we now let these be the location of the end nodes of a typical element we immediately find an expression to generate a stiffness matrix. Since in this process we have never had to construct
702
Solutions exact at nodes
an approximation for the dependent variables u it is immediately evident that at the end points the discrete values of the exact solution must coincide with any admissible approximation we choose. Thus, we always obtain exact solutions at these points. If we consider that all values of the forcing function are contained in f (i.e., no point loads at nodes), the terms in B(u) must be continuous between adjacent elements. At the boundaries the terms in B(u) include a flux term as well as displacements. As an example problem, consider the single differential equation d2u du ---5 dx + P dx + f - 0
(H.6)
with the associated weak form
]xl
du 1
v l dx2+Pdxx + f
dx-0
(H,7)
After integration by parts the weak form becomes
/x?" (
-
dv) ] [ (dUxU) + Pu dx + v f d x + v
dv ] xR - -~x u =0
(H.8)
XL
The adjoint differential equation is given by a*(v) =
d2v dx 2
and the boundary terms by B*(v) --
P
dv dx
=0
{ v} dv dx
(H.9)
(H.10)
and a(u) =
dx + P u
(H.11)
u
For the above example two cases may be identified: 1. P z e r o - where the adjoint differential equation is identical to the homogeneous equation in which case the problem is called self-adjoint. 2. P non-zero - where we then have the non-self-adjoint problem. The finite element solution for these two cases is often quite different. In the first case an equivalent variational theorem exists, whereas for the second case no such theorem exists.t In the first case the solution to the adjoint equation is given by v = Ax + B
(H.12)
t An integrating factor often may be introduced to make the weak form generate a self-adjoint problem; however, the approximation problem will remain the same. See Sec. 3.11.2.
References 703
which may be written as conventional linear shape functions in each element as X R ~
Nt~ = ~
X
;
X R ~
X L
X
NR -- ~
XR
~
;
X L
~
(H.13)
X L
Thus, for linear shape functions in each element used as the weighting function the interelement nodal displacements for u will always be exact (e.g., see Fig. 3.4) irrespective of the interpolation used for u. For the second case the exact solution to the adjoint equation is v - - A e Px + B - - A z + B
(H.14)
This yields the shape functions for the weighting function ZR
--
N L = ~ ; ZR
--
Z ZL
Z --
NR--~;
ZR
--
ZL
(H. 15)
ZL
which when used in the weak form again yield exact answers at the interelement nodes. After constructing exact nodal solutions for u, exact solutions for the flux at the interelement nodes can also be obtained from the weak form for each element. The above process was first given by Tong for self-adjoint differential equations. 1
1. P. Tong. Exact solution of certain problems by the finite element method. J. A I A A , 7:179-180, 1969.
Matrix diagonalization or lumping
Some of the algorithms discussed in this volume become more efficient if one of the global matrices can be diagonalized (also called 'lumped' by many engineers). For example, the solution of some mixed and transient problems are more efficient if a global matrix to be inverted (or equations solved) is diagonal [viz. Chapter 11, Eq. (11.94) and Chapter 16, Secs 16.2.4 and 16.4.2]. Engineers have persisted with purely physical concepts of lumping; however, there is clearly a need for devising a systematic and mathematically acceptable procedure for such lumping. We shall define the matrix to be considered as
A = f~ N a'cN d~2
(I. 1)
where c is a matrix with small dimension. Often c is a diagonal matrix (e.g., in mass or simple least square problems e is an identity matrix times some scalar). When A is computed exactly it has full rank and is not diagonal - this is called the consistent form of A since it is computed consistently with the other terms in the finite element model. The diagonalized form is defined with respect to 'nodes' or the shape functions, e.g., Na = N~I; hence, the matrix will have small diagonal blocks, each with the maximum dimension of e. Only when e is diagonal can the matrix A be completely diagonalized. Four basic lines of argument may be followed in constructing a diagonal form. The first procedure is to use different shape functions to approximate each term in the finite element discretization. For the A matrix we use substitute shape functions Na for the lumping process. No derivatives exist in the definition of A; hence, for this term the shape functions may be piecewise continuous within and between elements and still lead to acceptable approximation. If the shape functions used to define A are piecewise constants, such that l~la is a certain part of the element surrounding the node a and zero elsewhere, and such parts are not overlapping or disjoint, then clearly the matrix of Eq. (I.1) becomes nodally diagonal as fa
{ faa e dr2 -
0
a=b
a 7~ b
(I.2)
Such an approximation with different shape functions is permissible since the usual finite element criteria of integrability and completeness are satisfied. We can verify this using a patch test to show that consistency is still maintained in the approximation.
Matrixdiagonalization or lumping
N~
N~
a
a
(a)
(b) ~
Fig. 1.1 (a) Linear and (b) piecewise constant shape functions for a triangle.
The functions selected need only satisfy the condition l~a
-
-
/VaI with ~ / V a
- 1
(I.3)
a
for all points in the element and this also maintains a partition of unity property in all of f2. In Fig. I. 1 we show the functions Na and Na for a triangular element. The second method to diagonalize a matrix is to note that condition (I. 1) is simply a requirement that ensures conservation of the quantity c over the element. For structural dynamics applications this is the conservation of mass at the element level. Accordingly, it has been noted that any lumping that preserves the integral of c on the element will lead to convergent results, although the rate of convergence may be lower than with use of a consistent A. Many alternatives have been proposed based upon this method. The earliest procedures performed the diagonalization using physical intuition only. Later alternative algorithms were proposed. One suggestion, often called a 'row sum' method, is to compute the diagonal matrix from Aab --
{ ~Cf~aNTacNcdf2 a-- b 0 a :/: b
This simplifies to Aab --
( f~aNTacdf2 a - b 0
a =/: b
(1.4)
(I.5)
since the sum of the shape functions is unity. This algorithm makes sense only when the degrees of freedom of the problem all have the same physical interpretation. An alternative is to scale the diagonals of the consistent mass to satisfy the conservation requirement. In this case the diagonal matrix is deduced from A a b __
~ m ~-'~aNaTCNbdr2 L
0
a -- b
a~b
(1.6)
where m is selected so that
a
Aaa
-- f~ C d ~
(I.7)
The third procedure uses numerical integration to obtain a diagonal array without apparently introducing additional shape functions. Use of numerical integration to
705
706
Matrixdiagonalization or lumping
evaluate the A matrix of Eq. (I. 1) yields a typical term in a summation form (following Chapter 8) / ' 8
(NTcNb)~q Jq Wq
A a b - /.. NaTCNbdr2 - ~
(1.8)
q
~q refers to the quadrature point at which the integrand is evaluated, J is the jacobian volume transformation at the same point and Wq gives the appropriate quadrature weight. If the quadrature points for the numerical integration are located at nodes then, for standard shape functions (viz. Chapter 4), by Eq. (I.3) the diagonal matrix is given where
Aab --
s 0
Wa
a - b a =/: b
(1.9)
where Ja is the jacobian and Wa is the quadrature weight at node a. Appropriate weighting values may be deduced by requiting the quadrature formula to exactly integrate particular polynomials in the natural coordinate system. In general the quadrature should integrate a polynomial of the highest complete order in the shape functions. Thus, for 4-noded quadrilateral elements, linear functions should be exactly integrated. Integrating additional terms may lead to improved accuracy but is not required. Indeed, only conservation of c is required. For low order elements, symmetry arguments may be used to lump the matrix. It is, for instance, obvious that in a simple triangular element little improvement can be obtained by any other lumping than the simple one in which the total c is distributed in three equal parts. For an 8-noded two-dimensional isoparametric element no such obvious procedure is available. In Fig. 1.2 we show the case of rectangular elements of 4-, 8-, and 9-noded type and lumping by Eqs (I.5), (1.6) and (1.9). It is noted that for the 8-noded element some of the lumped quantities are negative when Eq. (1.5) or Eq. (1.9) is used. These will have some adverse effects in certain algorithms (e.g., time-stepping schemes to integrate transient problems) and preclude their use. In Fig. 1.3 we show some lumped matrices for triangular elements computed by quadrature [i.e., Eq. (1.9)]. It is noted here that the cubic element has negative terms while the quadratic element has zero terms. The zero terms are particularly difficult to handle as the resulting diagonal A matrix no longer has full rank and thus may not be inverted. Another aspect of lumping is the performance of the element when distorted from its parent element shape. For example, as a rectangular element is distorted and approaches a triangular shape it is desirable to have the limit triangular shape case behave appropriately. In the case of a 4-noded rectangular element the lumped matrix for all three procedures gives the same answer. However, if the element is distorted by a transformation defined by one parameter f as shown in Fig. 1.4 then the three lumping procedures discussed so far give different answers. The jacobian transformation is given by J = ab(1 - f ) (I. 10) and c is here taken as the identity matrix. The form (I.5) gives Aaa -
ab (1 - f / 3 ) ab (1 + f /3)
at top nodes at bottom nodes
(I.11)
Matrixdiagonalization or lumping 1
1
8 36
1
-
-
1
-
-
1
4
1
-
0 8 3-6
3-6
1
"~
l
4 ""
I -36
" 16
~
o~
I 36
04 3-6
8-node (I.6)
4-node All methods
1
O
I
36
9-node
All methods
I
1 ~
1
12
1 g
12
8-node (I.5) and (I.9) Fig. 1.2 Diagonalization of rectangular elements by three methods.
A-~M 1
0M
g1 MO
og1 M
p=l O(h)
o
o p=2 O(h 2)
o0 M
p=3 O(h 5)
Fig. 1.3 Diagonalization of rectangular elements by three methods. L fa.l. (1-f)a ,_
VT
J
r
J
b X
,iw
Fig. 1.4 Distorted 4-noded element.
w
M
707
708
Matrix diagonalization or lumping
the form (1.6) gives Aaa
ab(1 - f/2) ab(1 + f /2)
--
at top nodes at bottom nodes
(I.12)
and the quadrature form (1.9) yields Aaa
ab(1-f) ab(1 + f )
--
at top nodes at bottom nodes
(I.13)
The 4-noded element has the property that a triangle may be defined by coalescing 2 nodes and assigning them to the same global node in the mesh. Thus, the quadrilateral is identical to a 3-noded triangle when the parameter f is unity. The limit value for the row sum method will give equal lumped terms at the 3 nodes while method (1.6) yields a lumped value for the coalesced node which is two-thirds the value at the other nodes and the quadrature method (1.9) yields a zero lumped value at the coalesced node. Thus, methods (1.6) and (1.9) give limit cases which depend on how the nodes are numbered to form each triangular element. This lack of invariance is not desirable in computer programs; hence for the 4-noded quadrilateral, method (1.5) appears to be superior to the other two. On the other hand, we have observed above that the row sum method (1.5) leads to negative diagonal elements for the 8-noded element; hence there is no universal method for diagonalizing a matrix. A fourth but not widely used method is available which may be explored to deduce a consistent matrix that is diagonal. This consists of making a mixed representation for the term creating the A matrix. Consider a functional given by
uTcudf2
(1.14)
I-I 1 -- f ~ t~uTcu dr2
(I. 15)
I-I1 -- ~
The first variation of 1-II yields
Approximation using the standard form U ,~, f l - - N a f l a
yields
- Nu
(1.16)
P
31-I1 = 3fiT / o NTeN dS2fi
(1.17)
This yields exactly the form for A given by Eq. (I. 1). We can construct an alternative mixed form by introducing a momenta-type variable given by p = cu (I.18) A Hellinger-Reissner-type mixed form may then be expressed as 1-I2 -- f~ uTp dr2 - ~1 f
pTc_ 1p dr2
(I.19)
Matrix diagonalization or lumping
and has the first variation ~FI2- f ~uTp dr2 q - f ~pr ( u - c-lp)dr2
(1.20)
The term with variation on u will combine with other terms so is not set to zero; however, the other term will not appear elsewhere so can be solved separately. If we now introduce an approximation for p as p "~ ~
--
n bPb
-
-
n~
(1.21)
then the variational equation becomes ~H2 -- ~;~r f Nrn d ~ + ~ r ( f
nrN d ~ - fa nrc-ln df2~)
(1.22)
If we now define the matrices G =
J~ nN r dr2
H -
f nrc-lndf2
(1.23)
then the weak form is ~I-I2--[~fi T ~ r ]
([G
0
Gr
_HI(
~
~ }=(
0
O })
(1.24)
Eliminating ~ using the second row of Eq. (I.24) gives A = GTH-1G
(1.25)
for which diagonal forms may now be sought. This form has again the same options as discussed above but, in addition, forms for the shape functions n can be sought which also render the matrix diagonal.
709
This Page Intentionally Left Blank
Author index
Page numbers in bold me for pages at the end of chapters with names of author references. Abdulwahab, E, 469, 470, 497 Abramowitz, M., 161,184 Adams, M., 658, 663, 690, 691 Adee, J., 558, 562 Ahmad, S., 111,114, 120, 136, 331,354 Ainsworth, M., 478, 479, 484, 487, 492, 498 Alberty, J., 487, 498 Allwood, R.J., 445, 454 Alturi, S.N., 356, 380, 445,454 Alum, N.R., 660, 663 Andelfinger, U., 374, 381 Anderson, R.(2, 164, 184 Ando, Y., 88, 101 Archer, J.S., 566, 587 Argyris, J.H., 3,17, 111,118, 119, 120,137, 187, 188, 227, 228, 251,252, 261,262, 406, 427, 590, 628 Arlett, P.L., 229, 245, 261, 565, 575, 577, 587 Armstrong, C.(2, 265, 324, 325 Arnold, D.N., 367, 381, 390, 400, 425, 426, 442, 454 Arrow, K.J., 405, 427 Arya, S.K., 92, 94, 102 At-Abdulla, J., 445, 454 Atluri, S.N., 445, 454 Auricchio, E, 665, 667 Babu~ka, I., 96, 102, 330, 354, 356, 360, 380, 381, 442, 453, 478,479, 483,487,488,489,490, 494, 497, 498, 499, 500, 503, 514, 516, 522, 523, 524, 526, 527,539, 551,552, 553,555,558,560, 561, 562, 658, 663 Bachrach, W.E., 168, 184, 339, 355 Baehmann, P.L., 265, 286, 325 Bahrani, 229, 245,261 Baiocchi, C., 253, 262 Baker, T.J., 265, 307, 325 Balestra, M., 95, 102, 407, 427 Bampton, M.C.C., 573,587 Banerjee, P.K., 97, 102, 446, 454 Bank, R.E., 393,426, 478, 484, 487, 492, 497 Barlow, J., 461,496
Barsoum, R.S., 176, 185 Basu, U., 637, 661 Bathe, K.-J., 153, 184, 253, 263, 572, 583, 587, 614, 630, 666, 667 Batina, J., 540, 542, 543,561 Batina, J.T., 547, 561, 562 Baumann, C.E., 442, 453, 503, 514, 523 Bayless, A., 526, 560 Baynham, J.A.W., 356, 359, 380, 386, 402, 425 Bazeley, (2E, 37, 40, 52, 329, 354 Becker, E.B., 96, 102 Beer, (2, 97, 102, 171,172, 185 Beisinger, Z.E., 568,587 Belytschko, T., 168, 184, 339, 355, 469, 470, 497, 526, 527, 547, 558, 560, 561, 562, 606, 630, 654, 655, 662 Benz, W., 558, 562 Benzley, S.E.176, 186 Bercovier, H., 400, 426 Bercovier, M., 265,325 Beresford, EJ., 343, 345,355 Bettencourt, J.M., 600, 630 Bettess, P., 88,101, 171,172, 175,185, 398,426, 447, 455, 580, 588, 634, 657, 660, 662 Bey, K.S., 503, 514, 523 Biqaniq, N., 340, 342, 355, 648, 650, 652, 662 Biezeno, C.B., 3, 17, 61,100 Bijlaard, P.P., 187, 227 Binns, K.J., 250, 261 Biot, M.A., 648, 661 Bischoff, M., 374, 381 Blacker, T., 547, 562 Blacker, T.D., 265, 326, 469, 470, 497 Bochev, P.B., 407, 410, 411,427 Bociovelli, L.L., 568,587 Bogner, EK., 36, 52 Boley, B.A., 635,660 Bonet, J., 285,326, 558, 562 Booker, J.E, 251,261 Boroomand, B., 330, 354, 475,489, 497, 498
712
Author index Borouchaki, H., 303, 322, 327, 328 Bossak, M., 614, 630 Bottasso, C.L., 321,327 Boyer, A., 265,307, 308, 325 Braess, D., 374, 381 Brauchli, H.J., 207, 228, 466, 496 Brebbia, C.A., 446, 451,454 Brezzi, E, 360, 367, 381, 390, 407, 409, 425, 427, 442, 454 Brigham, E.O., 580, 588 Brown, C.B., 253, 262 Bruch, J.C., 253,262 Buck, K.E., 111,120, 137 Budyn, E., 527, 561 Bugeda, (2, 500, 505,523 Butterfield, 446, 454 Calvo, N.A., 265, 326 Campbell, D.M., 365, 381 Campbell, J., 91,101, 114, 121,134, 137, 207, 228, 466, 496 Canann, S.A., 286, 326 Cantin, (2,377, 382, 600, 630 Caravani, E, 583,588 Carey, (2E, 96, 102 Carlton, M.W., 652, 662 Carpenter, C.J., 171,185 Carslaw, H.S., 565, 586 Carstensen, C., 487, 498 Cavendish, J.C., 265, 280, 325, 326 Chadwick, P., 676, 682 Chan, A.H.C., 40, 53, 211,228, 330, 335, 354, 648, 650, 652, 653, 659, 662, 663 Chan, S.T.K., 252, 262 Chang, C.T., 648, 650, 657,661, 662 Chad, M.V.K., 245 Chen, C.M., 465, 496 Chen, H.-C., 578, 588 Chen, H.S., 88, 101 Chen Po-shu, 429, 453 Chessa, J., 527, 561 Cheung, B.M., 37, 40, 52, 329, 354 Cheung, Y.K., 3, 18, 210, 228, 229, 245, 248, 261, 565, 566, 587 Chi-Wang Shu, 442, 454 Chiba, N., 265,325 Chilton, L.K., 503, 514, 523 Chopp, D.L., 527, 560 Chopra, A.K., 565,566, 580, 583,587, 637, 641,661 Chu, T.Y., 251,261 Ciarlet, P.G., 38, 53, 188, 228, 504, 523 Clesbsch, R.F., 2, 17 Clough, R.W., 1, 2, 3, 17, 20, 37, 52, 187, 227, 398, 426, 566, 580, 583, 587, 644, 648, 661 Cockburn, B., 442, 454 Codina, R., 95, 102, 407, 414, 427, 428
Coffignal, G., 493,498 Collar, A.R., 3, 17 Collatz, L., 540, 561 Collins, T., 583, 588 Cook, R.D., 445,454 Coons, S.A., 141,184 Copps, K., 483, 487, 488, 489, 490, 498 Comes, G.M.M., 445,454 Courant, R., 3, 18, 20, 52, 94, 102, 407, 427 Cowper, (2R., 164, 184 Cox, H.L., 573,587 Craig, A., 573,587 Craig, A.W., 658,663 Crandall, S., 1, 17, 61,100, 563, 565, 586 Crank, J., 594, 629 Crochet, M., 152, 184 Crouzcix, M., 390, 426 Cruse, T.A., 176, 185 Curnier, A., 252, 262 Dahlquist, (2(2, 614, 630 Daniel, W.J.T., 639, 649, 659, 661 Daux, C., 527,560 de Arantese Oliveira, E.R., 38, 53, 330, 336, 354, 501, 523 de (2 Allen, D.N., 1, 17, 97, 102, 239, 261 de Vries, (2,252, 262 Delaunay, B., 304, 327 Demkowicz, L., 478,484, 497, 503, 514, 523 Demmel, J., 36, 52, 557, 562, 574, 588, 684, 691 Dennis, J.E., 666, 667 Desai, C.S., 252, 253, 262, 263 Dhondt, G., 265,326 Dfez, E, 478, 494, 498, 499 Doctors, L.J., 252, 262 Doherty, W.E, 133, 137, 343,355 Dohrmann, C.R., 407, 410, 411,427 Dolbow, J., 526, 527, 547,560, 561 Douglas, J., 367,381 Duarte, C.A., 526, 538, 539, 547, 552, 553,555,558, 560, 562 Duarte, C.A.M., 551,562 Duff, I.S., 557, 562 Duncan, W.J., 3, 17 Dungar, R., 445,454 Dunham, R.S., 359, 381 Dunne, EC., 108, 136 Egozcue, J.J., 478, 494, 498 Eiseman, ER., 316, 327 Elias, Z.M., 379, 382 Ely, J.E, 242, 261 Emson, C., 171,172, 175, 185 Engel, (2,442, 453 Engleman, M.S., 400, 426
Author index 713 Ergatoudis, J.G., 108, 111, 114, 116, 118, 120, 121, 134, 136, 137, 188, 217, 228 Eriksson, K., 596, 630 Escobar, J.M., 321,327 Evans, J.H., 652, 662 Evenson, D.A., 583,588 Farhat, Ch., 429, 435,453 Farin, 268, 290, 326 Faux, I.D., 268, 270, 296, 326 Felippa, C.A., 164,184, 405,427, 634, 639, 655,661, 662 Ferencz, I., 609, 630 Ferencz, R.M., 690, 691 Field, D.A., 265, 325 Finlayson, B.A., 1, 17, 61,100 Finn, N.D.L., 590, 629 Fish, J., 526, 560 Fix, G.J., 38, 40, 53, 141, 166, 184, 206, 228, 330, 336, 354 Fjeld, S., 188, 228 Fletcher, C.A.T., 96, 102 Flores, E, 547, 562 Formaggio, L., 265, 285,325 Forrest, A.R., 141,184 Forsythe, CtE., 540, 561 Fortin, M., 390, 405,406, 425, 426, 427 Fortin, N., 390, 425 Fox, R.L., 36, 52 Fraeijs de Veubeke, B., 36, 40, 52, 53, 119, 137, 330, 354, 359, 379, 381, 382, 490, 498 Franca, L.E, 95, 102, 407, 427, 590, 629 France, E/P., 652, 662 Frasier, G.A., 168, 184 Frazer, R.A., 61,100 Frazer, R.R., 3, 17 Freitag, L.A., 321,327 Freund, J., 95, 102 Frey, W.H., 265, 325 Fried, I., 91,101, 118, 137, 167, 184, 568, 572, 587, 590, 628 Friedmann, EE, 634, 638, 639, 661 Fujishiro, K., 265,325 Furukawa, T., 558, 562 Gago, J.P De S.R., 127, 128, 131,137, 478, 479, 497, 514, 523 Galerkin, B.Ct, 3, 17, 61, 62, 100 Gallagher, R.H., 187, 227, 356, 380, 445,454 Gangaraj, S.K., 483,487, 488, 489, 490, 498 Gantmacher, ER., 611,630 Gaul, L., 97, 102 Gauss, C.E, 3, 17 Gear, C.W., 614, 630 Gear, G.W., 589, 628 George, EL., 265, 303, 307, 322, 325, 327, 328
Georges, M.K., 265, 286, 325 Geradin, M., 634, 639, 661 Gethin, D.T., 303, 311,327 Ghaboussi, J., 343,355 GiD, 664, 666, 667 Gingold, R.A., 558, 562 Girault, V., 525, 559 Glowinski, R., 405,406, 427 Godbole, EN., 398, 426 Gong, N.G., 503, 515, 516, 517, 518, 519, 520, 523 Gonz~ilez-Yuste,J.M., 321,327 Goodier, J.N., 40, 42, 52, 101, 188, 195, 228, 242, 257, 261, 263, 378, 382 Gordon, W.J., 169, 185, 264, 324, 664, 667 Gorensson, E, 634, 639, 661 Gould, EL., 445,454 Gourgeon, H., 446, 455 Graichen, C.M., 316, 327 Grammel, R., 61,100 Gravouil, A., 527, 561 Gresho, EM., 400, 426 (;rice, K.R., 265, 286, 325 Griffiths, A.A., 176, 185 Griffiths, D., 97, 102 Griffiths, R.E., 330, 354 Gu, L., 526, 547, 560, 562 Guex, L., 429, 453 Gui, W., 503, 514, 523 Guo, B., 503, 507, 514, 523 Guo, B.Q., 503, 514, 523 Gupta, A.K., 681,682 Gupta, K.K., 572, 587, 639, 661 Gupta, S., 641,661 Gurtin, M.E., 594, 629 Hager, E, 474, 497 Hall, C.A., 169, 185, 264, 324, 664, 667 Hall, J.E, 641,661 Hammer, EC., 164, 184 Hansbo, E, 265, 285, 300, 325, 327, 478, 497 Hardy Cross, 2, 17 Hardy, O, 503, 514, 523 Hassan, O., 265, 307, 325 Hathaway, A.E, 316, 327 Hause, J., 316, 327 Hayes, L.J., 658, 662 Hearmon, R.ES., 197, 228 Hecht, E, 265, 307, 322, 325, 328 Heimsund, B., 469, 497 Hellan, K., 359, 381 Hellen, T.K., 163, 176, 184, 185 Hellinger, E., 365,381 Henrici, E, 589, 628 Henshell, R.D., 176, 185, 445, 454 Herrera, I., 446, 454, 455
714 Author index Herrmann, L.R., 229, 252, 261, 262, 359, 365, 38t), 381, 385, 386, 425, 462, 496 Hestenes, M.R., 405,427 Hibbitt, H.D., 176, 177, 186 Hilber, H., 614, 627, 630 Hilber, H.M., 111,120, 137 Hildebrand, EB., 34, 52, 76, 79, 97, 101, 102, 589, 628 Hill, T.R., 442, 454, 596, 630 Hine, N.W., 590, 629 Hinton, E., 91,101,207,228, 340, 342, 355, 398,426, 466, 496, 504, 523, 568,587, 634, 645,646, 648, 650, 655,660, 661, 662 Holbeche, J., 639, 661 Hood, P., 390, 426 Houbolt, J.C., 614, 630 HP Fortran home page, 667, 667 Hrenikoff, A., 1, 3, 17 Hsieh, M.S., 245,261 Huang, Y., 465,496 Huck, J., 634, 639, 661 Huebner, K.H., 251,261,262 Huerta, A., 478,494, 498, 499 Hughes, T.R.J., 95, 96, 1t)2, 372, 381, 397, 398,407, 426, 427, 442, 453, 590, 606, 609, 614, 627,629, 630, 634, 654, 655,660, 662 Hulbert, G.M., 95, 102, 407, 409, 427, 590, 629 Humpheson, C., 212, 228 Hurty, W.C., 573, 582, 587 Hurwicz, L., 405,427 Hurwitz, A., 611,630 Ibrahimbegovic, A., 346, 355, 578, 588 Idelsohn, S., 540, 541,542, 561 Idelsohn, S.R., 265,326, 540, 541,561 Iding, R., 501,523 Irons, B.M., 37, 40, 52, 53, 108, 111, 114, 116, 118, 120, 121,134,136,137, 141,155, 163, 164, 166, 177,184,186, 188,217,228,319,329,331,354, 590, 597, 629, 639, 661 Ito, Y., 303,327 Jaeger, J.C., 565,586 Jameson, A., 547, 561, 562 Javandel, I., 252, 262 Jennings, A., 572, 573,587 Jian, B.-N., 94, 102 Jin, H., 265, 285,325 Jirousek, J., 429, 446, 447, 448, 451,453, 455 Joe, B., 286, 316, 326, 327 Johnson, C., 478,497, 590, 596, 629, 630 Johnson, M.W., 38, 53 Johnston, B.E, 286, 326 Jones, W.E, 61,100 Jun, S., 558,562
Kantorovich, L.V., 72, 101 Karniadakis, GE., 442, 454 Kassos, T., 86, 101 Katona, M., 590, 629 Katz, I.N., 516, 524 kazarian, L.E., 652, 662 Kelley, D.W., 88, 101, 131,137, 171,185 Kelly, D.W., 447, 455, 478, 479, 484, 492, 497, 498, 514, 523, 634, 660 Kelsey, S., 3, 17 Key, S.W., 385,425, 568, 587 Kikuchi, E, 88, 101, 363,381 Kim, J.Y., 660, 663 Kitamura, M., 484, 498 Knupp, EM., 321,327 Koch, J.J., 3, 17 Kolsoff, D., 339, 355 Koshgoftar, M., 253, 263 Kosloff, D., 168, 184 Krizek, M., 465,496 Krok, J., 525,560 Kron, G., 3, 17, 435,453 Krylov, 72, 101 Kulasegaram, S., 558,562 Kvamsdal, T., 634, 639, 661 Kwasnik, A., 286, 326 Kwok, W., 285,326 Kythe, Prem K., 97, 102 Ladev~ze, E, 475, 484, 487, 493,497, 498, 518, 524 Ladkany, S.G., 445,454 Lambert, T.D., 589, 628 Lancaster, E, 526, 533,560 Larock, B.E., 252, 262 Larson, M.G., 442, 453 Laug, E, 303, 322, 327, 328 Laursen, T.A., 438, 453 Lawrenson, EJ., 250, 261 Leckie, EA., 566, 587 Ledesma, A., 648, 650, 652, 662 Lee, C.K., 286, 303,326, 327 Lee, K.N., 92, 94, 102 Lee, S.W., 469, 497 Lee, T., 469, 497 Lee, Y.K., 303,327 Lefebvre, D., 366, 371,381 Leguillon, D., 475,484, 493,497 Lekhnitskii, S.Ct, 188, 197, 228 Lesaint, E, 442, 454, 596, 630 Lesoinne, M., 429, 453 Leung, K.H., 648, 650, 655, 662 Levy, J.E, 398,426 Levy, N.177, 186 Lewis, R.W., 212, 228, 303,311,327, 590, 594, 599, 629, 634, 660 Li, B., 469, 496
Author index 715 Li, G.C., 253,263 Li, S., 558, 562 Li, X.D., 474, 497, 596, 623, 630 Liebman, H., 3, 17 Ligget, J.A., 446, 454 Lin, Q., 465,496 Lin, R., 469, 496 Lindberg, CtM., 566, 587 Liniger, W., 599, 614, 630 Liszka, T., 525, 541,547,560, 561 Liu, A., 316, 327 Liu, EL-E, 446, 454 Liu, W.K., 558, 562, 654, 655,662 Livesley, R.K., 11, 18 Lo, S.H., 265,266, 285, 286, 325, 326 Lohner, R., 265, 285, 303,325, 326, 327, 658, 663 Lomacky, O., 176, 185 Loubignac, C., 377, 382 Love, A.E.H., 188, 228 Lowther, C.J., 171,185 Lu, Y., 526, 547, 560, 562 Lucy, L.B., 558, 562 Luenberger, D.G., 405,406, 427 Luke, J.C., 253,262 Lyness, J.E, 245, 249, 261 Lynn, P.P., 92, 94, 102 Machiels, L., 494, 499 Maday, Y., 494, 499 Makridakis, C.G., 442, 453 Malkus, D.S., 390, 398, 426, 568, 587 Mallet, R.H., 36, 52 Mandel, J., 429, 453, 658, 663 Marqal, P.V., 177, 186 Marcum, D.L., 265, 307, 322, 325, 328 Mareczek, G., 111,120, 137, 252, 262 Marini, D., 442, 454 Marlowe, O.P., 164, 184 Martin, C.W., 265,324 Martin, H.C., 1, 3, 17, 187, 227, 252, 262 Martinelli, L., 547, 562 Mastin, C.W., 264, 324 MATLAB, 16, 18, 42, 53, 99, 102, 323,328 Matthies, H., 666, 667 Mavriplis, D.J., 322, 328, 547, 561 Mayer, P., 229, 245,261 Mazzei, L., 442, 453 McDonald, B.H., 245,261 McHenry, D., 1, 3, 17 McLay, R.W., 38, 53 Meek, J.L., 171,172, 185 Mei, C.C., 88, 101 Melenk, J.M., 526, 539, 551,560, 561 Melosh, R.J., 187, 227 Meyers, R.J., 265,326 Mikhlin, S.C., 38, 53, 81,101
Miller, A., 494, 499 Minich, M.D., 36, 52 Miranda, I., 609, 630 Mitchell, A.R., 97, 102, 330, 354 Mitchell, S.A, 265,326 Moes, N., 527, 560, 561 Mohammadi, B., 322, 328 Mohraz, B., 681,682 MOiler, P., 265, 285,325 Monaghan, 558, 562 Monk, P., 503, 514, 523 Montenegro, R., 321,327 Montero, G., 321,327 Moran, B., 527, 560 Morand, H., 634, 638, 639, 661 More, J., 666, 667 Morgan, K., 46, 53, 96, 102, 265,266, 270, 285,286, 325, 326, 407, 428, 447, 455, 504, 523, 658, 663 Morton, K.W., 540, 561, 589, 628 Mote, C.D., 131,137 Mullen, R., 654, 662 Mullord, P., 525, 560 Munro, E., 245, 261 Muskhelish, N.I., 188,228 Naga, A., 474, 497 Nagtegaal, J.C., 133, 137, 403, 427 Nakahashi, K., 303, 327 Nakazawa, S., 40, 53, 330, 354, 362, 381, 398, 404, 406, 426, 427 Nam-Sua Lee, 153, 184 Navert, U., 590, 596, 629 Nay, R.A., 525,560 Naylor, D.J., 91,101, 398, 426 Nayroles, B., 526, 538, 547, 560 Neitaanmaki, P., 465, 496 Newmark, N., 600, 606, 618, 630 Newmark, N.M., 1, 3, 17 Newton, R.E., 155,184, 565,587, 634, 645,648,660, 661 Nickell, R.E., 565,587, 594, 629 Nicolson, P., 594, 629 Nikuchi, N., 253, 262 Nishigaki, I., 265,325 Nithiarasu, P., 69, 71,101, 132, 137, 175, 185, 252, 262, 266, 326, 379, 382, 407, 414, 419, 421,427, 428, 519, 524, 527,560, 571,576, 580, 587, 590, 628, 634, 635, 660 Nitsche, J.A., 438, 453 Norrie, D.H., 252, 262 Oden, J.T., 81, 96, 101, 102, 207, 228, 363,381, 390, 425, 442, 453, 466, 478,479,484, 487,496, 497, 498, 503, 514, 518,523, 524, 526, 538,539, 547, 551,552, 553, 555, 558, 560, 562, 590, 628 Oglesby, J.J., 176, 185
716
Author index Oh, K.E, 251,262 Ohayon, R., 634, 639, 660, 661 Ohnimus, S., 483,498 Ohtsubo, H., 484, 498 Ofiate, E., 61,100, 500, 407, 427, 505,523, 540, 541, 542, 547, 551,552, 561, 562 Orkisz, J., 525, 540, 541,547, 560, 561 O'Rourke, J., 304, 327 Ortiz, E, 407, 414, 427 Osborn, J.E., 356, 380 Ostergren, W.J., 177, 186 Owen, D.R.J., 92, 94, 102, 245, 249, 261 Owen, S.J., 286, 326 Padlog, J., 187, 227 Paidoussis, M.E, 634, 638, 639, 661 Paraschivoiu, M., 494, 499 Parekh, C.J., 565, 587, 590, 629 Par6s, N., 494, 499 Parikh, E, 265,285,325 Parimi, C., 527, 561 Park, H.C., 469, 497 Park, K.C., 655, 659, 662, 663 Parkinson, A.R., 286, 326 Parks, D.M., 133, 137, 176, 185, 403, 427 Parlett, B.N., 547, 572, 587 Parthasarathy, V.N., 316, 327 Pastor, M., 211,228, 648, 650, 652, 662 Patera, A.T., 494, 499 Patil, B.S., 565, 576, 577, 587 Paul, D.K., 645, 646, 648, 650, 659, 661, 662, 663 Pavlin, V., 525, 559 Pawsey, S.E, 398, 426 Peano, A.G., 128, 137 Peck, R.B., 565,587 Peir6, J., 265, 285,286, 300, 303, 325, 326, 327, 442, 454 Pelle, J.E, 487, 493,497, 498 Pelz, R.B., 547, 561 Penzien, J., 566, 580, 583,587, 588 Peraire, J., 265, 266, 270, 285, 286, 325, 326, 494, 499, 504, 523 Perrone, N., 525,559 Phillips, D.V., 169, 184, 264, 324, 664, 667 Pian, T.H.H., 38, 53, 176, 185, 369, 381, 429, 445, 453, 454, 568, 587 Picasso, M., 483,498 Pierre, R., 418, 428 Pierson, K., 429, 453 Pilmer, R., 447 Pister, K.S., 359, 372, 381, 397, 426 Pitkaranta, J., 407, 409, 427, 590, 596, 629, 658, 663 Powell, M.J.D., 400, 405,426 Prager, W., 3, 18, 20, 52 Pratt, M.J, 268, 270, 296, 326 Preparata, EE, 304, 327
Press, W.H., 299, 326 Price, M.A., 265,324, 325 Przemieniecki, J.S., 11, 18 Puso, M.A., 438, 453 Qu, S., 40, 53, 362, 381 Rachowicz, W., 478,484, 497, 503, 514, 523 Radau, R., 164, 184 Ralston, A., 331,354, 684, 691 Ramm, E., 374, 381 Randolf, M.E, 400, 426 Rank, E., 286, 326 Rao, D.V., 251,261 Rashid, Y.R., 227 Rassineux, A., 303,327 Rausch, R.D., 547, 562 Raviart, E-A., 367, 381, 390, 426, 442, 454, 596, 630 Rayleigh, Lord, 3, 17, 35, 52, 76, 101 Razzaque, A., 40, 53, 329, 354 Rebay, S., 265, 307, 325 Reddi, M.M., 251,261 Redshaw, J.C., 188, 228 Reed, W.H., 442, 454, 596, 630 Reid, J.K., 557, 562 Reinsch, C., 572, 587, 666, 667, 684, 691 Reissner, E., 365,381 Rheinboldt, C., 478,479, 497, 500, 522 Rice, J.R., 133, 137, 176, 177, 185, 186, 403, 427 Richardson, L.E, 1, 3, 17, 39, 53 Richtmyer, R.D., 540, 561, 589, 628 Rifai, M.S., 373,381 Riks, E., 666, 667 Ritz, W., 3, 17, 35, 52, 76, 101 Roberts, (2, 634, 639, 661 Robinson, J. et al., 330, 354 Rock, T., 568, 587 Rodr~guez, E., 321,327, 483,498 Rohde, S.M., 251,262 Rougeot, E, 487, 497 Routh, E.J., 611,630 Roux, E-X., 429, 435,453 Rubinstein, M.E, 573, 582, 587 Rudin, W., 105, 136, 561 Sabin, M.A., 265,324, 325 Sabina, EJ., 446, 455 Saigal, S., 286, 326 Salkauskas, K., 526, 533,560 Salonen, E-M., 95, 102 Saltel, E., 265, 307, 322, 325, 328 Salvadori, M., 596, 630 Samuelsson, A., 3, 17 Sandberg, (2, 634, 639, 661 Sander, (2,330, 354 Sani, R.L., 400, 426
Author index 717 Satya, B.V.K.,407,428 Savign~,J.-M.,303,327 Scharp~ D.W.,lll, l18,120,137,251,252,261,262, 590,628 Schmit, L.A.,36,52 Schrefle~ B.A., 211,228, 634, 648, 650, 652, 660, 662 Schroede~W.J.,265,307,325 Schweingrube~M.,286,326 Schweizerho~ K.,666,667 Scoa, EC., 111, 114, 116, 120, 121, 134, 136, 137, 192,228 Scott, J.A.,557,562 Seed, H.B.,644,648,661 Sen, S.K.,445,454 Seveno, E.,265,325 Severn, R.T.,445,454 Shamos, M.I.,304,327 Shaw, K.G.,176,185 Sheffe~ A.,265,325 Shen, S.E, 171,185 Shepard, D.,526,533,539,560 Shephard, M.S.,265,286,307,325 Sherwin, S.J., 303,327, 442, 454 Shiomi, T.,211,228,648,650,652,662 Silves~ E, 118,137,171,185,245 Simkin, J.,245,261 Simo, J.C.,40,53,330,335,354,372,373,381,397, 407,426,427,634,660,666,667 Simon, B.R.,652,662 Sken, S.W.,61,100 Sloan, S.,492,498 Sloan, S.W.,400,426 Sloss, J.M.,253,262 Smolinski, E, 527,561 Snell, C.,525,560 Sokolnikoff, I.S.,101,188,195,228,680,682 Somme~M.,286,326 Soni, B.K.,265,326 Southwell, R.V.,1,2,3,17,69,97,100,379,382 Stab, 0.,303,327 Stanton, E.L., 36, 52 St~en, M.L., 286, 326 Stazi, EL., 527, 561 Stegun, I.A., 161,184 S~in, E.,483,498 Stenberg, R.,465,496 S~ang, G, 38,40,53, 141, 166, 184,206,228,330, 336,354,557,562,666,667,684,691 S~ouboulis, T.,483,487,488,489,490,498 Stroud, A.H.,164,184 Stummel, E, 330,354 Sukumar, N.,527,560,561 Sullivan, J.M.,286,326 Sumihara, K.,369,381 Sun, M.,503,514,523
Suzuki, M., 303,327 Synge, J.L., 3, 18 Szabo, B., 96, 102 Szabo, B.A., 86, 101, 516, 524 Szmelter, J., 20, 52, 283,326 Tabbara, M.,547,562 Tai, X.,469,497 Taig, I.C.,141,184 Takizawa, C.,265,325 Talbe~,J.A.,286,326 Tam, T.K.H.,265,324 Tanesa, D.V.,251,261 Tanne~ R.I.,265,285,325 Tautges, T.J.,265,326 Taylo~ C.,390,425,565,576,577,587 Taylo~ R.L.,33,40,42,52,53, 69,71,72,101,112, 114,118,132,133,137,175,185,216,228,249, 252,253,261, 262,266,299,326,330,335,343, 345,354,355, 356,357,359,362,372,379,380, 381,382,384, 386,397,398,402,407,414,419, 421,425,426, 427,428,442,454,501,519,523, 524,527,540, 541,551,552,560,561,562,571, 574,576,578, 580,587,588,590,595,614,624, 627,628,629, 630,634,635,641,642,648,649, 650,655,660, 661,662,664,665,666,667,684, 688,691 Teodorescu, E, 447,448, 455 Terzhagi, K., 211,228, 565, 587 Thames, EC., 264, 324 Thatcher, R.W., 171,185 Thomas, D.L., 583, 588 Thomasset, E, 405,427 Thompson, J.E, 264, 265, 316, 324, 326, 327 Thomson, H.T., 583,588 Tieu, A.K., 251,262 Timoshenko, S.E, 40, 42,52,101, 188, 195,228,242, 257, 261, 263, 378, 382 Ting, T.C.-T., 197, 228 Todd, D.K., 565, 587 Tong, E, 38, 53, 64, 100, 176, 185, 429, 445, 453, 454, 568, 587, 703,703 Tonti, E., 77, 81,101 Too, J., 398, 426 Topp, L.J., 1, 3, 17, 187, 227 Tortorelli, D.A., 660, 663 Touzot, C., 377, 382 Touzot, G., 526, 538, 547, 56t) Toyoshima, S., 404, 406, 427 Tracey, D.M., 176, 177, 185 Trefftz, E., 446, 454 Treharne, C., 590, 597, 629 Trowbridge, C.W., 245, 250, 261 Trujillo, D.M., 657, 662 Turner, M.J., 1, 3, 17, 187, 227
718 Author index Upadhyay, C.S., 483,487,488, 489, 490, 498 Usui, S., 527, 561 Utku, S., 525, 560 Uzawa, H., 405, 427 Vahdati, M., 265, 266, 270, 325, 504, 523 Vainberg, M.M., 81,101 Valliappan, S., 406, 427 Vanburen, W., 176, 185 Vardapetyan, L., 503, 514, 523 Varga, R.S., 3, 18, 69, 101 Varoglu, E., 590, 629 Vazquez, M., 95, 102, 407, 414, 427, 428 Verfurth, R., 478, 479, 483,497, 498 Vesey, D.G., 525, 560 Victory, H.D., 469, 496 Villon, E, 303, 327, 526, 538, 547, 560 Vilotte, J.E, 404, 406, 407, 427 Visser, W., 229, 261, 565, 586 Vogelius, M., 400, 426 Voronoi, G., 304, 327 Wachspress, E.L., 152, 153, 184, 340, 355 Wahlbin, L.B., 465,496 Walhom, E., 483,498 Walker, S., 446, 451,454 Walsh, EE, 176, 185 Wang, H., 527, 561 Wang, J., 469, 497 Wang, S., 492, 498 Warsi, Z.U.A., 264, 265,324 Washizu, K., 1, 17, 34, 35, 52, 88, 101,253,262, 357, 380, 594, 629 Wasow, W.R., 540, 561 Watson, D.E, 265, 307, 308, 325 Watson, J.O., 97, 102 Weatherill, N.E, 265, 307, 316, 322, 325, 326, 327 Weiner, J.H., 634, 660 Weiser, A., 478,484, 487, 492, 497 Welfert, B.D., 393, 426 Westerman, T.A., 478, 484, 497 Wexler, A., 245,261 Wiberg, N.-E., 431,453, 469, 470, 474, 497, 596, 623, 630 Wilkins, M.L., 3, 18 Wilkinson, J.H., 572, 587, 666, 667, 684, 691 Williams, EW., 583,588 Wilson, E.L., 133,137, 343, 345,346, 355, 565, 578, 583, 587, 588, 594, 614, 629, 630 Winslow, A.M., 229, 245,261 Witherspoon, EA., 252, 262 Wittchen, S.L., 265,286, 325 Wohlmuth, B.I., 432, 453 Wolf, J.A., 445,454
Wood, W., 614, 630 Wood, W.L., 589, 590, 594, 599, 609, 613, 628, 629 Wriggers, E, 666, 667 Wr6blewski, A.E, 429, 446, 45 l, 453 Wu, J., 283,326, 407, 413, 427, 469, 496, 504, 523 Wu, J.S-S., 652, 662 Wyatt, E.A., 171,185 Xie, Y.M., 648, 650, 652, 662 Xu, J., 469, 496 Xu, K., 547, 562 Yagawa, (2, 558, 562 Yamada, T., 558, 562 Yamaguchi, E, 270, 290, 326 Yamashita, Y., 265,325 Yan, N., 465, 496 Yang, H.T.Y., 547,562 Yerry, M.A., 265, 286, 325 Yoshida, Y., 445,454 Zarate, E, 547, 562 Zhang, Y.E, 558, 562 Zhang, Z., 469, 474, 496 Zhao, Q., 469, 496 Zheng, Y., 303, 311,327 Zhong, W.X., 330, 354 Zhu, J.Z., 283,326, 467,469, 477,483,489, 496, 498, 500, 503,504, 505,515,516, 517, 518,519,520, 522, 523 Zhu, Q.D., 465,469, 496 Zi, (2,527, 561 Zielinski, A.E, 429, 446, 447,450, 451,453, 455 Zienkiewicz, O.C., 3, 17, 18, 33, 37, 40, 46, 52, 53, 61, 69, 71, 72, 88, 90, 91, 92, 94, 95, 96, 100, 101,102, 108, 111,114, 116, 118, 120, 121,131, 132, 134,136,137, 164, 169, 171,172, 175,184, 185, 188, 192,210,211,212,216,217,228,229, 242, 245,248,249, 252, 261,262, 264, 265,266, 270, 283,285,299,324, 325, 326, 329, 330, 335, 354, 356, 357, 359, 362, 366, 371,379,380, 381, 382, 384, 386, 398,402, 404, 406, 407, 413, 414, 419, 421,425, 426, 427, 428, 442, 445,446, 447, 450, 454, 455, 467,469,475,477,479, 483,489, 496, 497, 498, 500, 503,504, 505, 514, 515, 516, 517, 518, 519,520, 521,522, 523, 524, 527,540, 541,542, 55 l, 552, 560, 561,562, 565,566, 568, 571,574, 576, 577,580, 587, 588, 590, 595,600, 614, 624, 628, 629, 630, 631,634, 635,641,642, 644, 645,646, 648,649, 650, 652, 653,655,657, 658, 659, 660, 661, 662, 663, 664, 665, 667 Ziukas, S., 469, 470, 497 Zlamal, M., 167, 184, 599, 614, 630
Subject index
Abstraction, 55 Accuracy assessment, numerical examples, 239-53 Acoustic problems, equations for, 635 Adaptive finite element refinement: about adaptive refinement, 500-3, 518-20 asymptotic convergence rate, 504-5 h-refinement, 501-3, 503-14 element subdivision, 501 hanging points, 501 L-shaped domain example, 509 machine part example, 509, 513 mesh regeneration/remeshing, 501 perforated gravity dam example, 513, 515 Poisson equation in a square domain example, 506-11 predicting element size, 503-5 r-refinement, 501-2 short cantilever beam example, 505, 507-9 stressed cylinder example, 505-6, 519 mesh enrichment, 504 p and hp-refinement, 501-3, 514-18 about p and hp-refinement, 51 4-16 L-shaped domain and short cantilever beam example, 51 6-18 permissible error magnitudes, 500 ADI (alternating direction implicit) scheme, 658 Adjoint differential equations: non-self-adjoint, 702-3 self-adjoint, 702-3 Advancing front method of mesh generation s e e Mesh generation, two dimensional, advancing front method Airy stress function, 378-9 Algorithm stability, 609-15 Algorithmic damping, 619 Alternating direction implicit (ADI) scheme, 658 Amplification matrix, 596 Anisotropic and isotropic forms for k, 231-2 Anisotropic materials, 394 elasticity equations, 197-200 Anisotropic seepage problem, 244-5, 247
Approximations: about approximations, 1 and displacement continuity, 20 history of approximate methods, 3 and transformation of coordinates, 12 s e e a l s o Convergence of approximations; Elasticity finite element approximations for small deformations; Function approximation; Least squares approximations; Moving least squares approximations/expansions; Tensor-indicial notation in the approximation of elasticity problems Arch dam in a rigid valley example, 216-17 Area coordinates, 117-18 Assembly and analysis of structures: boundary conditions, 6-7 electrical networks, 7-9 fluid networks, 7-9 general process, 5-6 step one, determination of element properties, 9-10 step two, assembly of final equations, 10 step three, insertion of boundary conditions, 10 step four, solving the equation system, 10 Assessment of accuracy, numerical examples, 239-53 Asymptotic behaviour and robustness of error estimators, 488-90 Asymptotic convergence rate, 504-5 Augmented lagrangian form, 406 Automatic mesh and node generation s e e Mesh generation Auxiliary functions, with complementary forms, 378-9 Axisymmetric deformation problems, 188-9, 235-7 B-bar method for nearly incompressible problems, 397-8 Babu~ka patch test, 490 Babu~ka-Brezzi condition, 363 Back substitution, simultaneous equations, 684 Base solution, patch test, 332 Basis functions s e e Shape functions
720
Subjectindex Beam, circular, subjected to end shear example, 209-10 Beam, rectangular, subjected to end shear example, 209 Bearings, stepped pad, 251 Biomechanics problem of bone-fluid interaction, 652 Blending functions, 169-70 Body forces, distributed, 26 Boundary conditions: about boundary conditions, 6-7, 191 Dirichlet, 59 and equivalent nodal forces, 29 errors from approximation of curved boundaries, 39 forced, 59 forced with natural variational principles, 81 and identification of Lagrange multipliers, 87-8 linear elasticity equations: on inclined coordinates, 192 normal pressure loading, 193 symmetry and repeatability, 192-3 natural (Neumann condition), 60 nodal forces for boundary traction example, 30-1 Boundary value problems: element, 483 Neumann, 484 Bounded estimators, 490--4 equilibrated methods, 494 Boussinesq problem, 174-6 CAD, with surface mesh generation, 286 Cavitation effects in fluids, 645-6 CBS (characteristic-based split) procedure, 407 Central difference approximation, multistep recurrence algorithms, 618 Characteristic-based split (CBS) procedure, 407 Collocation: collocation methods, 525 subdomain/finite volume mehod, 61,547 Taylor series collocation, 593-4 see also Point collocation Complementary forms see u n d e r Mixed formulations Completeness of expansions, 75 Computer procedures: about computer procedures, 664 see also FEAPpv (Finite Element Analysis Program personal version) Conical water tank example, 215-16 Consistency index, mesh generation, 274, 299 Consistent damping matrices, 566 Consistent mass matrix, 566 Constant stress state, 3-node triangle, 26-7 Constitutive relations, 195 Constrained parameters, 12 Constrained variational principles: discretization process, 84-6 -
enforcement with Lagrange multiplier example, 86 locking, 91 penalty function method, 88-9, 90-1 penalty method for constraint enforcement example, 90 perturbed lagrangian functional, 89-91 see also Lagrange multipliers Constraint and primary variables, 360 Continuity requirements: mapped elements, 143-5 mixed formulations, 358-9 Continuous and discrete problems, 1 Contravariant sets of transformations, 12 Contrived variational principles, 77 Convergence of approximations, 74-5 and completeness of expansions, 75 criterion of completeness, 75 h convergence, 75 p convergence, 75 ultraconvergence, 469 see also Superconvergence Convergence criteria and displacement shape functions: and constant strain conditions, 37 and functional completeness statements, 38 and rigid body motion, 37 for standard and hierarchical element shapes, 103 strains to be finite, 37 Convergence rate and discretization error rate, 38-9 Convergence requirements, patch test, 330-1 Coordinates: coordinate transformation, 11-12, 677-8 global, 12 local, 12 Coupled systems: about coupled systems, 631-4, 660 classes, 631-2 definitions, 631 different discretizations, need for, 632-3 partitioned single-phase systems- implicit-explicit partitions (Class I problems), 653-5 see also Fluid-structure interaction (Class 1 problem); Soil-pore fluid interaction (Class II problems); Staggered solution processes Crank-Nicholson scheme, 594, 596 Cubic elements, serendipity family, 113-14 conical water tank example, 215-16 rotating disc analysis example, 212-15 Cubic Hermite polynomials, 269 Cubic triangle, triangular elements family, 119 Curvilinear coordinates, patch test, 330 D'Alembert principle, linear damping, 565 Dam subject to external and internal water pressure example, 210-14 Darn/reservoir interaction, 641
Subject index 721 Damping: algorithmic damping, 619 and participation of modes, 583 s e e a l s o Dynamic behaviour of elastic structures with linear damping; Time dependence; Transient response by analytical procedures Darcy's law, 230 DATRI (FEAPpv sub program), 684-8 Degeneration/degenerate forms: about degeneration, 153-9 degenerate forms for a quadratic 27-node hexahedron example, 158-9 higher order degenerate elements, 155-9 quadratic quadrilateral degenerated triangular element example, 156-8 quadrilateral degenerated into a triangular element example, 153 Degrees of freedom: and matrices, 5 and total potential energy, 35 Delaunay triangulation s e e Mesh generation, three-dimensional, Delaunay triangulation Deviatoric stress and strain, 383-4 deviatoric form for elastic moduli of an isotropic material, 384 laplacian pressure stabilization, 408-9 pressure change, 384 Diagonality with shape functions, 105 Diagonalization or mass lumping, 568-70, 704-9 Diffuse finite element method, 526 Diffusion or flow problems, 230 Direct minimization, 36 Direct pressure stabilization approach to incompressible problems, 41 0-13 Direction cosines, 232 Dirichlet boundary conditions, 59, 231 Nitsche method example, 440 Discontinuity of displacement problems, 39-40 between elements, 32-3 Discontinuous Galerkin method, 442, 596 Discrete and continuous problems, 1 Discrete systems, standard methodology for, 2 Discretization: about discretization procedures, 2 constrained variational principles, 84-6 discretization error and convergence rate, 38-9 singularities problems, 38 finite element process, 233-4 of mixed forms, 358-60 partial, 71-4 three-dimensional curves, 297-9 Displacement, virtual, 28 Displacement approach: and bound on strain energy, 35-6 direct minimization, 36 and minimizing total potential energy, 34-6
Displacement discontinuity between elements problems, 32-3, 39-40 and patch tests, 39-40 Displacement formulation, 20 Displacement functions: about displacement functions, 21-4 rectangle with 4 nodes, 22-4 shape functions, 22 triangle with 3 nodes, 22-3 Displacement gradient, 676 Displacements, result reporting, 207-9 Distributed body forces, 26 Domain decomposition methods s e e Subdomain linking by Lagrange multipliers; Subdomain linking by perturbed lagrangian and penalty methods Driven cavity incompressibility example, 41 6-19 Dual mortar method, 433-4 Dummy and free index, 677 Dynamic behaviour of elastic structures with linear damping: about dynamic behaviour, 565--6 consistent damping matrices, 566 consistent mass matrix, 566 d'Alembert principle, 565 element damping matrix, 566 element mass matrix, 566 mass for isoparametric elements example, 568 plane stress and plane strain example, 567-8 s e e a l s o Eigenvalues and time dependent problems; Time dependence Effective stress concept, 646 Effective stresses with pore pressure, 211-14 Effectivity (error recovery) index 0,477 Eigenproblem assessment, 554 Eigenvalues and time dependent problems: about time dependent problems, 570-1 eigenvalues determination, 572-3 eigenvectors, 572 electromagnetic fields example, 575, 577 forced periodic response, 579 free dynamic vibration- real eigenvalues, 571-2 free responses- damped dynamic eigenvalues, 578 free responses - for first-order problems, 576-8 free vibration with singular K matrix, 573 general linear eigenvalue/characteristic value problem, 572 matrix algebra, 672-3 and modal orthogonality, 572 reduction of the eigenvalue system, 574 standard eigenvalue problem, 572-3 vibration of an earth dam example, 575-6 vibration of a simple supported beam example, 574-5
722 Subjectindex Eigenvalues and time dependent problems - c o n t . waves in shallow water example, 575-7 s e e a l s o Transient response by analytical procedures Elastic constitutive equations, 679-80 Elastic continua, stress and strain in: approximations, 20 and principle of virtual work, 20 solution principle, 19, 24-6 'weak form of the problem', 20 Elastic solution by Airy stress function, 378-9 Elastic structures s e e Dynamic behaviour of elastic structures with linear damping Elasticity finite element approximations for small deformations: approximate weak form, 202 constitutive equation, 201 displacement and strain approximation, 202-5 strains for 8-node brick example, 203-4 equation solution process, 201-2 stiffness and load matrices, 205 quadrature for 8-node brick element example, 205-7 virtual work expression, 201 Elasticity (linear) equations: anisotropic materials, 197-9 example, 199-200 boundary conditions: about boundary conditions, 191 on inclined coordinates, 192 normal pressure loading, 193 symmetry and repeatability, 192-3 constitutive relations, 195 displacement function: axisymmetric deformation, 189 plane stress and plain strain, 189 three-dimensions, 188-9 two-dimensions, 189 elasticity matrix of compliances, 195 elasticity matrix of moduli, 195 equilibrium equations, 190-1 initial strain, 200 isotropic materials, 195-6 material symmetry, 198 orthotropic materials, 197, 198 strain matrix, 189-90 thermal effects, 200-1 transformation of stress and strain, 194-5 Elasticity (linear) problems: about direct physical approaches, 19-20, 46-7 about linear elasticity problems, 187-8 accuracy assessment: beam subjected to end shear example, 40-2 circular beam subjected to end shear example, 42-5 convergence criteria, 37-8
convergence rate considerations, 38-9 displacement approach, 34-6 displacement function, 21-4 finite element solution process, 40 formulation of finite element characteristics, 20-31 generalization to whole region, 31-3 nodal forces for boundary traction example, 30-1 result reporting, 207-9 stiffness matrix for 3-node triangle example, 29-30 stress flow around a reinforced opening application, 45-7 two-dimensional: axisymmetric, 188 plane strain, 188 plane stress, 188 Elasticity (linear) problem examples: arch dam in a rigid valley, 216-17 beam subjected to end shear, 209 circular beam subjected to end shear, 209-10 conical water tank, 215-16 dam subject to external and internal water pressure, 210-14 hemispherical dome, 216 pressure vessel problem, 217 rotating disc analysis, 212-15 Elasticity matrix: compliances, 195 moduli, 195 Electrical networks, assembly, 7-9 Electrostatic field problems, 245-51 Element boundary value problem, 483 Element damping matrix, 566 Element mass matrix, 566 Element matrices, evaluation of, 148-50 Element properties determination, 9-10 Element shape functions s e e Shape functions Element subdivision, h-refinement methods, 501 Energy: and equilibrium, 678-9 minimization of an energy functional, 463 Equation systems: assembly, 10 solving, 10 Equilibrated methods, bounded estimators, 494 Equilibrated residual estimators, 483-4 Equilibrating form subdomains, 444-5 Equilibrium: and energy, 678-9 and total potential energy, 35 Equilibrium equations, linear elasticity, 190-1 'Equivalent forces' concept, 19-20 Errors: about errors/error definitions, 456, 494 bounds on quantifies of interest, 490--4 effectivity index 0,477
Subject index 723 error estimators: asymptotic behaviour and robustness, 488-90 explicit residual error estimator, 479-83 implicit residual error estimator, 483-7 recovery-based, 476-8 residual-based, 478-87 from approximation of curved boundaries, 39 from round-off, 39 Herrmann theorem, 462-5 and irregular scalar quantities, 456 local errors, 456 norms of errors, 457-9 optimal sampling points, 459-65 permissible error magnitudes, 500 recovery of gradients and stresses, 465-7 relative energy norm error, 458 RMS error, 500 singularity effects, 457-8 s e e a l s o Adaptive finite element refinement; Discretization error and convergence rate; Recovery by equilibrium of patches (REP); Residual-based error estimators; Superconvergence Euclidean metric tensor, 294 Euler equations, 78-80 and constrained variational principles, 85 Explicit methods, with time discretization, 592 Explicit residual error estimators, 479-83 Extended finite element method (XFEM), 527 External loads, potential energy, 34 FEAPpv (Finite Element Analysis Program- personal version): about FEAPpv, 664 DATRI sub program, 684-8 element library, 665-6 post-processor module, 666-7 pre-processing module: mesh creation, 664-6 solution module, 666 user modules, 667 Fick's law, 230 Field, electrostatic and magnetostatic, problems, 245-51 Finite element approximations s e e Elasticity finite element approximations for small deformations Finite element characteristics: direct formulation, 20-31 s e e a l s o Displacement functions; Nodal forces; Strain in elastic continua; Stress in elastic continua Finite element discretization, 233-4 Finite element mesh generation by mapping, 169-70 Finite element method/concept: displacement approach, 34-6 history of approximate methods, 3 with indicial notation, 680-2
scalar and vector quantities, 54 solution process, 40 s e e a l s o Generalized finite element method/concept Finite volume method/subdomain collocation, 61,547 Flow or diffusion problems, 230 Fluid flow problems, 251-3 Fluid networks, assembly, 7-9 Fluid-structure interaction (Class 1 problem): about fluid behaviour equations, 634-5 acoustic problems, 635 boundary conditions for the fluid, 635-7 free surface, 636 interface with solid, 636 linearized surface wave condition, 636 perfectly matched layers (PML), 637 radiation boundary, 636--7 cavitation effects in fluids, 645-6 discrete coupled system, 638 forced vibrations and transient step-by-step algorithms, 639-44 dam/reservoir interaction, 641 Routh-Hurwitz conditions, 643 stability of the fluid-structure time-stepping scheme, 642-4 free vibrations, 639-40 linearized dynamic equations, 634 special case of incompressible fluids, 644 added mass matrix, 644 standard Galerkin discretization, 638 weak form for coupled systems, 637-8 s e e a l s o Soil-pore fluid interaction (Class II problems) Fluid-structure systems, staggered schemes, 658-9 Forced boundary conditions, 59 with natural variational principles, 81 Forced periodic response, 579 Forming points, Delaunay triangulation, 304 Formulations s e e Irreducible formulations; Mixed formulations Forward elimination, simultaneous equations, 684 Fourier's law, 230 Fracture mechanics, solutions with mapping, 176-7 Frame methods of linking displacement frames: about frame methods, 442-4 hybrid-stress elements, 445 interior and exterior elements, 447 linking on equilibrating form subdomains, 444-5 equilibrium field example, 445 subdomains with standard elements and global functions, 451 Trefftz-type solutions, 445-51 using virtual work, 443-4 Free and dummy index, 677 Free surface flow and irrotational problems, 251-3
724 Subjectindex Function approximation: about function approximation, 527 interpolation domains and shape functions, 530-2 least squares fit scheme, 527-9 weighted least squares fit, 527-9 Functionals, 34 stationary, 34 Galerkin method/procedure/principle: diffuse elements, 547 solution of ordinary differential equations example, 547-9 discontinuous, 442, 596 and finite element discretization, 233-4 least squares (GLS) stabilization method, 94-5, 409-10 and variational principles, 80 s e e a l s o Weighted residual-Galerkin method Galerkin standard discretization, 638 Galerkin time discontinuous approximation, 619-25 Gauss quadrature, 160, 234 Gauss-Legendre quadrature points, 161,463-5 General linear eigenvalue/characteristic value problem, 572 Generalized finite element method/concept: about finite element generalization, 54-6, 95-7, 525 convergence, 74-5 partial discetization, 71-4 virtual work as 'weak' form of equilibrium equations, 69-71 s e e a l s o Constrained variational principles; Lagrange multipliers; Variational principles; Weighted residual-Galerkin method Generalized Newmark (GN) algorithms: GN22 algorithm, 608-9 GNpj algorithm, 606-9 stability, 609-12 Global coordinates, 12 Global derivatives, computation of, 146-7 GLS (Galerkin least squares) stabilization method, 94-5,409-10 GN algorithms s e e Generalized Newmark (GN) algorithms Gradient, with rates of flow, 230 Gradient matrix, 234 Green's theorem (integration by parts in two or three dimensions), 699-700 Gurtin's variational principle, 594 h convergence, 75 h-refinement s e e Adaptive finite element refinement Hamilton's variational principle, 594 Hanging points, h-refinement methods, 501 Heat conduction:
steady-state, equation in two-dimensions example, 55-6 steady-state Galerkin formulation with triangular elements example, 65-8 time problems, 237-8 weak form-forced and boundary conditions example, 59 Heat conduction-convection: steady state, equation in two-dimensions example, 56 steady-state Galerkin formulation in two-dimensions example, 68-9 Heat equation: in first-order form example, 82-3 with heat generation example, 73-4 Heat transfer solution by potential function example, 378 Hellinger-Reissner variation principle, 365 Helrnholz equation, least squares solution example, 93-4 Helmholz problem in two-dimensions example, 82 Helmholz wave equation, 565 Hemispherical dome example, 216 Hermite cubic spline/Hermite polynomials, 268-9 Hermitian interpolation function, 531 Herrmann theorem and optimal sampling points, 462-5 Hierarchic finite element method based on the partition of unity: about hierarchical forms, 549-52 and global functions, 551 and harmonic wave functions, 552 linear elasticity application, 553-7 polynomial hierarchial method, 552-3 quadratic triangular element example, 554-7 and singular functions, 552 solution of forms with linearly dependent equations, 557-8 Hierarchical shape functions: concepts, standard and hierarchical, 104--6 diagonality, 105 global and local finite element approximation, 131-2 improving of conditioning with hierarchical forms, 130-1 one-dimensional (elastic bar) problem, 105-6 one-dimensional hierarchic polynomials, 125-7 polynomial form, 106 triangle and tetrahedron family, 128-30 two- and three-dimensional elements of the brick type, 128 Hu-Washizu variational principle/theorem, 370, 395 Hybrid-stress elements, 445 Identical and similar algorithms, 599 Identity matrix, 232
Subject index 725 Implicit equations, 416 Implicit methods, with time discretization, 592 Implicit residual error estimators, 483-7 equilibrated residual estimator, 483-4 Incompressible elasticity, three-field (u-p-ev form), 393-8 anisotropic materials, 394 B-bar method for nearly incompressible problems, 397-8 enhanced strain triangle example, 394-6 Hu-Washizu-type variational theorem, 395 variational theorem, 394 Incompressible elasticity, two-field (u-p form), 384-93 bubble function, 386-7 locking (instability) prevention, 385,390 patch tests, multiple-element, 386-9 patch tests, single-element, 386-9 simple triangle with bubble- MINI element example, 390-3 driven cavity example, 418 stability (or singularity) of the matrices, 385 Incompressible problems: about incompressible problems, 383, 421-2 and deviatoric stress and strain, 383-4 driven cavity example, 41 6-19 reduced and selective integration and its equivalence to penalized mixed problems, 398-404 slow viscous flow application, 402-3 with weak patch test example, 403-4 simple iterative solution for mixed problems: Uzawa method, 404-7 tension strip with slot example, 419-21 Incompressible problems for some mixed elements failing the incompressibility patch test, 407-21 about the stability conditions, 407-8 characteristic-based split (CBS) procedure, 407 direct pressure stabilization, 410-13 3-node triangular element example, 412-13 driven cavity example, 418 implicit equations, 416 Galerkin least squares method, 409-10 driven cavity example, 417-18 incompressibility by time stepping, 413-16 Stokes flow equation, 413-14 laplacian pressure stabilization, 408-9 deviatoric stresses and strains, 408 Indicial notation: summation convention, 674-5 see also Tensor-indicial notation in the approximation of elasticity problems Infinite domains and elements: about infinite domains and elements, 170-2 Boussinesq problem, 174--6 convergence considerations, 173 electrostatic and magnetostatic problems, 250
mapping function, 172-6 quadratic interpolations, 174-5 Initial strain, and elasticity equations, 200 Integral/'weak' statements, 57-60 Integration see Numerical integration Integration by parts in two or three dimensions (Green's theorem), 699-700 Integration formulae: tetrahedron, 693 triangles, 692 Interpolating functions, 169 see also Shape functions Interpolation domains, 530-2 circular/spherical domains, 531 discontinuous interpolation, 532 Hermitian interpolation function, 531 and weighting functions, 531 Irreducible formulations, 56, 356-7, 359, 360 see also Mixed formulations Irregular scalar quantities, and errors, 456 Irrotational and free surface flow problems, 251-3 Isoparametric concepts, 554 Isoparametric expansions/elements, 145, 151-2 transformer example, 249-50 Isotropic and anisotropic forms for k, 231-2 Isotropic materials, elasticity equations, 195-6 Iterative solution, simultaneous equations, 688-91 Jacobian matrix, mapped elements, 146, 174 Jump discontinuites, 479-83 Kantorovich partial discretization, 72 Kronecker delta function/property, 108-9, 116, 433, 485 Lagrange multipliers: boundary conditions identification example, 87-8 and constrained variational principles, 84-6 constraint enforcement example, 86 identification of, 87-8 see also Subdomain linking by Lagrange multipliers Lagrange polynomials, 110-12 Lam6 constants, 680 Least squares approximations, 92-5 Galerkin least squares, stabilization, 94-5 and the Herrmann theorem, 462-3 interpolation domains and shape functions, 530-2 solution for Helmholz equation example, 93-4 and variational principles, 92 see also Moving least squares approximations/ expansions Least squares fit scheme, 527-8 fit of a linear polynomial example, 528-9 weighted least squares fit scheme, 529-30 Line (one-dimensional) elements, 119-20
726 Subjectindex Linear damping s e e Dynamic behaviour of elastic structures with linear damping Linear differential operators, 24-5 Linear elasticity s e e Elastic continua, stress and strain in; Elasticity (linear) equations; Elasticity (linear) problems Linear and non-linear relationships, 11 Linearization of vectors, 77 Linearized surface wave condition, fluid-structure interaction, 636 Linking subdomains by Lagrange multipliers s e e Subdomain linking by Lagrange multipliers Load matrix for axisymmetric triangular element with 3 nodes example, 237 Local coordinates, 12 Local errors, 456 Locking, constrained variational principles, 91 Lubrication problems, 251 Magnetostatic field problems, 245-51 Mapped elements: about mapping, 138-9 blending functions, 169-70 Boussinesq problem, 174-5 continuity requirements, 143-5 evaluation of element matrices, 148-50 finite element mesh generation, 169-70 fracture mechanics application, 176-7 geometric conformity of elements, 143 global derivatives, computation of, 146-7 infinite domains and elements, 170-6 interpolating functions, 169 isoparametric elements, 145 jacobian matrix, 146 one-to-one mapping, 141 order of convergence, 151-3 parametric curvilinear coordinates, 139-45 parent elements, 141 quadratic distortion, 142-3 shape functions for coordinate transformations, 139-43 singular elements by mapping, 176-7 subparametric elements, 145 surface integrals, 148 transformations, 145-50 uniqueness rules, 142-3 unreasonable element distortion problems, 141-2 variation of the unknown function problems, 143-5 volume integrals, 147 s e e a l s o Degeneration; Numerical integration Mass lumping or diagonalization, 568-70, 704-9 Material symmetry, 198 Matrices/matrix notation: about matrices, 54 and degrees of freedom, 5 evaluation of element matrices, 148-50
stiffness matrix, 4-5 transformation matrix, 12 Matrix algebra: addition, 669-70 definitions, 668-9 arrays, 668-9 columns, 669 eigenvalue problem, 672-3 inversion, 670 partitioning, 672 spectral form of a matrix, 673 subtraction, 669-70 sum of products, 671 symmetric matrices, 671 transpose of a product, 671 transposing, 670 Matrix singularity due to numerical integration, 167-8 Maxwell's equations, 245-9 Mesh enrichment, 504 Mesh generation: about manual, semi-automatic and automatic mesh generation, 264-6 adaptive refinement, 266 background mesh, 267-8 background meshes, 265-6 boundary curve representation, 267-70 geometrical characteristics of meshes, 266-7 Hermite cubic spline, 268 with mapping, 169-70 structured and unstructured meshes, 265 using quadratic isoparametric elements, 169-70 Mesh generation, surface meshes: about surface mesh generation, 286-7, 301-3 CAD applications, 286 composite cubic surface interpolation, 290-1 curve representation, 288 discretization of three-dimensional curves: node generation on the curves, 297-9 place boundary nodes to parametric plane, 299 element generation in parametric plane, 299-300 examples, 300-1 geometrical characteristics, 290-7 geometrical representation, 287-90 higher order surface elements, 301-2 major steps, 287 mesh control function in three dimensions, 290-3 parametric plane parameters, 293-7 spline surface, 290, 292 surface representation, 288-90 Mesh generation, three-dimensional, Delaunay triangulation: about Delaunay triangulation, 303-4, 306-7, 321-3 automatic node generation procedure, 311-12 Delaunay triangulation algorithm, procedure for, 308-11
Subject index 727 element transformation: two elements, 317, 318 three elements, 317, 318 four elements, 318, 319 five or more elements, 318, 320 forming points, 304 global procedure, 307-8 higher order elements, 321 mesh quality enhancement, 316-21 mesh smoothing, 320-1 node addition and elimination, 319-20 numerical examples, 321,322 surface mesh recovery procedure: boundary edge recovery, 312, 313, 314 boundary face recovery, 313, 315, 316, 317 edge swapping, 312 removal of added points, 315-16 Voronoi diagram, 304-6, 308-10 example, 305-6 properties, 304 Voronoi vertices, 309-10 Mesh generation, two dimensional, advancing front method: about the advancing front method, 266, 285-6 active sides and active nodes, 277 boundary node generation: algorithmic procedure, 271-5 procedure verification example, 275-7 consistency index, 274 diagonal swapping, 282-4 distorted elements, 282 element generation steps, 277-80 Euclidean metric tensor, 273-4 generation front, 277-8 geometrical transformation of the mesh, 271 transformation of a triangle example, 271 higher order elements, 283-5 mesh modification, 281-3 mesh quality enhancement for triangles, 280-3 mesh smoothing, 280 example, 281 node elimination, 282 triangular mesh generation, 270-80 Method of weighted residuals s e e Weighted residual-Galerkin method Minimization of an energy functional, 463 Mixed formulations: about mixed and irreducible formulations, 56, 356-7, 379 complementary forms with direct constraint: about directly constrained forms, 375 auxiliary function solutions, 378-9 complementary elastic energy principle, 377 complementary heat transfer problem, 376-7 elastic solution by Airy stress function, 378-9
heat transfer solution by potential function example, 378 continuity requirements, 358-9 count condition satisfying, 362 discretization of, 358-60 Hu-Washizu variational principle, 370 locking, 361-2 patch test, 362-3 physical discontinuity problems, 363 single-element test examples, 362-3 primary and constraint variables, 360 principle of limitation, 359 singular and non-singular matrices, 361 solvability requirement, 360-1 stability of mixed approximation, 360-3 and the variational principle, 357 Mixed formulations in elasticity, three-field: stability condition, 371-2 u - a - e mixed form, 370-1 u-o'-een form- enhanced strain formulation, 372-5 enhanced strain aspects, 374-5 sheafing strain effects, 374 Simo-Rifai quadrilateral example, 373-4, 375 Mixed formulations in elasticity, two-field: about two-field formulations, 363-4 Hellinger-Reissner variation principle, 365 Pian-Sumihara quadrilateral example, 368-70 Pian-Sumihara rectangle example, 367-8 u-a mixed form, 364-5 u-a mixed form stability, 365-70 Modal decomposition analysis, 580-3 Modal orthogonality, 572 Mortar/dual mortar methods, 432-4 for two-dimensional elasticity example, 436 Moving least squares approximations/expansions, 533-8 hierarchical enhancement, 538-40 partition of unity, 537 Shepard interpolation, 539-40 and symmetric functions, 533-5 and weighting functions, 533-5 Multidomain mixed approximations: about multidomain mixed approximations, 429, 451 s e e a l s o Frame methods of linking displacement frames; Subdomain linking by Lagrange multipliers; Subdomain linking by perturbed lagrangian and penalty methods Multigrid procedures, staggered solution processes, 658 Multistep methods/multistep recurrence algorithms s e e Time discretization, multistep recurrence algorithms Natural variational principles principles
see
Variational
728 Subjectindex Neumann boundary value problem, 484 Neumann (natural) boundary condition, 60 Newton-Cotes quadrature, 160 Nitsche method for subdomain linking, 438-40 Dirichlet boundary condition example, 440 Nodal forces equivalent to boundary stresses and forces, 26-31 3-node triangle, 26-7 boundary considerations, 29 external and internal work done, 28 internal force concept abandoned for generalization, 31-3 nodal forces for boundary traction example, 30-1 plane stress problem, 29 stiffness matrix for 3-node triangle example, 29-30 s e e a l s o Whole region generalization Norms of errors, 457-9 Numerical algorithms, 618-19 Numerical integration: computational advantages, 177-8 Gauss quadrature, 160-1 and matrix singularity, 167-8 minimum order for convergence, 165-6 Newton-Cotes quadrature, 160 one dimensional, 160-1 order for no loss of convergence rate, 166-7 rectangular (2D) or brick regions (3D), 162-4 required order, 164-8 triangular or tetrahedral regions, 164 s e e a l s o Mapped elements Oil fields: ground settlement, 652-3 oil recovery, 653 One dimensional elements, line elements, 119-20 One-to-one mapping, mapped elements, 141 Optimal sampling points and the Herrmann theorem, 462-5 Orthotropic materials, elasticity equations, 197, 198 p convergence, 75 p and ph-refinement s e e Adaptive finite element refinement Parametric curvilinear coordinates, 139-45 Parent elements, mapped elements, 141 Partial discretization, 71-4 finite element discretizations, 238 heat equation with heat generation example, 73-4 Kantorovich, 72 transient problems, 237-9 Partition of unity: and shape functions, 105, 537 s e e a l s o Hierarchic finite element method based on the partition of unity Partitioned single-phase systems - implicit-explicit partitions (Class I problems), 653-5
Pascal triangle, 110, 116 Patch recovery, superconvergent (SPR), 467-74, 490 Patch test: about the patch test, 329-30, 347-50 application to an incompatible element, 343-7 application to elasticity elements with 'standard' and 'reduced' quadrature, 337-43 for base solution example, 337-9 higher order test-assessment of order example, 340-3 for quadratic elements: quadrature effects example, 339-40 Babu~ka patch test, 490 base solution, 332 consistency requirement, 331 convergence requirements, 330-1,332, 334-5 curvilinear coordinates, 330 degree of robustness, 331,488-9 and discontinuity of displacement, 39-40 generality of a numerical patch test, 336 higher order patch tests, 336-7 assessment of robustness, 347 mapped curvilinear elements, 333 mixed formulations, 362-3 non-robust elements, 331 single element tests, 335 size of patch, 333 stability condition, 331 tests A and B, simple tests, 332-4 test C, generalized test, 334-5 weak patch test satisfaction, 333-4 PCG (preconditioned conjugate gradient), with iterative solutions, 690-1 Penalized mixed problems, and reduced and selective integration, 398-404 Penalty functions, constrained variational principles, 88-9 Penalty methods s e e Subdomain linking by perturbed lagrangian and penalty methods Perfectly matched layers (PML), 637 Periodic response, forced, 579 Permissible error magnitudes, 500 Perturbed lagrangian functional, constrained variational principles, 89-91 Perturbed lagrangian method s e e Subdomain linking by perturbed lagrangian and penalty methods Plane stress problem, 29 Plane triangular element with 3 nodes example, 235-6 PML (perfectly matched layers), 637 Point collocation: about point collocation, 61,540-2, 546-7 cross criterion method, 541 Galerkin weighting and finite volume methods, 546-9 with hierarchical interpolations, 543-6
Subject index 729 solution of ordinary differential equations example, 542-6 subdomain collocation/finite element method, 61, 547 Voronoi neighbour criterion method, 541-2 s e e a l s o Least squares approximations; Moving least squares approximations/expansions Polynomial hierarchial method based on the partition of unity, 552-3 Porous material, pore pressure effects, 211-14 Potential energy: external loads, 34 total, 34 Preconditioned conjugate gradient (PCG), with iterative solutions, 690-1 Prescribed functions of space coordinates, 564 Pressure vessel problem example, 217 Primary and constraint variables, 360 Principle of limitation, mixed formulations, 359 Principle of virtual work, 20 Prismatic problems, 72 Quadratic distortion, mapped elements, 142-3 Quadratic elements, serendipity family, 113 Quadratic interpolations, 174-5 Quadratic isoparametric elements, 169-70, 249 Quadratic triangle, triangular elements family, 119 Quasi-harmonic equations: about quasi-harmonic equations, 229 anisotropic and isotropic forms for k, 231-2 axisymmetric problem, 235-7 governing equations, 230-1 with time differential, 563-5 and torsion of prismatic bars, 240-2 two-dimensional plane, 235-6 weak form and variational principal, 233 r-refinement s e e Adaptive finite element refinement Rayleigh-Ritz process/procedure, 35 Recovery, definition, 456 Recovery based error estimators, 476-8 s e e a l s o Errors Recovery by equilibrium of patches (REP), 474--6, 490 Rectangle with 4 nodes, displacement function, 22-4 Rectangular (square) bar, transient heat conduction example, 242-4, 246 Rectangular (three-dimensional) prisms: Lagrange family, 120-1 serendipity family, 121-2 Rectangular (two-dimensional) elements: concepts, 107-9 Lagrange family, 110-12 serendipity family, 112-16 s e e a l s o Standard shape functions Recurrence algorithm, 603
Recurrence relations, 589 Reduced and selective integration and its equivalence to penalized mixed problems, 398-404 Relative energy norm error, 458 REP (Recovery by equilibrium of patches), 474-6, 490 Reproducing kernel (RPK) method, 558 Residual-based error estimators: about residual error estimators, 478 explicit residual error estimators, 479 deriving, example, 479-83 implicit residual error estimators, 483-7 equilibrated residual estimator, 483-4 jump discontinuites, 479-83 recovery processes, 480-1 Result reporting, displacements, strains and stresses, 207-9 Ritz process, 35 RMS error, 500 Robustness index, 488-90 Rotating disc analysis example, 212-15 Rotor blade, transient heat conduction example, 242, 246 Round-off errors, 39 Routh-Hurwitz conditions, fluid-structure interaction, 643 Routh-Hurwitz stability requirements for SS22/SS21 algorithms, 612 Scalar and vector quantities, 54 Seepage: anisotropic, 244-5, 247 fluid flow, 252 soil-pore fluid interaction equation, 647 transient seepage, 565 Self-adjointness/symmetry properties, variational principles, 81,357 Semi-discretization s e e Time dependence Serendipity family, cubic elements: conical water tank example, 215-16 rotating disc analysis, 212-15 Serendipity family, rectangular elements, 112-16, 121-2 corner shape functions, 115 cubic elements, 113-14 mid-side functions, 114-15 quadratic elements, 113 shape function generation, 114 Shape functions: about shape functions, 22, 103-4 and convergence criteria, 103 for coordinate transformations, 139-43 diagonality, 105 elimination of internal parameters before assembly, 132-3 and partition of unity, 105, 537
730
Subjectindex Shape functions - cont. standard and hierarchical concepts, 104-6 substructuring, 133-4 tetrahedral elements, 124-5 and the triangular element family, 118-19 s e e a l s o Displacement functions; Hierarchical shape functions; Standard shape functions Shepard interpolation, moving least squares expansions, 539-40 Similar and identical algorithms, 599 Simo and Rifai enhanced strain formulation, 373-4, 375 Simultaneous discretization, 590 Simultaneous linear equations: back substitution, 684 DATRI (FEAPpv sub program), 684-8 direct methods/solutions, 683-8 forward elimination, 684 iterative solution, 688-91 preconditioned conjugate gradient (PCG), 690-1 resolution process, 687 triangular decomposition, 684, 685 Single-step (SS) algorithms: SS 11 algorithm: example, 605 stability, 612-13 SS22 algorithm: example, 605 stability, 612-13 SS32/SS31 algorithms, stability of, 613-15 SS42/SS41 algorithms, stability of, 614 stability, 609-15 weighted residual finite element form SSpj, 601-6 s e e a l s o Time discretization, single-step algorithms, first and second order equations Singular elements by mapping, 176-7 Singularities, effects on errors, 458-9 Singularities problems, and convergence rate, 38 Smooth particle hydrodynamics (SPH) method, 558 Soil consolidation equations, 565 Soil-pore fluid interaction (Class II problems): about soil-pore fluid interaction, 645-8 biomechanics problem of bone-fluid interaction, 652 coupled equations format, 648 effective stress concept, 646 oil fields, ground settlement, 652-3 robustness requirements, 650 soil liquefaction examples, 650-2 special cases, 649-50 transient step-by-step algorithm, 648-9 Solutions exact at nodes, 701-3 Spectral radius, 619, 620 SPR (superconvergent patch recovery), 467-74, 490 SS algorithms see Single-step (SS) algorithms
Stability/stabilization: algorithm stability, 609-15 generalized Newmark (GN) algorithms, 609-12 incompressible problems, direct pressure stabilization, 410-13 laplacian pressure stabilization, 408-9 least squares (GLS) stabilization method, 94-5, 409-10 patch test stability condition, 331 staggered schemes, 658-9 s e e a l s o Incompressible problems for some mixed elements failing the incompressibility patch test; Single-step (SS) algorithms; Time discretization Staggered solution processes: about staggered solutions, 655 alternating direction implicit (ADI) scheme, 658 in fluid-structure systems and stabilization processes, 658-9 multigrid procedures, 658 in single phase systems, 655-8 Standard discrete systems: about, 1-3, 55 definition and unified treatment, 2, 10-11 linear and non-linear relationships, 11 system equations, 11 system parameters, 10-11 transformation of coordinates, 11-12 s e e a l s o Assembly and analysis of structures Standard shape functions: Kronecker delta, 108-9 standard and hierarchical concepts, 104-6 one-dimensional (line)elements, 119-20 two-dimensional elements, 107-19 completeness of polynomials, 109-10 Lagrange family, 110-12 rectangular element concepts, 107-9 rectangular element families, 110-16 serendipity family, 112-16 triangular element family, 116-19 three-dimensional elements, 120-5 rectangular prisms, Lagrange family, 120-1 rectangular prisms, serendipity family, 121-2 tetrahedral elements, 122-5 Stepped pad bearings, 251 Stiffness, direct stiffness process, 2-3 Stiffness matrix, 4-5 for axisymmetric triangular element with 3 nodes example, 236-7 Stokes flow equation, 413-14 Strain in elastic continua, 19, 24-5 and relationship with stress, 25 s e e a l s o elastic continua, stress and strain in; Elasticity (linear) problems Strain energy: strain energy bound, 36
Subject index 731 of a system, 34 Strain matrix equations, 189-90 Strain rate (virtual strain), 71 Strains, result reporting, 207-9 Stress in elastic continua, 19, 25-6 initial residual stresses, 25 and relationship with strain, 25 s e e a l s o elastic continua, stress and strain in; Elasticity (linear) problems Stress function, and tension of prismatic bars, 240-2 Stresses, result reporting, 207-9 Structural element and system, 3-5 Structure assembly, general process, 5-6 Structured and unstructured meshes, 265 Subdomain collocation/finite volume method, 61,547 Subdomain linking by Lagrange multipliers, 430-6 for elasticity equations, 434-6 mortar method for two-dimensional elasticity example, 431,436 for quasi-harmonic equations, 430-4 mortar/dual mortar methods, 432-4 treatment for forced boundary conditions, 432 Subdomain linking by perturbed lagrangian and penalty methods: about, 436-8 discontinuous Galerkin method, 442 multiple subdomain problems, 440-2 Nitsche method, 438-41 two domain problem example, 442 Subparametric elements, 145 Substructuring, 133-4 Superconvergence, 208, 459-65 about superconvergence, 459 Herrmann theorem and optimal sampling points, 462-5 one-dimensional example, 460-2 superconvergent patch recovery (SPR), 467-74, 490 for displacement and stresses, 474 SPR stress projection for rectangular element patch example, 470-4 Surface integrals, 148 Surface mesh generation s e e Mesh generation, surface meshes Symmetric operators, 357 Symmetry properties/self-adjointness, variational principles, 81 Symmetry and repeatability, with time dependence, 583 System equations, 11 System parameters, 10-11 Taylor series collocation, 593-4 Tension strip with slot incompressibility example, 419-21
Tensor-indicial notation in the approximation of elasticity problems: about the tensor-indicial notation, 674 coordinate transformation, 677-8 free and dummy index, 677 derivatives, 676--7 displacement gradient, 676 elastic constitutive equations, 679-80 equilibrium and energy, 678-9 finite element displacement approximation, 680-2 stiffness coefficient/tensor, 681 first and second rank cartesian tensors, 678 indicial and matrix notation relation, 682 indicial notation: summation convention, 674-5 indicial form, 675 intrinsic notation, 675 Lam6 constants, 680 tensor products, 676 tensorial relations, 676-7 Tetrahedral (three-dimensional) elements, 122-5 cubic shape functions, 124-5 quadratic shape functions, 124 volume coordinates, 122-4 Tetrahedron, integration formulae, 693 Thermal effects, elasticity equations, 200-1 Three-dimensional elements: about three-dimensional elements, 120, 125 rectangular prisms, Lagrange family, 120-1 rectangular prisms, serendipity family, 121-2 tetrahedral elements, 122-5 Time dependence: about time dependence, 563 and boundary conditions, 565 damped wave equation, 565 direct formulation of with spatial finite element subdivision, 563-70 Helmholz wave equation, 565 mass lumping or diagonalization, 568-70 and partial discretization, 237-9 prescribed functions of space coordinates, 564 quasi-harmonic equation with time differential, 563-5 soil consolidation equations, 565 symmetry and repeatability, 583 transient heat conduction equation, 565 s e e a l s o Dynamic behaviour of elastic structures with linear damping; Eigenvalues and time dependent problems; Transient response by analytical procedures Time discontinuous Galerkin approximation, 619-24 solution of a scalar equation example, 623-5 Time discretization: about discrete approximation in time, 589-90 general performance of numerical algorithms, 618-19
732 Subjectindex Time discretization, multistep recurrence algorithms: about multistep recurrence algorithms, 615 approximation procedures, 615-18 central difference approximation, 618 and recurrence relations, 589 three-point interpolation example, 617-18 two-point interpolation example, 617 Time discretization, single-step algorithms, first order equations, 590-600 amplification matrix, 596 conditionally stable/unconditionally stable algorithms, 592 consistency and approximation error, 594-6 Crank-Nicholson scheme, 594, 596 different weight functions problems, 592 discontinuous Galerkin process, 596 explicit/implicit solutions, 592 Gurtin's variational principle, 594 Hamilton's variational principle, 594 identical and similar algorithms, 599 initial value problems, 591 load discontinuities, 600 optimal value of 0,599 smoothing usage, 600, 601,602 stability, 596-9, 609-15 conditional/unconditional stability, 597-8 Taylor series collocation, 593-4 starting/non-starting schemes, 594 weighted residual finite element approach, 590-3 Time discretization, single-step algorithms, first and second order equations: about general single-step algorithms, 600--1 GN22 Newmark algorithm, 608-9 GNpj truncated Taylor series collocation algorithm, 606-9 mean predicted values, 60 predictor-corrector iteration, 609 recurrence algorithm, 603 Routh-Hurwitz stability requirements, 612 SS 11 algorithm: example, 605 stability, 612-13 SS22 algorithm: example, 605-6 stability, 612-13 SS32/SS31 algorithms, stability, 613-15 SS42/SS41 algorithms, stability, 614 stability, conditional/unconditional, 605 stability of general algorithms, 609-15 weighted residual finite element form SSpj, 601-6 Time-stepping procedures, 641 Torsion of prismatic bars, 240-2 hollow bimetallic shaft example, 242 rectangular shaft example, 242 stress function approach, 241 warping function approach, 240-1
Total potential energy: and equilibrium, 35 minimization by displacement approach, 34-6 Tractions, and virtual work, 70 Transformation of coordinates, 11-12 and approximations, 12 and constrained parameters, 12 contravariant sets, 12 stress and strain for linear equations, 194-5 Transformation matrix, 12 Transformations, 145-50 Transient heat conduction: rectangular bar example, 242-4 rotor blade example, 244, 246 Transient response by analytical procedures: about transient response, 579 damping and participation of modes, 583 frequency response procedures, 579-80 modal decomposition analysis, 580-3 s e e a l s o Time dependence Trefftz-type solutions for boundary linking, 445-51 Triangle with 3 nodes, displacement function, 22-3 Triangles, integration formulae, 692 Triangular decomposition, simultaneous equations, 684, 685 Triangular (two-dimensional) element family, 116-19 area coordinates, 117-18 cubic triangle, 119 quadratic triangle, 119 shape functions, 118-19 Truncated Taylor series expansion algorithm GNpj, 606-9 Two-dimensional elements s e e Rectangular (two-dimensional) elements; Triangular (two-dimensional) element family Two-dimensional plane problem, 235-7 load matrix for axisymmetric triangular element with 3 nodes example, 237 plane triangular element with 3 nodes example, 235-6 stiffness matrix for axisymmetric triangular element with 3 nodes example, 236-7 u-a-e mixed forms s e e u n d e r Mixed formulations Ultraconvergence, 469 Uzawa method, iterative solution process for mixed problems, 404-7 Variational principles: about variational principles, 76-8 contrived variational principles, 77 Euler equations, 78-80 forced boundary condition equations, 81 and the Galerkin method/process, 80 heat equation in first-order form example, 82-3 Helmholz problem in two-dimensions example, 82
Subject index 733 least squares approximations, 92-5 maximum, minimum, or saddle point?, 83-4 natural variational principles, 78-80, 81-3 self-adjointness/symmetry properties, 81,357 see also Constrained variational principles; Lagrange multipliers 9Variational theorem, 394 Vector algebra: about vector algebra, 694 addition, 694-5 direction cosines, 696 elements of area and volume, 697-8 length of a vector, 695-6 scalar products, 695 subtraction, 694-5 vector or cross product, 696-7 Vector linearization, 77 Vector potential, 245 Vibration: of an earth dam example, 575-6 free vibration with singular K matrix, 573 of a simple supported beam example, 574-5 also see u n d e r Fluid-structure interaction (Class 1 problem); Eigenvalues and time dependent problems Virtual displacement, 28 Virtual strain (strain rate), 71 Virtual work: principle, 20, 34 and tractions, 70 as 'weak form' of equilibrium equations, 69-71 Viscous flow problems, 251-3 Volume integrals, 147 Voronoi diagram, 304-6, 308-10 see also Mesh generation, three-dimensional, Delaunay triangulation Voronoi neighbour criterion point collocation, 541-2
Weak form: coupled systems, 637-8 integral/'weak' statements, 57-60 quasi-harmonic equations, 233 small elastic deformations, 202 and virtual work, 69-71 'weak form of the problem', 20 'Weak'/integral statements, 57-60 Weighted least squares approximation, 463 Weighted least squares fit scheme, 529-30 Weighted residual-Galerkin method: about the weighted residual method, 55, 60-2 approximation to integral formulations, 60-9 convergence, 74-5 Galerkin formulation with triangular elements example, 65-8 and integral/'weak' statements, 57-60 one-dimensional equation of heat conduction example, 62-5 and partial discretization, 72 partial discretization, 71-4 and point collocation, 61 residuals, 61 restrictions needed, 58 steady-state heat conduction in two-dimensions example, 65-8 steady-state heat conduction-convection in two-dimensions example, 68-9 and subdomain collocation, 61 virtual work as the 'weak form' of equilibrium, 69-71 weak form of the heat conduction equation example, 59-60 Weighting function choice, 701 Whole region generalization, 31-3 Work done principle/concept, 28 virtual work, 34
Warping function, and tension of prismatic bars, 240-2
XFEM (extended finite element method), 527
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Plate 2 Analysis of an arch dam in China
An early three dimensional analysis (1970). Analysis by OCZ and Cedric Taylor, Department of Civil Engineering, University of Wales Swansea.
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The Finite Element Method for Fluid Dynamics Sixth edition
Professor O.C. Zienkiewicz, CBE, FRS, FREng is Professor Emeritus at the Civil and Computational Engineering Centre, University of Wales Swansea and previously Director of the Institute for Numerical Methods in Engineering at the University of Wales Swansea, UK. He holds the UNESCO Chair of Numerical Methods in Engineering at the Technical University of Catalunya, Barcelona, Spain. He was the head of the Civil Engineering Department at the University of Wales Swansea between 1961 and 1989. He established that department as one of the primary centres of finite element research. In 1968 he became the Founder Editor of the International Journal for Numerical Methods in Engineering which still remains today the major journal in this field. The recipient of 27 honorary degrees and many medals, Professor Zienkiewicz is also a member of five academies - an honour he has received for his many contributions to the fundamental developments of the finite element method. In 1978, he became a Fellow of the Royal Society and the Royal Academy of Engineering. This was followed by his election as a foreign member to the US Academy of Engineering (1981), the Polish Academy of Science (1985), the Chinese Academy of Sciences (1998), and the National Academy of Science, Italy (Academia dei Lincei) (1999). He published the first edition of this book in 1967 and it remained the only book on the subject until 1971. Professor R.L. Taylor has more than 40 years' experience in the modelling and simulation of structures and solid continua including two years in industry. He is Professor in the Graduate School and the Emeritus T.Y. and Margaret Lin Professor of Engineering at the University of California at Berkeley. In 1991 he was elected to membership in the US National Academy of Engineering in recognition of his educational and research contributions to the field of computational mechanics. Professor Taylor is a Fellow of the US Association of Computational Mechanics - USACM (1996) and a Fellow of the International Association of Computational Mechanics- IACM (1998). He has received numerous awards including the Berkeley Citation, the highest honour awarded by the University of California at Berkeley, the USACM John von Neumann Medal, the IACM Gauss-Newton Congress Medal and a Dr.-Ingenieur ehrenhalber awarded by the Technical University of Hannover, Germany. Professor Taylor has written several computer programs for finite element analysis of structural and non-structural systems, one of which, FEAP, is used world-wide in education and research environments. A personal version, FEAPpv, available from the publisher's website, is incorporated into the book. Dr P. Nithiarasu, Senior Lecturer at the School of Engineering, University of Wales Swansea, has over ten years' experience in finite element based computational fluid dynamics research. He moved to Swansea in 1996 after completing his PhD research at IIT Madras. He was awarded the Zienkiewicz silver medal and prize of the Institution of Civil Engineers, UK in 2002. In 2004 he was selected to receive the European Community on Computational Methods in Applied Sciences (ECCOMAS) award for young scientists in computational engineering sciences. Dr Nithiarasu is the author of several articles in the area of fluid dynamics, porous medium flows and the finite element method.
The Finite Element Method for Fluid Dynamics Sixth edition O.C. Zienkiewicz, CBE, FRS
Professor Emeritus, Civil and Computational Engineering Centre University of Wales Swansea UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in Engineering, Barcelona
R.L. Taylor
Professor in the Graduate School Department of Civil and Environmental Engineering University of California at Berkeley Berkeley, California
P. Nithiarasu
Civil and Computational Engineering Centre School of Engineering University of Wales Swansea
ELSEVIER B~RWORTH HEINEMANN AMSTERDAM
9 BOSTON
PARIS 9 SAN DIEGO
9 HEIDELBERG
9 SAN FRANCISCO
9 LONDON
9 NEW YORK
9 SINGAPORE
9 SYDNEY
9 OXFORD 9 TOKYO
Elsevier Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 30 Corporate Drive, Burlington, MA 01803 First published in 1967 by McGraw-Hill Fifth edition published by Butterworth-Heinemann 2000 Reprinted 2002 Sixth edition 2005 Reprinted 2006 Copyright 9 2000, 2005, O.C. Zienkiewicz, R.L. Taylor and P. Nithiarasu. All rights reserved The right of O.C. Zienkiewicz, R.L. Taylor and P. Nithiarasu to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W 1T 4LP. Applications for the copyright holder's written permission to reproduce any part of this publication should be addressed to the publisher Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail:
[email protected]. You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting 'Customer Support' and then 'Obtaining Permissions' British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress
ISBN 0 7506 6322 7 Published with the cooperation of CIMNE, the International Centre for Numerical Methods in Engineering, Barcelona, Spain (www.cimne.upc.es)
For information on all Elsevier Butterworth-Heinemann publications visit our website at books.elsevier.com
Printed and bound in Great Britain by MPG Books Ltd., Bodmin, Cornwall
Dedication This book is dedicated to our wives Helen, Mary Lou and Sujatha and our families for their support and patience during the preparation of this book, and also to all of our students and colleagues who over the years have contributed to our knowledge of the finite element method. In particular we would like to mention Professor Eugenio Ofiate and his group at CIMNE for their help, encouragement and support during the preparation process.
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Contents
P refa c e Acknowledgements
1
2
3
xi xiii
Introduction to the equations of fluid dynamics and the finite element approximation 1.1 General remarks and classification of fluid dynamics problems discussed in this book 1.2 The governing equations of fluid dynamics 1.3 Inviscid, incompressible flow 1.4 Incompressible (or nearly incompressible) flows 1.5 Numerical solutions: weak forms, weighted residual and finite element approximation 1.6 Concluding remarks References
14 26 27
Convection dominated problems- finite element approximations to the convection-diffusion-reaction equation 2.1 Introduction 2.2 The steady-state problem in one dimension 2.3 The steady-state problem in two (or three) dimensions 2.4 Steady state - concluding remarks 2.5 Transients - introductory remarks 2.6 Characteristic-based methods 2.7 Taylor-Galerkin procedures for scalar variables 2.8 Steady-state condition 2.9 Non-linear waves and shocks 2.10 Treatment of pure convection 2.11 Boundary conditions for convection-diffusion 2.12 Summary and concluding remarks References
28 28 31 45 49 50 53 65 66 66 70 72 73 74
The characteristic-based split (CBS) algorithm. A general procedure for compressible and incompressible flow 3.1 Introduction 3.2 Non-dimensional form of the governing equations 3.3 Characteristic-based split (CBS) algorithm
79 79 81 82
1
4 11 13
viii
Contents
3.4 3.5 3.6 3.7 3.8 3.9 3.10
Explicit, semi-implicit and nearly implicit forms Artificial compressibility and dual time stepping 'Circumvention' of the Babu~ka-Brezzi (BB)restrictions A single-step version Boundary conditions The performance of two-step and one-step algorithms on an inviscid problem Concluding remarks References
92 95 97 98 100 103 104 105
4
Incompressible Newtonian laminar flows 4.1 Introduction and the basic equations 4.2 Use of the CBS algorithm for incompressible flows 4.3 Adaptive mesh refinement 4.4 Adaptive mesh generation for transient problems 4.5 Slow flows - mixed and penalty formulations 4.6 Concluding remarks References
110 110 112 123 131 131 136 136
5
Incompressible non-Newtonian flows 5.1 Introduction 5.2 Non-Newtonian flows - metal and polymer forming 5.3 Viscoelastic flows 5.4 Direct displacement approach to transient metal forming 5.5 Concluding remarks References
141 141 141 154 163 165 166
Free surface and buoyancy driven flows 6.1 Introduction 6.2 Free surface flows 6.3 Buoyancy driven flows 6.4 Concluding remarks References
170 170 170 189 191 193
Compressible high-speed gas flow 7.1 Introduction 7.2 The governing equations 7.3 Boundary conditions - subsonic and supersonic flow 7.4 Numerical approximations and the CBS algorithm 7.5 Shock capture 7.6 Variable smoothing 7.7 Some preliminary examples for the Euler equation 7.8 Adaptive refinement and shock capture in Euler problems 7.9 Three-dimensional inviscid examples in steady state 7.10 Transient two- and three-dimensional problems 7.11 Viscous problems in two dimensions 7.12 Three-dimensional viscous problems
197 197 198 199 202 203 205 206 212 217 226 227 240
Contents
7.13 7.14
Boundary layer-inviscid Euler solution coupling Concluding remarks References
241 242 242
8
Turbulent flows 8.1 Introduction 8.2 Treatment of incompressible turbulent flows Treatment of compressible flows 8.3 8.4 Large eddy simulation 8.5 Detached Eddy Simulation (DES) 8.6 Direct Numerical Simulation (DNS) 8.7 Concluding remarks References
248 248 251 264 267 270 270 271 271
9
Generalized flow through porous media Introduction 9.1 9.2 A generalized porous medium flow approach 9.3 Discretization procedure 9.4 Non-isothermal flows 9.5 Forced convection 9.6 Natural convection Concluding remarks 9.7 References
274 274 275 279 282 282 284 288 289
10 Shallow water problems 10.1 Introduction 10.2 The basis of the shallow water equations 10.3 Numerical approximation 10.4 Examples of application 10.5 Drying areas 10.6 Shallow water transport 10.7 Concluding remarks References
292 292 293 297 298 310 311 313 314
11 Long and medium waves 11.1 Introduction and equations 11.2 Waves in closed domains - finite element models 11.3 Difficulties in modelling surface waves 11.4 Bed friction and other effects 11.5 The short-wave problem 11.6 Waves in unbounded domains (exterior surface wave problems) 11.7 Unbounded problems 11.8 Local Non-Reflecting Boundary Conditions (NRBCs) 11.9 Infinite elements 11.10 Mapped periodic (unconjugated) infinite elements 11.11 Ellipsoidal type infinite elements of Burnett and Holford 11.12 Wave envelope (or conjugated) infinite elements 11.13 Accuracy of infinite elements
317 317 318 320 320 320 321 324 324 327 327 328 330 332
ix
x
Contents
11.14 11.15 11.16 11.17 11.18 11.19
Trefftz type infinite elements Convection and wave refraction Transient problems Linking to exterior solutions (or DtN mapping) Three-dimensional effects in surface waves Concluding remarks References
332 333 335 336 338 344 344
12 Shortwaves 12.1 Introduction 12.2 Background 12.3 Errors in wave modelling 12.4 Recent developments in short wave modelling 12.5 Transient solution of electromagnetic scattering problems 12.6 Finite elements incorporating wave shapes 12.7 Refraction 12.8 Spectral finite elements for waves 12.9 Discontinuous Galerkin finite elements (DGFE) 12.10 Concluding remarks References
349 349 349 351 351 352 352 364 372 374 378 378
13 Computer implementation of the CBS algorithm 13.1 Introduction 13.2 The data input module 13.3 Solution module 13.4 Output module References
382 382 383 384 387 387
Non-conservative form of Navier-Stokes equations Self-adjoint differential equations Postprocessing Integration formulae
389 391 392 395
Appendix E
Convection-diffusion equations: vector-valued variables
397
Appendix F
Edge-based finite element formulation
405
Appendix G
Multigrid method
407
Appendix H
Boundary layer-inviscid flow coupling
409
Appendix I
Mass-weighted averaged turbulence transport equations
413
Appendix Appendix Appendix Appendix
A B C D
Author index
417
Subject index
427
Preface
The major part of this book has been derived by updating the third volume of the fifth edition. However, it now contains three new chapters and also major improvements in the existing ones. Its objective is to separate the fluid dynamics formulations and applications from those of solid mechanics and thus to reach perhaps a different interest group. It is our intention that the present text could be used by investigators familiar with the finite element method in general terms and introduce them to the subject of fluid dynamics. It can thus in many ways stand alone. Although the finite element discretization is briefly covered here, many of the general finite element procedures may not be familiar to a reader introduced to the finite element method through different texts and therefore we advise that this volume be used in conjunction with the text on 'The Finite Element Method: Its Basis and Fundamentals' by Zienkiewicz, Taylor and Zhu to which we make frequent reference. In fluid dynamics, several difficulties arise. The first is that of dealing with incompressible or almost incompressible situations. These as we already know present special difficulties in formulation even in solids. The second difficulty is introduced by the convection which requires rather specialized treatment and stabilization. Here, particularly in the field of compressible, high speed, gas flow many alternative finite element approaches are possible and often different algorithms for different ranges of flow have been suggested. Although slow creeping flows may well be dealt with by procedures almost identical to those of solid mechanics, the high speed range of supersonic and hypersonic kind will require a very particular treatment. In this text we shall use the so-called Characteristic-Based Split (CBS) introduced a few years ago by the authors. It turns out that this algorithm is applicable to all ranges of flow and indeed gives results which are at least equal to those of specialized methods. We organized the text into 13 individual chapters. The first chapter introduces the topic of fluid dynamics and summarizes all relevant partial differential equations together with appropriate constitutive relations. Chapter 1 also provides a brief summary of the finite element formulation. In Chapter 2 we discuss convection stabilization procedures for convection-diffusion-reaction equations. Here, we make reference to methods available for steady and transient state equations and also one and multidimensional equations. We also discuss the similarity between various stabilization procedures. From Chapter 3 onwards the discussion is centred around the numerical
xii
Preface
solution of fluid dynamic equations. In Chapter 3, the CBS scheme is introduced and discussed in detail in its various forms. Its simplicity and universality makes it highly desirable for the study of incompressible and compressible flows and in the later chapters we shall indicate its widely applicable use. Though not all problems are necessarily solved using this method in this book, as work of several decades are reported here, the reader shall find the CBS method in general at least as accurate as other methods and that its performance is very good. For this reason we do not describe any other alternatives to make the reader's life simple. The topic of incompressible fluid dynamics is covered in Chapters 4, 5 and 6. Chapter 4 discusses the general Newtonian incompressible flows without reference to any special problems. This chapter could be used as a validating part of any fluid dynamics code development for incompressible flows. Chapter 5 discusses the non-Newtonian flows in general and metal forming and visco-elastic flows in particular. In Chapter 6 we discuss the special topics of gravity assisted incompressible flows which include treatment of free surfaces and buoyancy driven flows. Chapter 7 is devoted to compressible gas flows. Here, we discuss several special requirements for solving Navier-Stokes equations including phenomena such as shock capturing and adaptivity. Chapters 8 and 9 are new additions to the book. In Chapter 8 we discuss various basic turbulence modelling options available for both compressible and incompressible flows and in Chapter 9 we provide a brief description of flow through porous media. Chapter 10 discusses the shallow water flow and here application of the CBS scheme to a different incompressible flow approximation is considered. Although the flow is incompressible the approximations and variables involved produce a set of differential equations similar to those of compressible flows. Thus, the use of methods already derived for the solution of compressible flow is obvious for dealing with shallow water problems. Chapters 11 and 12 provide a detailed overview on the numerical treatment of long and short waves. Chapter 12 is a new chapter and both these chapters on waves are contributed by Professor Peter Bettess, University of Durham. The last chapter of this book is a brief outline on computer implementation. Further details, including source codes, are available from the author's personal home pages www.nithiarasu.co.uk and www.elsevier.com. We hope that the book will be useful in introducing the reader to the complex subject of computational fluid dynamics (CFD) and its many facets. Further, we hope it will also be of use to the experienced practitioner of CFD who may find the new presentation of interest to practical application.
Acknowledgements
The authors would like to thank Professor Peter Bettess for largely contributing the chapters on waves (Chapters 11 and 12), in which he has made so many achievements, and Dr Pablo Ortiz who with the main author was first to apply the CBS algorithm to shallow water equations and Chapter 10 of this text is partly contributed by him. Several other colleagues contributed to this text either directly or indirectly. Professors K. Morgan, N.P. Weatherill and O. Hassan, all from the University of Wales Swansea, Professors E. On~te and R. Codina, both from CIMNE, Barcelona, Professor J. Peraire from MIT and Professor R. Lrhner from George Mason University, USA, are a few to name. The third author thanks Professor P.G. Tucker, University of Wales, Swansea, and Dr S. Vengadesan, liT, Madras, for their constructive comments on the chapter on turbulence. The third author also thanks his graduate students Ray Hickey and Chun-Bin Liu for their assistance.
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..........................
................................
Introduction to the equations of fluid dynamics and the finite
element approximation
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The problems of solid and fluid behaviour are in many respects similar. In both media stresses occur and in both the material is displaced. There is, however, one major difference. Fluids cannot support any deviatoric stresses when at rest. Thus only a pressure or a mean compressive stress can be carried. As we know, in solids deviatoric stresses can exist and a solid material can support general forms of structural forces. In addition to pressure, deviatoric stresses can develop when the fluid is in motion and such motion of the fluid will always be of primary interest in fluid dynamics. We shall therefore concentrate on problems in which displacement is continuously changing and in which velocity is the main characteristic of the flow. The deviatoric stresses which can now occur will be characterized by a quantity that has great resemblance to the shear modulus of solid mechanics and which is known as dynamic viscosity (molecular viscosity). Up to this point the equations governing fluid flow and solid mechanics appear to be similar with the velocity vector u replacing the displacement which often uses the same symbol. However, there is one further difference, even when the flow has a constant velocity (steady state), convective acceleration effects add terms which make the fluid dynamics equations non-self-adjoint. Therefore, in most cases, unless the velocities are very small so that the convective acceleration is negligible, the treatment has to be somewhat different from that of solid mechanics. The reader should note that for self-adjoint forms, approximating the equations by the Galerkin method gives the minimum error in the energy norm and thus such approximations are in a sense optimal. In general, this is no longer true in fluid mechanics, though for slow flows (creeping flows) where the convective acceleration terms are negligible the situation is somewhat similar. With a fluid which is in motion, conservation of mass is always necessary and, unless the fluid is highly compressible, we require that the divergence of the velocity vector be zero. Similar problems are encountered in the context of incompressible elasticity
2 Introductionto the equations of fluid dynamics and the finite element approximation and the incompressibility constraint can introduce difficulties in the formulation (viz. reference 1). In fluid mechanics the same difficulty again arises and all fluid mechanics approximations have to be such that, even if compressibility is possible, the limit of incompressibility can be modelled. This precludes the use of many elements which are otherwise acceptable. In this book we shall introduce the reader to finite element treatment of the equations of motion for various problems of fluid mechanics. Much of the activity in fluid mechanics has, however, pursued a finite difference formulation and more recently a derivative of this known as the finite volume technique. Competition between finite element methods and techniques of finite differences have appeared and led to a much slower adoption of the finite element process in fluid dynamics than in structures. The reasons for this are perhaps simple. In solid mechanics or structural problems, the treatment of continua often arises in combination with other structural forms, e.g. trusses, beams, plates and shells. The engineer often dealing with structures composed of structural elements does not need to solve continuum problems. In addition when continuum problems are encountered, the system can lead to use of many different material models which are easily treated using a finite element formulation. In fluid mechanics, practically all situations of flow require a two- or three-dimensional treatment and here approximation is required. This accounts for the early use of finite differences in the 1950s before the finite element process was made available. However, as pointed out in reference 1, there are many advantages of using the finite element process. This not only allows a fully unstructured and arbitrary domain subdivision to be used but also provides an approximation which in self-adjoint problems is always superior to or at least equal to that provided by finite differences. A methodology which appears to have gained an intermediate position is that of finite volumes, which were initially derived as a subclass of finite difference methods. As shown later in this chapter, these are simply another kind of finite element form in which subdomain collocation is used. We do not see much advantage in using this form of approximation; however, there is one point which seems to appeal to some investigators. That is the fact the finite volume approximation satisfies conservation conditions for each finite volume. This does not carry over to the full finite element analysis where generally satisfaction of conservation conditions is achieved only in an assembly region of elements surrounding each node. Satisfaction of the conservation conditions on an individual element is not an advantage if the general finite element approximation gives results which are superior. In this book we will discuss various classes of problems, each of which has a certain behaviour in the numerical solution. Here we start with incompressible flows or flows where the only change of volume is elastic and associated with transient changes of pressure (Chapters 4 and 5). For such flows full incompressible constraints must be available. Further, with very slow speeds, convective acceleration effects are often negligible and the solution can on occasion be reached using identical programs to those derived for linear incompressible elasticity. This indeed was the first venture of finite element developers into the field of fluid mechanics thus transferring the direct knowledge from solid mechanics to fluids. In particular the so-called linear Stokes flow is the case where fully incompressible but elastic behaviour occurs. A particular variant of Stokes flow is that used in metal forming where the material can no longer be described by a constant
General remarks for fluid mechanics problems 3 viscosity but possesses a viscosity which is non-Newtonian and depends on the strain rates. Chapter 5 is partly devoted to such problems. Here the fluid formulation (flow formulation) can be applied directly to problems such as the forming of metals or plastics and we shall discuss this extreme situation in Chapter 5. However, even in incompressible flows, when the speed increases convective acceleration terms become important. Here often steady-state solutions do not exist or at least are extremely unstable. This leads us to such problems as vortex shedding. Vortex shedding indicates the start of instability which becomes very irregular and indeed random when high speed flow occurs in viscous fluids. This introduces the subject of turbulence, which occurs frequently in fluid dynamics. In turbulent flows random fluctuation of velocity occurs at all points and the problem is highly time dependent. With such turbulent motion, it is possible to obtain an averaged solution using time averaged equations. Details of some available time averaged models are summarized in Chapter 8. Chapter 6 deals with incompressible flow in which free surface and other gravity controlled effects occur. In particular we show three different approaches for dealing with free surface flows and explain the necessary modifications to the general formulation. The next area of fluid dynamics to which much practical interest is devoted is of course that of flow of gases for which the compressibility effects are much larger. Here compressibility is problem dependent and generally obeys gas laws which relate the pressure to temperature and density. It is now necessary to add the energy conservation equation to the system goveming the motion so that the temperature can be evaluated. Such an energy equation can of course be written for incompressible flows but this shows only a weak or no coupling with the dynamics of the flow. This is not the case in compressible flows where coupling between all equations is very strong. In such compressible flows the flow speed may exceed the speed of sound and this may lead to shock development. This subject is of major importance in the field of aerodynamics and we shall devote a part of Chapter 7 to this particular problem. In a real fluid, viscosity is always present but at high speeds such viscous effects are confined to a narrow zone in the vicinity of solid boundaries (the so-called boundary layer). In such cases, the remainder of the fluid can be considered to be inviscid. There we can return to the fiction of an ideal fluid in which viscosity is not present and here various simplifications are again possible. Such simplifications have been used since the early days of aerodynamics and date back to the work of Prandtl and Schlichting. 2 One simplification is the introduction of potential flow and we shall mention this later in this chapter. Potential flows are indeed the topic of many finite element investigators, but unfortunately such solutions are not easily extendible to realistic problems. A particular form of viscous flow problem occurs in the modelling of flow in porous media. This important field is discussed in Chapter 9. In the topic of flow through porous media, two extreme situations are often encountered. In the first, the porous medium is stationary and the fluid flow occurs only in the narrow passages between solid grains. Such an extreme is the basis of porous medium flow modelling in applications such as geo-fluid dynamics where the flow of water or oil through porous rocks occurs. The other extreme of porous media flow is the one in which the solid occupies only a small part of the total volume (for example, representing thermal insulation systems, heat exchangers etc.). In such problems flow is almost the same as that occurring in fluids without the solid phase which only applies an added, distributed, resistance to flow. Both extremes are discussed in Chapter 9.
4 Introduction to the equations of fluid dynamics and the finite element approximation Another major field of fluid mechanics of interest to us is that of shallow water flows that occur in coastal estuaries or elsewhere. In this class of problems the depth dimension of flow is very much less than the horizontal ones. Chapter 10 will deal with such problems in which essentially the distribution of pressure in the vertical direction is almost hydrostatic. For such shallow water problems a free surface also occurs and this dominates the flow characteristics and here we note that shallow water flow problems result in a formulation which is closely related to gas flow. Whenever a free surface occurs it is possible for transient phenomena to happen, generating waves such as those occurring in oceans or other bodies of water. We have introduced in this book two chapters (Chapters 11 and 12) dealing with this particular aspect of fluid dynamics. Such wave phenomena are also typical of some other physical problems. For instance, acoustic and electromagnetic waves can be solved using similar approaches. Indeed, one can show that the treatment for this class of problems is very similar to that of surface wave problems. In what remains of this chapter we shall introduce the general equations of fluid dynamics valid for most compressible or incompressible flows showing how the particular simplification occurs in some categories of problems mentioned above. However, before proceeding with the recommended discretization procedures, which we present in Chapter 3, we must introduce the treatment of problems in which convection and diffusion occur simultaneously. This we shall do in Chapter 2 using the scalar convection-diffusion-reaction equation. Based on concepts given in Chapter 2, Chapter 3 will introduce a general algorithm capable of solving most of the fluid mechanics problems encountered in this book. There are many possible algorithms and very often specialized ones are used in different areas of applications. However, the general algorithm of Chapter 3 produces results which are at least as good as others achieved by more specialized means. We feel that this will give a certain unification to the whole subject of fluid dynamics and, without apology, we will omit reference to many other methods or discuss them only in passing. For completeness we shall show in the present chapter some detail of the finite element process to avoid the repetition of basic finite element presentations which we assume are known to the reader either from reference 1 or from any of the numerous texts available. i~!~!i~i~!!~!~!~i~2ii~!!!ii~~ i ii i i2i ililili!li~~i i ili i i~~i ~~iililii!ii i~i ~~ ~~~i i ii!i ii !iii i i i!i!ii ii ii~ii iii i i i iii!ii i!iiiii~i ii i ~i i i iii~i !i i i!i i i!i i ~i iliJ~i!iii ii~i i i!!~ii iii~i!i!i!ii~i ii iii!i i!i i! i~~ii i !~ii~ilii!i i ii i i!i ii ii i i ii i!ii i!i!i i i i~i~i i i i i ~~, ~~ii iilii ii~iil!ii i~ii ii i!i i!iii~~ii iiii iii i!i il i!ii ililimiiili!ii|ii!ii~!ii~ii i i i!i i!iiliii i~i,i,ii~i!~!!~!~i~i!~ i i !i !i ili~~,,~i~ i!i !i!iilililiii i!i i !iili i i !~i ili!~!!i li i i i!i!~ii ii!ii~!i !i~i~iiilili !iiii i i
1.2.1 Stresses in fluids As noted above, the essential characteristic of a fluid is its inability to sustain deviatoric stresses when at rest. Here only hydrostatic' stress' or pressure is possible. Any analysis must therefore concentrate on the motion, and the essential independent variable is the velocity u or, if we adopt indicial notation (with the coordinate axes referred to as xi, i = 1, 2, 3), ui,
i--1,2,3
or
u=
EUl,
u2,
u3
l
(1.1)
This replaces the displacement variable which is of primary importance in solid mechanics.
The governing equations of fluid dynamics 5 The rates of strain are the primary cause of the general stresses, oij , and these are defined in a manner analogous to that of infinitesimal strain in solid mechanics as
9
I(OUi
,ij =
+
OUj)
This is a well-known tensorial definition of strain rates but for use later in variational forms is written as a vector which is more convenient in finite element analysis. Details of such matrix forms are given fully in reference 1 but for completeness we summarize them here. Thus, the strain rate is written as a vector (e) and is given by the following form -- [~11, ~22, 2~12] T = [~11, ~22, ~/12] T (1.3a) in two dimensions with a similar form in three dimensions -" [~ll, ~22, ~33, 2~12, 2~23, 2~31] T
(1.3b)
When such vector forms are used we can write the strain rate vector in the form = ,$u
(1.4)
where ,.q is known as the strain rate operator and u is the velocity given in Eq. (1.1). The stress-strain rate relations for a linear (Newtonian) isotropic fluid require the definition of two constants. The first of these links the deviatoric stresses ~-ij to the deviatoric strain rates:
1
Tij ~ O'ij -- ~(~ij O'kk --
2# ( ~ij -- l(~ij~kk)
(~.5)
In the above equation the quantity in brackets is known as the deviatoric strain rate, (~ij is the Kronecker delta, and a repeated index implies summation over the range of the index; thus
O'kk ~ fill -+" 0"22 + 0"33
and
~kk ~ ~ll + ~22 + ~33
(1.6)
The coefficient/z is known as the dynamic (shear) viscosity or simply viscosity and is analogous to the shear modulus G in linear elasticity. The second relation is that between the mean stress changes and the volumetric strain rate. This defines the pressure as 1
P = -- gO'kk = --I~kk-4- Po
(1.7)
where ~ is a volumetric viscosity coefficient analogous to the bulk modulus K in linear elasticity and P0 is the initial hydrostatic pressure independent of the strain rate (note that p and P0 are invariably defined as positive when compressive). We can immediately write the 'constitutive' relation for fluids from Eqs (1.5) and (1.7)
O'ij = Tij -- (~ij P
2#(~:ij - lt~ij~kk) + I~(~ij ~kk - t~ijPo k
(1.8a)
6 Introductionto the equations of fluid dynamics and the finite element approximation or 2 O'ij -- 2#~ij -t" t~ij (l'~ -- -~ #) ~kk -- t~ij PO
(1.8b)
The Lam6 notation is occasionally used, putting - g2 # = A
(1.9)
but this has little to recommend it and the relation (1.8a) is basic. There is little evidence about the existence of volumetric viscosity and, in what follows, we shall take n~kk = 0 (1.10) giving the essential constitutive relation as (now dropping the suffix on P0)
O'ij = 2# ( ~ij -- ~l(~ij~kk ) -- t~ij P ~ 7-ij - t~ij P
(1 l l a )
without necessarily implying incompressibility ~kk = 0. In the above,
Tij __ 2# (~ij- l(~ij~kk)= # [(OUi
OUj 2 On k "-[- -~Xi ) -- -~(~ij ~Xk]
(1.11b)
The above relationships are identical to those of isotropic linear elasticity as we will note again later for incompressible flow. However, in solid mechanics we often consider anisotropic materials where a larger number ofparameters (i.e. more than 2) are required to define the stress-strain relations. In fluid mechanics use of such anisotropy is rare and in this book we will limit ourselves to purely isotropic behaviour. Non-linearity of some fluid flows is observed with a coefficient # depending on strain rates. We shall term such flows 'non-Newtonian'. We now consider the basic conservation principles used to write the equations of fluid dynamics. These are: mass conservation, momentum conservation and energy
conservation.
1.2.2 Mass conservation If p is the fluid density then the balance of mass flow pui entering and leaving an infinitesimal control volume (Fig. 1.1) is equal to the rate of change in density as expressed by the relation
0p
0
Op
--~ + ~xi (pui) =-- -~ +
VT
(pu) = 0
(1.12)
where V T - [O/Oxl, O/Ox2, O/Ox3] is known as the gradient operator. It should be noted that in this section, and indeed in all subsequent ones, the control volume remains fixed in space. This is known as the 'Eulerian form' and displacements of a particle are ignored. This is in contrast to the usual treatment in solid mechanics where displacement is a primary dependent variable.
The governing equations of fluid dynamics 7 It is possible to recast the above equations in relation to a moving frame of reference and, if the motion follows the particle, the equations will be named 'Lagrangian'. Such Lagrangian frame of reference is occasionally used in fluid dynamics and briefly discussed in Chapter 6.
1.2.3 Momentum conservation" dynamic equilibrium .
.
.
.
--~ ...........................
.. . . . . . . . . . . . . . . . . .
: ...........................................
-
..............................
-
: : = =
...............
~
..................................
= : =
................................................
__.
=
=
...........
: . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In the j th direction the balance of linear momentum leaving and entering the control volume (Fig. 1.1) is to be in dynamic equilibrium with the stresses aij and body forces p g j . This gives a typical component equation
-O(puj) -f-
0
0
-~- -'~'~-_ [(puj)ui]-
-'-~'~-_ (O'ij ) -- pgj -- 0 uxi
uxi
(1.13)
or using Eq. (1.11 a),
O(puj)
T - t -
0
Orij
Op
=0
- -pgj
-'~-~iXi[ (pU j )Ui ] - ~ X i + "~Xj
(1.14)
with Eq. (1.11 b) implied. The conservation of angular momentum merely requires the stress to be symmetric, i.e. CYij -- O'j i
or
In the sequel we will use the term m o m e n t u m angular forms.
Ti j -- 7"j i conservation
to imply both linear and
1.2.4 Energy conservation and equation of state We note that in the equations of Secs 1.2.2 and 1.2.3 the dependent variables are U i (the velocity components), p (the pressure) and p (the density). The deviatoric stresses,
•
J
(y)
dXl
=
x~.(x)
x3; (z)
Fig. 1.1 Coordinate direction and the infinitesimal control volume.
8
Introduction to the equations of fluid dynamics and the finite element approximation of course, are defined by Eq. (1.1 l b) in terms of velocities and hence are dependent variables. Obviously, there is one variable too many for this equation system to be capable of solution. However, if the density is assumed constant (as in incompressible fluids) or if a single relationship linking pressure and density can be established (as in isothermal flow with small compressibility) the system becomes complete and solvable. More generally, the pressure (p), density (p) and absolute temperature (T) are related by an equation of state of the form (1.15a)
p = p(p, T)
For an ideal gas this takes the form p =
P RT
(1.15b)
where R is the universal gas constant. In such a general case, it is necessary to supplement the governing equation system by the equation of energy conservation. This equation is of interest even if it is not coupled with the mass and momentum conservation, as it provides additional information about the behaviour of the system. Before proceeding with the derivation of the energy conservation equation we must define some further quantifies. Thus we introduce e, the intrinsic energy per unit mass. This is dependent on the state of the fluid, i.e. its pressure and temperature or (1.16)
e=e(T,p)
The total energy per unit mass, E, includes of course the kinetic energy per unit mass and thus E -- e + ~1 UiU i (1.17) Finally, we can define the enthalpy as h = e+ p P
or
H = h + ~ UiU i = E
+P P
(1.18)
and these variables are found to be convenient to express the conservation of energy relation. Energy transfer can take place by convection and by conduction (radiation generally being confined to boundaries). The conductive heat flux qi for an isotropic material is defined as OT qi - - - k (1.19) OXi
where k is thermal conductivity. To complete the relationship it is necessary to determine heat source terms. These can be specified per unit volume as qH due to chemical reaction (if any) and must include the energy dissipation due to internal stresses, i.e. using Eqs (1.1 l a) and (1.11 b), 0
0
OX i ( O ' i j U j ) - - -'~Xi ( T i j U j )
-
0 -~xj ( p u j )
(1.20)
The governing equations of fluid dynamics The balance of energy in a infinitesimal control volume can now be written as O(pE)
t-
0
O / OT'x ~, +
(pui E ) -
0
(pui)
--
0
(7"ijU
J )-
pgiui -- qH = 0
(1.21a) or more simply O(pE)
~-
0 (pui H )
-
0(0k
)
0 (7"ijUj) -- pgiui
-
-- qH -- 0
(1.21b) Here, the penultimate term represents the rate of work done by body forces.
1.2.5 Boundary conditions On the boundary of a typical fluid dynamics problem b o u n d a r y conditions need to be specified to make the solution unique. These are given simply as: (a) The velocities can be described as on Fu
Ui - - Ui
or traction as ti - - n j o i j
--
-ti
on 1-'t
(1.22a) (1.22b)
where l-'u ~ 1-'t - 1-'. Generally traction is resolved into normal and tangential components to the boundary. (b) In problems for which consideration of energy is important the temperature on the boundary is expressed as T-7' onFr (1.23a) or thermal flux qn -- -nik
OT = - k OT
Ox----~
~ - -- g/" on 1-'q
(1.23b)
where Fr U Fq - - F . (c) For problems of compressible flow the density is specified as p=~
onFp
(1.24)
1.2.6 Navier-Stokes and Euler equations The governing equations derived in the preceding sections can be written in a general conservative form as 0~ 0Fi 0Gi --~ + ~ x / + ~ + Q - 0 (1.25) in which Eq. (1.12), (1.14) or (1.21b) provides the particular entries to the vectors.
9
10 Introduction to the equations of fluid dynamics and the finite element approximation Thus, using indicial notation the vector of independent unknowns is P pul
--
(1.26a)
pu2 pu3 pE
the convective flux is expressed as
Fi "-
pui pu 1ui -4- pt~li pu2ui -Jr-pt~2i pu3ui q- pt~3i pHui
(1.26b)
,
similarly, the diffusive flux is expressed as 0
Gi
--
and the source terms as
Q --
--,'i-li --T2i --'i-3i
(1.26c) OT
0 pgl Pg2 Pg3
(1.26d)
pgiui -- qu with ( Ou i
Ou j
2
Ou k
"l'i' -- #[~,OXj "qt- ~ x i ) -- "3tSiJ "~Xk ] The complete set of Eq. (1.25) is known as the Navier-Stokes equation. A particular case when viscosity is assumed to be zero and no heat conduction exists is known as the 'Euler equation' (where 7"ij = 0 and qi = 0). The above equations are the basis from which all fluid mechanics studies start and it is not surprising that many alternative forms are given in the literature obtained by combinations of the various equations. 5 The above set is, however, convenient and physically meaningful, defining the conservation of important quantities. It should be noted that only equations written in conservation form will yield the correct, physically meaningful, results in problems where shock discontinuities are present. In Appendix A, we show a particular set of non-conservative equations which are frequently used. The reader is cautioned not to extend the use of non-conservative equations to problems of high speed flow.
Inviscid, incompressible flow In many actual situations one or another feature of the flow is predominant. For instance, frequently the viscosity is only of importance close to the boundaries at which velocities are specified. In such cases the problem can be considered separately in two parts: one as a boundary layer near such boundaries and another as inviscidflow outside the boundary layer. Further, in many cases a steady-state solution is not available with the fluid exhibiting turbulence, i.e. a random fluctuation of velocity. Here it is still possible to use the general Navier-Stokes equations now written in terms of the mean flow with an additional Reynolds stress term. Turbulent instability is inherent in the Navier-Stokes equations. It is in principle always possible to obtain the transient, turbulent, solution modelling of the flow, providing the mesh size is capable of reproducing the small eddies which develop in the problem. Such computations are extremely costly and often not possible at high Reynolds numbers. Hence the Reynolds averaging approach is of practical importance. Two further points have to be made concerning inviscidflow (ideal fluid flow as it is sometimes known). First, the Euler equations are of a purely convective form: (~tI~ c~F i - - ~ -~ ~ X / "q- o "- D
F i -- F i ( t I ))
(1.27)
and hence very special methods for their solutions will be necessary. These methods are applicable and useful mainly in compressible flow, as we shall discuss in Chapter 7. Second, for incompressible (or nearly incompressible) flows it is of interest to introduce a potential that converts the Euler equations to a simple self-adjoint form. We shall discuss this potential approximation in Sec. 1.3. Although potential forms are also applicable to compressible flows we shall not use them as they fail in complex situations.
In the absence of viscosity and compressibility, p is constant and Eq. (1.12) can be written as Oui = 0 (1.28)
cgxi
and Eq. (1.14) as
0U i
(~
+ ~--(ujui) -~ oxj
10p P (~Xi
gi -- 0
(1.29)
1.3.1 Velocity potential solution The Euler equations given above are not convenient for numerical solution, and it is of interest to introduce a potential, qS, defining velocities as
or Ul --
OOX1
U2 w
or OOX2
/g3 --
or OOX3
or g i ---
Oxi
(1.30)
11
12
Introductionto the equations of fluid dynamics and the finite element approximation If such a potential exists then insertion of Eq. (1.30) into Eq. (1.28) gives a single governing equation ~r
OXiOXi
= V2r
0
(1.31)
which, with appropriate boundary conditions, can be readily solved. Equation (1.31) is a classical Laplacian equation. For contained flow we can of course impose the normal velocity Un on the boundaries"
Un =
On
"-" Un
(1.32)
and, as we shall see later, this provides a natural boundary condition for a weighted residual or finite element solution. Of course we must be assured that the potential function ~bexists, and indeed determine what conditions are necessary for its existence. Here we observe that so far in the definition of the problem we have not used the momentum conservation equations (1.29), to which we shall now return. However, we first note that a single-valued potential function implies that
OXj OXi
=
(1.33)
OXi OXj
Defining vorticity as rotation rate per unit area
l (Oui OJiJ --" -2 OXj
C~Uj) OXi
(1.34)
we note that the use of the velocity potential in Eq. (1.34) gives
OJij :
0
(1.35)
and the flow is therefore named irrotational. Inserting the definition of potential into the first term of Eq. (1.29) and using Eqs (1.28) and (1.34) we can rewrite Eq. (1.29) as
o(o ) o[:
OXi - ~
Jr- ~
UjUj + -- + P P
]
= 0
(1.36)
in which P is a potential of the body forces given by
gi --
OP
OXi
(1.37)
In problems involving constant gravity forces in the x2 direction the body force potential is simply P = g x2 (1.38) Equation (1.36) is alternatively written as -~- + H +
-
(1.39)
Incompressible (or nearly incompressible) flows where H is the enthalpy, given as H lgiUi/2 + p/p. If isothermal conditions pertain, the specific energy is constant and Eq. (1.39) implies that 0@ 1 p - - ~ -F -~ uiui + -- + P = constant (1.40) P over the whole domain. This can be taken as a corollary of the existence of the potential and indeed is a condition for its existence. In steady-state flows it provides the wellknown Bernoulli equation that allows the pressures to be determined throughout the whole potential field once the value of the constant is established. We note that the governing potential equation (1.31) is self-adjoint (see Appendix B) and that the introduction of the potential has side-stepped the difficulties of dealing with convective terms. It is also of interest to note that the Laplacian equation, which is obeyed by the velocity potential, occurs in other contexts. For instance, in twodimensional flow it is convenient to introduce a stream function the contours of which lie along the streamlines. The stream function, ~, defines the velocities as =
and
U1 = OX2
u2 =
OqXl
(1.41)
which satisfy the incompressibility condition (1.28)
Obli 0 0 OXi--OX1 (0-~22)-I---~Xl (--0-~11)--0
(1.42)
For an existence of a unique potential for irrotational flow we note that co12 - 0 [Eq. (1.34)] gives a Laplacian equation 02~D
-- V 2 ~ = 0
OxiOxi
(1.43)
The stream function is very useful in getting a pictorial representation of flow. In Appendix C we show how the stream function can be readily computed from a known distribution of velocities.
We observed earlier that the Navier-Stokes equations are completed by the existence of a state relationship giving [Eq. (1.15a)]
p - p(p, T) In (nearly) incompressible relations we shall frequently assume that: (a) The problem is isothermal. (b) The variation of p with p is very small, i.e. such that in product terms of velocity and density the latter can be assumed constant. The first assumption will be relaxed, as we shall see later, allowing some thermal coupling via the dependence of the fluid properties on temperature. In such cases we
13
14 Introduction to the equations of fluid dynamics and the finite element approximation shall introduce the coupling iteratively. For such cases the problem of density-induced currents or temperature-dependent viscosity will be typical (see Chapters 5 and 6). If the assumptions introduced above are used we can still allow for small compressibility, noting that density changes are, as a consequence of elastic deformability, related to pressure changes. Thus we can write P d p = ~ dp
(1.44a)
where K is the elastic bulk modulus. This also can be written as 1 dp -- ~ dp
(1.44b)
cOp lap - ~ - c2 0 t
(1.44c)
or
with c = ~/K/p being the acoustic wave velocity. Equations (1.25) and (1.26a)-(1.26d) can now be rewritten omitting the energy transport equation (and condensing the general form) as 10p /~Ui ~C -~- + P~X/ = 0
Ouj 0 10p Ot ~t_ ~Oxi ( Uj Ui ) .3t_ P Ox j
1 07"ji
p OXi
gj -- 0
(1.45a)
(1.45b)
In three dimensions j = 1, 2, 3 and the above represents a system of four equations in which the variables are u j and p. Here
1 l,,(OUi OUj -p 7"ij = \ OXj "~ OXi
2t~ij OUk) 3
-~xk :
where u = #/p is the kinematic viscosity. The reader will note that the above equations, with the exception of the convective acceleration terms, are identical to those governing the problem of incompressible (or slightly compressible) elasticity (e.g. see Chapter 11 of reference 1).
Numer ! o!u !o.si
................... l is .............
eS dua!......................
1.5.1 Strong and weak forms We assume the reader is already familiar with basic ideas of finite element and finite difference methods. However, to avoid a constant cross-reference to other texts (e.g. reference 1), we provide here a brief introduction to weighted residual andfinite element
methods.
Numerical solutions: weak forms, weighted residual and finite element approximation The Laplace equation, which we introduced in Sec. 1.3, is a very convenient example for the start of numerical approximations. We shall generalize slightly and discuss in some detail the quasi-harmonic (Poisson) equation k
0X i
+Q-0
(1.46)
where k and Q are specified functions. These equations together with appropriate boundary conditions define the problem uniquely. The boundary conditions can be of Dirichlet type q5 = 4) on Fr (1.47a) or that of Neumann type
0r
qn = - k ~
=?/n
on l-'q
(1.47b)
where a bar denotes a specified quantity. Equations (1.46) to (1.47b) are known as the strong f o r m of the problem.
Weak form of equations
We note that direct use of Eq. (1.46) requires computation of second derivatives to solve a problem using approximate techniques. This requirement may be weakened by considering an integral expression for Eq. (1.46) written as v
-~x/
k~x~
+ Q dr2 = 0
(1.48)
in which v is an arbitrary function. A proof that Eq. (1.48) is equivalent to Eq. (1.46) is simple. If we assume Eq. (1.46) is not zero at some point xi in f2 then we can also let v be a positive parameter times the same value resulting in a positive result for the integral Eq. (1.48). Since this violates the equality we conclude that Eq. (1.46) must be zero for every xi in ~ hence proving its equality with Eq. (1.48). We may integrate by parts the second derivative terms in Eq. (1.48) to obtain k~-~x/
d~+
v Q d S2 -
v n i
k -~xi
dF=0
(1.49)
We now split the boundary into two parts, F~ and Fq, with F = F~ U 1-'q, and use Eq. (1.47b) in Eq. (1.49) to give ~x/
k~-x/
dr2 +
v Q dr2 +
v qn dF - 0
(1.50)
q
which is valid only if v vanishes on Fr Hence we must impose Eq. (1.47a) for equivalence. Equation (1.50) is known as the weak form of the problem since only first derivatives are necessary in constructing a solution. Such forms are the basis for the finite element solutions we use throughout this book.
15
16
Introductionto the equations of fluid dynamics and the finite element approximation
Weighted residual approximation
In a weighted residual scheme an approximation to the independent variable ~bis written as a sum of known trial functions (basis functions) Na (xi) and unknown parameters ~ba. Thus we can always write
(9 ~, ~ -
Nl(Xi)~91 nt- N2(xi)@ 2 -Jr'''' : ~ Na(xi)dpa -- N(xi)dp a=l
(1.51)
where and
N - - [ N I , Nz,...Nn]
(1.52a)
t~--- [@1, q~2, ...@n] T
(1.52b)
In a similar way we can express the arbitrary variable v as
U ~ U -- Wl (xi)~31 .2f_W2(xi)~)2 -4-... -- ~ Wa(xi)l) a -- W ( x i ) v a=l
(1.53)
in which Wa are test functions and ~a arbitrary parameters. Using this form of approximation will convert Eq. (1.50) to a set of algebraic equations. In the finite element method and indeed in all other numerical procedures for which a computer-based solution can be used, the test and trial functions will generally be defined in a local manner. It is convenient to consider each of the test and basis functions to be defined in partitions ~'2e of the total domain f2. This division is denoted by
(1.54)
~ ~h -- U ~"~e
and in a finite element method ~'2 e a r e known as elements. The very simplest uses lines in one dimension, triangles in two dimensions and tetrahedra in three dimensions in which the basis functions are usually linear polynomials in each element and the unknown parameters are nodal values of 0. In Fig. 1.2 we show a typical set of such linear functions defined in two dimensions. In a weighted residual procedure we first insert the approximate function q~into the governing differential equation creating a residual, R (xi), which of course should be zero at the exact solution. In the present case for the quasi-harmonic equation we obtain
R--
Ox i
k ~-~ -~Xi a
+ Q
(1.55)
and we now seek the best values of the parameter set @a which ensures that
]~ WbR df2 = O; b - l , 2 , . . . , n
(1.56)
Note that this is the term multiplying the arbitrary parameter ~b. As noted previously, integration by parts is used to avoid higher-order derivatives (i.e. those greater than or
Numerical solutions: weak forms, weighted residual and finite element approximation
Y
.....
,,,,
. .
.
.
,
.
.
.
.
.
.
.
.
,,,
i
.
.
.
.
.
.
.
.
.
.
~
:
Fig. 1.2 Basis function in linear polynomials for a patch of trianglular elements. equal to two) and therefore reduce the constraints on choosing the basis functions to permit integration over individual elements using Eq. (1.54). In the present case, for instance, the weighted residual after integration by parts and introducing the natural boundary condition becomes k a~-~x /
WbQdS2+
dS2+
q
WbflndF=O
(1.57)
The Galerkin, finite element, method
In the Galerkin method we simply take equations
Wb =
Nb which gives the assembled system of
n
g b a @ a -+- f b
- - O;
b -- 1, 2 , . . .
,n
-
r
(1.58)
a--1
where r is the number of nodes appearing in the approximation to the Dirichlet boundary condition [i.e. Eq. (1.47a)] and Kba is assembled from element contributions K~a with
Kb~a--
L ONa
ONa --~-x/k-~x/dr2
(1.59)
e
Similarly,
fb is computed from the element as
f[ -- L Nb QdS2+ f NbClndF e
(1.60)
eq
To impose the Dirichlet boundary condition we replace @aby @afor the r boundary nodes. It is evident in this example that the Galerkin method results in a symmetric set of algebraic equations (e.g. Kba = Kab). However, this only happens if the differential equations are self-adjoint (see Appendix B). Indeed the existence of symmetry provides a test for self-adjointness and also for existence of a variational principle whose stationarity is sought. ~
17
18 Introduction to the equations of fluid dynamics and the finite element approximation 1 N1 y
D 1
2
3
.~X
Xl
I
(b) Shape function for node 1.
(a) Three-node triangle.
Fig. 1.3 Triangular element and shape function for node 1.
It is necessary to remark here that if we were considering a pure convection equation
0~
u i -~ix i --F a -" o
(1.61)
symmetry would not exist and such equations can often become unstable if the Galerkin method is used. We will discuss this matter further in the next chapter.
Example 1.1 Shape functions for triangle with three nodes
A typical finite element with a triangular shape is defined by the local nodes 1, 2, 3 and straight line boundaries between nodes as shown in Fig. 1.3(a) and will yield the shape of Na of the form shown in Fig. 1.3(b). Writing a scalar variable as ~) - - 0~1 -~" OZ2 Xl "q- 0~3 X2
(1.62)
we may evaluate the three constants by solving a set of three simultaneous equations which arise if the nodal coordinates are inserted and the scalar variable equated to the appropriate nodal values. For example, nodal values may be written as
~;~ -
~, + ~
x~ + ~ 3 x 1
~2 _ al + a2 Xl2 + a3 x~
(1.63)
@3 = Ol1 _+_ Ol2 X~ -+- OL3 X3
We can easily solve for al, a2 and a3 in terms of the nodal values ~l, ~2 and ~3 and obtain finally q~ "- ~
1
[(al + blXl -+- ClX2)~ 1 ~- (a2 + b2Xl nt- c2x2)@ 2 --t- (a3 + b3xl Jr c3x2)(~ 3]
(1.64) in which
23 32 al -- x 1X2 m Xl x2 b l
=
-
c, = x? -
(1.65)
Numerical solutions: weak forms, weighted residual and finite element approximation 19 where x a is the i direction coordinate of node a and other coefficients are obtained by cyclic permutation of the subscripts in the order 1, 2, 3, and 1
x~
x21
2A = det 1
X2
x~ = 2. (area of triangle 123)
1
x~
x3
(1.66)
From Eq. (1.64) we see that the shape functions are given by Na = (aa "+" ba Xl -+- Ca X 2 ) / ( 2 A ) ;
(1.67)
a = 1, 2, 3
Since the unknown nodal quantities defined by these shape functions vary linearly along any side of a triangle the interpolation equation (1.64) guarantees continuity between adjacent elements and, with identical nodal values imposed, the same scalar variable value will clearly exist along an interface between elements. We note, however, that in general the derivatives will not be continuous between element assemblies. 1
Poisson equation in two dimensions: Galerkin formulation with triangular elements Example 1.2
The relations for a Galerkin finite elemlent solution have been given in Eqs (1.58) to (1.60). The components of Kba and fb can be evaluated for a typical element or subdomain and the system of equations built by standard methods. For instance, considering the set of nodes and elements shown shaded in Fig. 1.4(a), to compute the equation for node 1 in the assembled patch, it is only necessary to compute the K~a for two element shapes as indicated in Fig. 1.4(b). For the Type 1 element [left element in Fig. 1.4(b)] the shape functions evaluated from Eq. (1.67) using Eqs (1.65) and (1.66) gives N1
=
1 -
--x2"
h,
N2
m
X l.
---~, N3
__ x 2 n X l
h
thus, the derivatives are given by:
ON _ 0Xl
• ON
0
ON
1
-
~
"ON ON_
and
Ol(!
-
Ox2
1
ON
1
0U
0 1
Similarly, for the Type 2 element [fight element in Fig. 1.4(b)] the shape functions are expressed by N1-
1 -xl"
h' N2
Xl - - X2
X2
h
h
20
Introduction to the equations of fluid dynamics and the finite element approximation
|
3 il
4
5
/ /1 /
9
/
/
(a) 'Connected' equations for node 1.
3
2
>q
3
I
2
~.
h
d
(b) Type 1 and Type 2 element shapes in mesh.
Fig. 1.4 Lineartriangularelementsfor Poissonequationexample. and their derivatives by
ON1
1
ON1 '
0
-
ON
yx;-
ON2
~
ON3
-
-h 1
and
~ o
ON
~
OU2
-
~ ON3 ~
-
-
1
and
1
-~
Evaluation o f the matrix K~a and f~ for Type 1 and Type 2 elements gives (refer to Appendix D for integration formulae)
[
Ke,f.e --
2
1
0
0
-1
1
-1
--I -1 2
1 ~le ~2e
@3e
and
Ke@e: ~ kl I - 11 0
-1
0
2 -1 -1 1
] @2e ~le ~3e
Numerical solutions: weak forms, weighted residual and finite element approximation respectively. The force vector for a constant Q over each element is given by
fe - g Ql h2{ 11 } 1
for both types of elements. Assembling the patch of elements shown in Fig. 1.4(b) gives the equation with non-zero coefficients for node 1 as (refer to references 1 and 10 for assembly procedure)
k [4
-1
-1
-1] ~4
-1
-'l- Q h 2
~6
=0
Using a central difference finite difference approximation directly in the differential equation (1.46) gives the approximation
h2
-1
-1
-1
-1]
~4 ~6
+Q-0
and we note that the assembled node using the finite element method is identical to the finite difference approximation though presented slightly differently. If all the boundary conditions are forced (i.e. ~b = qS) no differences arise between a finite element and a finite difference solution for the regular mesh assumed. However, if any boundary conditions are of natural type or the mesh is irregular differences will arise, with the finite element solution generally giving superior answers. Indeed, no restrictions on shape of elements or assembly type are imposed by the finite element approach.
Example 1.3
In Fig. 1.5 an example of a typical potential solution as described in Sec. 1.3 is given. Here we show the finite element mesh and streamlines for a domain of flow around a symmetric aerofoil.
Example 1.4
Some problems of specific interest are those of flow with a free surface. 11'12'13Here the governing Laplace equation for the potential remains identical, but the free surface position has to be found iteratively. In Fig. 1.6 an example of such a free surface flow solution is given. 12 For gravity forces given by Eq. (1.38) the free surface condition in two dimensions (Xl, x2) requires
12(b/1/'/1~ b/2U2)~ g x 2
-- 0
21
22 Introduction to the equations of fluid dynamics and the finite element approximation
Fig. 1.5 Potential flow solution around an aerofoil. Mesh and streamline plots.
Numerical solutions: weak forms, weighted residual and finite element approximation
Moving
grid.,
7/ /
Vj
I
ro
Ro
I
I
,
R0/~0 =
~.6
s~ ro = 0.8
~ = 5o.5~
Fig. 1.6 Free surface potential flow, illustrating an axisymmetric jet impinging on a hemispherical thrust reverser (from Sarpkaya and Hiriart12).
Solution of such conditions involves an iterative, non-linear algorithm, as illustrated by examples of overflows in reference 11.
1.5.2 A finite v o l u m e a p p r o x i m a t i o n Many choices of basis and weight functions are available. A large number of procedures are discussed in reference 1. An approximation which is frequently used in fluid mechanics is the finite volume process which many consider to be a generalized finite difference form. Here the weighting function is often taken as unity over a specified subdomain f2b and two variants are used: (a) an element (cell) centred approach; and (b) a node (vertex) centred approach. Here we will consider only a node centred approach with basis functions as given in Eq. (1.51) for each triangular subdomain and the specified integration cell (dual cell) for each node as shown in Fig. 1.7. For a solution of the Poisson equation discussed above, integration by parts of Eq. (1.56) for a unit Wb gives ni ~xi dF - 0
Q dr2 b
b
(1.68)
23
24 Introduction to the equations of fluid dynamics and the finite element approximation
Fig. 1.7 Finite volume weighting. Vertex centred method (~b).
for each subdomain f2b with boundary I'b. In this form the integral of the first term gives f
a dr2
(1.69)
Q f2b
b
when Q is constant in the domain. Introduction of the basis functions into the second term gives
~ dr" ,~ 7xi ~ fFO~)fF(~(/)f~Na~a ^
ni
ni
b
dr
dr
ni
=
b
(1.70)
b
requiting now only boundary integrals of the shape functions. In order to make the process clearer we again consider the case for the patch of elements shown in Fig. 1.4(a).
Example 1.5 Poisson equation in two dimensions: finite volume formulation with triangular elements
The subdomain for the determination of equation for node 1 using the finite volume method is shown in Fig. 1.8(a). The shape functions and their derivatives for the Type 1 and Type 2 elements shown in Fig. 1.8(b) are given in Example 1.2. We note especially that the derivatives of the shape functions in each element type are constant. Thus, the boundary integral terms in Eq. (1.70) become
ONo
n i - - ~ xi
dr
-
~
~
b
e
n e dr
ON~ Ox---~,
e
on I-'e
where e denotes the elements surrounding node a. Each of the integrals will be an I1, 12, 13 for the Type 1 and Type 2 elements shown. It is simple to show that the integral Ill --
JFF n~ d r e
=
J~4c n~ d r
+
f6
n~ d r
- x6 - x4
Numerical solutions: weak forms, weighted residual and finite element approximation 5
4
/ 4
3 le
ii
7"
l
7
8
(a) 'Connected' area for node 1.
5
6/
13
4
(b) Type 1 and Type 2 element boundary integrals. Fi 9. 1.8 Finite volume domain and integrations for vertex centred method.
w h e r e x 4, x 6 are m i d - e d g e c o o r d i n a t e s of the triangle as s h o w n in Fig. 1.8(b). Similarly, for the x2 derivative w e obtain
12 :
fF
n ~ d F --
e
fc
f6
n~dr +
n~dr
- x4 - x6
Thus, the integral
ONai d F 11 : fr ni-~x e
--
ONa (x 6 x~) + aXl _
OUa (x? - x 6)
25
26
Introductionto the equations of fluid dynamics and the finite element approximation The results 12 and 13 are likewise obtained as =
ONa (x 4 _ x5 ) -F ONa
=
ONa (x 5 _ x6 ) +
- Xr
ONa (x 6 -
x~)
Using the above we may write the finite volume result for the subdomain shown in Fig. 1.8(a) as
k [4 - 1
-1
-1
-1]
@4 + Oh 2 _ 0 ~6
We note that for the regular mesh the result is identical to that obtained using the standard Galerkin approximation. This identity does not generally hold when irregular meshes are considered and we find that the result from the finite volume approach applied to the Poisson equation will not yield a symmetric coefficient matrix. As we know, the Galerkin method is optimal in terms of energy error and, thus, has more desirable properties than either the finite difference or the finite volume approaches. Using the integrals defined on 'elements', as shown in Fig. 1.8(b), it is possible to implement the finite volume method directly in a standard finite element program. The assembled matrix is computed element-wise by assembly for each node on an element. The unit weight will be 'discontinuous' in each element, but otherwise all steps are standard.
We have observed in this chapter that a full set of Navier-Stokes equations can be written incorporating both compressible and incompressible behaviour. At this stage it is worth remarking that 1. More specialized sets of equations such as those which govern shallow water flow or surface wave behaviour (Chapters 10, 11 and 12) will be of similar forms and will be discussed in the appropriate chapters later. 2. The essential difference from solid mechanics equations involves the non-selfadjoint convective terms. Before proceeding with discretization and indeed the finite element solution of the full fluid equations, it is important to discuss in more detail the finite element procedures which are necessary to deal with such non-self-adjoint convective transport terms. We shall do this in the next chapter where a standard scalar convective-diffusivereactive equation is discussed.
References
i i ii i i
~iiiiiiiiiiiiiii}iiiiiiiiiiiiiiiiii iiiiDiii!iiiiii! iiiiii!iiii!iii!iiiii!i!ii!ii!iiiiiiiiiiii iiiiiiiiii iiiiiiii !iiiiiii iiiii i iiiiiiiiiiiiiii!iiiiiiii iiii!iiiiiiiiiiiiiiiiiiiiiiii ii!ii
1. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Elsevier, 6th edition, 2005. 2. H. Schlichting. Boundary Layer Theory. Pergamon Press, London, 1955. 3. C.K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 1967. 4. H. Lamb. Hydrodynamics. Cambridge University Press, Cambridge, 6th edition, 1932. 5. C. Hirsch. Numerical Computation oflnternal and External Flows, volumes 1 & 2. John Wiley & Sons, New York, 1988. 6. P.J. Roach. Computational Fluid Mechanics. Hermosa Press, Albuquerque, New Mexico, 1972. 7. L.D. Landau and E.M. Lifshitz. Fluid Mechanics. Pergamon Press, London, 1959. 8. R. Temam. The Navier-Stokes Equation. North-Holland, Dordrecht, 1977. 9. I.G. Currie. Fundamental Mechanics of Fluids. McGraw-Hill, New York, 1993. 10. R.W. Lewis, P. Nithiarasu and K.N. Seetharamu. Fundamentals of the Finite Element Method for Heat and Fluid Flow. Wiley, Chichester, 2004. 11. P. Bettess and J.A. Bettess. Analysis of free surface flows using isoparametric finite elements. International Journal for Numerical Methods in Engineering, 19:1675-1689, 1983. 12. T. Sarpkaya and G. Hiriart. Finite element analysis of jet impingement on axisymmetric curved deflectors. In J.T. Oden, O.C. Zienkiewicz, R.H. Gallagher and C. Taylor, editors, Finite Elements in Fluids, volume 2, pages 265-279. John Wiley & Sons, New York, 1976. 13. M.J. O'Carroll. A variational principle for ideal flow over a spillway. International Journal for Numerical Methods in Engineering, 15:767-789, 1980.
27
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•i!i•i!•i!••i!i!•!i
Convection dominated problemsfinite element approximations to the convection-diffusion-reaction
equation
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In this chapter we are concerned with steady-state and transient solutions for equations of the type 0tI )
OF/
OCT/
Ot + ~ x / + ~ x / + Q - 0
(2.1)
where in general 4I, is the basic dependent, vector-valued variable, Q is a source or reaction term vector and the flux matrices F and G are such that Fi -- F i ( @ ) Gi "- Gi ( k
(2.2a)
and in general
Q - Q(xi, ~ )
(2.2b)
In the above, X i and i refer, in the indicial manner, to Cartesian coordinates and quantities associated with these. A linear relationship between the source and the scalar variable in Eq. (2.2b) is frequently referred to as a reaction term. The general equation (2.1) can be termed the transport equation with F standing for the convective and G for the diffusive flux quantities. Equation (2.1) is a set of conservation laws arising from a balance of the quantity with its fluxes F and G entering and leaving a control volume. Such equations are typical of fluid mechanics which we have discussed in Chapter 1. As such equations may also arise in other physical situations this chapter is devoted to a general discussion of their approximate solution. The simplest form of Eqs (2.1), (2.2a) and (2.2b) is one in which the unknown is a scalar. Most of this chapter is devoted to the solution of such equations. Throughout this book we shall show that there is no need dealing with convection of vector type
Introduction
quantities. Thus, for simplicity we may consider the form above "~ ~
r
Q ~
F i -+ Fi = Ui (fi
Q(xi, q~)
G i -->" G i =
-
k
or
(2.3)
~
OXi
We now have in Cartesian coordinates a scalar equation of the form Ot I
OX i
OX i k
+ Q - 0
(2.4)
In the above equation, Ui is a known velocity field and q~is a scalar quantity being transported by this velocity. However, diffusion can also exist and here k is the diffusion coefficient. The term Q represents any external sources of the quantity q5 being admitted to the system and also the reaction loss or gain which itself is dependent on the function ~b. A simple relation for the reaction may be written as Q - c q5
(2.5)
where c is a scalar parameter. The equation can be rewritten in a slightly modified form in which the convective term has been differentiated as
oui --~ -]- Ui --~xi + (~ Ox----~t
o Ox i
--~xi ) + Q -- 0
(2.6)
We note that the above form of the problem is self-adjoint with the exception of a convective term which is underlined. The reader is referred to Appendix B for the definition of self-adjoint problems. The third term in Eq. (2.6) disappears if the flow itself is such that its divergence is zero, i.e. if
OU/ Oxi
= 0
(2.7)
In what follows we shall discuss the scalar equation in much more detail as many of the finite element remedies are only applicable to such scalar problems and are not directly transferable to the vector form. In the CBS scheme, which we shall introduce in Chapter 3, the equations of fluid dynamics will be split so that only scalar transport occurs, where the treatment considered here is sufficient. From Eqs (2.6) and (2.7) we have "-~ -~- Ui Ox-----~
Ox i k~x~
+ Q = 0
(2.8)
With the variable ~bapproximated in the usual way: (2.9)
29
30 Convectiondominated problems the problem may be presented following the usual (weighted residual) semi-discretization process as M& + H ~ + f -- O
(2.101
where
Mab -- f~ WaNbdf2 nab = fa -- f
Wa Ui -~xi + OXi "~-X/J d a Wa a da + fF Waqn df2 q
Now even with standard Galerkin weighting the matrix H will not be symmetric. However, this is a relatively minor computational problem compared with inaccuracies and instabilities in the solution which follow the arbitrary use of the weighting function. This chapter will discuss the manner in which these difficulties can be overcome and the approximation improved. We shall in the main address the problem of solving Eq. (2.8), i.e. the scalar form, and to simplify matters further we shall start with the idealized one-dimensional equation:
---~ -~- U ox
Ox k
+Q=O
(2.11t
The term 05OU/Ox has been removed for simplicity, which of course is true if U is constant. The above reduces in steady state to an ordinary differential equation: U
d~ dx
d (kd~b)+Q_O dx \ dx or
(2.12)
s162 + Q = o
in which we shall often assume U, k and Q to be constant. The basic concepts will be evident from the above and will later be extended to multidimensional problems, still treating ~bas a scalar variable. Indeed the methodology of dealing with the first space derivatives occurring in differential equations governing a problem, which lead to non-self-adjointness, opens the way for many new physical situations. 1 The present chapter will be divided into three parts. Part I deals with steady-state situations starting from Eq. (2.12), Part II with transient solutions starting from Eq. (2.11) and Part III with treatment of boundary conditions for convective-diffusive problems where use of 'weak forms' is shown to be desirable. Although the scalar problem will mainly be dealt with here in detail, the discussion of the procedures can indicate the choice of optimal ones which will have much beating on the solution of the general case of Eq. (2.1). The extension of some procedures to the vector case is presented in Appendix E.
The steady-state problem in one dimension
Part I" Steady-state problems
2.2.1 General remarks We consider the discretization of Eq. (2.12) with
-- N(~
~ ~) "-- Z Nasa
(2.13)
where Na are shape functions and q~represents a set of still unknown parameters. Here we shall take these to be the nodal values of qS. The weighted residual form of the one-dimensional problem is written as (see Chapter 1)
d (kaY) ~x
J~ Wa [U d~ dx
]
dx
+ Q dr2 - 0
(2.14)
Integrating the second term by parts gives
Wa U
---h-T-k~d~'~+
+ Q d~ +
q
Wa Ondr
=0
(2.15)
^
where
or On=-k--~
onr'q
and ~=q~
onFe
is assumed. For a typical intemal node 'a' the approximating equation becomes
KabOb+ fa
= 0
(2.16)
where Kab - -
fa =
fo L
/0
Wa
U
dNb dx
dx +
fo L d Wak dNb dx dx dx
W a Q d x + WaOn
IFq
(2.17)
and the domain of the problem is 0 < x _< L. For linear shape functions (Fig. 2.1), Galerkin weighting (Wa - Na) and elements of equal size h, we have for constant values of U, k and Q (see Appendix D)
Ke - - 2g [-1 1 11 I f e -1Q }h2{ - 1
k I - 11 -11 1
+h
31
32 Convectiondominated problems Na
a-1
~, h
a
J..,. h
Na+ l
I
a+l
J . , _ .h
a+2 .
Fig. 2.1 A linear shape function for a one-dimensional problem.
which yields a typical assembled equation (after multiplying by h / k ) for node 'a'
( - P e - 1)~)a_ 1 "q- 2q~a q-
(Pe-
1)~a+ 1 +
Qh 2
=0
(2.18)
where Pe =
Uh
(2.19)
2k
is the element Peclet number. Incidentally, for the case of constant Q the above is identical to the usual central finite difference approximation obtained by putting (for node a) dq~ ~)a+l- ~)a-1 ,~ (2.20a) dx 2h and
d2* dx 2
~a+l -- 2~a -~- ~a-1 h2
(2.20b)
The algebraic equations (2.18) are obviously non-symmetric and their accuracy deteriorates as the parameter Pe increases i.e. when convective terms are of primary importance. Indeed as Pe --+ c~, the solution is purely oscillatory and bears no relation to the underlying problem. This may be ascertained by considering Eq. (2.18) for different element Peclet numbers and it is easy to show that with the standard Galerkin procedure oscillations in ~a Occur when IPel > 1 (2.21) To illustrate this point we consider a simple example.
Example 2.1
One-dimensional convection diffusion (Q = 0)
The domain of the problem considered is 0 < x < L and the boundary conditions are both of Dirichlet type and given by dp(O) = 1
and
r
-- 0
We approximate the solution using nine equal size linear elements and the Galerkin form of Eq. (2.15)
[/0L(da
]
The s t e a d y - s t a t e
1
:
Convection-Diffusion: I
0.9
- - -' . . . . . .
o.8
: ......
0.7
: ......
Pe
i ...... '- .....
: '- . . . . .
! ..... ; . . . . .
". . . . . .
" .....
- - -: . . . . . .
',- . . . . .
0.4
'
,
- ....
0.2
0.6 ~
0.5 0.4
.....
0.3
' - ....
0.1
0.2 - -
0
0.2
0.4
~L
0.6
0.1
1
0.8
Pe
Convection-Diffusion: 1.5
0
0
s
,i .
I
"L:
1 s
.
.
.
.
, i
,
|
.
.
"
i
i
0.2
I
"
.
0
1
i
"
0
-
012
!
.
.
11
i .
i
- i i
"
.
1
, i
.
:
|
. . . . . '
!
.
I
' s
.
.
.
,,
.
i . . .
i ' ,i
1
1 l-
s
i
:;
,
I
1
.
.
. r ,
.1.. 1
--0.6
. i i
,.
,..~
,
,
x//.
.
' . . . . . .
;,...,,.
,
T 014
I..I 1
,' ....
.
.... !
. . . . . .
.
t
,..,:
-
,
i
i .
.
I . . . I .
"
- .
-i
I .
'...:..'..,..,
i
11
,
.i
',1 .
"1 . . . .
. . . . . ,. ,
0.8
. i
I
- - -"'-: . . -
0.1 0.6
.
1
,
' ,
s
i, .
.1..I l
,..
.....
1
I
1
x/L
.... . . 1
"
....
~ :
i
L
1
I
.
i .
i
-1-
i
. . . . . . . . . . . . . .
0.3
i
.
-i
, I
i
9
,
I
1
=
iI
"11
'1
,
...,
i
. ;.,
1
i, "
1
0.4
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.4
Pe
||
,
1
::
1 .1.
.
,
1
i, ~
......
0.6
I
i . . . . . . . . i
-~ - ' . . . . .
. . . .
,
~- 0 . 5 ,
.
. .....
0.8 0.7
0.2
0.8
i
0.9
al a
a
0
0.6
~L
Convection-Diffusion:
,
0
0.4
1
,
0.5
0.2
= 2.5 ,
1
o.7
" .....
--';
=
0.9
i ......
0.5
0
Pe
1
',. . . . . .
0.3
in one d i m e n s i o n
Convection-Diffusion:
= 0
:. . . . . .
- - -', . . . . . .
0.6
:
problem
1 1"
' ,i
.
1l .
.
.i1
. . . . . s
9 f
. . . . .
1
1 : -
....
,,,:
1
. . . . .
ii,
~.8
Fig. 2.2 Approximations to Uddpldx - kd 2 ~ / d x 2 = 0 for ~b(O) = 1 and (;b(L) = 0 for various Peclet numbers. Solid line - exact solution" dotted line with triangular symbol- standard Galerkin solution.
The solution shown in Fig. 2.2 with curves labelled with triangles gives results for element Peclet numbers [given by Eq. (2.19)] Pe = 0, 1, 2.5 and co (the solution for this problem with Pe = co is only possible for an odd number of elements). We see that as the Pe number increases above 1 the solution becomes oscillatory and progressively departs from the smooth exact solution (solid line in Fig. 2.2)
eUx/k _ 1-
eUL/k
e uL/g
Of course the above is partly a problem of boundary conditions. When diffusion is omitted (k = 0) only a single boundary condition can be imposed and when the diffusion is small we note that the downstream boundary condition (~b(L) = 0) is felt in only a very small region of a boundary layer. The above example clearly demonstrates that the Galerkin method cannot be used to solve problems in which convective terms are large compared with those of diffusion. Of course, one can consider replacing the weight functions Na by other more general
i
33
34 Convectiondominated problems ones Wa. Indeed for the linear one-dimensional steady-state problem we can always find weight functions which give exact solutions at the interelement nodes.
Example 2.2
Weight function for exact nodal solutions
Here we consider the problem of Example 2.1 where the weak form including Q is given by v
[
Udx
dx
1
~-
+ Q dx-0
where x] < x < x2 denotes the domain. After integration by parts for all derivatives on ~bwe obtain dx
/xl
-+in which
dv - U dx
Qdx+v
v
(
Uq~ - kdq~ -dx
)1
x2+ ~ dv k~ Xl
i
x, - 0 x~
d (kdV) dx ~xx
is the adjoint equation for the original problem. We note that the presence of the first derivative term makes the problem non-self-adjoint (see Appendix B). Motivated by the fact that the propagation of information is in the direction of the velocity U, finite difference practitioners were the first to overcome the bad approximation problem of the central difference method. They used one-sided finite differences to approximate the first derivative. 4-7 Thus in place of Eq. (2.20a) and with positive U, the approximation was put as
de
~a --~a--1
(2.22) dx h changing the central finite difference form of the governing equation approximation [as given by Eq. (2.18)] to
(-2Pe
-
1)qSa_1 q- (2 +
2Pe)()
a -
~a+l "+"
Qh 2
= 0
(2.23)
With this upwind difference approximation, non-oscillatory solutions are obtained through the whole range of Peclet numbers as shown in Fig. 2.3 by curves labelled c~ = 1. Now exact nodal solutions are obtained for pure diffusion (Pe = 0) and for pure convection (Pe = c~); however, results for other values of Pe are generally not accurate. How can such upwind differencing be introduced into a finite element scheme and generalized to more complex situations? This is the problem that we now address and indeed show that, for linear one-dimensional elements, this form of finite element solution can also result in exact nodal values for all Peclet numbers.
2.2.2 Petrov-Galerkin methods for upwinding in one dimension The first possibility is that of the use of a Petrov-Galerkin type of weighting in which Wa ~ Na. 8-11 Such weighting was first suggested by Zienkiewicz et al. 8 in 1976 and
The steady-state ....
,a, . . . .
- - o-
-
------c
problem in one dimension
S t a n d a r d Galerkin a = 0 Petrov--Galerkin a - 1.0 (full u p w i n d difference)
Petrov-Galerkin a = aopt Exact
Pe = UN2k = 0
1.0
Z~=O=O (Allexact) i n
r
Pe= 1.0
Pe = 2 . 5
(Exact)
:
;
;,,
L
;,:'
Exact
L
p
'',, .] rl
Fig. 2.3 Approximations to Udd21dx- k d 2~/dx 2 = 0 with ~(0) = 0 and ~(L ) = 1 for various Peclet numbers.
used by Christie et al. 9 In particular, again for elements with linear shape functions Na, shown in Fig. 2.1, we shall take weighting functions constructed as shown in Fig. 2.4 so that Wa - ga -~- ogW: (2.24) where Wa is such that (to obtain finite difference equivalent) f~ Wa dx e
h
(2.25t
35
36 Convectiondominated problems
! Na i I ~
L
~'ll'~,,i,'II"I"~I"I"'~,i,,,i,,
h
r'-
A
P]
!
! !
!
I
o
!
; or
i~ , i i !
~
**
os
i ! !
i
Fig. 2.4 Petrov-Galerkinweight function W
a "-
B
Na q- 0/. W a.
Continuous and discontinuous definitions.
the sign depending on whether U is a velocity directed toward or away from the node. With this approximation Eq. (2.15) becomes
U
+ Q
\ dx
- (Ua +
dx
d--x
(2.26)
W;)On I = 0 Fq
Various forms of Wa are possible, but the most convenient is the following simple definition which is, of course, a discontinuous function (continuity requirements are discussed below) hUdNa W2 = (2.27) 2 ]UI dx where [U] denotes absolute value. With the above weighting functions the approximation from Eq. (2.15) for a typical node a becomes
[ - P e ( a + 1)
-
1]~a_ 1 "~- [2 + 2a(Pe)]~a -+- [-Pe(ce - 1)
--
1]~a_F1 "~-
Qh 2
=0 (2.28)
where Q is assumed constant for the whole domain and equal length elements are used. Immediately we see that with a = 0 the standard Galerkin approximation is recovered [Eq. (2.18)] and that with a = 1 the full upwind form [Eq. (2.23)] is available, each giving exact nodal values for purely diffusive or purely convective cases respectively (Fig. 2.3). Now if the value of a is chosen as O~ - -
O~op t
--
coth
IPel
IPel
(2.29)
The steady-state problem in one dimension 1.0 0.8 0.6
~ o
~
I
I
2
!
3
I
4 Pe
I
I
5
6
7
Fig. 2.5 Critical (stable) and optimal values of the 'upwind' parameter oL for different values of Pe= Uh/2k.
then exact nodal values will be given for all values of Pe. The proof of this is given in references 9 and 12 for the present, one-dimensional, case where it is also shown that if OL >
OLcrit =
1
1
(2.30)
IPel
oscillatory solutions never arise. Figure 2.5 shows the variation of OLopt and
OLcrit with
Pe. Although the proof of optimality for the upwinding parameter was given for the case of constant coefficients and constant size elements, nodally exact values will also be given if c~ = C~optis chosen for each element individually. We show some typical solutions in Fig. 2.613 for a variable source term Q = Q(x), convection coefficients U = U(x) and element sizes. Each of these is compared with a standard Galerkin solution, showing that even when the latter does not result in oscillations the accuracy is improved. Of course in the above examples the Petrov-Galerkin weighting must be applied to all terms of the equation. When this is not done (as in simple finite difference upwinding) totally wrong results will be obtained, as shown in the finite difference results of Fig. 2.7, which was used in reference 14 to discredit upwinding methods. The effect of c~ on the source term is not apparent in Eq. (2.28) where Q is constant in the whole domain, but its influence is strong when Q = Q (x).
Continuity requirements for weighting f u n c t i o n s The weighting function W a (or Wa) introduced in Fig. 2.4 can of course be discontinuous as far as the contributions to the convective terms are concerned [see Eq. (2.17)], i.e.
L
Wa
~d(UNb) dx dx
or
WaU
dx
dx
Clearly no difficulty arises at the discontinuity in the evaluation of the above integrals. However, when evaluating the diffusion term, we generally introduce integration by
37
38
Convection dominated problems ~x 10 4
d2r
d~
-~x 2 +200--=x
dx
O<x
"~ 0.3
-0.3
(a)
I.
I
I
I
I
,1
I
I
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 Normalized length
1.3 [] 0=1/2
,.. t~ -~ 0.8 o
OO=0
"1o o
"B 0 . 3 -
.
-0.3-
L !
I
I
I
I
I
I
I
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
Normalized length
(b)
Fig. 2.21 Propagation of a steep wave by Taylor-Galerkin process: (a) explicit methods C = 0.5, step wave at Pe = 12 500; (b) explicit methods C= 0.1, step wave at Pe = 12 500.
For a multidimensional problem, a degree of anisotropy can be introduced and a possible expression generalizing Eq. (2.133) is
[s =
CLap
where
v/=
h 2 IV/VjJ
Ivl
(2.134)
oe OXi
Other possibilities are open here and numerous papers have been devoted to the subject of 'shock capture'. We will return to this problem in Chapter 7 where its importance in the high speed flow of gases is paramount.
69
70
Convection dominated problems
A
~
.
_
A
V
v
c=o.~, c~=o
c=o.~,c~=~ Fig. 2.22 Propagationof a steep front in Burger's equation with solution obtained using different values of Lapidus C v = CLap.
Part II1" Boundary conditions
ii~iiii!ii~i!i}i}i~i~i~~~~~i i~i~}iii~!i~ii}!ii~i~i~i}i~~ii~iii}!ii~!~ii~ii!}!ii~ii!ii~i!iii~iiii~}i~iiii~ii~ii~ii~~i!i!i!ii!~ii~ii~ii~ii~ii~iii~i~c~ii~i~!i!i~!i!i~i~i}iii~iiii!i!~i!iiiiiii!ii!~~~i~i!iiii!i~i}i~i}iiii~!iiii!i!ii!iiii!iii!ii{iiiii In pure convection problems the equation reduces to
or ~_ui ~x/4Or -~-Q -- 0
(2.135)
It is clear that only the 'inlet' values of q~can be set. By inlet we mean those where
Ui ni
--0.4-0.2
i
0
-0.4
(c) R e = 1000.
,
-0.2
0
0.2 0.4 0.6 Horizontal velocity
* I
0.8
(d) R e = 5000.
Fig. 4.4 Incompressible flow in a lid-driven cavity. Horizontal velocity distribution at different Reynolds numbers along the mid-vertical line.
In Fig. 4.9, the unstructured mesh used and the contours of horizontal velocity component and pressure are given. The mesh is finer near the walls and coarser away from the walls. The pressure and velocity distributions shown in Figs 4.9(b) and (c) are in good agreement with the available data. In Fig. 4.10, the numerical data are compared against the experimental data in the recirculation zone. As seen the agreement between the numerical and experimental datal~ is excellent.
Example 4.3 Steady flow past a sphere
The next problem considered is fully three dimensional and shows flow past a sphere. The computational domain is a rectangular imaginary box of length 25 D, where D is the diameter of the sphere, with the downstream boundary located at 20D from the centre of the sphere. The four side walls are located at a distance of 5 D from the centre of the sphere. All four confinement walls are assumed to be slip walls with normal velocity equal to zero. The inlet velocity is assumed to be uniform and the no-slip condition prevails on the sphere surface. This problem is solved using the fully explicit form of the CBS scheme.
117
118
Incompressible Newtonian laminar flows 0.6,
i
i
,
0.6
,
0.4
cBs t
0.4 l
-~-~8020
CBS
-~~o0.2
>
0
0
>a~ -0.2
>
-0.4
-0.4
-0 6 "0
, 0.2
, , 0.4 0.6 Horizontal distance
0.8
-0.6
1
0'.2
(a) Re = 0.
0.6
.
.
.
0'.6
0'.8
(b) Re = 4 0 0 .
.
0.6
0.4
0.4 cBs
@9 0.2
O mO
>
0~.4
Horizontal distance
GhiaetaL
*
o
02
cBs
~
GhiaetaL
*
0.
. uO
>
-0.2
_
-0.4 -0"60
-0.4 i 0.2
0.4 0.6 Horizontal distance (c) Re = 1000.
0.8
1
-0 6 "0
, 0.2
, , 0.4 0.6 Horizontal distance
0.8
1
(d) Re = 5 0 0 0 .
Fig. 4.5 Incompressible flow in a lid-driven cavity. Vertical velocity distribution at different Reynolds numbers along the mid-horizontal line.
For this problem, an unstructured grid containing 953 025 tetrahedral elements has been used. This mesh is generated using the PSUE mesh generator. TM12 Figure 4.11 (a) shows a portion of the surface mesh and Fig. 4.11 (b) shows a sectional view. The mesh is refined close to the sphere surface and in the rear where recirculation is expected. Figure 4.12 shows the contours of the Ul component of the velocity and the pressures computed at Reynolds numbers of 100 and 200. The coefficient of pressure, Cp, values on the surface along the flow axis are shown in Fig. 4.13. The non-dimensional Cp is calculated as Cp = 2 ( p - Pref) (4.10) where eref is a reference pressure (free stream value). Note that the results used for comparison were generated using very fine structured meshes. TM14 It should also be noted that all the results differ from each other close to the separation zone.
Example 4.4
Transient flow past a circular cylinder
This is a popular test case for validating the transient part of numerical schemes. Many other problems of interest can also be solved for transient accuracy but in this section only flow over a single circular cylinder is considered.
Use of the CBS algorithm for incompressible flows 119
(a) Unstructured mesh.
(c) u3 contours.
(b) Ul contours.
(d) Pressure contours.
Fig. 4.6 Incompressibleflow in a 3D lid-driven cavity. Mesh and contours at Re = 400.
Problem definition is standard. The inlet flow is uniform and the cylinder is placed at the centreline between two slip walls. The distances from the inlet and slip walls to the centre of the cylinder are 4D, where D is the diameter of the cylinder. Total length of the domain is 16D. A no-slip condition is applied on the cylinder surface. The initial values of horizontal velocity were assumed to be unity and the vertical component of velocity was assumed to be zero all over the domain.
120 Incompressible Newtonian laminar flows Re = 400
1
0.1
'Explicit-Semi-impli.---
0.1
,
Re = 5000
,
' Explicit'--Semi-impli. --..
0.01
0.01
0.001 E
0.001 0.0001
0.0001 1e-05
1e-05
I e-06
1e-06 I e-O;
,
0
1000
2000
3000
4000
5000
6000
1e-07
0
5(~00
No. time steps
10C)00 15()00 NO. time steps
20()00
25 ()00
(b) Re = 5000.
(a) Re = 400.
Fig. 4.7 Lid-driven cavity. Steady-state convergence histories for Re = 400 and 5000. Comparison between fully explicit (artificial compressibility) and semi-implicit schemes.
Experimental Ul and u2 = 0
\ UlU
o
p=O
Fig. 4.8 Incompressible flow past a backward facing step. Geometry and boundary conditions.
(a) Unstructured mesh.
(b) Ul velocity contours.
(c) Pressure contours. Fig. 4.9 Incompressible flow past a backward facing step. Mesh and contours of ul velocity and pressure,
Re = 229.
Use of the CBS algorithm for incompressible flows !
[]
I.
\
Exp. [] ~ CBS-AC ........
2.5 (=
n =.
i
I
i
n
0
II
~
0 r
=-
-o 1.5 m
b
=. "
=.-
nl
,i
,4,..o r .m
i .i
.=
r .B
i
1=
>
" | i
1 .i
.
0.5
0
2
.. i
4
.,
~
"
.a
[] !
|. ._
=.-
g
/
|
_
|
10 6 8 Horizontal velocity
|
12
=,.
14
16
Fig. 4.10 Incompressible flow past a backward facing step. Comparison between experimental 1~ and numerical data, Re = 229.
(a) Unstructured surface mesh. Fig. 4.11
(b) Unstructured mesh, cross-section.
Incompressible flow past a sphere. Unstructured mesh.
121
122
Incompressible N e w t o n i a n laminar flows
: :
: iiiiii!!i~ ii!!::i!ili !::!ii i:.!ii::i:i::ili i ::: ::i~ iiii~!!i ii i :: ~' :~::~i::ii!!ii~ii~i~i~:~!i~ii!~:!iiii::i:~i!ii~i~i~!!i~ii!i~ii!~i::i~i~iii!ii!iii!~i~i~!~ii~i~ii~iiii::~
................................ .................. ~:~,:~:.............. ~ :::::~~i~::~i~i~:~:~i~!i~`i!ii~i!~!ii~i~ii~~
~:~i~~'ii:
~i.
.
: : ~ii~i~. ii::::~. :: i i ~,.": .~i~!~'~:~'.....
i~
................ ~:~:~:~:~:!~!~:~:~~i!!i~:~:~:~:~:~~i :~:~~i ~i ~i ~i :~:~:~ ~i
:::!i ':!i!i !i i i i!i i i i i i!i i i i !i i i i i i i i~i i :'.i!.i!!i~ii .!:~i..... ::: ~::i~i::i~:i:.i~i:!ii~:i~!~i~i~iii:,i:i:ii~iii~iii:!ii:i:i:::iii:~. ,~ii~i::i if! :i~:.i:;.i:.~::~iii'~',i:,':t~;ii:...... ,iii'~ii!i:,ii
9
iii!
..........
,,
;i........
~!;iii;,,i:,iiiii~:iiiii~i~iiiii~iiii,i:!iiiii!ii!ii,i!i!!i
.....
i:i::,i!iii!,,i,~,iiiiiii:,!~,!i!%i!i~i~,iiiii~iii,,i .::: 9 :~ iiii:.::iiiiiiiii!iiiii'i!!iiiiiii!i!i!i!i':il
,:.: ~.;~~:; i:,:::,:::',il ............................................................. iiiiiiiiiiiiii!~iiii~:ii~i iiii!!iliiiii~
:
(b) Pressure contours, Re = 100.
(a) u~ contours, Re= 100.
........ ~........................................... :~',~ :!:"~:: ~ !:: ~::i!':'~!!~i ' i!! i i~!!i!!!i!i',i !~!~i!:i~i i!i !i ~!~iiii!i~i!i!i~i! i~! ......'~:,::::":::J:';::......'~"~i ~!:::::::::::::::::: i~!:,!i',~i!~'i,i:.::~: .::.::::::~::..-.:: i:!i~i~!~i!~i~:.::..:..:..:. i~:~:~i!~i~!~i~i!~i~!i!~ ................. ..:~..:..:.: 9 ............
....... ,.-~.
.....
!i'~i: ! !i~i~i'!i!i!
.::.:~~.:..::i~:.:
.i~!!:.~!~!i!i!,!. ii: ::ii :~i:::;i.....l
(d) Pressure contours, Re = 200.
(c) Ul contours, Re = 200. Fig. 4.12
Incompressible flow past a sphere. Contours of ul velocity and pressure.
I
~
0.6
i
i
G u ,cat PrdsAs?t:~;97~
•
/
I
0.6
0.2
ro~.0"2
-0.2
-0.2
-0.6
-0.6
-1
0
i
90
i
i
180 270 Theta (degrees) (a) Re= 100.
360
"'~
-1
I
0
i
i
Present scheme G. ulcat and Asian (1997)
90
I
•
/ ~/
I
180 270 Theta (degrees) (b) Re = 200.
360
Fig. 4.13 Incompressible flow past a sphere. Coefficient of pressure distribution on the surface along the flow direction.
Adaptive mesh refinement 123 Figure 4.14 shows the mesh used in two dimensions and the solution obtained. As seen the mesh close to the cylinder is very fine in order to capture the boundary layer and separation. Figures 4.14(b) and (c) show the time history of vertical velocity component at the mid-exit point and drag coefficient. Both the histories are in good agreement with many reported results.
4.2.3 Quasi-implicit solution We have already remarked in Chapter 3 that the reduction of the explicit time step due to viscosity can be very inconvenient and may require a larger number of time steps. The example of the cavity is precisely in that category and at higher Reynolds numbers the reader will certainly note a very large number of time steps are required before results become reasonably steady. In quasi-implicit form, the viscous terms of the momentum equations are treated implicitly. Here the time step is governed only by the relation given in Eq. 4.7. We have rerun the problems with a Reynolds number of 5000 using the quasi-implicit solution15 which is explicit as far as the convective terms are concerned. The solution obtained is shown in Fig. 4.15 for Re = 5000.
We have discussed the matter of adaptive refinement in Chapters 13 and 14 of reference 1 in some detail. In that volume generally an attempt is made to keep the energy norm error constant within all elements. The same procedures concerning the energy norm error can be extended of course to viscous flow especially when this is relatively slow and the problem is nearly elliptic. However, the energy norm has little significance at high speeds and here we revert to other considerations which simply give an error indicator rather than an error estimator. Two procedures are available and will be used in this chapter as well as later when dealing with compressible flows. References 1674 list some of the many contributions to the field of adaptive procedures and mesh generation in fluid dynamics.
4.3.1 Second gradient- (curvature) based refinement Here the meaning of error analysis is somewhat different from that of the energy norm and we follow an approach where the error value is constant in each element. In what follows we shall consider first-order (linear) elements and the so-called h refinement process in which increased accuracy is achieved by variation of element size. The p refinement in which the order of the element polynomial expression is changed is of course possible. Many studies are available on hp refinements where both h and p refinements are carried out simultaneously. This has been widely studied by Oden et al.27, 28, 44, 45 but we believe that such refinements impose many limitations on mesh generation and solution procedures and as most fluid mechanics problems involve an explicit time marching algorithm, the higher-order elements are not popular.
124 IncompressibleNewtonian laminar flows
(a) Unstructured mesh. 0.4o" ~ ,-, E o o
0.30.2-
o
0
o
i ~
t
i
_~
I
0.1
>
"~ -0.1 ._o 1:
>
-0.2 -0.3 -0.4 0
1.55
1.5
50
100 150 200 Non-dimensional real time (b) Vertical velocity fluctuation at the exit mid-point.
II
1
1.45
1.4
1.35
1.3
I
0
I
50
I
I
100 150 Non-dimensional real time (c) Drag history.
Fig. 4.14 Transient flow past a circular cylinder, Re = 100.
200
Adaptive mesh refinement 125 Pressure contours
Streamlines
ul velocity distribution along mid-vertical line i
i
-0.4
-0.2
l
i
0.2 Ul
0.4
'Ghia' m 'Present'---
0.8
0.6
# 0.4
0.2
0
Fig. 4.15
-0.6
0
I
I
0.6
0.8
1
Lid-driven cavity. Quasi-implicit solution for a Reynolds number of 5000.
The determination of error indicators in linear elements is achieved by consideration of the so-called interpolation error. Thus if we take a one-dimensional element of length h and a scalar function q~, it is clear that the error in q5is of order O (h 2) and that it can be written as (see reference 21 for details) e
--
d2~ d2~ h #p_ ~h = ch2_d~xZ ~ ch 2 dx z
where ~bh is the finite element solution and c is a constant.
(4.11)
126
Incompressible Newtonian laminar flows x2
Exact
Linear
A v
lw
nl
n2
A v
n I , n 2, nodes
h, element size > Xl
Fig. 4.16
Interpolation error in a one-dimensional problem with linear shape functions.
If, for instance, we further assume that q5 = q5h at the nodes, i.e. that the nodal error is zero, then e represents the values on a parabola with a curvature of d 2q5h/dx 2. This allows c, the unknown constant, to be determined, giving for instance the maximum interpolation error as (see Fig. 4.16) emax
-- l h 2 d2~bh 8 dx 2
(4.12)
or an RMS departure error as 1
h2
eRMS --
d 2~bh dx 2
(4.13)
In deducing the expressions (4.12) and (4.13), we have assumed that the nodal values of the function ~b are exact. As is shown in reference 1 this is true only for some types of interpolating functions and equations. However, the nodal values are always more accurate than those noted elsewhere and it would be sensible even in one-dimensional problems to strive for equal distribution of such errors. This would mean that we would now seek an element subdivision in which h 2d2~h dx 2
=
C
(4.14)
To appreciate the value of the arbitrary constant C occurring in expression (4.14) we can interpret this as giving a permissible value of the limiting interpolation error and simply insisting that h2d2~ h dx 2 where
e p --- C
< ep
-
is the user-specified error limit.
(4.15)
Adaptive mesh refinement 127 If we consider the shape functions of ~bto be linear then of course second derivatives are difficult quantities to determine. These are clearly zero inside every element and infinity at the element nodes in the one-dimensional case or element interfaces in two or three dimensions. Some averaging process has therefore to be used to determine the curvatures from nodally computed ~bvalues. However, before discussing the procedures used for this, we must note the situation which will occur in two or three dimensions. The extension to two or three dimensions is of course necessary for practical engineering problems. In two and three dimensions the second derivatives (or curvatures) are tensor valued and given as
OXi OXj
(4.16)
and such definitions require the determination of the principal values and directions. These principal directions are necessary for element elongation which is explained in the following section. The determination of the curvatures (or second derivatives) of ~bh needs of course some elaboration. With linear elements (e.g. simple triangles or tetrahedra) the curvatures of ~bh which are interpolated as q5h =Nq~
(4.17)
are zero within the elements and become infinity at element boundaries. There are two convenient methods available for the determination of curvatures of the approximate solution which are accurate and effective. Both of these follow some of the matter discussed in Chapter 13 of reference 1 and are concerned with recovery. We shall describe them separately.
4.3.2 Local patch interpolation In the first method we simply assume that the values of the function such as pressure or velocity converge at a rate which is one order higher at nodes than that achieved at other points of the element. If indeed such values are more accurate it is natural that they should be used for interpreting the curvatures and the gradients. Here the simplest way is to assume that a second-order polynomial is used to interpolate the nodal values in an element patch which uses linear elements. Such a polynomial can be applied in a least square manner to fit the values at all nodal points occurring within a patch which assembles the approximation at a particular node. For triangles this rule requires at least five elements that are assembled in a patch and this is a matter easily achieved. The procedure of determining such least squares is given fully in Chapter 13 of reference 1 and will not be discussed here. However, once a polynomial distribution of, say, q~ is available then immediately the second derivatives of that function can be calculated at any point, the most convenient one being of course the point referring to the node which we require. On occasion, as we shall see in other processes of refinement, it is not the curvature which is required but the gradient of the function. Again the maximum value of the gradient, for instance of ~b, can easily be determined at any point of the patch and in particular at the nodes.
128 IncompressibleNewtonian laminar flows
4.3.3 Estimation of second derivatives at nodes In this method we assume that the second derivative is interpolated in exactly the same way as the main function and write the approximation as (02q5)
OXiOXj
h
(
02q~ ) *
(4.18)
= N OxiOxj
This approximation is made to be a least square approximation to the actual distilbution of curvatures, i.e. OXi OXj
d ~2 = 0
(4.19)
and integrating by parts to give
(OXiOXj
OXiOXj) d~'2-- -M- (
0NT 0N d~.~) ~
Oxi Oxj
(4.20)
where M is the mass matrix given by M - ~ NTN dr2
(4.21)
which of course can be 'lumped'.
4.3.4 Element elongation Elongated elements are frequently introduced to deal with 'one-dimensional' phenomena such as shocks, boundary layers, etc. The first paper dealing with such elongation was presented as early as 1987 by Peraire et al. 21 But the possible elongation was limited by practical considerations if a general mesh of triangles was to be used. An alternative to this is to introduce a locally structured mesh in shocks and boundary layers which connects to the completely unstructured triangles. This idea has been extensively used by Hassan et al., 46' 49 Zienkiewicz and W u 43 and Marchant et al. 58 in the compressible flow context. In both procedures it is necessary to establish the desired elongation of elements. Obviously in completely parallel flow phenomena no limit on elongation exists but in a general field the elongation ratio defining the maximum to minimum size of the element can be derived by considering curvatures of the contours. Thus the local error is proportional to the curvature and making h 2 times the curvature equal to a constant, we immediately derive the ratio hmax/hmin. In Fig. 4.17, X~ and X2 are the directions of the minimum and maximum principal values of the curvatures. Thus for an equal distribution of the interpolation error we can write for each node* h2nl~22 1-
I-c
(4.22)
*Principal curvatures and directions can be found in a manner analogous to that of the determination of principal stresses and their directions. Procedures are described in standard engineering texts.
Adaptive mesh refinement 129 X1 X2
~
S hmin = hmax
Fig. 4.17 Elementeloncjation~ and minimum and maximumelementsizes.
which gives us the stretching ratio s as
s =
hmax = min
O2~b
(4.23)
With the relations given above, we can formulate the following steps to adaptively refine a mesh: 1. Find the solution using an initial coarse mesh. 2. Select a suitable representative scalar variable and calculate the local maximum and minimum curvatures and directions of these at all nodes. 3. Calculate the new element sizes at all nodes from the maximum and minimum curvatures using the relation in Eq. (4.22). 4. Calculate the stretching ratio from the ratio of the calculated maximum to minimum element sizes [Eq. (4.23)]. If this is very high, limit it by a maximum allowable value. 5. Remesh the whole domain based on the new element size, stretching ratios and the direction of stretching. To use the above procedure, an efficient unstructured mesh generator is essential. We normally use the advancing front technique operating on the background mesh principle 21 in most of the examples presented here.* The information from the previous solution in the form of local mesh sizes, stretching ratio and stretching direction are stored in the previous mesh and this mesh is used as a background mesh for the new mesh. In the above steps of anisotropic mesh generation, to avoid very small and large elements (especially in compressible flows), the minimum and maximum allowable sizes of the elements are given as inputs. The maximum allowable stretching ratio is also supplied to the code to avoid bad elements in the vicinity of discontinuities. It is generally useful to know the minimum size of element used in a mesh as many flow solvers are conditionally stable. In such solvers the time step limitation depends very much on the element size. *Another successful unstructured mesh generator is based on Delaunay triangulation. The reader can obtain more information by consulting references 53-55, 64, 66, 69-73.
130
IncompressibleNewtonian laminar flows The procedure just described for an elongated element can of course be applied for the generation of isotropic meshes simply by taking the maximum curvature at every point. The matter to which we have not yet referred is that of suitably choosing the variable q~to which we will wish to assign the error. We shall come back to this matter later but it is clear that this has to be a well-representative quantity available from the choice of velocities, pressures, temperature, etc.
4.3.5 First derivative- (gradient) based refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
: .......
The nature of the fluid flow problems is elliptic in the vicinity of the boundaries often forming so-called viscous boundary layers. However, at some distance from the boundaries the equations become almost hyperbolic. For such hyperbolic problems it is possible to express the propagation type error in terms of the gradient of the solution in the domain. In such cases the error can be considered as
o4,
h--~ - C
On
(4.24)
where n is the direction of maximum gradient and h is the element size (minimum size) in the same direction. The above expression can be used to determine the minimum element size at all nodes or other points of consideration in exactly the same manner as was done when using the curvature. However, the question of stretching is less clear. At every point a maximum element size should be determined. One way of doing this is of course to return to the curvatures and find the curvature ratios. Another procedure to determine the maximum size of element is described by Zienkiewicz and W u . 43 In this the curvature of the streamlines is considered and hmax is calculated as
hmax _ "O
or)
o (...
E
(u
.__.
o'1 m._ oJ (-..
II
('o
~ .......
"
H=l
t=7.s~ -
"
'-
, \
"~//////////// I
,,.,,:', ...'-,...J
'
"
t=5.0
~--t=7.5
f
'
~
,
.
9
9
**
.-
-
#
/
.=H=l
\ 7'
t-O
9
$ 9 IL
_
~, "1
. |
-
Fig. 10.3 /
Propagation of waves due to dam break (CLap = 0). Forty elements in analysis domain. C = ~ / q h - - 1, At=0.25. u
and the almost vertical waves progress into the two domains. This problem, somewhat similar to those of a shock tube in compressible flow, has been solved quite successfully even without artificial diffusivity.
ExamplelO.3
Bore
The final example of this section, Fig. 10.4, shows the formation of an idealized 'bore' or a steep wave progressing into a channel carrying water at a uniform speed caused by a gradual increase of the downstream water level. Despite the fact that the flow speed is 'subcritical' (i.e. velocity < ~/-~), a progressively steepening, travelling shock clearly develops. 10.4.2
Two-dimensional
periodic tidal motions
The extension of the computation into two space dimensions follows the same pattern as that described in compressible formulations. Again linear triangles are used to interpolate the values of h, U1 and/-/2. The main difference in the solutions is that of emphasis.
Example 10.4 Periodic wave
The first example of Fig. 10.5 is presented merely as a test problem with periodic surface elevation at the inlet. Here the frictional resistance is linearized and an exact solution
Examples of application 40 elements 2 11 Prescribed
water level
-1o
history
1.0
uI
0.5-
v
-0.5 Distance Fig. 10.4 A 'bore' created in a stream due to water level rise downstream (A). Level at A, r/= 1 - cos 7r t130 (0 < t < 30), 2 (30 < t). Levels and velocities at intervals of five time units, A t -- 0.5.
known for a periodic response 58 is used for comparison. This periodic response is obtained numerically by performing some five cycles with the input boundary conditions. Although the problem is essentially one dimensional, a two-dimensional uniform mesh was used and the agreement with analytical results is found to be quite remarkable.
Example 10.5 Bristol channel
In the second example we enter the domain of more realistic applications. 4, 5,52-54,59 Here the 'test bed' is provided by the Bristol Channel and the Severn Estuary, known for some of the highest tidal motions in the world. Figure 10.6 shows the location and the scale of the problem. The objective is here to determine tidal elevations and currents currently existing (as a possible preliminary to a subsequent study of the influence of a barrage which some day may be built to harness the tidal energy). Before commencement of the analysis the extent of the analysis domain must be determined by an arbitrary, seaward, boundary. On this the measured tidal heights will be imposed. This height-prescribed boundary condition is not globally conservative and can also produce undesired reflections. These effects sometimes lead to considerable errors in the calculations, particularly if long-term computations are to be carded out (like, for instance, in some pollutant dispersion analysis). For these cases, more general open
301
302
Shallow w a t e r problems
I'(.
AIOI
.................................
!
E ,2,o
~-
~
=
A
t=TI2
~>5
v
Analytical y = 0, :12 Ay y=O,•
TI40
Computed At =
8 0
I
I
I
~
I
I ~
x
L
I0~'---9 Analytical
,-.
A v
t=3TI4
5
y = 0, .+.2 Ay } y = 0, +_Ay
Computed At= TI40
Z
O v
Z"
~
~0
-5
i
i
I
I
:~
I
I
i
~
I
I
~-'~
I
x= L
R
/////
///
//.
/ ///
//
L = 22~Lx
Fig. 10.5 Steady-state oscillation in a rectangular channel due to periodic forcing of surface elevation at an inlet. Linear frictional dissipation? 2
Examples of application .i
E
Stockpole Quay
om m
t-l,_
Beachley w
Swansea
Newp~, j ~
Port Talbot
Worms Head
Porthcawl
E
Ilfracombe
E
/1" Clevedon
Barry Lynmouth
,,.,= ...,,,
Cardiff
,'
Weston-Super-Mare
Minehead Watchet Hinkley Pt w
Q)
I
0
I
I
I
I
I
50 km
I
Fig. 10.6 Locationmap. BristolChanneland SevernEstuary.
boundary conditions can be applied, as, for example, those described in references 35 and 36. The analysis was carried out on four meshes of linear triangles shown in Fig. 10.7. These meshes encompass two positions of the external boundary and it was found that the differences in the results obtained by four separate analyses were insignificant. The mesh sizes ranged from 2 to 5 km in minimum size for the fine and coarse subdivisions. The average depth is approximately 50 m but of course full bathygraphy information was used with depths assigned to each nodal point. The numerical study of the Bristol Channel was completed by a comparison of performance between the explicit and semi-explicit algorithms. 53 The results for the coarse mesh were compared with measurements obtained by the Institute of Oceanographic Science (lOS) for the M2 tide, 59 with time steps corresponding to the critical (explicit) time step (50 s), four times (200 s) and eight times (400 s) the critical time step. A constant real friction coefficient (Manning) of 0.038 was adopted for all of the estuary. Coriolis forces were included. The analysis proved that the Coriolis effect was very important in terms of phase errors. Table 10.1 represents a comparison between observations and computations in terms of amplitudes and phases for seven different points which are represented in the location map (Fig. 10.6), for the three different time steps described above. The maximum error in amplitude only increases by 1.4% when the time step of 400 s is used with respect to the time step of 50 s, while the absolute error in phases ( - 13 o) is two degrees more than the case of 50 s ( - 11 o). These bounds show a remarkable accuracy for the semi-explicit model. In Fig. 10.8 the distribution of velocities at different times of the tide is illustrated (explicit model).
303
e-o
9
E _m o o
"o o c(:o b,. to
2.i u_
o
o
E
o -(3 o c,~. co ,,-
.LJ 0
to
t~
o
,).-
co
ffl co
E
o o
(o (,o
o
o cOd O0 CO u..
r-. (D
(1)
E
8 c-
nO
.C o
o =~
00
0 0~ 9 o.
o
cD
D,.
~-- c~ c'o ,~- LO CO D,. O 0 0 ~
~'-~o o Or)
o
co
cO
rL.. O.J (I;
c-c'-
-0 c-cO
co r--
o
(I.;
c-
E
+.~ rE . '..., ,
"------- A ------- g E
0
I
.,,
l
5
, "-...1..."
l
10
/
l
15 t(h)
I
20
25
30
Water elevations for points A, B, E
8 ~6 _
/
A
~----
Computed Measured
10 ~-
c 4 _ .o 2
LU 0
8
MWL
;
-2 12
.... Computed
~'9
v
>
/'k
14
16
18 20 t(h)
22
24
26
61""5 12
'l 14
16
Point B
J.
i
18 20 t(h)
I 22
r-24
26
Point E Computed and measured elevations (d)
Fig. 10.9 Continued.
10.4.3 Tsunami waves A problem of some considerable interest in earthquake zones is that of so-called tidal waves or tsunamis. These are caused by sudden movements in the earth's crust and can on occasion be extremely destructive. The analysis of such waves presents no difficulties in the general procedure demonstrated and indeed is computationally cheaper as only relatively short periods of time need be considered. To illustrate a typical possible tsunami wave we have created one in the Severn Estuary just analysed (to save problems of mesh generation, etc., for another more likely configuration).
Example 10.7
Tsunami wave in Severn
Estuary
Here the tsunami is forced by an instantaneous raising of an element situated near the centre of the estuary by some 6 m and the previously designed mesh was used
307
308 Shallowwater problems (Fig. 10.7, FL). The progress of the wave is illustrated in Fig. 10.10. The tsunami wave was superimposed on the tide at its highest level - though of course the tidal motion was allowed for. This example was included in reference 5. One particular point only needs to be mentioned in this calculation. This is the boundary condition on the seaward, arbitrary, limit. Here the Riemann decomposition of the type discussed earlier has to be made if tidal motion is to be incorporated and note taken of the fact that the tsunami forms only an outgoing wave. This, in the absence of tides, results simply in application of the free boundary condition there. The clean way in which the tsunami is seen to leave the domain in Fig. 10.10 testifies to the effectiveness of this process.
10.4.4 Steady-state solutions On occasion steady-state currents such as may be caused by persistent wind motion or other influences have to be considered. Here once again the transient form of explicit computation proves very effective and convergence is generally more rapid than in compressible flow as the bed friction plays a greater role. The interested reader will find many such steady-state solutions in the literature.
ExamplelO.8 Steadystatesolution
In Fig. 10.11 we show a typical example. Here the currents are induced by the breaking of waves which occurs when these reach small depths creating so-called radiation stresses.6, 30,60 Obviously as a preliminary the wave patterns have to be computed using procedures to be given later. The 'forces' due to breaking are the cause of longshore currents and rip currents in general. The figure illustrates this effect on a harbour. It is of interest to remark that in the problem discussed, the side boundaries have been 'repeated' to model an infinite harbour series. 6~
Example lO.9 Supercriticalflow
Another type of interesting steady-state (and also transient) problem concerns supercritical flows over hydraulic structures, with shock formation similar to those present in high-speed compressible flows. To illustrate this range of flows, the problem of a symmetric channel of variable width with a supercritical inflow is shown here. For a supercritical flow in a rectangular channel with a symmetric transition on both sides, a combination of a 'positive' jump and 'negative' waves, causing a decrease in depth, appears. The profile of the negative wave is gradual and an approximate solution can be obtained by assuming no energy losses and that the flow near the wall turns without separation. The constriction and enlargement analysed here was 15 ~ and the final mesh used was of only 6979 nodes, after two remeshings. The supercritical flow had an inflow Froude number of 2.5 and the boundary conditions were as follows: heights and velocities prescribed in inflow (left boundary of Fig. 10.12), slip boundary on walls (upper and lower boundaries in Fig. 10.12) and free variables on the outflow boundary (fight side of Fig. 10.12). The explicit version with local time step was adopted. Figure 10.12 represents contours of heights, where 'cross'-waves and 'negative' waves are contained. One can observe the 'gradual' change in the behaviour of the negative wave created at the origin of the wall enlargement.
Examples of application
Time = 0
Fig. 10.10 Severntsunami. Generation during high tide. Water height contours (times after generation).
309
310
Shallow water problems
%
"
. . . .
-
---.. " - - " . "I..:.
'
'
,
-
,
,=,
_
,,
'
,Q ,
Fig. 10.11 Wave-inducedsteady-stateflow past a harbour.3~
Fig. 10.12 Supercritical flow and formation of shock waves in symmetric channel of variable width contours of h. Inflow Froude number = 2.5. Constriction" 15~
A special problem encountered in transient, tidal, computations is that of boundary change due to changes of water elevation. This has been ignored in the calculation
Shallow water transport
/•Boundary
at time tn
~
'
-
Boundaryat time t n +
Atn
!1
Fig. 10.13 Adjustmentof boundarydue to tidal variation. presented for the Bristol Channel-Severn Estuary as the movements of the boundary are reasonably small in the scale analysed. However, in that example these may be of the order of 1 km and in tidal motions near Mont St Michel, France, can reach 12 km. Clearly on some occasions such movements need to be considered in the analysis and many different procedures for dealing with the problem have been suggested. In Fig. 10.13 we show the simplest of these which is effective if the total movement can be confined to one element size. Here the boundary nodes are repositioned along the normal direction as required by elevation changes Aq. If the variations are larger than those that can be absorbed in a single element some alternatives can be adopted, such as partial remeshing over layers surrounding the distorted elements or a general smooth displacement of the mesh.
~i~ii~i~i!i~i~i!~i~i~i~i~~~i~i~~i~i~i~i~i~i~i~i!i~~ih~~i~i~~~!~~~~~~~~~!~ ~~~~~~~i~i~i~!~i~i~i~!~i~!~i~i~i~i~!~i~i~i~i~~~~~~ i~ Shallow water currents are frequently the carrier for some quantities which may disperse or decay in the process. Typical here is the transport of hot water when discharged from power stations, or of the sediment load or pollutants. The mechanism of sediment transport is quite complex 61 but in principle follows similar rules to that of the other equations. In all cases it is possible to write depth-averaged transport equations in which the average velocities Ui have been determined independently. A typical averaged equation can be written - using for a scalar variable [e.g. temperature (T) ] as the transported quantity - as
O(hT) C~
Jr-
O(hEtiT) OX i
0 (OT) hk ~xi
OX i
+R- 0
for i -- 1, 2
(10.24)
where h and ~i are the previously defined and computed quantifies, k is an appropriate diffusion coefficient and R is a source term. A quasi-implicit form of the characteristic-Galerkin algorithm can be obtained when diffusion terms are included. In this situation practical horizontal viscosity ranges (and
311
312
Shallowwater problems diffusivity in the case of transport equations) can produce limiting time steps much lower than the convection limit. To circumvent time step restriction imposed by the diffusion term, a quasi-implicit computation, requiring an implicit computation of the viscous terms, is recommended. The application of the characteristic-Galerkin method for any scalar transport equation is straightforward, because of the absence of the pressure gradient term. The computation of the scalar h T is analogous to the intermediate momentum computation, but now a new time integration parameter 03 is introduced for the viscous term such that 0 < 0 3 __ r > 2, k 1 = 107r Robin boundary conditions, plane wave
incident along x axis outer domain (2) 2 > r > 3, k 2 = 67r upper half plane wave basis finite elements, lower half plane analytical solution 3.4 dof per wavelength, L 2 error 0.4%, left real, right imaginary. Reproduced by permission of Elsevier.
on the surface of the water. Around the cylinder, the water is of depth h l up to a circular region of radius r2. Then for r > r2, the depth is h2. In Fig. 12.8, the outer region is deeper than the one around the cylinder (hi < h2). However, the theory remains the same if h~ > h2. The physical domain of this problem is infinite in extent. This means that it must be truncated at a finite distance from the scatterer to enable a numerical simulation. An analytical solution for the problem in terms of Bessel and Hankel functions was developed in reference 48. The problem was analysed for a range of different k values. The errors are summarized in Table 12.1. E"2 is the error in the L2 norm. The parameter 7-is the number of degrees of freedom per wavelength. Figure 12.8 shows contours of the real and imaginary components of the wave potential around the cylinder for kl -- 107r and k2 -- 67r. The results are similar to those of Huttunen et al. obtained using the ultra weak variational formulation.
12.7.2 Refraction caused by flows .................
.-..::::- .......................................................................
The problem of waves refracted by flows is more difficult than the case of refraction due to changing wave speed. The topic is dealt with briefly in Chapter 11. Recently Astley and Gamallo 5~ 51 have applied plane wave basis type methods to wave refraction in the presence of a known flow field in one and two dimensions. In the one-dimensional case the flow is along a duct with changing cross-sectional area. So the flow speed varies with position along the duct. x is the distance along the duct and A(x) is the corresponding cross-sectional area of the duct. The sound speed, Co(X), the density, Table 12.1
Plane wave scattering by a rigid circular cylinder, k l -- 2k2
kz
271"
471"
67/"
87r
107r
127r
1471"
167r
187r
207]"
7"
25.9
12.9
8.6
6.5
5.2
4.3
3.7
3.2
2.8
2.6
E'21%]
0.002
0.007
0.02
0.1
0.5
2.5
0.9
0.4
0.6
1.1
.--
,
369
370 Shortwaves
p(x), and the velocity, u0(x), along the duct have been previously found using the one-dimensional nozzle equation. The velocity is given as u(x) = dr where r is the acoustic velocity potential. The governing equation for small acoustic perturbations, derived from the linearized momentum and continuity equations, is then the convected wave equation
apodx
P~
-t- c 2 r
c~xx
_
co
~xx
c~
(12.50)
In the two-dimensional scheme the wavenumber varies in accordance with the prevailing flow. In this case uniform flow in the x direction, with Mach number M was considered in a duct, extending from x = 0 to x = L and of width a. The resulting convected wave equation in non-dimensionalized form is
o~y +
(1 - M 2 ) 02r ~ x
- 2ikM~x +
k2r - 0
(12.51)
The specific boundary conditions considered are as follows (1 - M 2 ) 0r ~ x- _ i k M r - -cos(mTry)
or
(1 - M 2 ) ~x - i k M r
0r Oy
= -ikr
=0
on
on on
x - 0
x = L/a
y--0,1
(12.52) (12.53) (12.54)
A polar plot of the wavenumber as a function of direction at any point takes the form of an ellipse. This is illustrated in Fig. 12.9. The local plane wave basis is also
ky/k
1/(1-M)
1/(1+M)
i
o
kx/k
(a) The continuous set of wavenumber vectors.
270 (b) A finite set of wavenumber vectors uniformly distributed.
Fig. 12.9 Wavenumber ellipse for the convective wave equation, Gamallo and Astley.sl Reproduced by permission from reference 51, copyright John Wiley & Sons Ltd.
Refraction
Trial basis function
l
Vlj(_~) = Nt(x,Y)exp(-ik(O/).x)
,, L
,
Local approximation function exp(-ik_(ej).x_)
"
"
'
I' I
Nodal interpolation function NI~
!
"-.. ,~. .............
". . . . . .
.~::::............
1 ~:,. I
, ~
"'-....~.::.~. . . . . . . . . . . . . .
Finite element mesh
~.. ........
Node I (a) Construction of the PUFEM trial basis Mean flow streamline FE mesh /~A
node
/
f~-
Local wavenumber ellipse aligned with the flow
(b) The local wavenumber ellipse Fig. 12.10 PUFEM basis, Gamallo and Astley. sl Reproduced by permission from reference 51, copyright John Wiley & Sons Ltd.
shown in Fig. 12.10. Astley and Gamallo report on the accuracy of the partition of unity elements. In summary they state that 'Clearly all of the PUM models offer a huge improvement over conventional low order FEM.' Astley and Gamallo report the same general experiences as in the zero flow case for such types of element, namely reduction in number of degrees of freedom compared with conventional elements and large condition numbers. The problem of completely general flow fields is more difficult. The plane waves are no longer solutions to the wave pattern in the presence of a
371
372
Short waves
0.8
0.6L~
Duct
"-- 0.4 ~ ~ , ~ , ~
0"2t t pinner'~ -o
t 1 i~ ', ---"~'---~--.-~ o skr,.... 'a9 / .~or, . " !
0.4 p ~
i
15
o.=~ o., o . ~
~
0,21
~.~ ~. ~.~.
~
x
:
......
S p i n ~
t
(a) Computational domain
/
.~,;Y" k,
i
,, 'o
.......d:6"'~
~.;\~ i. . . . . . .
o's . . . .
.....
,
\,
' .
",
: .o
lls ....... x
2
2s
(b) Mean flow Mach numbers
Fig. 12.11 Non-uniformaxisymmetricduct, Gamallo and Astley.51 Reproducedby permission from reference 51, copyright John Wiley & Sons Ltd.
general flow field. However, they are still a legitimate solution space, though with no guarantee of completeness. The refracted plane wave solutions can still be used on the assumption that they are a good approximation if the flow is not varying rapidly, and is fairly constant locally. Gamallo and Astley 5~ have applied the same elements to a general flow field through a non-uniform axisymmetric duct. Figure 12.11 shows the geometry, and Fig. 12.12 shows results obtained using PUFEM and Q-FEM (quadratic finite elements). They report that accurate solutions are obtained using 12 000 degrees of freedom with PUFEM, whereas 100000 degrees of freedom are needed for the quadratic conventional finite elements. If the conventional finite element mesh was made any coarser, pollution errors were encountered.
Spectral finite elements are elements which exploit choices in the selection of the finite element node locations and integration schemes. Conventional consistent mass matrices lead to a large though sparse system mass matrix. This is expensive in time stepping schemes, since even if the time stepping scheme is explicit the mass matrix has to be factorized and a back substitution carried out at each time step. If the mass matrix is lumped, the formulation is no longer strictly consistent and the results may be unreliable. If it were possible to integrate exactly the mass matrix by sampling only at the nodes within an element, then the mass matrix though theoretically consistent would be diagonal and explicit time stepping schemes would be much more economical. If
1.
Il
l !ii ill i ii ii :i !i~ I'
N i: i::~ li .~
o.5i
00 . . . .
~
0.5 . . . .
,
1
1.5
i)
~::~I~
2
2.5
X
(a) PUFEM, 12000 degrees of freedom (1500 points and 8 directions)
_15
-l:~llg
-
iti~:
I .
::!ii!i o.5' 0
0
"
.i,
~
-
! ii ,I
.
:..
i'i
ii~ ~!! 0.5
1
1.5
2
2.5
X
(b) Q-FEM, 100000 degrees of freedom
Fig. 12.12 Non-uniform axisymmetric duct. Acoustic pressure field (real part) for c~ = 25, Gamallo and Astley.51 Reproducedby permission from reference 51, copyright John Wiley & Sons Ltd.
Spectral finite elements for waves
a linear finite element is considered, then the general terms in the mass matrix are quadratic. If a Newton-Cotes formula is used to integrate the mass matrix exactly, then three sampling points are needed in one dimension. As there are only two nodes in a linear element, the number of nodes is inconsistent with an exact integration using a Newton-Cotes integration scheme. If Gauss-Legendre integration is used, then two sampling points are needed to exactly integrate the mass matrix. But these are not at the ends of the element and so the element cannot enforce continuity. However, the N point Gauss-Lobatto scheme integrates over the range - 1 to + 1, and samples at the two end points of the range and N - 2 internal points. The scheme integrates exactly powers of x up to 1 + 2(N - 2) - 1 = 2N - 3, for N sampling points. The scheme comes close to integrating the mass matrix exactly. For example, in a one-dimensional quintic finite element with six nodes, the highest power of x in the mass matrix would be 10. The corresponding Gauss-Lobatto scheme would integrate exactly powers of x up to 9. Such integration formulas appear to have been first adopted by Fried and Malkus. 52 They achieved orders of integration, using Gauss-Lobatto schemes which were sufficiently accurate to integrate the stiffness matrix exactly, though not the mass matrix. The Gauss-Chebyshev scheme is also close to integrating the mass matrix exactly. In both these types of element the internal nodes of an element are not equally spaced, but are located at the positions dictated by the integration formulas. The Gauss-LegendreLobatto scheme is given by
f
+l
1
2
f ( x ) d x ~,
n(n-
n-1
1)
+ y ~ wjf(xj)
[/(1) + / ( - 1 ) ]
j=2
(12.55)
where xj is the (j - 1)th zero of Pn'-I (X), where P(x) is a Legendre polynomial and the weights, w j, are given by Wj =
r
(12.56)
~
n ( n - 1)LPn_,(xj)J
2
The corresponding Gauss-Chebyshev-Lobatto points are given by 52-54 7ri -cos ~ (12.57) N where N is the number of integration points. The Gauss-Legendre-Lobatto elements have been investigated by Mulder and others 55-57 and the Gauss-Chebyshev-Lobatto elements by Dauksher, Gottlieb, Hesthaven and others. 53, 54,58-64 There are numerous references to these methods and the above citations are only given as examples of recent work. They are not meant to be exhaustive. In general the numerical investigations demonstrate that spectral elements outperform conventional finite elements in transient wave problems. They have lower dispersion and better phase properties. The Gauss-Chebyshev formulation has also been used for direct collocation approaches to wave problems. This will not be discussed here but references are available. 58-64 In their basic form the spectral elements are not applicable to problems meshed with triangles or tetrahedra. In order to do this, it is necessary to generalize the GaussLobatto and Gauss-Chebyshev-Lobatto schemes to triangles. It is notoriously difficult
373
374
Short waves
Fig. 12.13 Electromagnetic waves scattered by fighter aircraft. Projection of finite element mesh onto surface of aircraft. Results due to Hesthaven and Warburton and published in reference 63. Results reproduced with permission from Elsevier.
to generalize integration schemes from straight line segments, squares and cubes, to triangles and tetrahedra. Although there are proofs that efficient symmetrical integration schemes for triangles and tetrahedral exist there are no general (open ended) formulas for them, but only special results for certain numbers of points. 65' 66 Considerable effort has gone into generalizing the Gauss-Lobatto schemes for triangles and tetrahedra. The corresponding points are called Fekete points. The Fekete points are defined as the points which maximize the determinant of the Vandermonde matrix relating the coordinates of the sampling points to the polynomial coefficients. For large numbers of points the interpolating polynomials may become oscillatory and the Vandermonde matrix can become ill-conditioned. The Fekete points are closely related to the Lebesgue points, but are easier to determine. 67 Kamiadakis and Sherwin 68 give a very comprehensive treatment of spectral finite elements applied to CFD.
This method is frequently used in the time domain, although it can be applied to virtually any finite element problem and not just the Helmholtz wave equation. In the
Discontinuous Galerkin finite elements (DGFE) 375
discontinuous Galerkin method, elements are linked together by constraints, which approximately satisfy continuity of various quantities between elements. The jumps between elements are of the same magnitude as the truncation error. The most natural way of applying the constraints is probably the method of Lagrange multipliers. However, this has two disadvantages: the introduction of additional equations to solve and the indefiniteness of the resulting matrix. In practice the constraints are applied using the element variables themselves. This can be done using some kind of penalty formulation. The mathematical details of the method are explained in reference 5. A comprehensive set of papers on the method edited by Cockburn e t al. 69 are available. The paper by Zienkiewicz e t al. 49 gives an approachable introduction to the method. The method has been applied to a large range of problems, including waves. 7~Important applications to electromagnetic wave scattering problems are given by Hesthaven and Warburton. 71' 72 They explain in considerable detail the techniques needed to extract the full potential of the method. Some of the points are explained below. A significant property of the discontinuous Galerkin finite element method is that the associated mass matrix is local to each element. Furthermore, the affine nature of straight sided triangles and tetrahedra implies that their mass matrices differ only by a multiplicative constant. In practical computations, the relatively small reference triangle (or tetrahedron) mass matrix can be inverted in preprocessing, leading to an extremely efficient method. In the conventional finite element method the assembled stiffness and mass matrices are large sparse matrices, linking together all adjacent degrees of freedom. In the DGFE method the system stiffness matrix need not be formed, and only its product with the vector of field variables from the previous iteration need be retained. This greatly reduces the storage requirement of the method, which becomes O ( N ) , where N is the number of variables in the problem. The other features of the method, as developed by Hesthaven and Warburton and others, are that high-order finite elements are used. They show a 56 node tetrahedron, for example, and use tetrahedra with up to 286 nodes. The node locations are selected using special procedures, and the shape functions are formed using Lagrange polynomial interpolation. Error bound results, Eq. (12.1), show that for a given number of degrees of freedom in a wave problem, it is better to use higher-order polynomials. The use of higher-order elements involves the use of dense, local, reference element operator matrices. However, in the case of straight sided triangle meshes only one set of reference operator matrices is required. For a given operation, say differentiation, the reference element matrix is only loaded into the cache once and the field data in all elements can be differentiated with this one matrix and then physical derivatives are computed using the chain rule. Coupling this approach with standard, optimized, linear algebra mathematics libraries, leads to extremely efficient codes. Exact analytic integration is used where necessary. The authors analyse their scheme for consistency and accuracy. They present a number of results including the scattering of a plane wave by a sphere of radius a at k a - 10. Warburton 7~ also gives results for the scattering of electromagnetic waves by an F15 fighter aircraft. Results are shown in Figs 12.13 and 12.14. Eskilsson and Sherwin 7~ discuss the modelling of the shallow water equations using the discontinuous Galerkin method. They consider as an example the modelling of the
376 Shortwaves
Fig. 12.14 Electromagnetic waves scattered by fighter aircraft. Contours of electrical field on surface of aircraft. Results due to Hesthaven and Warburton and published in reference 63. Results reproduced with permission from Elsevier.
Port of Visby on the Baltic Sea. Figure 12.15 shows the mesh of elements which was used and the water depths, and Fig. 12.16 shows a snapshot of the surface elevations after 500 seconds.
1500 3
3
1000 ~'
i,j1 500
Berth no. 5 1. Generating boundary 2. Open boundary 3. Wall boundary
- direction /
0-
~
I
0
~
,
,
5
....
1obo ....
5 bo'
(a) Mesh and boundary conditions. Fig. 12.15 Harbour layout, Eskilsson and Sherwin.7~ Reproduced by permission from reference 70, copyright John Wiley & Sons Ltd.
Discontinuous G a l e r k i n f i n i t e e l e m e n t s (DGFE) 1500 d i i i i i i i i i i i!i!i-i l- 2
~i~ii - 3 -4 -5 -6 -7 -8 -9 -10
1000
500 -
0
Fig. 12.15
i
I 0
i
i
i
i
I 500
i
i
i
x (b) Depth.
i I 1000
i
i
i
i
I 1500
Continued.
Fig. 12.16 Snapshot of surface elevation after 500 seconds, Eskilsson and Sherwin. 7~ Reproduced by permission from reference 70, copyright John Wiley & Sons Ltd.
377
378
Short waves
,i'i!ii,'li!i!ii!i!ili~ilii~!:i'~!i,~' i~i~'i!i~il'iii,l'~,iiiliiiiii,i'i~i~ili'~ i'ii~,i'~i,~iii,~ii,i,~i~,ii,~' i!i~i,'~ilii!iii~i~iii,~,~,~::~ii~,'i~~ii,!~!i~i!~i~i~/~i!~~i~i~i~i~i~i~i~i~~ii~~ii~!i~ii!~i i~iii!iiii!!i!ii!ii~!!i!iiii!~ii~ii!~iii!i!i!i!ii~!~ii~iii~ii~ii~i~i~ii~!iii!i~ii~ii!ii~iiiiii!~iiiiiii~i~i,'!i~i~i~'ii~'ii~~i'~i',il,'i,'i!!~i i,'ili~:'i,~~i'i,!~'i,~' i,~i,!~i,~' i,~i,~i'',i~'i~i,~i,~'i,~ii,~:' The field of short wave modelling is currently the focus of intense research activity, not only using finite element, but also boundary integral and other domain- and boundarybased methods. At the m o m e n t there are a n u m b e r of promising algorithms, some of which have been described above. The most powerful finite element-based m e t h o d appears to be the discontinuous Galerkin method. Certainly this has achieved the solution o f problems containing the largest n u m b e r o f wavelengths. The other finite element-based algorithms do not seem to be quite so powerful. However, given the high level of research activity, this m a y change. The chief competing boundary-based m e t h o d seems to be the fast multipole method, based on a more efficient formulation of the boundary integral method.
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1. P. Bettess, O. Laghrouche and E. Perrey-Debain, editors. Short-wave scattering. Theme Issue, Philosophical Transactions Royal Society, Series A, volume 362, number 1816, 15 March 2004. 2. M. Ainsworth, P. Davies, D. Duncan, P. Martin and B. Rynne, editors. Topics in Computational Wave Propagation. Lecture Notes in Computational Science and Engineering, volume 51. Springer, Berlin, 2003. 3. T.J.R. Hughes. Multiscale phenomena: Green's functions, the Dirichlet to Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Meth. Appl. Mech. Eng., 127:387-401, 1995. 4. T. Stroubolis, I. Babu~ka and K. Copps. The design and analysis of the Generalised Finite Element Method. Comput. Meth. Appl. Mech. Eng., 181:43-69, 2000. 5. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: The Basis and Fundamentals. Elsevier, Amsterdam, 6th edition, 2005. 6. F. Ihlenburg and I. Babu~ka. Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. Int. J. Num. Meth. Eng., 38:3745-3774, 1995. 7. I. Babu~ka, E Ihlenburg, T. Stroubolis and S.K. Gangaraj. A posteriori error estimation for finite element solutions of Helmholtz' equations. Part I: the quality of local indicators and estimators. Int. J. Num. Meth. Eng., 40(18):3443-3462, 1997. 8. I. Babu~ka, E Ihlenburg, T. Stroubolis and S.K. Gangaraj. A posteriori error estimation for finite element solutions of Helmholtz equations. Part II: estimation of the pollution error. Int. J. Num. Meth. Eng., 40(21):3883-3900, 1997. 9. M. Ainsworth. Discrete dispersion relation for hp-finite element approximation at high wave number. SlAM J. Numer. Analysis, 42(2):553-575, 2004. 10. K. Morgan, O. Hassan and J. Peraire. A time domain unstructured grid approach to the simulation of electromagnetic scattering in piecewise homogeneous media. Comp. Meth. Appl. Mech. Eng., 152:157-174, 1996. 11. K. Morgan, P.J. Brookes, O. Hassan and N.P. Weatherill. Parallel processing for the simulation of problems involving scattering of electromagnetic waves. Comp. Meth. Appl. Mech. Eng., 152:157-174, 1998. 12. O.C. Zienkiewicz and P. Bettess. Infinite elements in the study of fluid structure interaction problems. In J. Ehlers et al. editors, Lecture Notes in Physics, 58. Springer-Verlag, Berlin, 1976. 13. P. Bettess and O.C. Zienkiewicz. Diffraction and refraction of surface waves using finite and infinite elements. Int. J. Num. Meth. Eng., 11:1271-1290, 1977.
References 379 14. R.J. Astley and W. Eversman. A note on the utility of a wave envelope approach in finite element duct transmission studies. J. Sound Vibration, 76:595-601, 1981. 15. R.J. Astley. Wave envelope and infinite elements for acoustical radiation. Int. J. Num. Meth. Fluids, 3:507-526, 1983. 16. E. Chadwick, P. Bettess and O. Laghrouche. Diffraction of short waves modelled using new mapped wave envelope finite and infinite elements. Int. J. Num. Meth. Eng., 45:335-354, 1999. 17. J.M. Melenk and I. Babu~ka. The partition of unity finite element method. Basic theory and applications. Comp. Meth. Appl. Mech. Eng., 139:289-314, 1996. 18. J.M. Melenk and I. Babu~ka. The partition of unity finite element method. Int. J. Num. Meth. Eng., 40:727-758, 1997. 19. P.M. Morse and H. Feshbach. Methods of Theoretical Physics, volumes 1 and 2. McGraw-Hill, New York, 1953. 20. T. Huttunen, P. Monk and J.P. Kaipio. Computational aspects of the ultra weak variational formulation. J. Comput. Phys., 182:27-46, 2002. 21. T. Huttunen, P. Monk, F. Collino and J. P. Kaipio. Computation aspects of the ultra weak variational formulation. SIAM J. Scientific Computing, 25(5): 1717-1742, 2004. 22. P. Mayer and J. Mandel. The finite ray element method for the Helmholtz equation of scattering: first numerical experiments. Report Number CU-CAS-00-20, College of Engineering, University of Colorado, Boulder, Colorado, USA. UCD/CCM Report 111, URL: http:// www-math.cudenver.edu/ccrn/reports.html, 1997. 23. A. de La Bourdonnaye. High frequency approximation of integral equations modelizing scattering phenomena. Mod. Math. Anal. Numer., 28(2):223-241, 1994. 24. A. de La Bourdonnaye. Convergence of the approximation of wave functions by oscillatory functions in the high frequency limit. C. R. Acad. Paris Sdr. L 318:765-768, 1994. 25. O. Laghrouche and P. Bettess. Short wave modelling using special finite elements - towards an adaptive approach. In J.R. Whiteman, editor, The Mathematics of Finite Elements and Applications X, pp. 181-194. Elsevier, 2000. 26. O. Laghrouche and P. Bettess. Short wave modelling using special finite elements. J. Comput. Acoustics, 8(1): 189-210, 2000. 27. P. Ortiz and E. Sanchez. An improved partition of unity finite element model for diffraction problems. Int. J. Num. Meth. Eng., 50:2727-2740, 2001. 28. O. Laghrouche, P. Bettess and R.J. Astley. Modelling of short wave diffraction problems using approximating systems of plane waves. Int. J. Num. Meth. Eng., 54:1501-1533, 2002. 29. O. Laghrouche, P. Bettess, E. Perrey-Debain and J. Trevelyan. Plane wave basis for wave scattering in three dimensions. Comm. Num. Meth. Eng., 19:715-723, 2003. 30. P. Bettess, J. Shirron, O. Laghrouche, B. Peseux, R. Sugimoto and J. Trevelyan. A numerical integration scheme for special finite elements for Helmholtz equation. Int. J. Num. Meth. Eng., 56:531-552, 2002. 31. E. Perrey-Debain, O. Laghrouche, P. Bettess and J. Trevelyan. Plane wave basis finite elements and boundary elements for three dimensions. Phil. Trans. Roy. Soc. - Theme Issue, 362(1816):561-578, 2004. 32. R. Sugimoto and P. Bettess. Coupling of mapped wave infinite elements and plane wave basis finite elements for the Helmholtz equation in exterior domains. Comm. Num. Meth. Eng., 19: 761-777, 2003. 33. C. Farhat, I. Harari and L. Franca. The discontinuous enrichment method. Report CU-CAS-00-20, Center for Aerospace Structures, University of Colorado, 2000. 34. C. Farhat, I. Harari and L. Franca. A discontinuous finite element method for the Helmholtz equation. Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2000, Barcelona, Spain, 11-14 September 2000, 1-15, 2000. 35. C. Farhat, I. Harari and L. Franca. The discontinuous enrichment method. Comput. Meth. Appl. Mech. Eng., 190:645-679, 2001.
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Short waves
36. B. Despr6s. Sur une formulation variationelle de type ultra-faible. Comptes Rendus de l'Acad~mie des Sciences- Series L 318:939-944, 1994. 37. O. Cessenat and B. Despr6s. Application of an ultra weak variational formulation of elliptic PDEs to the two dimensional Helmholtz problem. SIAM J. Num. Anal., 35(1):255-299, 1998. 38. J. Jirousek and A. Wr6blewski. T-elements: state of the art and future trends. Arch. Comp. Meth. Eng. State Art Rev. 3, 323-434, 1996. 39. I. Herrera. Trefftz method: a general theory. Numerical Methods for Partial Differential Equations, 16(6):561-580, 2000. 40. J. Jirousek. Basis for development of large finite elements locally satisfying all field equations. Comput. Meth. Appl. Mech. Eng., 14:65-92, 1978. 41. M. Stojek. Least-squares Trefftz-type elements for the Helmholtz equation. Int. J. Num. Meth. Eng., 41:831-849, 1998. 42. I. Herrera. Boundary Methods: anAlgebraic Theory. Pitman, London, 1984, ISBN 0 273 08635 9. 43. Y.K. Cheung, W.G. Jin and O.C. Zienkiewicz. Solution of Helmholtz equation by the Trefftz method. Int. J. Num. Meth. Eng., 32:63-78, 1991. 44. M. Stojek. Finite T-elements for the Poisson and Helmholtz equations. Ph.D. thesis number 1491, Ecole Polytechnique Federale de Lausanne, 1996. 45. J.C.W. Berkhoff. Refraction and diffraction of waver waves: derivation and method of solution of the two-dimensional refraction-diffraction equation. Delft Hydraulics Laboratory, Report and Mathematical Investigation, W154, 1973. 46. P. Ortiz. Finite elements using plane wave basis. Phil. Trans. Roy. Soc., 362(1816):525-540, 2004. 47. P. Bettess. Special wave basis finite elements for very short wave refraction and scattering problems. Comm. Num. Meth. Eng., 20:291-298, 2004. 48. O. Laghrouche, P. Bettess, E. Perrey-Debain and J. Trevelyan. Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed. Comput. Meth. Appl. Mech. Eng., 194:367-381, 2004. 49. O.C. Zienkiewicz, R.L. Taylor, S.J. Sherwin and J. Peir6. On discontinuous Galerkin methods. Int. J. Num. Meth. Eng., 58:1119-1148, 2003. 50. R.J. Astley and P. Gamallo. Special short wave elements for flow acoustics. Comput. Meth. Appl. Mech. Eng., to appear. 51. P. Gamallo and R.J. Astley. The partition of unity finite element method for short wave acoustic propagation on nonuniform potential flows. Int. J. Num. Meth. Eng., to appear. 52. I. Fried and D.S. Malkus. Finite element mass matrix lumping by numerical integration without convergence rate loss. Int. J. Solids Struct., 11:461-466, 1976. 53. W. Dauksher and A.E Emery. Accuracy in modeling the acoustic wave equation with Chebyshev spectral finite elements. Finite Elements in Analysis and Design, 26:115-128, 1997. 54. W. Dauksher and A.E Emery. An evaluation of the cost effectiveness of Chebyshev spectral and p-finite element solutions to the scalar wave equation. Int. J. Num. Meth. Eng., 45(8):1099-1114, 1999. 55. W.A. Mulder. Spurious modes in finite-element discretisations of the wave equation may not be all that bad. Appl. Num. Math., 30:425-445, 1999. 56. W.A. Mulder. Higher-order mass-lumped finite elements for the wave equation. J. Comput. Acoustics, 9(2):671-680, 2001. 57. M.J.S. Chin-Joe-Kong, W.A. Mulder and M. Van Veldhuizen. Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation. J. Eng. Math., 35:405-426, 1999. 58. J.S. Hesthaven, P.G. Dinesen and J.P. Lynov. Spectral collocation time-domain modeling of diffractive optical elements. J. Comput. Phys., 155:287-306, 1999. 59. J.S. Hesthaven. Spectral penalty methods. Appl. Num. Math., 33:23-4 1, 2000. 60. J.S. Hesthaven and C.H. Teng. Stable spectral methods on tetrahedral elements. SIAM J. Sci. Comput., 21 (6):2352-2380, 2000.
References 381 61. B. Yang, D. Gottlieb and J.S. Hesthaven. Spectral simulations of electromagnetic waves scattering, J. Comput. Phys., 134:216-230, 1997. 62. J.S. Hesthaven and D. Gottlieb. Stable spectral methods for conservation laws on triangles with unstructured grids. Comput. Methods Appl. Mech. Eng., 175:361-381, 1999. 63. D. Gottlieb and J.S. Hesthaven. Spectral methods for hyperbolic problems. J. Comput. Appl. Math. 128:83-131, 2001. 64. J.S. Hesthaven. From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM. J. Numer. Anal., 35(2):655-676, 1998. 65. R. Cools and P. Rabinowitz. Monomial cubature rules since 'Stroud': a compilation. J. Comput. and Appl. Math., 48:309-326, 1993. 66. R. Cools. Monomial cubature rules since 'Stroud': a compilation- Part 2. J. Comput. and Appl. Math., 112:21-27, 1999. 67. M.A. Taylor, B.A. Wingate and R.E. Vincent. An algorithm for computing Fekete points in the triangle. SlAM J. Numer. Anal., 38(5): 1707-1720, 2000. 68. G.E.M. Karniadakis and S.J. Sherwin. Spectral/hp Element Methods for CFD. Oxford University Press, Oxford, 1999. 69. B. Cockburn, G.E. Karniadakis and C.-W. Shu. Discontinuous Galerkin Methods. Springer, Berlin, 2000. 70. C. Eskilsson and S.J. Sherwin. A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations. Int. J. Num. Meth. Fluids, 45(6):605-624, 2004. 71. T. Warburton. Application of the discontinuous Galerkin method to Maxwell's Equations using unstructured polymorphic hp-finite elements. In B. Cockburn et al., editors, Discontinuous Galerkin Methods, pp. 451-458. Springer, Berlin, 2000. 72. J.S. Hesthaven and T. Warburton. Nodal high-order methods on unstructured grids - I time domain solution of Maxwell's equations. J. Comp. Phys., 181:186-221, 2002.
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Computer implementation of the CBS algorithm
In this chapter we shall consider some essential steps in the computer implementation of the CBS algorithm on structured or unstructured finite element grids. Only linear triangular elements will be used and the notes given here are intended for a two-dimensional version of the program. The sample program listing and user manual along with several solved problems are available to download from the website http://www.nithiarasu.co.uk or the publisher's site, http://books.elsevier.com, free of charge. The program discussed can be used to solve the following different categories of fluid mechanics problems: 1. Compressible viscous and inviscid flow problems. 2. Incompressible viscous flows. 3. Incompressible flows with heat transfer. With further simple modifications, many other problems such as turbulent flows, solidification, mass transfer, free surfaces, etc. can be solved. The procedures presented here are largely based on the theory presented in Chapter 3. The language employed is FORTRAN. It is assumed that the reader is familiar with FORTRAN 1'2 and finite element procedures discussed in this volume. We call the present program CBSflow since it is based on the CBS algorithm discussed in Chapter 3 of this volume. We prefer to keep the compressible and incompressible flow codes separate to avoid any confusion. However, an experienced programmer can incorporate both parts into a single code without much memory loss. Each program listing is accompanied by some model problems which helps the reader to validate the codes. In addition to the model inputs to programs, a complete user manual is available to users. Any error reported by readers will be corrected and the program will be continuously updated by the authors. The modules are (1) the data input module with preprocessing and continuing with (2) the solution module and (3) the output module. The program CBSflow contains the listing for solving transient Navier-Stokes (or Euler-Stokes) equations iteratively. Here there are many possibilities such as fully explicit forms, semi-implicit forms and quasi-implicit forms and fully implicit forms as discussed in Chapter 3. We concentrate
The data input module 383 mainly on the first two forms which require small memory and simple solution procedures compared to other forms. In both the compressible and incompressible flow codes, only non-dimensional equations are used. The reader is referred to the appropriate chapters (Chapters 3, 4 and 5) for different non-dimensional parameters. In Sec. 13.2 we shall describe the essential features of data input to the program. Here either structured or unstructured meshes can be used to divide the problem domain into finite elements. Section 13.3 explains how the steps of the CBS algorithm are implemented. In that section, we briefly remark on the options available for shock capturing, various methods of time stepping and different procedures for equation solving. In Sec. 13.4, the output generated by the program and postprocessing procedures are considered.
iiiiil~ilili3~ iiiiil~iiiiThe lililiIil~ili~iii~d~a iliiiifli!li!liiii!liRpUt i!ii~!i!il:i~iliil~liliili!imlilililiOdi i!iilii~Uii i!l~ii~i~~~~~~~.iii ~ii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ i iiiii!iifiiilii i iiiilili ilil This part of the program is the starting point of the calculation where the input data for the solution module are prepared. Here an appropriate input file is opened and the data are read from it. The mesh generators are provided separately. By suitable coupling, the reader can implement various adaptive procedures as discussed in Chapters 4 and 6. Either structured or unstructured mesh data can be given as input to the program. Readers are referred to the web page for further details.
13.2.1 Mesh data - nodal coordinates and connectivity Once the nodal coordinates and connectivity of a finite element mesh are available from a mesh generator, they are allotted to appropriate arrays. The coordinates are allotted to coord(i, a) with i defining the appropriate Cartesian coordinates xl(i = 1) and x2 (i = 2) and a defining the global node number. Similarly the connectivity is allotted to an array intma(b, l). Here b is the local node number and I is the global element number. It should be noted that the material code normally used in heat conduction and stress analysis is not used but can be introduced if necessary.
13.2.2 Boundary data In CBSflow we mostly use the edges to store the information on boundary conditions. Some situations require boundary nodes (e.g. pressure specified at a single node) and in such cases corresponding node numbers are supplied to the solution module.
13.2.3 Other necessary data and flags In addition to the mesh data and boundary information, the user needs to input a few more parameters used in flow calculations. For example, compressible flow computations need the values of non-dimensional parameters such as the Mach number,
384 Computer implementation of the CBS algorithm Reynolds number, Prandtl number, etc. Here the reader may consult the non-dimensional equations and parameters discussed in Sec. 3.1, Chapter 3, and in Chapter 4. Several flags for boundary conditions, shock capture, etc. also need to be given as inputs. For a complete list of such data and flags, the reader is referred to the user manual and program listing at the author's web page.
13.2.4 Preliminary subroutines and checks A few preliminary subroutines are called before the start of the time iteration loop. Establishing the surface normals, element area (for direct integration), mass matrix calculation and lumping and some data initialization subroutines are necessary before starting the time loop.
The solution module primarily contains the following steps (explicit time stepping) pre-processing do iter = i, n u m b e r of time steps call a l o t i m ! allot a p p r o p r i a t e time step v a l u e call shock ! calculate shock capturing viscosity call stepl ! intermediate momentum call step2 ! calculate density/pressure call step3 ! correct momentum call e n e r g y ! e n e r g y e q u a t i o n call p r e s s ! r e l a t e d e n s i t y and p r e s s u r e u s i n g energy cal I b o u n d ! apply boundary conditions cal i c h e c k ! c h e c k s t e a d y state c r i t e r i o n e n d d o !iter post-processing
The time iteration is carried out over the steps of the CBS algorithm and over many other subroutines such as the local time step and shock capture calculations. As mentioned, the energy can be calculated after the velocity correction. However, for a fully explicit form of solution, the energy equation can be solved in step 1 along with the intermediate momentum variable if preferred. Most of the routines within the time loop are further subdivided into several other subroutines. For instance convection and diffusion are treated using separate routines within the first step.
13.3.1 Time step In general, three different ways of establishing the time steps are possible. In problems where only the steady state is of importance, so-called 'local time stepping' is used
Solution module (Chapter 3). Here a local time step at each and every nodal point is calculated and used in the computation. When we seek accurate transient solution of any problem, the so-called 'minimum step' value is used. Here the minimum of all local time step values is calculated and used in the computation. Another and less frequently used option is that of giving a 'fixed' user-prescribed time step value. Selection of such a quantity needs considerable experience from solving several flow problems. The time loop starts with a subroutine where the above-mentioned time step options are available ( a l o t i m ) . If the last option of the user-specified fixed time step is used, the local time steps are not calculated.
13.3.2 Shock capture The CBS algorithm introduces naturally some terms to stabilize the oscillations generated by the convective acceleration. However, for compressible high-speed flows, these terms are not sufficient to suppress the oscillations in the vicinity of shocks and some additional artificial viscosity terms need to be added (Chapter 7). We provide two different forms of artificial viscosities based on the second derivative of pressure in the program. Another possibility is to use anisotropic shock capturing based on the residual of individual equations solved. However, we have not included the second alternative in the program as the second derivative-based procedures give quite satisfactory results for all high-speed flow problems. In the first method implemented, we need to calculate a pressure switch (Chapter 7) from the nodal pressure values. We calculate the switch for internal node as (Fig. 13.1)
S1 =
14pl -- P2 -- P3 -- P4 -- Psi IPl - P21 + IPl - P31 + IPl
-
Pal +
(a) Fig. 13.1 Typical element patches: (a) interior node; (b) boundary node.
IPl -
(b)
Psi
(13.1)
385
386 Computerimplementation of the CBS algorithm and for the boundary node we calculate (Fig. 13. l b) 15pl S1 --
-- 2p2
--
P3
-
-
2p41
21pl - P21 + IPl - P31 + 21pl - P41
(~3.2)
The nodal quantifies calculated in the manner explained above are averaged over individual elements. In the next option available in the code, the second derivative of pressure is calculated from the smoothed nodal pressure gradients (Chapter 4) by averaging. Other approximations to the second derivative of pressure are described in Chapter 4. The user can employ those methods to approximate the second derivative of pressure if desired.
13.3.3 CBS algorithm. Steps In the subroutine s t ep 1 we calculate the temperature-dependent viscosity at the beginning according to Sutherland's relation (see Chapter 7). The averaged viscosity values over each element are used in the diffusion terms of the momentum equation and dissipation terms of the energy equation. The diffusion, convective and stabilization terms are integrated over elements and assembled appropriately into the RHS vector. The integration is carried out directly. Finally the RHS vector is divided by the lumped mass matrices and the values of intermediate momentum variables are established. In step two, in explicit form, the density/pressure values are calculated. The subroutine s t e p 2 is used for this purpose. Here the option of using different values of 01 and 02 is available. In explicit form 02 is identically equal to zero and 01 varies between 0.5 and 1.0. For compressible flow computations, the semi-implicit form with 02 greater than zero has little advantage over the fully explicit form. For this reason we have not included the semi-implicit form for compressible flow problems in the program. For incompressible flow problems, the semi-implicit form is given. In this 01, as before, varies between 0.5 and 1 and 02 is also in the same range. Now it is essential to solve the pressure equation implicitly in s t e p 2 of the algorithm. Here in general we use a conjugate gradient solver as the coefficient matrix is not necessarily banded. The third step is the one where the intermediate momentum variables are corrected to get the real values of the intermediate momentum. In all three steps, mass matrices are lumped if the fully explicit form of the algorithm is used. As mentioned in earlier chapters, this is the best way to accelerate the steady-state solution along with local time stepping. However, in problems where transient solutions are of importance, either a mass matrix correction as given in Chapter 2 or a simultaneous solution using a consistent mass matrix may be necessary.
13.3.4 Boundary conditions As explained before, the boundary edges are stored along with the elements to which they belong. Also in the same array i s 2 de ( 2, 3 ) the flags necessary to inform the solution module which type of boundary conditions are stored. In this array i = 1, 2 correspond to the node numbers of any boundary side of an element, i = 3 indicates
References 387
the element to which the particular edge belongs and i - 4 is the flag which indicates the type of boundary condition (a complete list is given in the user manual available at the authors' and publisher's web page). Here j is the boundary edge number.
13.3.5 Solution of simultaneous equations - semi-implicit form .................................................................................................................................................................
........ ----. ............
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---~
The simultaneous equations need to be solved for the semi-implicit form of the CBS algorithm. Two types of solvers are provided. The first one is a banded solver which is effective when structured meshes are used. For this the half-bandwidth is necessary in order to proceed further. The second solver is a conjugate gradient solver. The latter can be used to solve both structured and unstructured meshes. The details of procedures for solving simultaneous equations can be found in reference 3.
13.3.6 Different forms of energy equation In compressible flow computations only the fully conservative form of all equations ensures correct position of shocks. Thus in the compressible flow code, the energy equation is solved in its conservative form with the variable being the energy. However, for incompressible flow computations, the energy equation can be written in terms of the temperature variable and the dissipation terms can be neglected. In general for compressible flows, Eq. (3.13) is used, and Eq. (4.6) is used for incompressible flow problems.
13.3.7 Convergence to steady state The residuals (difference between the current and previous time step values of parameters) of all equations are checked at every few user-prescribed number of iterations. If the required convergence (steady state) is achieved, the program stops automatically. The aimed residual value is prescribed by the user. The program calculates L2 norm of residual of each variable over the domain. The user can use them to fix the required accuracy.
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If the imposed convergence criteria are satisfied then the output is written into a separate file. The user can modify the output according to the requirements of postprocessor employed. Here we recommend the education software developed by CIMNE (GiD) for post- and preprocessing of data. 4 The facilities in GiD include two- and threedimensional mesh generation and visualization.
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1. I. Smith and D.V. Griffiths. Programming the Finite Element Method. Wiley, 3rd Edition, Chichester, 1998. 2. D.R. Willr. Advanced Scientific Fortran. Wiley, Chichester, 1995.
388 Computer implementation of the CBS algorithm 3. O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Elsevier, 6th edition, 2005. 4. GiD. International Center for Numerical Methods in Engineering, Universidad Polit6cnica de CatalunL 08034, Barcelona, Spain.
Non-conservative form of Navier-Stokes equations To derive the Navier-Stokes equations in their non-conservative form, we start with the conservative form omitting the body forces. Conservation of mass:
Op { Ot
O(pui) Op OUi Op --" ~ -~"t- Ui --0 OXi Ot P~X i
(A.1)
Conservation of momentum:
O(ujpui)
O(pui) cot
Oxj
07"ij
Op
-- Ox----j-t- ~x/ = 0
(A.2)
O(OT) O(ujp) O('rijuj) Oxl k-~x~ + Oxj - Oxj = 0
(1.3)
Conservation of energy:
O(pE) O(ujpE) ot + Oxj
Rewriting the momentum equation with terms differentiated as
Op OTij OXj F ~x/-- 0
Oili Op Ouj Op On i P--'~ -JI-Ui(-'~ ~- P~xj "JI-UJ-~xj) '~- pUJ oxj
(A.4)
and substituting the equation of mass conservation [Eq. (A.1)] into the above equation gives the reduced momentum equation
On i
1 070 10p p Oxj F P Oxi = 0
On i
--~ -F uj Oxj
(A.5)
Similarly as above, the energy equation [Eq. (A.3)] can be written with differentiated terms as
Op I
Ouj O(uip) Oxi
Op t " x ,,,
~
OE x
Oxi
)- 0
OE
+ U ox
0 (kCgT oxi
~) (A.6)
390 AppendixA Again substituting the continuity equation into the above equation, we have the reduced form of the energy equation
OE OE --~ + uj Oxj
1 O ( O__ff_T'~ l O(uip) p Oxi \k Oxi) -! P Oxi
10('rijuj) p Ox~
(A.7)
Some authors use Eqs (A.1), (A.5) and (A.7) to study compressible flow problems. However, these non-conservative equations can result in multiple or incorrect solutions in certain cases. This is true especially for high-speed compressible flow problems with shocks. The reader should note that such non-conservative equations are not suitable for simulation of compressible flow problems.
i i ,i'~..,'~' '~'~~''~'~i'~'i~'i~' ~ii'~i~'i'~i']i~'i~i~'i'~i '~~i'~'~'~i'~i'~i'~ii~'il'i~'i~ii'i!ii Self-adjoint differential equations Let us consider the following system of linear partial differential equation to demonstrate the property of self-adjointness A(u) = Lu + b = 0
(B.1)
where L is a linear differential operator. For the above equation to be self-adjoint the operator L requires L ~ r (L')') df2 = L "yr (L~) dr2 + b.t.
(B.2)
for any two functions ~ and 7. In the above equation b.t. stands for boundary integral terms.
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The drag force is the resistance offered by a body which is equal to the force exerted by the flow on the body at equilibrium conditions. The drag force arises from two different sources. One is from the pressure p acting in the flow direction on the surface of the body (form drag) and the second is due to the force caused by viscosity effects in the flow direction. In general the drag force is characterized by a drag coefficient, defined as
Cd =
D 1
(C.1)
2
A f ~PooUoo
where D is the drag force, A f is the frontal area in the flow direction and the subscript c~ indicates the free stream value. The drag force D contains the contributions from both the influence of pressure and friction, i.e. (C.2)
D = Dp -k- D f
where Dp is the pressure drag force and Df is the friction drag force in the flow direction. The pressure drag, or form drag, is calculated from the nodal pressure values. For a two-dimensional problem, the solid wall may be a curve or a line and the boundary elements on the solid wall are one dimensional with two nodes if linear elements are used. The pressure may be averaged over each one-dimensional element to calculate the average pressure over the boundary element. If this average pressure is multiplied by the length of the element, then the normal pressure acting on the boundary element is obtained. If the pressure force is multiplied by the direction cosine in the flow direction, we obtain the local pressure drag force in the flow direction. Integration of these forces over the solid boundary gives the drag force due to pressure D p. The viscous drag force D f is calculated by integrating the viscous traction in the flow direction over the surface area. The relation for the total drag force in the Xl direction may be written for a two-dimensional case as Dx~
- fA [(--P +
Tll)nl
-}- T12n2]
dAs
s
where n l and n2 are components of the surface normal n as shown in Fig. C. 1.
(c.3)
Appendix C 393
u~
v
Fig. C.1 Normalgradient of velocity close to the wall.
iiiiHiii~i~ iii!iiHiiii"~"' ii~'i~ii'i!"il'~i~ii~iiiii~'i'i':iiii~ii' !iiii!ieiii~~ i!i'~!ii!ilii~i!'iii~i~iiilililil!iiiiii!ii!i!il!i!~iiil!il~ili!i~!i lili!iii~ii!~~i~ "i~'ii~~i i~~ ~ ~ iiiH~ ~ ~ ~ }~ ~ iiiiiiiiiiii } ilii ~iiiiHiiiiiiii ~ iiiiii ~iiiiiiiiiiiii ~ iiiiiiiiiiiiiill iiiiiii i i The local coefficient of friction on a solid surface is defined as
"Fw
Cf
--
1 ~p~u2~
(C.4)
where "rw is the local shear stress and subscript c~ indicates a reference quantity. The wall shear stress at the centre of a boundary surface element (one dimensional in 2D and two dimensional in 3D problems) is calculated as 7-w = ~.t _ (-r.n)n where n is the surface normal and
(C.5)
./.t is the total viscous traction given as "r t = r n
(C.6)
where 7-are the deviatoric stresses. The pressure coefficients are normally calculated
as
Cp = 2 ( p i - Prey)
(C.7)
where the subscript r e f i s a reference value (often the value at inlet or free stream). iiiiiiiiiliiRiiiiiii ~i~i~i~iiil}ii~i~i~!~iiiiiiiiiiiii!iii }i! ii~i~i~i~iiiiiiiiiiiiiiiiii!iiiiiiiiii!iiiiil iiiii!ii}iiiiiii}}i~iiiiiiiiiiiii iiiiiiiiiiiiiiii i~iiiiiiiiii iiliiiiiiiiiiiil ii}ii[ii!iiiiii}[ii}!i[ii!i}iiiii}iiiiiiii[H!iiiiiiiii!iiiiiiiiiiii!iiii[!iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiiiiii!iiil~iiii!!iliiiiii[i[iiiiiiiiii!iiiiiiiiiiiiiiiiiiiiii ilil iiiiiii[}ilil}iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiilliiiiiiiii
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In most fluid dynamics and convection heat transfer problems, it is often easier to understand the flow results if the streamlines are plotted. In order to plot these streamlines, or flow pattern, it is first necessary to calculate the stream function values at the nodes. The stream function is defined by the following relationships,
Ul ---
or OX2
0r U2 -'- C~X1
(C.8)
394 AppendixC where ~ is the stream function. If we differentiate the first relation with respect to x2 and the second with respect to Xl and then sum, we get the differential equation for the stream function as (C.9) A solution to the above equation is straightforward for any numerical procedure. This equation is similar to Step 2 of the CBS scheme and an implicit procedure immediately gives the solution. Unlike the pressure equation of Step 2, the stream function of a solution needs to be calculated once only.
Integration formulae
Let a, b and c be the nodes of a triangular element. Integrating over the triangular area gives 1
A-
Xla
X2a
f dxldX2 "- 1 1 Xlb
XZb
1
Xlc
(D.1)
X2c
where A is the area of the triangle. For a linear triangular element (shape functions are the same as local coordinates), the integration of the shape functions can be written as
J~ d e
d !e !f !2A N a N b N f dr2 = (d + e + f + 2)!
(D.2)
On the boundaries
f r NdN[~ d r -- (d +d!e!l e + 1)!
(D.3)
Note that a - b is assumed to be the boundary side. The above equation is identical to the integration formula of a one-dimensional linear element. In the above equation I is the length of a boundary side.
Let a, b, c and d be the nodes of a linear tetrahedron element. Integrating over the volume gives
V --
dx 1 dx2 dx3
1 -
-
--'a
a
1 1
x1 Xlb
X2
X~
x~
x~
1
C
xt
C
X2
X3
1
xd
x2d
x3d
C
(D.4)
396
AppendixD where V is the volume of a tetrahedron. For linear shape functions, the integration formula can be written as
f ueuIUcUd"=(e+ e!f f!g!h!6V +g+h+3)!
(D.5)
On the boundaries
fF NaN e bf N cdF g -
e! f!g!2A (e+f+g+2)!
(D.6)
Note that the above formula is identical to the integration formula of triangular elements within the domain. In the above equation A is the area of a triangular face.
iii Convection-diffusion equations: vector-valued variables i!!ii~i~i ili}JJJ~:::~::!~J!J:!:::i~ili :iJ!{i!J:::: d{iiJii~i!iii~j~!!d{ii~~::/~i~~!!i::,:~~i~!iJii!iiJ{J::s~!~i{~i~i:~::::::::::::::::::::::: :: i~~!~!:~i:i:iJJ}JJJJJi iJlJiJi i::!:Ji!~i~si~ii::::~:i:!~!:J~!::~::::Ji:: :!:i~:~d{:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::: ~i~i::iiiis::::~::i{7;::!::!::ili ::~:;:~:~iiJl:,ii~:~:~::sif:i~!i~::i::;::::i!~J~JiJiJ~:iJ~:i,:i:::::::::::::::::::::::::::::::::::::: ::~sJi: ;!:#41i~i!{~J{i }:i::Jii i~ >,~d:::~Jiii~iii~i iiili::::: i~i!i:: sljl::::::::::::::::::::::::::::::: :)~iu:s:i::ili::~::::l{il{!Ji{i:::::: :::::::::::::::::::::::::::::::::::: ~:~:: :~i!!!:~i:i:{i~i:,:i!,:{!iiiii :~i:iiijli :~iiiii!ijJi!i{i~S~i iliii'!i~i~
The only method which adapts itself easily to the treatment of vector variables is that of the Taylor-Galerkin procedure. Here we can repeat the steps of Sec. 2.7 but now addressed to the vector-valued equation with which we started Chapter 2 [Eq. (2.1)]. Noting that now 9 has multiple components, expanding ,I~ by a Taylor series in time we have r )n+l
=
r ~n -~-
At--~--
.
At2 02@ I
+ 2
(E.1)
Ot2 n+O
where 0 is a number such that 0 < 0 < 1. From Eq. (2.1), n
= -- rL~X/
"~" -~-X/
(E.2a)
"~-
and differentiating ('~ OF i
Ot [
Q]
~G i
+~
+
n+O
(E.2b)
In the above we can write 0
OF i
0
OF i 0r
-O tggxi( ) ~~ ( o ~) where Ai
-----
~
-
0[A/(
OX i
-'~Xj -~- --~Xj ~t-
(E.2c)
OFi/Ot~ and if Q - Q ( ~ , x) and 0 Q / 0 ~ = S,
OQ Ot
OQ o ~ O~ Ot
(OFi
~i
Q)
=-svaz/+ ~7x/+
(E.2d)
398 AppendixE
We can therefore approximate Eq. (E. 1) as
A~I~n ~ r ~n+l __r ~n
[OFi
0 (O~i)
+b7
At2 ~X/ 0 [Ai ~Xj OFj ~ J Q] -~"T n { ( + 0Fj oqGj Q)} +s( + +
OGi
+
Q)] (E.3)
Omitting the second derivatives of Gi and interpolating the n + 0 between n and n + 1 values we have
AgI) ~ (I~n+l __ It)n
[Ore -- -At/~X/"+-
i O'qI- ~ Q ] (n[--] [ C At~ J i ] ) ~ 0 ~n+l
"q- ---2---[~X/{Ai(
+
n(1 --0)
\~Xj "]"
..~..T[_~xi{Ai(_~xj_.[ -
n-t-
1
(E.4)
(OFj -q- Q)] n(1 - 0)
At this stage a standard Galerkin approximation is applied which will result in a discrete, semi-implicit, time-stepping scheme. As the explicit form is of particular interest we shall only give the details of the discretization process for 0 = 0. Writing as usual
q,~N@ we have
(J~NTN dr2) A~
_ _At [ ~ Nx ( OFi\_~x~ -b -~xi AtfNT
2
0
OFj "~-Q ) ) d f 2
~X/(Ai(~
At~NTs(OFJ 2
Q) d~
\~xj +
Q) d~ 1
(E.5)
,
This can be written in a compact matrix form similar to Eq. (2.107) as M A ~ = - A t [ ( C + Ku + K ) ~ + f]" in which, with
0,~
Gi - -kij Oxj
(E.6a)
Appendix E 399 we have (on omitting the third derivative terms and the effect of S) matrices of the form of Eq. (2.108), i.e. C-
L
ON N TAi ~ dr2
K u - L 0NT (AiAj ~_~t)ON
K-
L GgNT
ON
--~xi kiJ~xj d~2
(E.6b)
f = L ( N T + ~AtA i cgNT'~ OX i ,] Q dr2 + boundary terms
M - L NTN d~ With 0 = 1/3 it can be shown that the order of approximation increases and for this scheme a simple iterative solution is possible. We note that with the consistent mass matrix M the stability limit for 0 = 1/3 is increased to C = 1. Use of 0 -- 1/3 apparently requires an implicit solution. However, similar iteration to that used in Eq. (2.117) is rapidly convergent and the scheme can be used quite economically. Jiii~ii~iiiiiiiiiiiiiiii~E~i~iiiiiiiii~ii~ii~iiiiii~iiiiiii~i~r~iii~iiiiiiiiiiii~iiii~iiiiiiiiiiii~iiiii~iiiiiii~iii~ii~ii~iii~iiiiii~i~iiiiiii~iiiii~i~iiiiiii~iiiiiiiii~i~i~iii~ii~ii~i~ii~iiiiiii~i~i~iiiiiii~iiiiii~iii~
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There are of course various alternative procedures for improving the temporal approximation other than the Taylor expansion used in the previous section. Such procedures will be particularly useful if the evaluation of the derivative matrix A can be avoided. In this section we shall consider two predictor-corrector schemes (of Runge-Kutta type) that avoid the evaluation of this matrix and are explicit. The first starts with a standard Galerkin space approximation being applied to the basic equation (2.1). This results in the form Md,~, - M ,I~9 = Pc + P o + f = ~ dt
(E.7)
where again M is the standard mass matrix, f are the prescribed 'forces' and ec(~I')
- L
NT ~x/dr2 cgFi
(E.8a)
NT~Gi ~x/dff2
(E.8b)
represents the convective 'forces', while PD(@) are the diffusive ones.
-- L
400 AppendixE
If an explicit time integration scheme is used, i.e. MAcI, _= M(~, "+~
- ~n) =
At~3n(~n)
(E.9)
the evaluation of the fight-hand side does not require the matrix product representation and Ai does not have to be computed. Of course the scheme presented is not accurate for the various reasons previously discussed, and indeed becomes unconditionally unstable in the absence of diffusion and external force vectors. The reader can easily verify that in the case of the linear one-dimensional problem "' "n 1 the fight-hand side is equivalent to a central difference scheme with ~I'nl and 'I'i+ only being used to find the value of ,1,7+1, as shown in Fig. E.l(a). The scheme can, however, be recast as a two-step, predictor-corrector operation and conditional stability is regained. Now we proceed as follows: Step 1. Compute cI:)n+l/2 using an explicit approximation of Eq. (E.9), i.e. ~n+l/2
__ t~n _~_
At M _ I ~ n 2
(E.10)
and Step 2. Compute t~ n+l inserting the improved value of of Eq. (E.9), giving t ~ n + l __ ~ n Jl-
t~ n+l/2
in the fight-hand side
AtM-I~ n+1/2
(E.11)
This is precisely equivalent to the second-order Runge-Kutta scheme being applied to the ordinary system of differential equations (E.7). Figure E.l(b) shows in the one-dimensional example how the information 'spreads', i.e. that now ~n+l will be dependent on values at nodes i - 2, . . . , i + 2. It is found that the scheme, though stable, is overdiffusive and numerical results are poor. An alternative is possible, however, using a two-step Taylor-Galerkin operation. Here we return to the original equation (2.1) and proceed as follows: Step 1. Find an improved value of cI,n+ 1/2 using only the convective and source parts. Thus ,~,n+l/2 _ ,~n _~At2 ( ~0Fin + )Qn (E.12a) which of course allows the evaluation of F n+l/2. We note, however, that we can also write an approximate expansion as
F7+1/2
-Fn+
A t OF n
20t At
n
A l A n Or
-Fi2 0Fj 0Gj
'
Q)n
(E.12b)
Appendix E 401
t~
(a) Single-step explicit
,i,n+,,
,n
i-1
~
i
,
i+1
......
x
....
i-1 (b) Standard predictor--corrector
i
+ 1/2At r
i+1
tTtn+z~tt l~O~I~0~~I
....
n
i-1 (c) Local prediction-corrector (two-step Taylor-Galerkin)
tn
i
tn +
x
1/2At
x
i+1
Fig. E.I Progressionof information in explicit one- and two-step schemes.
This gives An (OFj
aGj
Q)n
=
2
At
(Fn+l/2 ',--i
--
FT)
(E.12c)
Step 2. Substituting the above into the Taylor-Galerkin approximation of Eq. (E.5) we have
Mmt~ = _ AtILNT(OFi + ~oqGi 0 (Fn+l/2 x / + O )n dr2 + LNT ~X/,,__ i -- F n) dr2 +
L NTS(Fn+l/2 i - Fn i ) dr2 ] (E.lZd)
and after integration by parts of the terms with respect to the Xi derivatives we obtain simply
n d ~ + L N T[Q + S (Fn+l/2 - Fn)] dS2 M A ~ = - A t { L ~0NT (FT/+1/2+ G/)
+ fF NT---i (Fn+I/2 + Gni )n/dF }
(E.13)
We note immediately that: 1. The above expression is identical to using a standard Galerkin approximation on Eq. (2.1) and an explicit step with Fi values updated by the simple equation (E. 12a).
402 AppendixE 2. The final form of Eq. (E.13) does not require the evaluation of the matrices Ai resulting in substantial computation savings as well as yielding essentially the same results. Indeed, some omissions made in deriving Eq. (E.6a) did not occur now and presumably the accuracy is improved. A further practical point must be noted: 3. In non-linear problems it is convenient to interpolate Fi directly in the finite element manner as Fi -- N~i
rather than to compute it as Fi (~}). Thus the evaluation of F n+l/2 need only be made at the quadrature (integration) = n+l/2
points within the element, and the evaluation of 9 by Eq. (E.12a) is only done on such points. For a linear triangular element this reduces to a single evaluation of ,~n+l/2 and F n+l/2 for each element at its centre, taking of course ~n+l/2 and F n+l/2 as the appropriate interpolation average there. In the simple one-dimensional linear example the information progresses in the manner shown in Fig. E.l(c). The scheme, which originated at Swansea, can be appropriately called the Swansea two step, and has found much use in the direct solution of compressible high-speed gas flow equations. We have shown some of the results obtained by this procedure in Chapter 7. However in Chapter 3 we have discussed an alternative which is more general and has better performance. It is of interest to remark that the Taylor-Galerkin procedure can be used in contexts other than direct fluid mechanics. The procedure has been used efficiently by Morgan et al. in solving electromagnetic wave problems.
E.2.1 Multiple wave speeds When ~bis a scalar variable, a single wave speed will arise in the manner in which we have already shown at the beginning of Chapter 2. When a vector variable is considered, the situation is very different and in general the number of wave speeds will correspond to the number of variables. If we return to the general equation (2.1), we can write this in the form C~
0~I)
C~ i
+ A i ~ x/ + ~
+ Q = 0
(E.14)
where Ai is a matrix of the size corresponding to the variables in the vector ~. This is equivalent to the single convective velocity component A - U in a scalar problem and is given as 0Fi Ai = 0 ~
(E.15)
This in general may still be a function of cI,, thus destroying the linearity of the problem.
Appendix E 403 Before proceeding further, it is of interest to discuss the general behaviour of Eq. (2.1) in the absence of source and diffusion terms. We note that the matrices Ai can be represented as (E. 16)
Ai "- Xi Ai X~- 1
by a standard eigenvalue analysis in which Ai is a diagonal matrix. If the matrices Xi are such that Xi = X
(E.17)
which is always the case in a single dimension, then Eq. (E. 14) can be written (in the absence of diffusion or source terms) as O~ -
Ot -
-]-- X A i
X _ 1 0r ~ .. = 0
OXi
(E. 18)
Premultiplying by X -~ and introducing new variables (called Riemann invariants) such that t~ -- x - l ~ I ~
(E.19)
we can write the above as a set of decoupled equations in components 4) of q~ and corresponding A of A: -O~ & - ]- Ai ~O~ x/= 0
(E.20)
each of which represents a wave type equation of the form that we have previously discussed. A typical example of the above results from a one-dimensional elastic dynamics problem describing stress waves in a bar in terms of stresses (a) and velocities (v) as
&r
Ov
Ov Ot
1 0o pox
ot
=o =0
This can be written in the standard form of Eq. (2.1) with
crlp} The two variables of Eq. (E.19) become ~1 = O " - CV
where c =
~/E/p
and
~)2 = O" "+" CI)
and the equations corresponding to (E.20) are 0~1
0~)1
ot + c 0~2
Ot
=~ 0~2
c-- x = 0
representing respectively two waves moving with velocities +c.
404 Appendix E
Unfortunately the condition of Eq. (E.17) seldom pertains and hence the determination of general characteristics and therefore decoupling is not usually possible for more than one space dimension. This is the main reason why the extension of the simple, direct procedures is not generally possible for vector variables. Because of this we shall in Chapter 3 only use the upwinding characteristic-based procedures on scalar systems for which a single wave speed exists and this retains justification of any method proposed.
ii!i
i
Edge-based finite element formulation The edge-based data structure has been used in many recent finite element formulations for flow problems. As mentioned in Sec. 7, Chapter 7, this formulation has many advantages such as smaller storage, etc. To explain the formulation we shall consider the Euler equations and a few assembled linear triangular elements on a two-dimensional finite element mesh as shown in Fig. E 1. From Eq. (1.25) we rewrite the following Euler equations 0tI)
0F i
Ot + ~xi
--
(F.1)
0
where 9 are the conservative variables. If the element-based formulation for the above equation omits the stabilization terms, the weak form can be written as
f N k AAt~ d ~ 2 = - f ( N k ) TOFi ~x~ d r
(E2)
In a fully explicit form of solution procedure, the left-hand side becomes M(A cI,/At) and here M is the consistent mass matrix (see Chapter 3). We can write the RHS of the above equation for an interior node I [Fig. C. 1(a)] by interpolating Fi in each element and after applying Green's theorem as
~ Eel
(NkF/k)
=
E
Eel
3
OX i
E
(F~ + F/J + F/r)
(F.3)
where A E is the area and I, J and K are the three nodes of the element (triangle) E. This is an acceptable added approximation which is frequently used in the TaylorGalerkin method (see Chapter 2). In another form, the above RHS can be written as [Fig. F. 1(a)]
A1 ON1 ~(F/+ 3 OX i
F1 + F2) -t
A2 ON1
3
~(F/+
F2 + F3) -i
OX i
A3 ON1
~(F/+
3 Oxi
F3 + F1) (E4)
where A1, A2 and A3 are the areas of elements 1, 2 and 3 respectively. For integration over the boundary on the RHS, we can write the following in the element formulation N I (NkF/k) dFn8 B~I
B
B6I
406 AppendixF 2
2
3
1
(a)
(b)
Fig. F.I Typical patch of linear triangular elements: (a)inside node; (b) boundary node. where n is the boundary normal. The above equation can be rewritten for the node I in Fig. F. 1(b) as F B1 (2F[ + F~)na + FB2(2F/ + V~)n2 (F.6) 6 --6 where FB1 and FB2 are appropriate edge lengths and subscripts 1 and 2 indicate the edges in Fig. F. 1(b). The above Eqs (F.3) and (E5) can be reformulated for an edge-based data structure. In such a procedure, Eq. (E3) can be rewritten as (for node I)
L
l EON' ( F / + F [ ' ) I ~ON'(NkF/k) d a - - ~ m s ~ ( [Ae30xi (F.7) E6I E S--1 E~_IIs where m~ is the number of edges in the mesh which are directly connected to the node I and the summation ~-'~EEIIs extends over those elements that contain the edges I I~.
The coefficient in Eq. (E7) is
c:'s= It can be easily verified that
Ae
tONI]
-T L x/J E=Ils E ms
S=I
C/I" - 0
(F.8)
(F.9)
for all i. The user can now readily verify that the above equation is identically equal to the standard element formulation of Eq. (F.4) if we consider the node I in Fig. F. 1(a). For the boundary nodes, however, Eq. (F.9) is not satisfied and thus the element formulation is not reproduced. For the boundary edges, in addition to Eqs. (F.6) and (F.7) the following addition is necessary
-
[1-'--un, B1
1-'B2 ]
+ --unz r[
(F.10)
Alternatively the above contribution may be added to Eq. (E6) to complete the edge formulation.
"'iiiiHiiii'i'ii''ii i'i'iiiiiiiiii i'''ii iii,,i,ii,ii,,i i'i'iii' iiiiiii'iiiii'iiii iiiil iiiiiiiiiiiiiiii'''i''i'!iiiiii' ii'iiiiiii Multigrid method It is intuitively obvious that whenever iterative techniques are used to solve a finite element or finite difference problem it is useful to start from a coarse mesh solution and then to use this coarse mesh solution as a starting point for iteration in a finer mesh. This process repeated on many meshes has been used frequently and obviously accelerates the total convergence rate. This acceleration is particularly important when a hierarchical formulation of the problem is used. We have indeed discussed such hierarchical formulations in Chapter 4 of The Finite Element Method: Its Basis and Fundamentals (Zienkiewicz, Taylor and Zhu) and the advantages are pointed out there. The simple process which we have just described involves going from coarser meshes to finer ones. However, it is not useful if no return to the coarser mesh is done. In hierarchical solutions such returning is possible as the coarser mesh matrix is embedded in the finer one with the same variables and indeed the iteration process can be described entirely in terms of the fine mesh solution. The same idea is applied to the multigrid form of iteration in which the coarse and fine mesh solution are suitably linked and use is made of the fact that the fine mesh iteration converges very rapidly in eliminating the higher frequencies of error while the coarse mesh solution is important in eliminating the low frequencies. To describe the process let us consider the problem of
LO=f
in
f2
(G.1)
which we discretize incorporating the boundary conditions suitably. On a coarse mesh the discretization results in KCq~c = r c
(G.2)
which can be solved directly or iteratively and generally will converge quite rapidly if q~c is not a big vector. The fine mesh discretization is written in the form
=f:
(G.3)
and we shall start the iteration after the solution has been obtained on the coarse mesh. Here we generally use aprolongation operator which is generally an interpolation from which the fine mesh values at all nodal points are described in terms of the coarse mesh values. Thus qS[ = Pq~C_1 -k- AqS[
(G.4)
408 AppendixG where Ath f is the increment obtained in direct iteration. If the meshes are nesting then of course the matter of obtaining P is fairly simple but this can be done quite generally by interpolating from a coarser to a finer mesh even if the points are not coincident. Obviously the values of the matrices P will be close to unity whenever the fine mesh points lie close to the coarse mesh ones. This leads to an almost hierarchical form. Once the prolongation to (~f has been established at a particular iteration i the fine mesh solutions can be attempted by solving K f A~:
- ff - R {
(G.5)
where the residual R is easily evaluated from the actual equations. We note that the solution need not be complete and can well proceed for a limited number of cycles after which a return to the coarse mesh is again made to cancel out major low frequency errors. At this stage it is necessary to introduce a matrix Q which transforms values from the fine mesh to the coarse mesh. We now write for instance q~c = Qq~/f
(G.6)
where one choice for Q is, of course, pT. In a similar way we can also write R; = QR /
(G.7)
where Ri are residuals. The above interpolation of residuals is by no means obvious but is intuitively at least correct and the process is self-checking as now we shall start a coarse mesh solution written as "C "" KC(t~i+l - ~b c) = R iC
(G.8)
At this stage we solve for t~icq_1 using the values of previous iterations of q~c and putting the collected residuals on the fight-hand side. This way of transferring residuals is by no means unique but has established itself well and the process is rapidly convergent. In general more than two mesh subdivisions will be used and suitable operators P and Q have to be established for transition between each of the stages. The total process of solution is vastly accelerated and proceeds well as shown by the many papers cited in Chapter 7.
....
Boundary layer-inviscid flow coupling A few references on the topic of boundary layer-inviscid flow coupling are given in Chapter 6. In this appendix we shall briefly explain a simple procedure of this flow coupling procedure. To understand the process of coupling the Euler and integral boundary solutions we shall consider a typical flow pattern around a wing as shown in Fig. E. 1. Both turbulent and laminar regimes are shown in this figure. We summarize the procedure as follows: Step 1. Solve the Euler equations in the domain considered around the aerofoil. Here any mesh can be used independently of the mesh used for the boundary layer solution. The solution thus obtained will give a pressure distribution on the surface of the wing. Step 2. Solve the boundary layer using an integral approach over an independently generated surface mesh. If the surface nodes do not coincide with the Euler mesh, the pressure needs to be interpolated to couple the two solutions. The laminar portion near the boundary (Fig. E.1) is calculated by the 'Thwaites compressible' method and the turbulent region is predicted by the 'lag-entrainment' integral boundary layer model. Step 3. The Euler and integral solutions are coupled by transferring the outputs from one solution to the other. As indicated in Fig. H. 1, direct and semi-inverse couplings can be used for different regions. The semi-inverse coupling is introduced here mainly to stabilize the solution in the turbulent region close to separation. Figure H.2 shows the flow diagrams for the present boundary layer-inviscid coupling. Further details on the Thwaites compressible method and semi-inverse coupling can be found in the references discussed in Sec. 6.12, Chapter 7 (Le Balleur and coworkers). In Fig. H.2, Cp is the coefficient of pressure; s the coordinate along the surface; ~ the boundary layer thickness; 0 the momentum thickness; Cf the skin friction coefficient; H the velocity profile shape parameter; p the density; VN the transpiration velocity; K* is a factor developed from stability analysis; the subscript v marks the viscous boundary layer region; 6* the displacement thickness; the superscript i indicates inviscid region and the superscript m indicates the current iteration.
410 Appendix H Semi-inverse
~" , , 2 T ' , I I ~ F ~ ~9 '
Turbulent
II Transition Laminar ~
sss
... ~ o ~ ~,=, ~
" .
Turbulent i|
. ....
. . . . . . . . . . . . . . . )~',-~ . . . . . . . . . . . . Direct ',
- . . . . . . . . . . . . . . . . . . .
Semi-inverse
Semi-inVerse"~
Fig. H.1 Flow past an aerofoil. Typical problem for boundary layer-inviscid flow coupling.
The following are useful relations for some of the above quantities: H--if,
=
1
n, "
pvu~
K*=
/3= 4 1 - M
27r0'
2 (H.1)
where n is the normal direction from the wing surface. We have the following equations to be solved in the integral boundary layer lagentrainment model. Continuity
d----s- = d H
Ce - H~
~
- (H +
~
1)--uv ~
(H.2)
Momentum dO __ C f
ds
2
(H+2-M
(H.3)
~ 2) - 0- du, uo ds
Lag-entrainment
0
F 2.8 ( - F/ ds LH + H1
dCe
Ws)EQ
0.5
0 duo
0 duo (1 + 0.2M 2) ] uo ds (1 + 0.075M 2) (1 + O..i~-)
(H.4)
where F is a function of Ce and Cf and given as 0.02Ce + C 2 + F
(0.01 + Ce)
3 (H.5)
Appendix H 411
J
Unstructured grids or multiblock Euler inviscid method
Cp,8
Direct calculation d PVN=~(Pv UvS*)
T
i_
Lag-entrainment boundary layer viscous method
6, O, Cf, H
I-
Direct calculation
(a)
Unstructured grids or multiblock Euler inviscid method
Cp,S
Direct calculation
pvN
pv~n+1=pv~n+ K*[ e-du~
v
Uv ds
du i m Uv
Lag-entrainment boundary layer viscous method VN
---~6"'- I
Direct calculation
(b) Fig. H.2 Couplingtechniques: (a) direct; (b) semi-inverse.
In the above equations,/-/and
-
Hi are
the velocity profile shape parameters defined as
1 / ~ ( u1)- - -
H--O
Uv
n.,
H1 - -
0
(H.6)
412 AppendixH C e is the entrainment coefficient; uv the mean component of the streamwise velocity at the edge of the boundary layer; M the Mach number; CT the shear stress coefficient; A the scaling factor on the dissipation length; the subscripts EQ and EQo denote respectively the equilibrium conditions and equilibrium conditions in the absence of secondary influences on the turbulence structure. Once the above equations are solved, the transpiration velocity VN is calculated as shown in Fig. H.2 and is added to the standard Euler boundary conditions on the wall and plays the role of a surface source. The coupling continues until convergence. In practice, in one coupling cycle, several Euler iterations are carried out for each boundary layer solution.
Mass-weighted averaged turbulence transport equations iiiiii iiiiii~ii$~iiiiiiJ Jiiiii iiiiiiiiiiiiFiii iiiiiiiii iiiiiiii iiiiii
iiiliiiiiiiiiiii
In this subsection we provide two turbulence models commonly employed in the compressible flow calculations. Before discussing these models we write the Reynolds stress term and turbulent heat flux term in terms of turbulent eddy viscosity as
~ij -" IdT
( OLIi OUj ~Xj ~ OXj
2 0U k (~ij ) 2 30Xk --
5ptcrij
(I.1)
and
Cp OT q~ = -Izr Prr Oxj
(I.2)
One of the following turbulence models may be employed to calculate the turbulent viscosity.
5palart-AIImaras model
In this model the turbulent eddy viscosity is calculated as ~T
YT -- ~ ~- ~fol p
(1.3)
where fol
=
X3
x 3 + c31
(1.4)
with b'
x = -
//
(1.5)
414
Appendix I The viscosity variable ~ is calculated from Of,
Of,
leo {
Col[1 - ft2]~S~' + -O" ~X j
=
E
-- Cwl fw -- "-~ ft2
l[ l
(u-+- ~,) ~
+ Cb2 -~Xj
(I.6)
+ ftl A~2
The parameters used in the above equation are written as // --
K 2 y 2 fv2
CO +
X
fv2 "- 1--
f w -- g g
-
-
1 +Xfol
1 + cw3 g6 + cw6 3
r + cw2(r 6
r =min
-
-
r)
~K~d 2 , 1 0
ft; - ct3 e x p ( - c t 4 X 2) ft 1 --" Ctl gt exp
gt
--
CUt
-ct2 ~
2 2
[y2 + gt Yt ] (I.7)
0.1, ~3t Ax
min
where y is the distance from a given node to the nearest wall, ~ is the vorticity given as
"--
On3
OU2
OX2
OX3
+
OUl
OU3
OX3
OXl
"[-
OU2
OUl
OXl
OX2
(I.8)
Aft is the difference in velocity between the point and trip, yt is distance from a node to a trip point or curve, ~t is the vorticity magnitude at trip point or curve, Ax is the surface grid spacing at trip. Other constants used in the model are Cbl = 0.1355, Cb2 = 0.622, cr = 2/3, K = 0.41, cwl = Cbl/K 2 + (1 + Cb2)/cr, Cw2 = 0.3, Cw3 = 2, Col = 7.1, Ctl = 1, Ct2 = 2, Ct3 - " 1.1 and C t 4 = 2. The major difference between the model given here and the one used in Sec. 8.2 is that here we have a trip curve (3D) or trip point (2D) to trigger turbulence. The trip curve is often defined at a 3% distance from the leading edge of a solid surface. However, the model without a trip curve is widely employed as explained in Chapter 8. - w model
The basic idea of the n - a; model arises from the fact that vorticity is directly proportional to n2fl, i.e. /~2
- c~ 1
(I.9)
Appendix I 415 where c is a constant. The eddy viscosity may therefore be written as #r =/9-
(I.10)
W
The transport equations for ~ and w may be written as
~(p/~) -~- ~-X/(p/~Ui) =
+ ~X/(~/jUj) -- ~*p/'~W
(I.11)
and
~(9( ~ )
(9 ( ~ ) + ~x-Tx
- ~xiCg(#W~xi~ + o~WC9 ~-~/(~j) - ~:
(i.12)
where #~ = # + #r/o,~ and #~ = # + #r/tr~. The constants are c~ = 5 / 9 , / 3 = 3 / 4 0 , / 3 * = 9/100, cr~ = a~ = 2. The turbulence models discussed in this section can be non-dimensionalized as discussed in Sec. 8.2 if necessary.
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Author index
Page numbers in bold are for pages at the end of chapters with names of author references. Abarbanel, S., 327, 346 Abbott, M.B., 292, 314 Abe, K., 250, 271 Adey, R.A., 54, 56, 77 Ainsworth, M., 123, 138, 349, 351,378 Aizinger, V., 297, 316 Akin, J.E., 82, 93, 106, 191,196 Alexander, J.M., 143, 144, 166 Alkire, R.L., 284, 291 Allmaras, S.R., 252, 270, 272, 273 Almeida, R.C., 41, 75 Altan, A.T., 143, 166 Alvarez, G.B., 41, 75 Amsden, A.A., 172, 186, 194 Apsley, D.D., 250, 262, 271,273 Armstrong, R.C., 160, 162, 163, 169 Ashida, H., 340, 341,348 Asian, A.R., 118, 137 Astley, R.J., 317, 327, 328, 330, 331,332, 333, 334, 335,336, 344, 346, 347, 348, 352, 369, 370, 371, 372, 379, 380 Aswathanarayana, P.A., 82, 106 Atkinson, B., 143, 166 Atkinson, J.D., 41, 75 Austin, D.I., 297, 308, 315, 316 Ayers, R., 197, 222, 224, 225, 227, 228, 244 Baaijens, EET., 144, 167 Babau, A.V.R., 123, 129, 139 Babuska, I., 81, 97,105, 319, 332, 345, 347, 349, 35 l, 353, 354, 355,357, 378, 379 Bach, E, 172, 193 Bai, K.J., 177, 195 Baker, T.J., 221,246 Balaji, EA., 143, 167 Baldwin, B.S., 203, 204, 245 Bando, K., 324, 325, 327, 328, 329, 345, 346 Barai, E, 332, 347 Barbone, EE., 332, 347
Barnett, M., 254, 272 Barrett, K.E., 34, 74 Barton, J.T., 208, 209, 245 Batchelor, C.K., 4, 27 Baum, J.D., 123, 138 Baumann, C.E., 52, 76 Bay, F., 143, 167 Bayliss, A., 321,324, 345 Bayne, L.B., 197, 221,227, 230, 244 Beck, R.F., 177, 195 Beddhu, M., 172, 193, 194 Behr, M., 297, 315 Bejan, A., 191,196, 274, 275,276, 288, 290, 291 Bellet, M., 152, 168, 172, 193 Belytschko, T., 41, 75, 152, 168 Benque, J.P., 56, 77 Benson, D.J., 163,169 Bercovier, H., 136, 140 Bercovier, M., 56, 77 Berenati, R.E, 283, 290 B6renger, J.P., 327,345, 346 Berkhoff, J.C.W., 318, 336, 344, 347, 364, 380 Bermudez, A., 56, 77 Bettess, J.A., 21, 23, 27, 323, 327, 328, 345 Bettess, P., 21, 23, 27, 297, 308, 315, 316, 318, 320, 323,324, 325,327,328, 329, 331,332, 336, 339, 343,344, 345, 346, 347, 348,349, 351,352, 354, 355,356, 357, 365,368, 369, 378, 379, 380 Bhandari, D.R., 143, 147, 166 Bhargava, P., 143, 152, 167, 168 Bilger, R.W., 143, 166 Billey, V., 123, 137, 221,246 Bishop, A.R., 203, 244 Biswas, G., 52, 77, 82, 107 Blasco, J., 82, 107 Bonet, J., 143, 152, 154, 165, 167, 168, 169 Boris, J.P., 68, 78, 270, 273 Bornside, D.E., 162, 163, 169 Borouchaki, H., 123, 129, 131,139, 140, 233, 246
418 Author index Bosch, G., 256, 272 Bourgault, Y., 123, 139 Bova, S., 297, 315 Bradshaw, P., 241,247 Braess, H., 172, 186, 194 Brakalmans, W.A.M., 152, 168 Brebbia, C.A., 82, 106, 297, 314 Bremhorst, K., 253, 257, 272 Brezzi, E, 42, 75, 99, 109 Briard, P., 256, 257, 272 Brinkman, H.C., 284, 291 Brook, D.L., 68, 78 Brookes, P.J., 118, 137, 197, 221,244, 352, 378 Brooks, A.N., 39, 46, 47, 74, 76 Brooman, J.W.E, 241,247 Brosilow, C.B., 283,290 Brown, R.A., 160, 162, 163, 169 Budiansky, B., 152, 168 Burnett, D.S., 328, 329, 330, 335,346 Burton, A.J., 165, 169 Butler, M.J., 334, 347 Calvo, N., 175, 195 Cambier, L., 235, 239, 246 Cao, Y., 177, 195 Card, C.C.M., 143,166 Cardle, J.A., 52, 76 Carew, E.O.A., 154, 169 Carey, G., 297, 315 Carey, G.E, 41, 52, 75, 76 Carotenuto, A., 82, 108 Carter, J.E., 229, 230, 246 Casciaro, R., 220, 246 Castro, J.M., 144, 167 Castro-Diaz, M.J., 123, 131,139, 233,246 Caswell, B., 143, 166 Cavendish, J.C., 123, 139 Cebeci, T., 251,272 Cecchi, M.M., 297, 315 Cervera, M., 45, 76 Cessenat, O., 359, 360, 380 Chadwick, E., 352, 379 Chan, A.H.C., 274, 275, 290 Chan, S.T., 82, 107 Charpin, F., 247 Chartier, M., 297,314 Chastel, Y., 143, 167 Chaudhuri, A.R., 82, 107 Chen, A.J., 228,246 Chen, C.-H., 172, 186, 194 Chen, C.K., 191,196 Chen, C.Y., 191,196 Chen, H.S., 177, 195, 327, 336, 337, 338, 346 Chen, W.L., 250, 271 Cheng, P., 275,285, 290, 291 Cheng, S.I., 118, 137
Cheng, T.W., 113, 137 Chenot, J.L., 143, 144, 152, 167, 168, 172, 193 Cheung, Y.K., 363, 380 Chiam, T.C., 327, 328, 329, 346 Chin-Joe-Kong, M.J.S., 373,380 Chippada, S., 297, 315 Choi, H.G., 172, 186, 194 Chorin, A.J., 82, 83, 106 Chow, R.R., 24 l, 247 Christie, I., 34, 35, 37, 74, 299, 316 Christon, M.A., 82, 107 Chung, T.J., 143, 147, 166 Cipolla, J.L., 334, 347 Clark, EJ., 327, 328, 343,346, 348 Cockburn, B., 374, 381 Codina, R., 42, 43, 45, 47, 52, 57, 59, 75, 76, 78, 82, 91, 93, 97, 99, 102,107,108, 112, 123,136,137, 176,195, 203,205,218, 219, 229, 232, 245, 246 Colella, P., 207, 210, 245 Collino, E, 354, 357, 359, 364, 365,379 Comini, G., 82, 106 Connor, J.J., 297, 314 Constantineseu, G., 256, 270, 272 Cook, J.L., 172, 186, 194 Cools, R., 374, 381 Copps, K., 349, 357, 378 Cornfield, G.C., 143, 166 Coupez, T., 172, 193 Courant, R., 34, 74 Coyette, J.P., 328, 332, 334, 346, 347 Craggs, A., 317, 344 Cremers, L., 328, 332, 334, 346, 347 Crepon, M., 297, 314 Crochet, M.J., 154, 157, 168, 169 Cullen, M.J.P., 297,314 Currie, I.G., 4, 27 Dalsecco, S., 297, 303, 313, 315 Dancy, H., 327, 345 Darcy, H., 274, 290 Daubert, O., 297, 315 Dauksher, W., 373,380 David, E., 285,291 Davidson, L., 256, 270, 272, 273 Davies, A.R., 154, 169 Davis, J., 297, 314 Dawson, C., 297, 315, 316 Dawson, C.N., 297, 316 Dawson, C.W., 177, 195 Dawson, P.R., 143, 166 de La Bourdonnaye, A., 354, 379 De Roquefort, T.A., 191,196 de Sampaio, P.A.B., 52, 76, 123, 138 de Sampmaio, P.A.B., 47, 48, 76 De Vahl Davis, G., 191,196 de Villiers, R., 297, 315
Author index 419 Del Guidice, S., 82, 106 Delgiudice, S., 191,196 Demkowicz, L., 123, 138, 199, 213, 244, 245, 332, 346, 347 Denham, M.K., 115, 117, 137, 256, 257,272 Depr6s, B., 359, 360, 380 Derby, J.J., 113, 137 Derviaux, A., 123, 137 Desaracibar, C.A., 152, 168 Deshpande, M.D., 113, 137 Devloo, E, 123, 137 Dewitt, D.E, 191,196 Dinesen, EG., 373, 380 Ding, D., 154, 169 do Carmo, E.G.D., 41, 75 Dompierre, J., 123, 139 Donea, J., 37, 41, 42, 45, 69, 74, 75, 78, 82, 106, 152, 168, 172, 185, 186, 194 Dou, H-S., 159, 160, 162, 163, 169 Douglas, J. Jr, 56, 77, 297, 314 Downie, M.J., 343,348 Dreyer, D., 332, 347 Duncan, D., 349, 351,378 Duncan, J.H., 180, 181,195 Dupont, I., 297, 314 Dupont, S., 154, 169 Durany, J., 56, 77 Durbin, EA., 256, 272 Dutra Do Carmo, E.G., 205,245 Eiseman, ER., 123, 140 Ellwood, K., 101,109 Emery, A.E, 373,380 Emson, C., 324, 325,327, 328, 329, 345, 346 Engleman, M.S., 136, 140 Engquist, B., 325,345 Ergun, S., 276, 290 Esche, S.K., 152, 168 Eskilsson, C., 375,376, 377, 381 Evans, A., 123,138 Eversman, W., 330, 333,346, 347, 352, 379 Ewing, R.E., 56, 77 Fan, S., 254, 272 Fan, Y., 169 Fares, E., 256, 272 Farhat, C., 357, 379 Farmer, J., 172, 193 Feng, Y.T., 172, 193 Fenner, R.T., 143, 166 Feshbach, H., 318,344, 354, 379 Field, D.A., 123,139 Fix, G.J., 297, 314 Flanagan, D.E, 152, 168 Fleming, C.A., 297, 308,315 Fletcher, C.A., 40, 74
Forchheimer, E, 284, 291 Foreman, M.G.G., 297,314, 315 Formaggia, L., 197, 221,223, 243 Fortin, M., 123, 136, 139, 140 Fortin, N., 136, 140 Fourment, L., 143, 167 Franca, L., 357,379 Franca, L.E, 41, 75 Frey, EJ., 123, 139 Frey, W.H., 123, 139 Fried, I., 373,380 Fyfe, K.R., 328, 332, 346 Gale~o, A.C., 205,245 Gallagher, R.H., 34, 74, 191,196 Gamallo, E, 369, 370, 37 l, 372, 380 Gangaraj, S.K., 319, 345, 351,378 Garcfa, J., 172, 179, 194 Garg, V.K., 277, 290 Geers, T.L., 324, 326, 327, 335,345 George, EL., 123, 129, 13 l, 139, 140, 233,246 Gerdes, K., 327, 332, 346, 347 Germano, M., 269, 273 Ghia, K.N., 113, 131,136 Ghia, U., 113, 131,136 Ghosh, S., 144, 167 GiD, 387, 388 Giraldo, EX., 297, 316 Girodroux-Lavigne, E, 241,247 Giuliani, S., 82, 106, 152, 168, 187, 195 Givoli, D., 324, 326, 335,336, 345 Gladwell, G.M.L., 317,344 Godbole, EN., 134, 140, 143, 149, 151,166 Gottlieb, D., 327, 346, 373,374, 376, 381 Goudreau, G.L., 163, 169 Goussebaile, J., 56, 77 Gowda, Y.T.K., 82, 107, 191,196 Gray, W., 297, 315 Gray, W.G., 301,316 Gray, W.R., 297, 314 Green, A., 222, 246 Green, J.E., 241,247 Green, EJ., 123,139 Gregoire, J.E, 56, 77 Gresho, EM., 82, 107, 136, 140 Griffiths, D.E, 34, 35, 37, 74 Griffiths, D.V., 382, 387 Grinstein, EE, 270, 273 Gtil~at, 0., 118, 137 Gunzberger, M., 32 l, 324, 345 Guymon, G.L., 40, 75 Habashi, W.G., 123, 139 Hackbusch, W., 220, 246 Hagstrom, T., 325,345 Hall, C.D., 297,314
420 Authorindex Halleux, J.I., 152, 168 Hallquist, J.O., 163, 169 Halpern, E, 297, 314 Haltiner, G.L., 292, 314 Hansbo, E, 187, 196, 205,245 Hara, H., 340, 34 l, 348 Harari, I., 41, 75, 332, 347, 357, 379 Harbani, Y., 56, 77 Hardy, O., 123, 138 Hariharan, S.I., 325,345 Harlow, EH., 195 Hassager, O., 172, 193 Hassan, O., ll8, 123, 128, 129, 137, 138, 139, 140, 197, 216, 22 l, 222, 224, 225,227, 228,230, 233, 234, 238,240, 243, 244, 246, 256, 266, 268,272, 273, 352, 353,378 Hauguel, A., 56, 77 Hause, J., 123, 140 Havelock, T.H., 325,345 Hearn, G.E., 343,348 Hecht, E, 56, 77, 123, 128, 129, 131,139, 233,246 Heinrich, J.C., 34, 46, 52, 74, 76, 77, 143, 147, 166, 191,196, 297, 308, 314, 315 Hermansson, J., 187, 196 Herrera, I., 362, 380 Herrmann, L.R., 40, 75 Hervouet, J., 297, 315 Hesthaven, J.S., 327, 346, 373, 374, 375, 376, 380, 381 Hetu, J.E, 123, 13 l, 138 Higdon, R.L., 325,345 Hill, T.R., 7 l, 78 Hindmarsh, A.C., 82, 107 Hino, T., 179, 180, 195 Hinsman, D.E., 297, 314 Hirano, H., 299, 316 Hiriart, G., 21, 27 Hirsch, C., 4, 27, 198, 199, 200, 244 Hirt, C.W., 172, 176, 186, 193, 194 Hoffman, J.D., 203,244 Holdo, A.E., 256, 262, 263,272 Holford, R.L., 328, 329, 330, 335,346 Homsy, G.M., 288, 291 Hood, E, 34, 74, 19 l, 196 Houston, J.R., 338, 339, 348 Hsu, C.T., 275,290 Hsu, M.-H., 172, 186, 194 Huang, G.C., 136, 140, 146, 148, 151,152, 167 Huerta, A., 37, 42, 45, 74, 75, 82, 107, 172, 185, 186, 187, 194, 196 Hu6tink, J., 144, 167 Huetnik, J., 152, 168 Hughes, T.J.R., 39, 4 l, 42, 46, 47, 52, 74, 75, 76, 134, 140, 205,207,245, 349, 378 Hulbert, G.M., 4 l, 75 Hulbert, H.E., 297, 314
Huttunen, T., 354, 357, 359, 360, 361,362, 364, 365, 379 Huyakorn, P.S., 34, 46, 74 Hydraulic Research Station, 301,303,316 Idelsohn, I.R., 172, 179, 194 Idelsohn, S.R., 52, 76, 77, 172, 175, 179, 194, 195 Ihlenburg, E, 319, 324, 345, 351,378 Iida, M., 172, 186, 194 Inaba, H., 285,291 Inagaki,, 299, 316 Incropera, EP., 191,196 Irons, B.M., 143, 166 Isaacson, E., 34, 74 Ito, H., 297, 315 Jain, P.C., 143, 166 Jaluria, Y., 191,196 Jameson, A., 98, 109, 172, 179, 180, 193, 195, 203, 221,245, 246 Jami, A., 297, 315 Jang, Y.J., 250, 271,272 Janson, C., 177, 195 Jenson, G., 177, 195 Jiang, B.N., 52, 76 Jiang, M.Y., 172, 194 Jin, H., 123, 139 Jin, W.G., 363, 380 Jirousek, J., 362, 380 Johan, Z., 41, 47, 52, 75, 76, 205,245 Johansson, S.H., 256, 272 Johnson, C., 47, 52, 76, 205,245 Johnson, R.H., 143, 166 Johnson, W., 144, 167 Johnson, W.E., 176, 195 Jones, J., 197, 221,243, 244 Jones, J.W., 118, 137 Jones, W.P., 253,272 Jou, W.H., 270, 273 Jue, T.C., 82, 93, 106, 191,196 Kaipio, J.P., 354, 357, 359, 360, 361,362, 364, 365, 379 Kakita, T., 152, 168 Kallinderis, Y., 228, 246 Kalro, V., 262, 273 Kamath, M.G., 123, 129, 140 Kanehiro, K., 340, 341,348 Karniadakis, G.E.M., 374, 381 Kashiyama, K., 297, 315 Kawahara, M., 82, 106, 172, 173, 186, 193, 194, 297, 299, 303,314, 315, 316 Kawka, M., 152, 168 Kelly, D.W., 37, 46, 47, 74, 299, 316, 320, 336, 345, 347 Kennedy, J.M., 152, 168
Author index 421 Keunings, R., 154, 169 Keyhani, M., 289, 291 Kim, J., 113,137, 271,273 Kim, Y.H., 177, 195 Kinzel, G.K., 152, 168 Kleiber, M., 152, 168 Kobayashi, S., 143, 166, 167 Kodama, T., 297, 303,315 Kolbe, R.L., 270, 273 Kolmogorov,A.N., 248, 271 Kong, L., 197, 242 Koshyk, J.N., 327,346 Kulacki, F.A., 289, 291 Kumar, K.S.V., 123, 129, 139 Kumar, S.G.R., 82, 1116 Kuo, J.T., 297,314 Labadie, C., 56, 77 Labadie, G., 297, 303, 313,315 Laghrouche, O., 349, 351, 352, 354, 355, 356, 357, 365, 368, 369, 378, 379, 381) Lahoti, G.D., 143,166 Laitone, E.V., 187, 196 Lakshminarayana, B., 254, 272 Lal, G.K., 143, 167 Lam, C.K.G., 253,257, 272 Lamb, H., 4, 10, 27, 317, 318, 344 Landau, L.D., 4, 27 Lapidus, A., 67, 78, 203,245 Larsen, J., 327,345 Larsson, L., 177, 195 Larwood, B., 118, 137 Latteaux, B., 297, 303, 313,315 Lau, L., 343,348 Laug, P., 123, 129, 131,139 Launder, B.E., 250, 251,253, 271, 272 Lauriat, G., 285,288, 291 Laval, H., 82, 106 Lax, P.D., 61, 66, 78 Le Balleur,, 241,247 Le Quere, P., 191,196 Lee, C.H., 143, 166 Lee, J.H.W., 297, 298, 302, 315 Lee, J.K., 144, 152, 167, 168 Lee, R.L., 82, 106 Lee, S.C., 191,196 Lee, T.H., 177, 195 Lee, W.I., 172, 176, 194 Legat, V., 123, 139, 154, 169 Leonard, B.P., 37, 74 Lesaint, P., 71, 78 Leschziner, M.A., 250, 271,272 Levine, E., 56, 77 Lewis, R.W., 21, 27, 123, 129, 140, 172, 176, 186, 190, 191,194, 195, 196, 274, 290, 292, 314 Lick, W., 311,316
Lien, ES., 250, 271 Lifshitz, E.M., 4, 27 Lighthill, M.J., 241,247, 317, 334, 343, 344 Lin, P.X., 56, 77 Liou, J., 56, 77 Liu, A.W., 162, 163, 169 Liu, C-B., 256, 272 Liu, Y.C., 123, 136,137,14t), 146, 148, 151,152,167, 197, 243 Lo, D.C., 172, 186, 194 Ltihner, R., 56, 64, 65, 77, 78, 123,137,138,139, 172, 178, 179,194, 197,207,208,220, 228,242, 243, 246, 299, 316 Lopez, S., 220, 246 Lucas, T., 177, 195 Lynch, D.R., 297, 301,314, 315, 316 Lynov, J.P., 373,380 Lyra, P.M.R., 197,244 Lyra, P.R.M., 52, 76, 82,1t)8, 123,138,216,217,246 Macaulay, G.J., 328, 332, 334, 346, 347 McCarthy, J.H., 177, 195 MacCormack, R.W., 203, 204, 245 Majda,, 325,345 Maji, P.K., 52, 77, 82, 11)7 Makinouchi, A., 152, 168 Malamataris, N., 101,109 Malett, M., 205,245 Malkus, D.S., 134, 14t), 373,380 Malone, T.D., 297, 314 Mandel, J., 354, 357, 379 Manzari, M.T., 82,11)8, 197,216, 217, 221,244, 246, 266, 268,273 Marchal, J.M., 154, 169 Marchant, M.J., 123, 128, 138, 139, 197, 221,233, 238,244 Marcum, D.L., 123, 129, 140 Margolin, L., 270, 273 Marini, L.D., 42, 75 Marshall, R.S., 191,196 Martin, J.C., 175, 194 Martin, P., 349, 351,378 Martinelli, L., 172, 179, 180, 193, 195 Martinez, M., 297, 315 Martinez-Canales, M.L., 297, 316 Massarotti, N., 82, 11)8, 190, 191,196, 277, 290 Massoni, E., 152, 168 Matallah, H., 154, 169 Mathur, J.S., 82, 96, 108, 112, 136 Matthieu, C., 256, 270, 272 Mavripolis, D.J., 98, 11)9 Maxant, M., 56, 77 Maxworthy, T., 188, 190, 196 Mayer, P., 354, 357, 379 Mead, H.R., 241,247 Medale, M., 172, 186, 194
422
Authorindex Mei, C.C., 177, 195, 317, 327, 336, 337, 338, 344, 346 Melenk, J.M., 353, 354, 355,357, 379 Mellor, P.B., 144, 167 Melnok, R.E., 241,247 Minev, P.D., 82, 107 Mitchell, A.R., 34, 35, 37, 46, 74, 342, 348 Mitsoulis, E., 160, 169 Mittal, S., 235, 239, 247 Mohammadi, B., 123, 129, 131,139, 233,246 Moin, P., 113, 137, 271,273 Monk, P., 354, 357, 359, 360, 361,362, 364, 365,379 Morgan, K., 47, 52, 56, 64, 65, 76, 77, 78, 82, 91, 96, 99, 106, 107, 108, 112, 118, 123, 125, 128, 129,136,137,138,139, 197, 203,204,205,207, 208,213,214, 215,216, 217,218,219, 220, 221, 222, 223,224, 225,227,228, 230, 233,238,240, 242, 243, 244, 245, 246, 256, 266, 268,272, 273, 296, 297,298, 299, 301,308,314, 316, 352, 353, 378 Morinishi, K., 235,239, 247 Morse, P.M., 318, 344, 354, 379 Morton, K.W., 54, 56, 77, 203,245 Moser, R., 271,273 Mulder, W.A., 373,380 Muttin, F., 172, 193 Nakayama, T., 82, 106 Nakazawa, D.W., 37, 45, 46, 47, 74, 75 Nakazawa, S., 56, 77, 97, 109, 134, 136, 140, 143, 166, 299, 316 Nakos, D.E., 177, 195 Narayana, P.A.A., 82, 106, 107, 123, 129, 139, 191, 196 N~ivert, U., 47, 52, 76 Navon, I.M., 297, 314, 315 Navti, S.E., 172, 186, 194 Nesliturk, A., 41, 75 Neta, B., 326, 345 Newton, R.E., 317, 324, 344 Nguem, N., 52, 76 Nichols, B.D., 172, 176, 193 Nickell, R.E., 143, 166 Nicolaides, R.A., 220, 246 Nield, D.A., 274, 275, 276, 290 Nishida, Y., 235,239, 247 Nithiarasu, P., 21, 27, 47, 52, 57, 59, 76, 78, 82, 83, 93, 94, 95, 96,107,108,109, 112, 113, 116, 123, 131,136,139, 159,169, 172, 186, 190, 191,194, 196, 205,206, 207, 217, 218, 219, 229, 232,245, 246, 256, 272, 275, 277, 281,282, 290 Nitsche, J.A., 72, 78 Noble, R., 222, 246 Noblesse, E, 177, 195 Nonino, C., 191,196
O'Brien, J.J., 297, 314 O'Carroll, M.J., 21, 27 Oden, J.T., 123, 134, 137, 138, 139, 140, 143, 147, 166, 197, 199, 213,244, 245 Oh, S.I., 143,166 Ohmiya, K., 82, 106 Olsson, E., 256, 272 Ofiate, E., 44, 45, 49, 52, 75, 76, 77, 143, 147, 152, 153,166,167,168, 172, 175,176, 178, 179,194, 195 Oran, E.S., 270, 273 Ortiz, P., 57,59, 78, 82,107, 108, 229, 232, 246, 297, 298, 301,303,316, 355, 364, 365,379, 380 Pagano, A., 241,247 Paisley, M.E, 203,245 Pal, M., 332, 347 Palit, K., 143,166 Palmeiro, B., 123, 138 Palmeiro, E, 123, 137 Papanastasiou, T.C., 101,109 Pastor, M., 82,108, 164, 165,169, 274, 275,290, 297, 316 Patil, B.S., 319, 345 Patnaik, B.S.V.E, 82, 107 Patrick, M.A., 256, 257, 272 Patrik, M.A., 115, 117, 137 Peiro, J., 123, 137, 138, 197, 203,204, 213,216, 221, 223,243, 245, 246, 365,375,380 Pelletier, D.H., 123, 131,138 Peraire, J., 56, 64, 77, 78, 82, 101,108, 123, 125, 128, 129,137,138,139, 197,203,204, 207, 208,213, 214, 215, 216, 217, 221,223,233,240, 242, 243, 244, 245, 246, 256, 272, 293,296, 297, 298, 301, 302, 308, 314, 315, 352, 353,378 Periaux, J., 123, 137, 221,246 Peric, D., 172, 193 Perrey-Debain, E., 349, 351,357, 378, 379 Perrier, P., 221,246 Perry-Debain, E., 365,368,369, 380 Peseux, B., 357, 379 Phan-Thien, N., 157, 159, 160, 162, 163, 169 Philips, T.N., 160, 169 Pica, A., 297,315 Pin, ED., 175,195 Pinsky, P.M., 335,347 Pironneau, O., 56, 77 Pitk~anta, J., 47, 52, 76, 99, 109 Pittman, J.E, 45, 75, 143, 144, 166 Ponthot, J.-Ph., 172, 186, 194 Posse, M., 56, 77 Postek, E., 143, 167 Prasad, V., 289, 291 Price, J.W.H., 143, 166 Probert, E.J., 123, 128, 138, 139, 197, 221,227, 228, 229, 233, 238, 240, 244, 246
Author index 423 Proft, J., 297, 316 Prudhomme, S., 123, 139 Pulliam, T.H., 208, 209, 245 Qu, S., 97,109 Quartapelle, L., 82, 106 Quecedo, M., 297, 316 Rabier, S., 172, 186, 194 Rabinowitz, P., 374, 381 Rachowicz, W., 123, 138, 139 Raithby, G.D., 82, 106 Raj, K.H., 143, 167 Ramaswamy, B., 82, 93,106, 172, 173, 186, 191,193, 194, 196 Rannacher, R., 82, 106 Raveendra, V.V.S., 123, 129, 140 Raven, H., 177, 195 Raviart, P.-A., 71, 78 Ravindran, K., 172, 176, 186, 194, 195, 281,290 Ravisanker, M.S., 82, 106 Rebelo, N., 143,166 Reed, W.H., 71, 78 Rees, M., 34, 74 Reinhart, L., 56, 77 Ren, G., 82, 106 Reynen, J., 52, 76 Rice, J.G., 82, 106 Richez, M.C., 297,314 Richtmyer, R.D., 203,245 Rider, W.J., 270, 273 Rimon, Y., 118, 137 Rivara, M.C., 220, 246 Roach, P.J., 4, 27 Rodi, W., 256, 272 Rodriguez-Ferran, A., 172, 186, 194 Rojek, J., 82, 108, 143, 164, 165, 167, 169 Runchall, A.K., 34, 74 Russel, T.E, 56, 77 Russo, A., 42, 75 Rynne, B., 349, 351,378 Sacco, C., 172, 179, 194 Saghafian, M., 262, 273 Sai, B.V.K., 191,196 Sai, B.V.K.S., 47, 52, 76, 82, 91, 99, 106, 107, 108, 205, 217, 218, 219, 245 Said, R., 118, 137, 197, 221,244 Saidi, M.S., 262, 273 Sakhib, E, 41, 75 Salasnich, 297, 315 Saltel, E., 123, 129, 131,139 Sanchez, E., 355,365,379 Sani, R.L., 136, 140 Sarpkaya, T., 2 l, 27 Sarrate, J., 187, 196
Sastri, V., 56, 77 Satofuka, N., 235,239, 247 Scavounos, Pd., 177, 195 Sch~ier, U., 176, 195 Schlichting, H., 3, 27, 262, 272 Schmidt, W., 203, 245 Schmitt, V., 247, 267, 268, 273 Schneider, G.E., 82, 106 Schnipke, R.J., 82, 106 Schoombie, S.W., 342, 348 Schrefler, B.A., 274, 275,290 Schreurs, P.J.G., 152, 168 Schr6der, W., 256, 272 Schroeder, W.J., 123, 140 Scott, V.H., 40, 75 Secco, E., 297,315 Seetharamu, K.N., 21, 27, 82, 93, 106, 107, 123, 129, 139, 190, 191,196, 275,282, 290 Seki, N., 285, 291 Selvam, R.P., 262, 272 Semeniuk, K., 327,346 Shakib, E, 47, 52, 76, 205, 245 Shanker, P.N., 113, 137 Shephard, M.S., 123, 140 Shepherd, T.G., 327, 346 Sherwin, S.J., 365,374, 375, 376, 377, 380, 381 Shih, T.H., 257, 272 Shimizaki, Y., 143, 166 Shimura, M., 82, 106 Shin, C.T., 113, 131,136 Shin, S., 172, 176, 194 Shiomi, T., 274, 275,290 Shirron, J., 357, 379 Shirron, J.J., 332, 347 Shu, C.-W., 375,381 Sibson, R., 123,139 Silva, R.S., 41, 75 Sinha, S.K., 277, 290 Slavutin, M., 332, 347 Smagorinsky, J., 269, 273 Smith, A.M.O., 251,272 Smith, I., 382, 387 Smith, J.W., 113, 137 Smith, M.D., 160, 162, 163, 169 Smolinski, P., 41, 75 Sod, G., 207, 208, 245 Soding, H., 177, 195 Sommerfield, A., 321,345 Soni, B.K., 123, 140 Sorensen, K.A., 197, 221,244, 256, 272 Sosnowski, W., 152, 168 Souli, M., 172, 186, 194 Spalart, P.R., 252, 270, 272, 273 Spalding, B., 251,272 Spalding, D.B., 34, 74 Squire, H.B., 241,247
424 Author index Squires, K., 256, 270, 272 Srinivas, M., 82, 106 SSC Program Ltd., 222, 246 Stagg, K.G., 292, 314 Staniforth, A.N., 297, 315 Stansby, EK., 172, 185, 186, 194, 262, 273 Steger, J.L., 203,245 Stewart, J.R., 197, 217, 220, 243 Stojek, M., 362, 363, 380 Storti, M.A., 52, 76 Stoufflet, B., 221,246 Strada, M., 191,196 Strelets, M., 270, 273 Stroubolis, T., 319, 345, 351,378 Strouboulis, T., 123, 137, 349, 357, 378 Subramanian, E., 286, 291 Subramanian, G., 123, 129, 140 Sugawara, T., 340, 341,348 Sugimoto, R., 357, 379 Suli, E., 56, 77 Sun, J., 160, 162, 163, 169 Sundararajan, T., 82, 93,107, 123,129,139, 143,167, 190, 196, 275,277, 282, 290 Sung, J., 172, 186, 194 Szepessy, A., 47, 52, 76, 205,245 Szmelter, J., 82, 101,108, 123, 131,138, 241,247 Takeuchi, N., 297, 315 Tam, A., 123, 139 Tam, Z., 343,348 Tamaddonjahromi, H.R., 154, 169 Tang, L.Q., 113,137 Tanner, R.I., 143,166, 169 Taylor, C., 82, 106, 172, 186,194, 297, 314, 319, 345 Taylor, L.K., 172, 193, 194 Taylor, M.A., 374, 381 Taylor, R.L., 2, 4, 5, 14, 17, 19, 21, 23, 27, 30, 40, 52, 54, 74, 80, 81, 82, 92, 97, 105, 108, 109, 111, 112, 123, 127, 131,134, 136,136,140, 143, 152, 164, 165,166,169, 180,195,319,320,321,323, 327, 331,338, 339, 345, 350, 354, 362, 365,375, 378, 380, 387, 388 Temam, R., 4, 27 Teng, C.H., 373,380 Teng, W.-H., 172, 186, 194 Tezduyar, T., 297, 315 Tezduyar, T.E., 207,245 Tezduyar, T.T.I., 56, 77 Thareja, R.R., 197, 217, 220, 243 Thomas, C., 94, 109 Thomas, C.G., 206, 207, 245 Thompson, E.G., 143,166 Thompson, J.E, 123, 140 Thompson, L.L., 335,347 Tien, C.L., 275,284, 290, 291 Tong, P., 40, 74
Tong, T.W., 286, 291 Tottenberg, U., 220, 246 Townsend, E, 154, 169 Toyoshima, S., 134, 136, 140, 152, 167 Tremayne, D., 222, 246 Trevelyan, J., 357, 365,368, 369, 379, 380 Trevisan, O.V., 288, 291 Tsang, T.T.H., 113, 137 Tsubota, K., 299, 316 Tucker, EG., 250, 256, 262, 270, 271,272, 273 Turkel, E., 321,345 Turner-Smith, E.A., 197, 243 Tutar, M., 256, 262, 263,272 Tzabiras, G.D., 172, 194 Upson, C.D., 82, 106 Ushijima, S., 172, 186, 194 Usmani, A.S., 123, 129, 140 Utnes, T., 82, 106 Vadyak, J., 203,244 Vafai, K., 275, 284, 290, 291 Vahdati, M., 123, 125, 128, 129, 137, 197, 213, 214, 215, 216, 243 Vallet, M.G., 123, 139 van der Lugt, J., 152, 168 van Estorff, O., 332, 347 Van Veldhuizen, M., 373, 380 Vazquez, C., 56, 57, 59, 77, 78 V~quez, M., 47, 52, 76, 82, 93, 97, 99, 102,107,108, 112, 123,136,137,205,217,218,219,229,232, 245, 246 Veldpaus, EE., 152, 168 Verhoeven, N.A., 197, 221,244 Vilotte, J.P., 134, 136, 140 Vincent, R.E., 374, 381 von Neumann, J., 203,245 Walker, K.L., 288, 291 Waiters, K., 154, 169 Wang, C., 250, 272 Wang, H.H., 297, 314 Wang, N.M., 152, 168 Warburton, T., 325,345, 375,381 Ward, S., 228, 246 Wargedipura, A.H.S., 152, 168 Watson, D.E, 123, 129, 139 Weatherill, N.E, 52, 76, 82, 96, 108, 112, 118, 123, 129,136,137,138,139,140, 197, 221,222, 224, 225,227, 228,230, 233,238,243, 244, 246, 266, 268, 273, 352, 378 Webster, M.E, 154, 169 Weeks, D.J., 241,247 Wehausen, J.V., 177, 195 Welch, J.E., 195 Wendroff, B., 61, 66, 78
Author index 425 Wesswling, E, 220, 246 Westermann, T.A., 123, 138 Wheeler, M.E, 297, 315 Whitaker, S., 275,290 Whitfield, D.L., 172, 193, 194 Whitham, G.B., 317, 318, 344 Wieting, A.R., 197, 217, 220, 243 Wifi, A.S., 152, 168 Wilcox, D.C., 25 l, 272, 273 Will6, D.R., 382, 387 Williams, A.J., 160, 169 Williams, R.T., 292, 297, 314, 315 Wingate, B.A., 374, 381 Wolfstein, M., 34, 74, 25 l, 272 Wong, K.K., 343,348 Wood, R.D., 143, 144, 152, 154, 166, 167, 168 Wood, R.W., 152, 168 Woodward, E., 297, 314 Woodward, E, 207, 210, 245 Wriggers, E, 172, 186, 194 Wr6blewski, A., 362, 380 Wu, J., 82, 109, 123, 128, 130, 131, 138, 139, 231, 233, 236, 246, 297, 315 Yagewa, G., 143, 147, 166 Yahia, D.D.D., 123, 139 Yang, B., 373,381 Yang, C., 172, 178, 179, 194 Yang, Z., 257, 272 Yeckel, A, 113, 137 Yiang, C.B., 82, 106 Yoo, J.Y., 172, 186, 194
Yoshida, T., 297,315 Yoshimura, T., 340, 341,348 Young, A.D., 241,247 Young, D.L., 172, 186, 194 Yovanovich, M.M., 82, 106 Yu, C.C., 52, 76, 191,196 Zalesiak, S.T., 68, 78 Zhang, Y., 123, 129, 140 Zhou, J.G., 172, 185, 186, 194 Zhu, J.Z., 2, 4, 5, 14, 17, 19, 21, 23, 27, 80, 81, 92, 97, 105, 111,112, 123, 127, 131,134, 136, 136, 137, 152,167, 180,195, 197,213,243,245,319, 320, 321,323,327, 331,338,339, 345, 350, 354, 362, 375,378, 387,388 Ziegler, C.K., 311,316 Zienkiewicz, O.C., 2, 4, 5, 14, 17, 19, 21, 23, 27, 30, 34, 35, 37, 40, 45, 46, 47, 52, 54, 56, 57, 59, 64, 65, 74, 75, 76, 77, 78, 80, 81, 82, 91, 92, 93, 95, 97, 99, 101, 102, 105, 107, 108, 109, 111, 112, 123, 125,127, 128, 129, 130, 131,134, 136,136, 137,138,139,140, 143, 144, 146, 147, 148, 149, 151,152, 153, 154, 162, 164, 165,166,167,168, 169, 180, 190, 191, 195,196, 197,203,204,205, 207,208, 213, 214, 215,216, 217,218, 219,221, 223,229, 231,232, 233,236, 242, 243, 245, 246, 274,275,277,290, 292, 296, 297,298,299, 301, 302,303,308,314, 315, 316, 317, 318, 319,320, 321,323,324,327, 328, 329, 331,336, 338, 339, 344, 345, 346, 347, 348, 350, 352, 354, 362, 363, 365,375,378, 380, 387,388 Zolesio, J.E, 172, 186, 194
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Subject index Acoustic wave velocity, 14 Adaptive mesh generation, incompressible Newtonian laminar flows, for transient problems, 131,133 Adaptive mesh refinement, incompressible Newtonian laminar flows: about adaptive mesh refinement, 123 choice of variables, 130-1 element elongation, 128-30 estimation of second derivatives at nodes, 128 first derivative- (gradient) based refinement, 130 lid-driven cavity example, 131-2 local patch interpolation: superconvergent values, 127 second gradient- (curvature) based refinement, 123-7 interpolation errors, 125-6 principal values and directions, 127 see also Compressible high-speed gas flow, adaptive refinement and shock capture in Euler problems Aerofoil, potential flow solution example, 21-2 Anisotropic shock capturing, with high-speed gas flow, 205 Arbitrary-Lagrangian-Eulerian (ALE) methods, free surface flows, 172, 185-9 Artificial compressibility: and the character-based split (CBS) algorithm, 95-6 and dual time stepping, 96 Artificial diffusion concept, 40 Astley's shape function, with conjugated infinite elements, 331 Babu~ka-Brezzi restriction (BB), 81 circumvention of, 97-8 Balancing diffusion, 46 streamline balancing diffusion, 47 Bore, shallow water example, 300-1 Boundary conditions: with the CBS algorithm, 100-3
convection-diffusion-reaction equation, 72-3 Dirichlet type, 15 governing equations, 9 Neumann type, 15 radiation, 62-4 Boundary layer-inviscid flow coupling, 410-12 Boussinesq approximation, porous medium flow, 279, 285 Boussinesq assumption, with turbulent flows, 250 Brinkman extensions, porous medium flow, 284 Bristol channel, shallow water example, 301-5 Buoyancy driven incompressible flows: about buoyancy driven flows, 189-91 flow in an enclosure example, 191-3 Grashoff number, 190 Prandtl number, 191 Rayleigh number, 191 Burger equation, 68-70 Cauchy-Poisson free surface condition, waves, 339 Character-based split (CBS) algorithm: about the CBS algorithm and compressible and incompressible flow, 79-81,104-5 about the split, 82-3 artificial compressibility, 95-6 in transient problems (dual time stepping), 96 boundary conditions: application of real boundary conditions, 101-3 fictitious boundaries, 100-1 prescribed traction boundary conditions, 101 solid boundaries in inviscid flow (slip conditions), 101 solid boundaries with no slip, 101 Chorin split, 82-3 circumvention of the Babu~ka-Brezzi (BB) restrictions, 97-8 forms/schemes: about forms, 92 evaluation of time limits, 93-5 fully explicit form, 92 quasi- (nearly) implicit form, 93
428
Subjectindex Character-based split (CBS) algorithm- cont. semi-implicit form, 93 governing equations for, 79-82 with high-speed gas flow, 202-3 inviscid problem, performance of two- and single-step algorithms, 103-5 mass diagonalization (lumping), 91-2 with metal forming, transient, 164 with Porous medium flow, 280 with shallow water problems, 297-8 single step version, 98-9 spatial discretization and solution procedure, 86-91 split A, 86-91 split B, 91 temporal discretization, 83-5 split A, 84-5 split B, 85 with viscoelastic flows, 163 see a l s o Computer implementation of the CBS algorithm; Incompressible Newtonian laminar flow Chrzy bed friction, long and medium waves, 320 Chrzy coefficient, 294 Chorin split, 82-3 see also Character-based split (CBS) algorithm Cnoidal and solitary waves, 340-2 Coefficients of pressure and friction, postprocessing, 393 Compressible high-speed gas flow basics: about high-speed gas flow, 197-8 boundary conditions, subsonic and supersonic: Euler equation, 200-1 Navier-Stokes equations, 201-2 boundary layer-inviscid Euler solution coupling, 241 Euler equation examples: inviscid flow past an RAE2822 airfoil, 208-10, 211 isothermal flow through a nozzle in one dimension example, 207, 209 Riemann shock tube - transient problem in one dimension example, 207, 208 two-dimensional transient supersonic flow over a step example, 207-8, 210 governing equations: ideal gas law, 198 internal energy, 198 Navier-Stokes, 198 total specific energy, 198-9 numerical approximations and the CBS algorithm, 202-3 shock capture methods: about shock capture, 203-4 anisotropic viscosity shock capturing, 205 residual-based methods, 205
second derivative-based methods, 204-5 structured meshes, 197 variable smoothing, 205-6 subsonic inviscid flow past an NACA0012 airfoil, 206 see also Turbulent flows, compressible material Compressible high-speed gas flow, adaptive refinement and shock capture in Euler problems: about adaptive refinement, 212 h-refinement process and mesh enrichment, 212-13 h-refinement and remeshing in steady-state two-dimensional problems, 213-17 remeshing in steady-state two-dimensional problems examples: hypersonic inviscid flow past a blunt body example, 214-16 inviscid flow with shock reflection from a solid wall example, 214, 215 inviscid shock interaction example, 217, 220 supersonic inviscid flow past a full circular cylinder example, 217-19 Compressible high-speed gas flow, steady state three-dimensional inviscid examples: complete aircraft flow patterns: inviscid engine intake example, 221-2, 224 inviscid flow past full aircraft example, 221,223 multigrid approaches, 220-1 parallel computation, 221 recasting element formulations in an edge form, 217 THRUST- the supersonic car, 222-6 Compressible high-speed gas flow, transient twoand three-dimensional problems: exploding pressure vessel example, 226, 228 shuttle launch, 227, 229, 230 Compressible high-speed gas flow, viscous problems in three dimensions, 240 hypersonic viscous flow past a double ellipsoid, 240-1 Compressible high-speed gas flow, viscous problems in two dimensions: about viscous flow, 227-8, 231 adaptive refinement in both shock and boundary layer, 230, 234-5 special adaptive refinement for boundary layers and shocks, 230-3, 236--8 transonic viscous flow past an NACA0012 aerofoil, 235-9 viscous flow past a plate example, 229-30, 232 Compressible and incompressible flow see Character-based split (CBS) algorithm Computer implementation of the CBS algorithm: about computer implementation, 382-3 data input module, 383--4 boundary data, 383
Subject index 429 mesh data- nodal coordinates and connectivity, 383 necessary data and flags, 383-4 preliminary subroutines and checks, 384 output module, 387 solution module, 384-7 boundary conditions, 386-7 CBS algorithm; steps, 386 convergence to steady state, 387 different forms of energy equation, 387 shock capture, 385-6 solution of simultaneous equations semi-implicit form, 387 time step, 384-5 see also Character-based split (CBS) algorithm Conservation laws, 28-30 Conservation of mass: about conservation of mass, 1-2 governing equations, 6-7, 81 incompressible Newtonian laminar flow, 110 Conservation of momentum: for CBS algorithm, 80-1 dynamic equilibrium, 7 incompressible Newtonian laminar flow, 111 Constitutive equation, viscoelastic flows, 158 Continuity equation: with free surface flows, 173 viscoelastic flow, 157 Convection-diffusion equations: vector-valued variables: multiple wave speeds, 402-4 Swansea two step operation, 402 Taylor-Galerkin method, 397-9 two-step predictor-corrector methods, 399-400 two-step Taylor-Galerkin operation, 400-2 Convection-diffusion-reaction equation: about convection-diffusion equations, 28-30, 73 boundary conditions, 72-3 Galerkin process/method/procedure, 72-3 conservation laws, 28-30 convective flux quantities, 28 diffusive flux quantities, 28 Galerkin weighting, 30 pure convection treatment, 70-2 transport equation, 28 velocity field, 29 see also Steady state convection-diffusion equation in one dimension; Steady state convection-diffusion equation in two (or three) dimensions; Transient convection-diffusion equation; Waves and shocks, non-linear Convective acceleration effects, 1 Convective flux quantities, 28 Coriolis accelerations/parameters, with shallow water problems, 293-5,303
Dam break, shallow water example, 299-300 Darcy-Rayleigh number, porous medium flow, 279 Darcy's law/flow regime, porous medium flow, 274-5, 286-8 Deborah number, with viscoelastic flows, 158, 160, 162 Detached Eddy Simulation (DES), 270 Deviatoric stresses/deviatoric strain rates, 1, 5 Differential equations, self-adjoint, 391 Diffusion: artificial diffusion concept, 40 balancing diffusion, 46 cross-wind diffusion, 47 negative diffusion concept, 40 streamline balancing diffusion, 48-9 see also Convection-diffusion-reaction equation Diffusive flux quantities, 28 Direct Numerical Simulation (DNS), 270-1 Dirichlet type boundary conditions, 15 Discontinuous enrichment method, short waves, 357-9 Discontinuous Galerkin Finite Elements (DGFE), 374-7 Discretization procedure, porous medium flow, 279-82 Drag force calculation, postprocessing, 392-3 Dual time stepping, and artificial compressibility, 96 Dynamic viscosity, 1 Eddies, large eddy simulation, 267-9 eddy viscosity, 269 Kolmogorov cascade/constant, 269 Smagorinsky's model/constant, 269 standard SGS model, 269-70 Edge-based finite element formulation, 405--6 Eigenvalues, problems with waves in closed domains, 319 Elastic bulk modulus, 1, 14 Elastic springback, in non-Newtonian flows, 152-4 Electromagnetic scattering, 352-3 Energy conservation: for CBS algorithm, 80 and equation of state, 7-9 Enrichment functions, short waves, 358 Enthalphy, 8, 13 Equation of state and energy conservation, 7-9 Equations of fluid dynamics, 1-4 see also Governing equations of fluid dynamics; Inviscid, incompressible flow Ergun correlation, porous medium flow, 276-7 Euler equations, 9-11 compressible high speed gas flow examples, 206-11 with high-speed gas flow, 200-1
430 Subjectindex Euler problems s e e Compressible high-speed gas flow, adaptive refinement and shock capture in Euler problems Euler solutions, boundary layer-inviscid Euler solution coupling, 241 Eulerian form, 6 Eulerian methods, free surface flows, 176-84 Explicit characteristic-Galerkin procedures, 56-62 Extrusion: transient extrusion problem, 150-4 s e e a l s o Metal forming, transient, direct displacement approach; Non-Newtonian flows - metal and polymer forming; Viscoelastic flows Fekete points, 374 Finite element methods/approximation s e e Weighted residual and finite element methods Finite increment calculus (FIC): in multidimensional problems, 48-9 stabilization of the convection-diffusion equation, 43-4 Finite volume approximation/technique/methodology, 2, 23-4 Poisson equation in two dimensions example, 24-6 Forcheimmer extensions, porous medium flow, 284 Forming, steady-state problems, 144-7 Free surface incompressible flows: about free surface flows, 170-2 Arbitrary-Lagrangian-Eulerian (ALE) methods: about the ALE method, 172, 185-6 implementation of the ALE method, 186-7 solitary wave propagation example, 187-9 Eulerian methods: about Eulerian methods, 172, 176 hydrostatic adjustment, 178-9 mesh updating or regeneration methods, 176-84 sailing boat example, 184-5 ship motion problem, 177 submarine example, 181-3 submerged hydrofoil example, 179-82 Lagrangian methods, 171-5 continuity equation, 173 Kroneker delta, 172 model broken dam problem example, 173-5 momentum equation, 173 Frequency domain solutions, short waves, 350 Friction coefficients, postprocessing, 393
with transients, 51-3 Galerkin procedure: and boundary conditions for convection-diffusion, 72-3 simple explicit characteristic, 56-62 Galerkin scheme/equations, short waves, 367 Galerkin spatial approximation, 56, 59 and CBS algorithm, 80 Galerkin weighting, convection-diffusion-reaction equation, 30, 32 Gas flow s e e Compressible high-speed gas flow Gauss-Chebyshev-Lobatto scheme, 373-4 Gauss-Legendre integration, 355, 373 Gauss-Lobatto scheme, 373-4 Givoli's procedure, NRBCs with waves, 326 GLS s e e Galerkin least squares (GLS) approximation Governing equations of fluid dynamics: Babu~ka-Brezzi restriction, 81 balance of energy, 9 boundary conditions, 9 for character-based split (CBS) algorithm, 79-82 compressible flow, 11 conservation of energy and equation of state, 7-9, 80 non-dimensional form, 82 conservation of mass, 6-7 non-dimensional form, 81 conservation of momentum, 7, 80 non-dimensional form, 81 deviatoric stresses/deviatoric strain rates, 5 enthalphy, 8 Euler equations, 9-11 Eulerian form, 6 gradient operator, 6 indicial notation, 4 intrinsic energy, 8 inviscid flow, 11 Kroneker delta, 5, 80 Lam6 notation, 6 Navier-Stokes equations, 9-11 non-dimensional form (for CBS algorithm), 81-2 rates of strain, 5 stress-strain rate relations, 5 stresses in fluids, 4-6 turbulence/turbulent instability, 11 volumetric viscosity, 5-6 Gradient operator, 6 Grashoff number, with buoyancy driven flows, 190 Green's function, 43
Galerkin, finite element, method, 17-18 Galerkin formulation with two triangular elements example, 19-21 Galerkin least squares (GLS) approximation/method: in multidimensional problems, 48-9 in one dimension, 41-2
h-refinement: and mesh enrichment, 212-13 and remeshing in steady-state two-dimensional problems, 213-17 Hankel functions, 338 and infinite elements, 328
Subject index 431 with Trefftz type infinite elements, 333 Higdon boundary condition, short waves, 365 Hydrofoil, submerged, example, free surface flows, 179-82 Hydrostatic adjustment, free surface flows, 178-9 Ideal gas law, with high-speed gas flow, 198 Incompressibility constraint difficulties, 1-2 Incompressible flows, 13-14 about free surface and buoyancy driven flows, 170 about incompressible flows, 3 acoustic wave velocity, 14 elastic bulk modulus, 14 s e e a l s o Buoyancy driven incompressible flows; Character-based split (CBS) algorithm; Free surface incompressible flows Incompressible Newtonian laminar flow: about laminar flow, 110, 136 basic equations: conservation of energy, 111 conservation of mass, 110 conservation of momentum, 111 with CBS algorithm: fully explicit artificial compressibility form, 112 incompressible flow in a lid-driven cavity example, 113-20, 125 quasi-implicit form, 123 semi-implicit form, 112-23 steady flow past a backward facing step example, 115-21 steady flow past a sphere example, 117-22 transient flow past a circular cylinder example, 118-24 mixed and penalty discretization/formulations, 134-6 slow flows, analogy with incompressible elasticity, 131,134 s e e a l s o Adaptive mesh refinement, incompressible Newtonian laminar flows Incompressible non-Newtonian flows: about non-Newtonian effects, 141, 165 s e e a l s o Extrusion; Metal forming, transient, direct displacement approach; Non-Newtonian flows - metal and polymer forming; Viscoelastic flows Indicial notation, 4 Infinite elements: about infinite elements, 327 accuracy of, 332 Burnett and Holford ellipsoidal type infinite elements, 328-30 mapped periodic (unconjugated) infinite elements, 327-8 Hankel functions, 328 Trefftz type infinite elements, 332-3 Hankel functions, 333
wave envelope (conjugated) infinite elements, 330-2 Astley's shape function, 331 Integration formulae: linear tetrahedron, 395-6 linear triangles, 395 Interpolation errors, 125-6 Intrinsic energy, 8 Inviscid, incompressible flow, 11-13 irrotational flow, 12 stream function, 13 velocity potential solution, 11-13 Irrotational flow, 12 Kolmogorov cascade/constant, 269 Kolmogorov length scale, 248 Korteweg-de Vries wave equation, 342 Kroneker delta, 5 and CBS algorithm, 80 with free surface flows, 172 with viscoelastic flows, 157 Lagrangian methods, free surface flows, 171-5 Lam6 notation, 6 Laminar flow s e e Incompressible Newtonian laminar flow Lapidus type diffusivity, 67-8 Local Non-Reflecting Boundary Conditions (NRBCs), long and medium waves, 324-7 Long waves s e e Waves, long and medium Mass-weighted averaged turbulence transport equations, 413-15 Maxwell equation, with viscoelestic flows, 156 Medium waves s e e Waves, long and medium Mesh enrichment: and h-refinement process, 212-13 with high-speed gas flow problems, 212-13 Mesh refinement s e e Adaptive mesh refinement, incompressible Newtonian laminar flows Mesh updating, free surface flows, 176-8 Metal forming, transient, direct displacement approach, 163-5 and the CBS algorithm, 164, 165 impact of circular bar example, 165 Microlocal discretization, 354 Mixed and penalty discretization/formulations, 134-6 Modelling errors, short waves, 351 Mollifying/smoothing discontinuities, 39 Momentum equation: free surface flows, 173 viscoelastic flow, 157 Monotonically Integrated LES (MILES), 270 Multigrid method, 407-8
432
Subjectindex NACA0012 aerofoil: inviscid problems of subsonic and supersonic flow, 103-5, 206 transonic viscous flow past the aerofoil, 235-9 Navier-Stokes equations, 9-11, 26 with compressible high-speed gas flow, 198 derivation of non-conservative form, 389-90 with high-speed gas flow, 198, 201-2 with turbulent flows, 248, 249 Neumann type boundary conditions, 15 Newton-Cotes formula, 372-3 Newtonial dynamic viscosity, 157 Newtonion laminar flow s e e Adaptive mesh refinement, incompressible Newtonian laminar flows; Incompressible Newtonian laminar flow Non-Newtonian flows - metal and polymer forming: about viscosity, 141-2 elastic springback, 152-4 flow formulation, 143 Oswald de Wahle law, 142 prescribed boundary velocities, 143 steady-state problems of forming, 144-7 kinetic energy and work considerations, 147 steady state rolling example, 147-8 transient problems with changing boundaries, 147-52 punch indentation example, 149, 151-2 transient extrusion problem, 150--4 viscoelastic fluids, 152-4 viscoplastic fluids, 142 viscoplasticity and plasticity, 141-4 s e e a l s o Metal forming, transient, direct displacement approach; Viscoelastic flows Non-self-adjoint equations, 1 Oldroyd-B model, 156 ONERA-M6 wing, turbulent flow past example, 266-8 Ortiz formulation, short waves, 365 Oswald de Wahle law, 142 Partition of Unity Finite Elements (PUFEs), 357 Peclet number, 32, 33 Perfectly Matched Layers (PMLs), NRBCs with waves, 326-7 Petrov-Galerkin methods: with transients, 51-2, 60-1 for upwinding in one dimension, 34-9, 42 Poisson equation in two dimensions: finite volume formulation with triangular elements example, 24-6 Galerkin formulation with two triangular elements example, 19-21 Pollution error, waves in closed domains, 319 Polymeric liquids, 154
Porous medium flow: about flow through porous media, 274-5 Boussinesq approximation, 279 Brinkman extensions, 284 with CBS scheme, 280 continuity equation, 277, 278-9 Darcy-Rayleigh number, 279 Darcy's law, 274-5 discretization procedure, 279-82 energy equation, 277, 278, 279 Ergun correlation, 276-7 forced convection, 282-3 heat transfer in a packed channel example, 283-4 Forcheimmer extensions, 284 generalized approach, 275-9 momentum equation, 277, 278, 279 natural convection: about natural convection, 284 Boussinesq approximation, 285 buoyancy driven convection in an axisymmetric enclosure example, 288-9 buoyancy driven convection in a packed enclosure example, 285-6 buoyancy driven flow in a saturated cavity example, 286-8 constant porosity medium, 285-6 Darcy flow regime, 286-8 non-dimensional scales, 277-9 non-isothermal flows, 282 porosity definition, 276 semi- and quasi-implicit forms, 281-2 Postprocessing: coefficients of pressure and friction, 393 drag force calculation, 392-3 stream function, 393-4 Prandtl number, 82 with buoyancy driven flows, 191 Pressure coefficients, postprocessing, 393 PUFEs (Partition of Unity Finite Elements), 357 Radiation, boundary conditions, 62-4 RAE2822 airfoil, inviscid flow past example, 208-11 Rayleigh number, with buoyancy driven flows, 191 Refraction s e e Waves, short Regeneration methods, free surface flows, 176-8 Reynolds averaged Navier-Stokes equations, 251 Reynolds number: with turbulent flows, 248, 249 with viscoelastic flows, 158, 160 Reynolds stress, with turbulent flows, 250 Riemmann invariants, with shallow water transport, 313 Riemmann shock tube - high-speed gas flow example, 207 River Severn bore, shallow water example, 305-7 Robin boundary conditions, short waves, 366
Subject index 433 Sailing boat example, free surface flows, 184-5 Self-adjoint differential equations, 17, 391 SGS s e e Sub-Grid Scale (SGS) approximation/method Shallow water: about shallow water problems, 4, 292 drying areas, 310-11 equations for shallow water, basis of, 293-7 Ch6zy coefficient, 294 Coriolis accelerations/parameters, 293-5 Helrnholtz equation, 296 mass conservation with full incompressibility, 293 notation, 294 numerical approximation, 297-8 Characteristic-Based-Split (CBS) algorithm, 297-8 Taylor--Galerkin approximation, 298 Steady state solutions examples: steady state solution, 308, 310 supercritical flow, 308, 310 transient one-dimensional examples: bore, 300-1 dam break, 299-300 solitary wave, 299 transport of shallow water, 311-13 characteristic-Galerkin method/procedure, 311-12 depth-averaged transport equations, 311 Riemmann invariants, 313 tsunami wave in Severn Estuary example, 307-9 two dimensional periodic tidal motion examples: Bristol channel, 301-5 periodic wave, 300-2 River Severn bore, 305-7 Ship motion problem, free surface flow methods, 177 Shocks s e e Waves and shocks, non-linear Shuttle launch, high-speed gas problem flow example, 227, 228, 230 Slow flows, analogy with incompressible elasticity 131, 134 Smagorinsky's model, 269 Sommerfield radiation condition, 363 Spalart-Allmaras (SA) model, turbulence transport equations, 252-3, 413-14 Split, the s e e Character-based split (CBS) algorithm Sponge layers, NRBCs with waves, 326-7 Steady state convection-diffusion equation in one dimension: about the steady state problem, 31-2, 49-50 artificial diffusion concept, 40 balancing diffusion in one dimension, 39-40 continuity requirements for weighting functions, 37-9 convection diffusion example, 32-4 discretization, 31
finite increment calculus (FIC) stabilization, 43--4 Galerkin least squares approximation (GLS), 41-2 Galerkin weighting, 31-2 Green's function, 43 higher order approximations, 44-5 mollifying/smoothing discontinuities, 39 Peclet number, 32, 33 Petrov-Galerkin methods for upwinding, 34-9 sub-grid scale (SGS) approximation, 42-3 variational principle, 40-1 weight function for exact solution example, 34 Steady state convection-diffusion equation in two (or three) dimensions: about two or three dimensions, 45, 49-50 balancing diffusion, 46 finite increment calculus (FIC), 48-9 Galerkin least squares (GLS), 48-9 streamline (Upwind) Petrov-Galerkin (SUPG) weighting, 45-8 Stokes flow, 2-3 Stokes waves, 342-3 Stream function, 13 postprocessing, 393-4 Streamline balancing diffusion, 47-8 Streamline (Upwind) Petrov--Galerkin (SUPG) weighting, 45-8 Stresses in fluids, governing equations, 4-6 Strong and weak forms s e e Weighted residual and finite element methods Structures meshes, with compressible high-speed gas flow, 197 Sub-grid scale (SGS) approximation/method, 42-3 Submarine example, free surface flows, 181-3 Submerged hydrofoil example, free surface flows, 179-82 Supercritical flow, shallow water example, 308, 310 SUPG (Streamline (Upwind) Petrov-galerkin) weighting, 45-8 Swansea two step operation, convection-diffusion equations, 402 Taylor-Galerkin method/procedures, 52, 65-6 with shallow water problems, 298 used for vector-valued variables, 387-9 Tetrahedron, linear, integration formulae, 395-6 THRUST the supersonic car, Euler solution example, 222--6 Time domain solutions, short waves, 350 Transient convection-diffusion equation: about transients, 50--3 advection of a Gaussian cone in a rotating fluid, 62 boundary conditions - radiation, 62--4 characteristic directions, 51 characteristic-Galerkin method/procedures, 54-6, 61-3 discretization procedures, 51-3 -
434 Subjectindex Transient convection-diffusion equation - cont. explicit characteristic-Galerkin procedures, 56-62 Galerkin least squares (GLS) method, 51-3 mathematical background, 50-1 mesh updating and interpolation methods, 53-4 Petrov-Galerkin method, 51-2, 60-1 and the steady-state condition, 66 Taylor-Galerkin methods/procedures, 52, 65-6 see a l s o Waves and shocks, non-linear Transport equation, 28 Transport of shallow water, 311-13 Trefftz type finite elements for short waves, 362-4 Triangles, linear, integration formulae, 395 Tsunami wave in Severn Estuary example, 307-9 Turbulence transport equations, mass-weighted averaged: about turbulence models, 413 Spalart-Allmaras model, 413-44 Turbulent flows, basics: about turbulent flows, 248-9 Boussinesq assumption, 250 instability, 11 Kolmogorov length scale, 248 large eddy velocity scale, 249 Reynolds stress, 250 time averaging, 249-50 turbulent eddy/kinematic viscosity, 250-1 Turbulent flows, compressible material: detached Eddy Simulation (DES), 270 Direct Numerical Simulation (DNS), 270-1 energy conservation equation, 265 large eddy simulation, 267-9 Kolmogorov constant, 269 Smagorinsky's model, 269 standard SGS model, 269-70 mass conservation equation, 264 mass-weighted (Favre) time averaging, 265-6 momentum equation, 265 Monotonically Integrated LES (MILES), 270 turbulent flow past an ONERA-M6 wing example, 266-8 Turbulent flows, incompressible material: diffusion Prandtl number, 252, 253 governing equations, non-dimensional: i( - e model, 255 one-equation model, 255 Spalart-Allmaras model, 255 turbulent flow solution, 254-5 Reynolds averaged Navier-Stokes equations, 251 shortest distance to a solid wall, 256 solution procedure/examples: CBS scheme recommended, 256 turbulent flow past a backward facing step example, 256-9 unsteady turbulent flow past a circular cylinder example, 259-64
Spalart-Allmaras (SA) (one equation) model, 252-3 standard K - e (two equation) model, 253-4 Wolfstein i( - l (one equation) model, 251-2 Ultra weak formulation, short waves, 359-62 Ultra Weak Variational Formulation (UWVF), 361-2 Upwinding: optimal streamline upwinding, 59-60 using Petrov-Galerkin methods, 34-9 Vandermonde matrix, 374 Velocity field, 29 Viscoelastic flows: about viscoelastic flows, 154--7 and the CBS algorithm, 163 flow past a circular cylinder example, 159-63 governing equations: constitutive equation, 158 continuity, 157, 158 Deborah number, 158, 160, 162 Kroneker delta, 157 momentum, 157, 158 Newtonial dynamic viscosity, 157 Reynolds number, 158, 160 Maxwell equation, 156 Oldroyd-B model, 156 polymeric liquids, 154 Viscosity: polymers and hot metals, 141 secant velocity, 141 viscoelastic fluids, 152-4 viscoplastic fluids, 142 viscosity-strain rate dependence, 141-2 viscous flow problems, 3 volumetric viscosity, 5 see also Compressible high-speed gas flow, viscous problems .... Volumetric viscosity, 5-6 Wave envelope (conjugated) infinite elements, 330-2 Wave propagation, solitary wave example, free surface flows, 186-9 Waves, long and medium: about long and medium waves, 317-18, 344 bed friction, 320 Chrzy bed friction, 320 convection of waves, 333-5 linking finite elements to exterior solutions (DtN mapping): about linking, 336 Hankel functions, 338 to boundary integrals, 336-7 to series solutions, 337-8
Subject index local Non-Reflecting Boundary Conditions (NRBCs): about NRBCs, 324-6 Givoli's procedure, 326 Perfectly Matched Layers (PMLs), 326-7 sponge layers, 326-7 modelling difficulties, 320 refraction of waves, 333-5 three-dimensional effects in surface waves: about surface waves in deep water, 338-40 Cauchy-Poisson free surface condition, 339 cnoidal and solitary waves, 340-2 free surface condition, 338, 338-9 Korteweg--de Vries equation, 342 large amplitude waves, 340 Stokes waves, 342-3 transient problems, 335 unbounded problems, 324 waves in closed domains, finite element models, 318-19 eigenvalue problem, 319 pollution error, 319 waves in unbounded domains, 321-3 diffraction and refraction problems, 320-1 incident waves, domain integrals and nodal values, 323 radiation condition, 321-2 radiation problem, 321 scattering problem, 321 wave diffraction, 321-3 s e e a l s o Infinite elements; Shallow water Waves and shocks, non-linear, 66-70 Burger equation, 69-70 development of shock, 67-8 Lapidus type diffusivity, 67-8 propagation speeds, 67 steep wave modelling, 67-9 Waves, short: about short waves, 349-51 frequency domain solutions, 350 time domain solutions, 350 Discontinuous Galerkin finite elements (DGFE), 374-7 electromagnetic scattering, transient solution, 352-3 finite elements incorporating wave shapes: about finite elements with short waves, 352-4 discontinuous enrichment method, 357-9 enrichment functions, 358 Gauss-Legendre integration points, 355 microlocal discretization, 354 Partition of Unity Finite Elements (PUFEs), 357 shape functions using products of polynomials and waves, 354-7
shape functions using sums of polynomials and waves, 357 Sommerfield radiation condition, 363 Trefftz type finite elements for waves, 362--4 ultra weak formulation, 359--62 Ultra Weak Variational Formulation (UWVF), 361-2 modelling developments, 351 modelling errors, 351 refraction: about refraction, 364 acoustic velocity potential, 370 convected wave equation, 370 Galerkin scheme, 367 Higdon boundary condition, 365 Ortiz formulation, 365 plane scattered by stepped cylinder example, 368-9 plane wave basis finite elements, 366 refraction caused by flows, 369-72 Robin boundary conditions, 366 wave speed refraction, 364-9 weighted residual scheme, 366 spectral finite elements for waves, 372--4 Fekete points, 374 Gauss-Chebyshev-Lobatto scheme, 373-4 Gauss-Legendre integration, 373 Gauss-Lobatto scheme, 373-4 Newton-Cotes formula, 372-3 Vandermonde matrix, 374 T-complete systems, 363-4 Weak form of equations, 15 s e e a l s o Weighted residual and finite element methods Weighted residual and finite element methods: about strong and weak forms, 14-15 boundary conditions, Neumann type, 15 elements, 16 examples: free surface potential flow, 21-3 Poisson equation in two dimensions: Galerkin formulation with two triangular elements, 19-21 potential flow solution around an aerofoil, 21-2 shape functions for triangle with three nodes, 18-19 Galerkin, finite element, method, 17-18 nodal values, 16 self-adjoint differential equations, 17 test functions, 16 weak form of equations, 15 weighted residual approximation, 16-17 Wolfstein x-1 model, 251-2
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Plate 1 Analysis of subsonic flow around an aircraft (Dassault Falkon) Courtesy of Prof. Ken Morgan, School of Engineering, University ofWales Swansea. Source: J. Peraire, J. Peiro and K. Morgan, Multigrid solution of the 3-D compressible Euler equations on unstructured tetrahedral grids, Int. J. Num. Meth. Eng., 36, 1029-1044, 1993.
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Plate 2 Supersonic car, THRUST SSC, surface mesh and pressure contours Details near the nose. Part of the car is shown on the book cover. Courtesy of Prof. Ken Morgan, School of Engineering, University of Wales Swansea (Nodes: 39, 528 and Elements: 79060). Source: K. Morgan, 0. Hassan and N.P. Wetherill, Why didn't the supersonic car fly?, Mathernat~csToday, Bulletin of the Institute of Mathematics and its Applications, 35, 110-1 14, August 1999.
Plate 3 Wave elevation pattern behind a ship C60 hull for Froude number 0.238. Courtesy of Prof. E. Oiiate, CIMNE, Barcelona. Source: E. Oiiate, J. Garcia and S.R. Idelsohn, Ship hydrodynamics. In E. Stein, R. de Borst and 1I.R. Hughes, editors, Chapter 18, Encyclopediaof Computational Mechanics. John Wiley, 2004.
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Pressure contours. Courtesy of Prof. 0. Hassan, School of Engineerlng, University of Wales Swansea. Source: 0. Hassan, L.B. Bayne, K. Morgan and N.P. Wetherill, An adaptive unstructured mesh method for transient flows involving moving boundaries, ECCOMAS'98. John Wiley.
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Plate 5 3D dam-break. Wave interacting with two circular cylinders Courtesy of Prof. R. L~hner, George Mason University, USA. Reference: C. Yang, R. L~hner and S.C.Yim. Development of a CFD simulation method for extreme wave and structure interactions, 24th International Conferenceon Offshore Mechanics and Arctic Engineering, 12-17 June 2005, Halkidiki, Greece. Problem definition and free surface evaluation.
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/ (a) Simulation of waves hitting a breakwater in a harbour using the particle finite element method (PFEM)I, 2
(b) Simulation of the sinking of a tanker ship using the particle finite element (PFEM)I, 2
Plate 6 Particle finite element method Courtesy of Prof. E. Ohate, ClMNE, Barcelona. References: (1) E. OEate, S.R. Idelsohn, E Del Pin and R. Aubry. The particle finite element method. An overview, Int. J. Comp. Meth., 1:267-307, 2004. (2) S.R. Idelsohn, E. Ohate and F. Del Pin. The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. Int. J. Num. Meth. Eng., 61:964-989, 2004.