Unification of Finite Element Methods (Mathematics Studies)

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UNIFICATION OF FINITE ELEMENT METHODS

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NORTH-HOLLAND MATHEMATICS STUDIES

UNIFICATION OF FINITE ELEMENT METHODS

Edited by

H. KARDESTUNCER University of Connecticut Storrs Connecticut U.S.A.

1984

NORTH-HOLLAND-AMSTERDAM . NEW YORK. OXFORD

94

I S B N : 0 444 X 7.5100

Piihli,slier.s:

ELSE 'IER SCIENCE PUBLIS IERS R . V . P . O . Box IY91 1000 BZ Anisterdam The Netherlands Sole rli.strihritorsfbr tlie U .S . A . trtrrl Cirtiot!ii:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC 52 Vantlerbilt Avenue NewYork. N.Y. 10017 U.S.A.

Library of Congresi Cataloging in Publication Data

Main entry under t i t l e : Unification of f i n i t e element methods. (North-Holland mathematics studies) Bibliography: p 1. Finite element method. 2. Argyris, J. B. (John E.), 1916I. Kardestuncer, m e t t i n . 11. Series. TA347.F5U55 1984 620' .001'515353 84-6006 ISBN 0-444-87519-0 ( U . 6 . )

.

.

PRINTED IN T H E NETHERLANDS

7th

Dedicated to

Professor John H. Argyris for his pioneering and continuing contributions to the finite element methods

Alliance of Industry and Academe


vii

PROFESSOR JOHN H. ARGYRIS A man who unifies engineering and mathematics with elegance My first encounter with Professor John H. Argyris’ work occurred during my graduate studies at MIT in the mid fifties. His elegant treatment of Bernoulli’s virtual work and energy principles mounted on Menabrea’s il minimo lavoro with Castigliano’s two theorems, St. Venant’s theories of torsion, Maxwell’s reciprocity principles, Lord Rayleigh’s variational principles, Muller-Breslau’s and Otto Mohr’s unit load ideas, etc. gave me the impression that this man belonged to the last century. Yet the methodology presented (stiffness and flexibility methods in structural analysis) was so new that it was unknown to my fellow students and did not even exist in the curriculum. A few years later, I learned that he was the holder of the prestigious Chair of Aeronautical Structures at the University of London where he was also Professor of Aerospace Sciences and at the same time was Director of the Institute of Statics and Dynamics and Director of the Computer Center at the University of Stuttgart. I began to wonder if perhaps there were more than one J.H. Argyris, and whose work was I studying? The more I studied his work and the more I learned of his accomplishments the more convinced I was that the man must be older than I thought; perhaps he was born a century before the last. However, when I finally met him in 1961, I was sure that he must be the grandson of the man whose work was so inspiring me and guiding my doctoral dissertation at the Sorbonne. A citizen of Great Britain, a resident of West Germany, John H. Argyris was born August 19, 191 6, in the Land of’the Gods. A child prodigy who graduated from the Technical University of Athens at the age of eighteen, he received the all-German Prize of Deutscher Stahlbauverband during his postgraduate studies in Munich when h e was only twenty years old. Many believe that it was not merely coincidental that Sir Isaac Newton was born o n Christmas day of 1642, the same day (with acceptable approximation based o n the theories presented in this volume) that another genius, Galileo Galilei, had died. I am curious to know what genius it was who died on August 19, 1916.

I had every intention here to write more about Professor Argyris and his work but the more I wrote the more I became convinced that my writing could in no way reflect the accomplishments of this great man. H i s life can not be told in an essay; his work can not be assessed in an article; his abundant energy can not be formulated as an energy functional. He is beyond and above all that most of us know of him. H. Kardestuncer


MAIN DISTINCTIONS OF PROFESSOR JOHN H. ARGYRIS,D.Sc., Dr.h.c.mult. 1937

Dip1.-1ng.D.E.

Munich

1954

D.Sc. (Eng.)

University of London

1955

Fellow R.Ae.S.

Royal Aeronautical Society, London

1962

Honorary Associate Hon. A.C.G.I.

City Guilds Institute, London

1970

Laura h.c.dott.Ing.

University of Genoa, Special Distinction on the 100th Anniversary of the Faculty of Applied Mechanics and Ship Building

1971

George Taylor Prize


1971

Silver Medal


1972

Principal Editor

Computer Methods of Applied Mechanics and Engineering (Journal)

1972

dr.techn.h.c. and jus docendi

University of Norway, Trondheim

1973

Corresponding member

Academy of Sciences of Athens (Positive Sciences)

1974

Honorary Fellow

Groupe pour l'Avancement des MCthodes Numeriques de l'hgenieur (GAMNI), Paris

1975

von Karmin Medal

Highest Scientific Award, American Society of Civil Engineers, New York

1976

Honorary Fellow Hon.F.C.G.1.

City Guilds Institute, London

1979

Member A.S.C.E.

American Society of Civil Engineers, New York

Main Distinctions of Professor John H. Argyris

X

1979

Copernicus Medal

Highest Award in Natural Sciences Polish Academy of Sciences, Warsaw

1980

Gold Medal

of the Land Baden-Wurttemberg

1980

Honorary Professor

Northwest Polytechnical University, Xian, People’s Republic of China

1981

Timoshenko Medal

Highest Scientific Award, American Society of Mechanical Engineers, New York

1981

Life Member A.S.M.E.

American Society of Mechanical Engineers

1981

Member

The New York Academy of Sciences, New York

1982

I.B. Laskowitz Award with Gold Medal

Highest Astronautical Award of the New York Academy of Sciences

1983

Fellow of the AIAA

Highest Grade of Membership, American Institute of Aeronautics and Astronautics, New York

1983

Dr .Ing .E.h.

University of Hanover, Honorary Doctorate

1983

Honorary Professor

Technical University of Peking (Beijing)

1983

Honorary Life Member

New York Academy of Sciences, New York

1983

World Prize in Culture and Election as Personality of the Year 1984

Centro Studi e Ricerche delle Nazioni Accademia Italia, Salsomaggiore Terme

1984

Honorary Professor

Qinghua University, Beijing

1940

340 scientific publications

and continuing

xi

ACKN'OWLEDGMENTS The UFEM series could not take place without the generous help of the following friends, organizations, and societies. Their encouragement, support, and sharing of the ideals of the conference are sincerely appreciated and gratefully acknowledged.

Organizing Committee Members H. Clark, Hon. Chairman (UConn) H. Kardestuncer, Chairman (UConn) W.W. Bowley (UConn) J.J. Connor (MIT) H.A. Koenig (UConn) A. Phillips (Yale) R.J. Pryputniewicz (WPI)

H. Allik (BBN) W.W. Bowley (UConn) F. Camaretta (Sikorsky) A.D. Carlson (NUSC) M.K.V. Chari (General Electric) L. Collatz (Hamburg, Germany) J.H. Connor (MIT) A.C. Eringen (Princeton) S. Gordon (Electric Boat)

Session Chairmen H.A. Koenig (UConn) R. Lalkaka (United Nations) H. Mayer (Hamilton Standard) D.H. Norrie (Calgary) T. Onat (Yale) A. Phillips (Yale) T.H.H. Pian (MIT) J.A. Roulier (UConn)

Local Arrangements J.J. Farling (UConn, Conf. & Inst.) G.D. Smith (UConn) G.M. Wallace (UConn)

Analysis & Technology, Inc. AVCO Lycoming Corp. Bolt Beranek & Newman, Inc. Control Data Corp. Electric Boat General Electric Hamilton Standard Conf. & Inst. (UConn)

Advisory Board Members I. Babugka (Maryland) L. Collatz (Hamburg, Germany) A.C. Eringen (Princeton) R.H. Gallagher (Arizona) J.T. Oden (Texas) T.H.H. Pian (MIT) O.C. Zienkiewicz (Swansea, U.K.)

Participatmg Societies AIAA ASME CC-ASCE

Sponsoring Organizations Kaman Aerospace Corp. Naval Underwater Systems Center Northeast Utilities Perkin Elmer Pratt & Whitney Aircraft Sikorsky Aircraft UConn Foundation UConn Research Foundation


LIST OF CONTRIBUTORS J.H. Argyris (l), Institut fur Statik und Dynamik der Luft- und Raumfahrtkonstruktionen, University of Stuttgart, Stuttgart, Fed. Rep. Germany. J.F. Abel (47), Department of Structural Engineering, Cornell University, Ithica, New York, U.S.A. S.N. Atlun’ (65), CACM, Georgia Institute o f Technology, Atlanta, Georgia, U.S.A. 1. Bubufku (97), Institute of Physical Science and Technology, University of Maryland, College Park, Maryland, U.S.A. K . 4 Buthe (123), Department o f Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. J. Bieluk (1 49), Department of Civil Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania, U.S. A. J.H. Bramble (1 67), Department of Mathematics, Cornell University, Ithica, New York, U.S.A. C A . Brebbiu (185), The Institute of Computational Mechanics, Ashurst Lodge, Southampton, England M.A. Celiu (303), Civil Engineering Department, Princeton University, Princeton, New Jersey, U.S.A. A. Chuudhury (1 23), Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. J. Sf. Doltsinis (l), Institut fur Statik und Dynamik der Luft- und Raumfahrtkonstruktionen, University of Stuttgart, Stuttgart, Fed. Rep. Germany. J. F. Hajjur (47), Department of Structural Engineering, Cornell University, Ithica, New York, U.S.A. T -Y. Hun (47), Department of Structural Engineering, Cornell University, Ithica, New York, U.S.A. A.R. Ingruffeu (47), Department of Structural Engineering, Cornell University, Ithica, New York, U.S.A. K. Izudpunah (97), Computational Mechanics Center, Washington University, St. Louis, Missouri, U.S.A. H. Kurdestuncer (207), Department of Civil Engineering, University o f Connecticut, Storrs, Connecticut, U.S.A. R. C MucCumy (149), Department of Civil Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania, U.S.A. D. S. Mulkus (235), Mathematics Department, Illinois Institute of Technology, Chicago, Illinois, U.S.A. A. Needleman (249), School of Engineering, Brown University, Providence, Rhode Island, U.S.A. 7: Nishioku (65), CACM, Georgia Institute of Technology, Altlanta, Georgia, U.S.A.

xiv

List

0.f

Contributors

A.K. Noor (275), NASA Langley Research Center, The George Washington University, Hampton, Virginia, U.S.A. E. T. Olsen (235), Mathematics Department, Illinois Institute of Technology, Chicago, Illinois, U.S.A. J.E. Pusciuk (1 67), Brookhaven National Laboratory, Upton, New York, U.S.A. R. Pemcchio (47), Department of Structural Engineering, Cornell University, Ithica, New York, U S A . A. Philpott (321), Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. G.F. Pinder (303), Civil Engineering Department, Princeton University, Princeton, New Jersey, U.S.A. R.J. Pryputniewicz (207), Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts, U.S.A. G. Strung (321), Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. B. Szubo (97), Computational Mechanics Center, Washington University, St. Louis, Missouri, U.S.A.

xv

PREFACE The 7th UFEM gathering, like its predecessors, advances further toward its goal of accomplishing “a unified method” in computational mechanics. No matter how powerful a methodology might be for a certain class of problems, it often presents shortcomings for others. Since engineering problems today are very complex and contain subregions with completely different physical and geometrical characteristics, certainly no single method is capable of handling the entirety of the problem. Consequently, the identification of various methodologies, each suitable for a particular subregion, and their unification have recently been in the minds of many researchers. The flow chart in Fig. 1 indicates three stages of such a unifica.tion: unified formulation, unified means, and unified methods.

Fig. 1. Flowchart for the Unification of Methods in Mechanics.

Preface

XVi

The components of the first stage of this unification are illustrated in Fig. 2. -_____ FORMULATIONS IN MECHANICS

t d---'-TENSORIAL E o u A ~ l o S

[

EMPIRICAL EOUATIONS

L_

, , /

/

, -

,

,

i

/ ' '

INTEGRAL EOUATIONS

_

-

-5-

,

,'

_

~

- .DIFFERENTIAL EOUATIONS ~

L

~--__

ANALYTICAL SOLUTIONS

-.-

Fig. 2. Unification of Formulations on Mechanics.

Many of the papers presented here address various stages of unification, and we believe that in the near future commercial or in-house codes will be developed to accomplish this task. The possibility of unifying various numerical methodologies using interactivegraphics has been investigated by John ABEL and his co-workers. Their work is fostering the unification concept with a unified means which interconnects analysis methods and design parameters. They are not only improving man-machine communication but communication between methodologies employed in different regions of the domain and stages of processing. ATLURI and NISHIOKA emphasize the unification (hybridization) of various methodologies (numerical, analytical, and experimental) in engineering for the solution of complex problems (e.g. crack propagation in 3-D domain with irregular geometry and material properties) for which none of the existing methodologies alone is sufficient. The authors have, in fact, been unifying these methodologies in their earlier work and they advocate the necessity of unification. The problems in this presentation, drawn from the field of fracture mechanics, demonstrate the use of more than one methodology (in time and space) for their solution. Undoubtedly, one can easily apply concepts presented in this paper to other problems. Intermethod compatibilities and error bounds, however, remain to be explored. In the opinion of the editor, the concepts presented here are firm enough ground to stand on when reaching for further goals in unification. Dealing primarily with problems for which energy functionals exist, BABUSKA and his co-workers present h-, p-, and h-p versions of the finite element methods.

Preface

xvii

They claim that error measures in stresses often do not follow monotonic behavior of the error measure in the energy norm. To overcome this difficulty, they introduce an extraction function and demonstrate the selection of such a function during adaptive post-processing. A numerical example accompanying the presentation uses the extraction technique. Contact problems, in particular between nonlinear deformable bodies subject to large deformation with sticking, sliding and separating, are among the most difficult problems in solid mechanics. BATHE and CHAUDHARY present a solution algorithm that they have developed for two-dimensional contact problems. They believe that alongside finite differences, finite elements, and surface integral techniques, there is still room for more reliable and effective algorithms to analyze general problems in this field. Numerical results for two problems - a pipe buried in soil and a traction of a rubber sheet embedded in a rigid channel - accompany the paper. BIELAK and MACCAMY unify variational finite element methods with the boundary integral equation method using the former in the interior of the domain and the latter at the exterior. They apply the methodology to a two dimensional electromagnetic interface problem: the interaction between air and a dielectric obstacle subject to two different sets of Maxwell’s equations. In t h s problem, a homogeneous differential equation defined over an infinite domain interfaces with a nonhomogeneous differential equation defined over a finite domain. After reviewing the fundamental principles b e h n d various approximate methods, BREBBIA embarks on the unification of finite elements and boundary elements. While acknowledging the power and potential of the former, he points out certain advantages of the latter and maintains that the complexity of the problems at hand necessitates combining (unifying) many methodologies. He refers to these as “the discrete element methods” and cites some recent attempts coinciding with the philosophy behind UFEM gatherings. KARDESTUNCER and PRYPUTNIEWICZ explore the possibility of unifying finite element modeling with laser experimentation in two different stages of the, procedure. The first part deals with evaluation of the stiffness and/or flexibility matrix coefficients for irregular (geometrically as well as physically) elements by experiment. The second part deals with determining the unknown values of the function by lasers. This, in turn, leads t o a reduction in the order of the stiffness matrix and to an increase in the accuracy of the results. Measurement techniques and numerical examples accompany the presentation. The main theme of the paper by MALKUS and OLSEN centers on the question of whether the NCR (Nagtegaal redundant constraint) element which fails to satisfy the LBB condition can be used for incompressible media. The NRC element is a quadrilateral macroelement with four triangles and has been used successfully by

xviii

Preface

others for problems involving inelastic deformation. Here, the authors discuss why NRC elements violate the LBB condition for convergence and how this condition can be removed so that the NRC element can be used for plane and axisymmetric incompressible flows, In order to demonstrate this, they present error estimates for the element when it is used in Stokesian flow. In his work, NEEDLEMAN applies finite element techniques to necking instabilities subject to classical and nonclassical constitutive relationships. The presentation is accompanied by a numerical analysis of tension tests using constitutive descriptions for polycrystalline metal. He presents remarkably good agreement between the FEM analysis and experimental results and claims that this is the result of incorporating into FEM modeling the constitutive relations for polycrystalline metals arising due to crystallographic texture. He also points out that localized shear stresses play a significant role in texture development. Two recent advances toward the unification of various methodologies in one physical problem are presented by A.K. NOOR. They are (i) a hybrid method based on the combination of the direct variational techniques with perturbation methods, and (ii) a two-stage direct variational technique. Advantages of both forms of unification are illustrated for nonlinear steady-state thermal and structural problems. The author also points out other combinations of various methodologies and research areas for more effective solution of nonlinear problems. Comparative numerical studies accompany the presentation. PHILPOTT and STRANG idealize the internal fiber of a human patella as a plane truss and then, using linear programming techniques, they try to optimize the system to accomplish minimum weight. After presenting the standard procedure for a fixed geometry problem, they develop an algorithm for problems with variable geometry, indeed an interesting and difficult task. Most truss problems consist of members with zero loads which in turn introduce degeneracy during the optimization procedure. The authors attempt in particular to deal with this difficulty of optimization. The alternating-direction collocation (ADC) method is presented by CELIA and PINDER with particular application to multi-dimensional transport equations. These authors enhance the ADC procedure by adding a small number of quintic elements along the principle direction of flow governed by the convection-dominated transport equation. A numerical example confined to a rectangular region and a flow chart for the enhanced ADC procedure accompany the presentation. All of these invited presentations have been written specifically to honor Professor Argyris with the understanding that they also follow as much as possible the spirit of the conference.

Preface

XiX

On behalf of the Organizers and Advisory Board members, I would like to thank our distinguished speakers, session chairmen, and participants for their kind cooperation. Some prepared their papers under a severe deadline, some traveled long distances, and some took time off from their demanding tasks to be here today. The result is most gratifying. The Editor


xxi

CONTENTS Dedication to Professor John H. Argyris Main Distinctions of Professor John H. Argyris Acknowledgments List of Contributors Preface

Chapter 1

Chapter 2

Chapter 3 Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

On the Natural Approach to Flow Problems J.H. Argyris & J. St. Doltsinis (University of Stuttgart, Fed. Rep. Germany)

V

ix xi xiii xv

1

Interactive Computer Graphics for Finite Element, Boundary Element, & Finite Difference Methods J.F. Abel, A . R Ingraffea, R. Perucchio, T.-Y. Han, & J.F. Hajjar (Cornell University)

47

Hybrid Methods of Analysis S.N. Athri & T. Nishioka (GeorgiaInstitute of Technology)

65

The Postprocessing Technique in the Finite Element Method. The Theory & Experience I. Baburka (University ofMaryland), K. Izadpanah, & B. Szabo (Washington University)

97

On Finite Element Analysis of Large Deformation Frictional Contact Problems K . 4 Bathe & A . Chaudhary (MIT)

123

Mixed Variational Finite Element Methods for Interface Problems J. Bielak & R. C. MacCamy (Carnegie-Mellon University)

149

Preconditioned Iterative Methods for Nonselfadjoint or Indefinite Elliptic Boundary Value Problems J.H. Bramble (Cornell University) & J.E. Pasciak ( Brookhaven National Laboratory)

167

On the Unification of Finite Elements & Boundary Elements C.A. Brebbia (ICM, Southampton, England)

185

xxii

Chapter 9

Con ten t S

Unification of FEM with Laser Experimentation H. Kardestuncer (University of Connecticut) & R.J. Pryputniewicz (WorcesterPolytechnic Institute)

207

Chapter 10 Linear Crossed Triangles for Incompressible Media D.S.Malkus & E.T. Olsen (IIT, Chicago)

235

Chapter 11 The Numerical Analysis of Necking Instabilities A, Needleman (Brown University)

249

Chapter 12 Recent Advances in the Application of Variational Methods t o Nonlinear Problems A .K. Noor (NASA - Langley)

275

Chapter 13' Collocation Solution of the Transport Equation using a Locally Enhanced Alternating Direction Formulation M.A. Celia & G.F. Pinder (Princeton University)

303

Chapter 14 Numerical & Biological Shape Optimization A. Philpott & G. Strang (MIT)

32 1

Index

345

Unification of Finite Element Methods H. Kardestuncer (Editor) @ Elsevier Science Publishers B.V.(North-Holland), 1984

1

CHAPTER 1 ON THE NATURAL APPROACH TO FLOW PROBLEMS J. H. Argyris & J. St, Doltsinis

The paper surveys recent work on fluid dynamics performed a t the ISD, University of Stuttgart. I t i s i n particular directed to a natural description o f the flow phenomena and includes also a consideration o f thermally coupled problems. The derivation o f the relevant finite element equations when referred to natural quantities i s outlined and examples of application are given. For a discussion on the associated modern developments in numerical solution techniques the reader may consult 1281

.

1.

INTRODUCTION

The present paper surveys recent work on fluid dynamics performed a t the ISD, University of Stuttgart. The paper serves i n the main as a survey on modern developments i n finite element methods for fluid motion, and i s particularly devoted to a natural description o f the relevant phenomena. Its main attention i s focused on incompressible media. First draft of the theory has been presented a t a lecture given a t the Conference on Finite Elements in Water Resources i n Hanover i n 1982. In section 2, the natural terminology [l, 21 i s introduced and methodically applied to the formulation o f field quantities characteristic of fluid motion, such as the scalar pressure field and the vectorial velocity field. The condition of conservation of mass i s derived i n natural terms and natural measures for the stress and the rate of deformation are connected by the appropriate constitutive relations. Aiming a t the analysis of fluid motion coupled with thermal phenomena, the natural approach i s subsequently extended to the considemtion o f the temperature field and the heat flow 13

1.

Section 3 indicates the tmnsition to finite domains as a foundation for the development o f the finite element theory o f the flow problem. The streamline upwind/PetrovGalerkin formulation o f [ 4 1 may be used for the discretisation technique i n connection with either the strict fulfilment o f the incompressible statement or with the penaltyapproach to the condition o f incompressibility. In a subsequent step the finite element discretisation o f the thermally coupled fluid flow problem i s considered and the governing equations are established. For typographical brevity, we omit in t h i s paper a discussion of numerical integration schemes i n the time domain. Also the important task of an effective solution o f the

J.H. Argyris & J. St. Doltsinis

2

equations governing the flow problem i s not handled in the present contribution. For this purpose the reader i s referred to the presentation in1281

.

The theory presented in the paper i s applied in section 4 to the numerical analysis of some typical examples of viscous fluid motion. Thus, the convection dominated flow over a step i s considered for the two- and the three-dimensional case, and the solution o f thermally coupled flows i s demonstrated on the BBnard instability phenomenon i n a fluid between two planes of different temperatures. The interested reader may consult [281 for an analysis of cavity flows involving free and forced heat convection with a change from liquid to solid phase of the material.

2.

ON THE NATURAL APPROACH TO FLUID MOTION

2.1 Natural approach In the natural methodology of continuum mechanics, a l l considerations are established on or derived from an infinitesimal tetrahedron element which replaces the classical parallelopiped applied in the traditional cartesian point o f view. For comparison purposes both elements are shown in fig. 2.1 together with the associated coordinate systems. An elegclnt application of the tetrahedron element demands the use of supernumerary or homogeneous reference systems. One of these m y be defined by the directions of the six edges of the tetrahedron. The natural formulation of the mechanics and thermomechanics of solids [5, 3, 2 1 may be based on the Lagmngean approach in the sense that the tetrahedron constitutes then a moving and deforming material element. In our present considerations o f fluid motion, however, we prefer to adopt the Eulerian description L6, 71 in which the tetrahedron represents a fixed geometrical element in space. Before developing the natural concept we first review alternative representations of a vector i n three-dimensional space 13, 21 and illustrate then the argument on the twodimensional case depicted in fig. 3.2. Consider the vector r defined by cartesian en tries

In the natural terminology the vector r may, on the one hand, be composed from nonunique independent vectorial contributions taken along the tetrahedral edges

This forms the so-called component description of a vector. On the other hand, we may introduce as measures the unique orthogonal projections o f the vector f onto the natural coordinate axes,

This forms the total description o f a vector.

On the Natural Approach t o Flow Problems

3

Considering fig. 2.2, we observe that for a given component representation vector its artesian form r i s deduced by the transformation

re

of the

with the matrix

gNb =

[codq;)]

The total natural entries

4.=d)...,1;

I'=f,2, 3

(2.5)

of (2.3) are then obtained through the relation

where (2.4) has been used. n e symmetric matrix

establishes the direct connection between total and component definitions. I t i s evident that due to the redundancy of the natural quantities, the above transformations are not invertible

.

Finally, we note that the scalar product o f two arbitrary vectors Q and given in one of the equimlent forms

6 may be

as i s easily confirmed with the aid of (2.4) and (2.6)

2.2 Pressure field We proceed next to the description of a scalar field e.g. the pressure p i n the fluid which for arbitrary non-steady conditions i s a function of the time t and the position vector X We express this by

.

and examine the consequences of the different representations o f the vector W on the description of the scalar field. Positions may be defined by component coordinates X c dependence i n (2.9) the pressure gradient then reads

, cf.

(2.2). From the associated

where the chain rule confirms the transformation (2.6) between total natural and cartesian specification o f a gradient vector and justifies the total notation of (2.10).


4

Actually, &Cp) comprises the rates of change of p i n the natural directions (and hence the corresponding orthogonal projections of the gradient vector), Consider next the transformation rule (2.4) leading from the component to the artesian definition of the gradient vector,

Here the component vector

%,'PI

i s clearly

(2 12) 0

In fact, s f p ) merely comprises the formal deriwtives of p with respect to a nonunique dependence on X t and represents component contributions to the gradient vector. Applying next the chain rule to the gradient of (2.10) we obtain the relation

(2 13) which agrees with the transform tion of (2.6) between component no tura I and tota I na tural quaniities. In conclusion we list the inwriance of the expression

which furnishes the increment of the scalar p associated with a change of spatial location, and may be verified by the chain rule, or via an appropriate interpretation of (2.8).

2.3 Velocity field The extension of the above terminology to the description of a vector field i s stmightforword. Consider for instance the velocity field,

v = d t ,d

(2.15)

which i s i n general unsteady. The acceleration of a certain particle may be obtained by the so-called material differentiation of the velocity vector with respect to time as

(2.1 6) The f i r s t term of the expression in (2.16) represents the loca I deriw tive of the velocity with respect to time and i s to be evaluated a t a fixed location. The second term represents the contribution of convection and i s dependent on the gradient of the velocity field.

On the Natural Approach to Flow Problems

5

,

Disregarding for the time being a particular representation o f the velocity vector its gradient may be measured with respect to one of the different specifications of the location vector X Taking component natural coordinates Xc we obtain i n analogy to (2.10) the total natural gradient

.

In (2.17) the Cartesian gradient

may be related i n analogy to (2.1 1) to the component gradient of V

, (2.19)

which represents an extension of (2.12) and i s derived from a functional dependence of Y on Applying once more the chain rule to (2.17) we deduce

at,

(2.20) which relates directly the total to the component natural gradient and represents an extension of (2.13) to a vector field Y We also note the invariance of the expression

.

(2.21) which isanalogous to (2.14) and represents the convective acceleration term of (2.16). Here Y symbolises one o f the three differently defined representations of the velocity vector. The inwriance of (2.21) may be confirmed by the chain rule. We next turn our attention to the particle acceleration o f (2.16) and observe that i t may be represented by one of the alternative forms adopted for the description of the velocity vector y Thus, i n component natural terms we have

.

(2.22) with the total gradient matrix of V,

(2.23)

J. H. Argyris & J. St. Doltsinis

6

The cartesian form of the acceleration on the other hand may be expressed as (2.24) Here V comprises the artesian components of the velocity, cf. (2.1). Here the cartesian gradient matrix

(2.25) should not be confused with the expression o f (2.18), which i s not limited to a particular representation of the vector Y The total natural formulation of the acceleration i s given by

.

(2.26) with the associated component gradient matrix of

5

We observe that relations (2.4) and (2.6) between the alternative vector specifications also apply to the acceleration, as confirmed by (2.24) and (2.26).

2.4 Continuity condition We proceed to the natural formulation of the continuity condition. To this end, consider in fig. 2.3 the infinitesiml tetrahedron element defined by the lengths of the six edges

1 = r i d 14

8

18 1' i r j = rt?

Q = ~...J ,

(2.28)

.

with a volume c/ When determining the flow of mass through the element as induced by component natural velocities we stipulate the column matrix

containing the rate of change of a l l component velocities along the edges

9.

' s

Consider next a component natural velocity characterised by the intensity and assume for the time being an incompressible fluid with density f , Under these condithrough the centre of the face A' not tions mass permeates a t a rate f5'4' containing 1'' into the element and i s discharged a t a rate f ( ~ + dA''~ ~ c through the opposite face at a distance I"/J from the point of input. The balance of inputoutput of fluid m s s due to the component natural velocity ye" i s seen to be simply

On t h e Natural Approach to F l o ~Problcwis )

7

(2.30) where

(2.31)

denotes the rate o f change of Gd along 4.Generalising to a l l natural directions 4. and applying the column matrix eC of (2.29) we obtain the rate of mass supply by summation o f the individual contributions defined by (2.30) as arising for a l l component Hence the condition of conservation o f mass for an incomnatural velocities of U; pressible fluid i s given by

.

where

i=

1, 2,

3

(2.33)

i s the artesian counterpart of mC. Note also the summation vectors

(2 .34)

In the case of a compressible fluid, expression (2.30) for a typical rate of the component mass supply must account for a change of the density f along o( Consequently, the condition of conservation o f mass (2.31) i s modified into

.

Here the density gradients

(2 .36)

(2 e37) correspond to the definitions of (2.10) and (2.1 1 )

.

J.H. Arg-vris & J. St. Doltsinis

8

2.5 Rate of deformation and stress A specification of the behaviour of fluid flow demands the introduction of suitable stress and deformation measures. In classical continuum mechanics (see e.g. [83), the rate o f deformation i s defined by the symmetric part of the Cartesian gradient o f the instantaneous velocity field. With reference to (2.18) we may thus write

(2.38)

The associated instantaneous material spin i s then

and i s defined by the antisymmetric part. The rate of deformation of the material

i s correspondingly specified by the symmetric part of the Cartesian velocity gradient matrix. In what follows we refer to the column matrix

as the Cartesian rate of deformation. Natural measures of the rate o f deformation were originally defined by reference to the deformation of the fluid material instantaneously occupying the tetrahedron element [9, 71, They may be expressed i n terms of the natural definitions of the velocity gradient [lo] Thus, the tom1 natural rate of deformation i s given by

.

4

comprises the rates of extension of the material along the six The column matrix natural directions [5] I t may be related to the Cartesian definition of the rate of def o r m tion via

.

(2.43)

9

On tlie Natural Approach to Flow Probleins

c

.

where the detailed structure of the transformation matrix may be found in 191 The component natural rate of deformation i s defined i n analogy to the total one i n (2.42) as

(2.44)

The column matrix

6,may be related uniquely to the total natural rate of deformation

by

(2 .45) where the transformation matrix

i s also given in [ 91 and presumes that component velocities vary only along the direction of their action [ 101 In this case, o f (2.44) and & , o f (2.29) are identical.

.

4

For stresses we must adopt a corresponding definition to the rate o f deformation so that their scalar product satisfies the condition of invariance for the virtual rate of work. Thus the column matrix

=

[ G"

6 ' '

C3'

h

GIL

f i b z 3& G " ]

Q .47)

comprises the Cauchy stresses i n their Cartesian form and corresponds to the rate of deformation o f (2.41). The natural component stresses

6

correspond following 19, 5 1 to the total natural rate of deformation while the total natural stresses

dt

o f (2.42),

Q .49) correspond to the component natural rate o f deformtion of (2.44) (c.f. fig. 2.4). The invariance of the virtual rate of work may now be expressed as

(2 .50) Bearing i n mind (2.43) and (2.45) we easily confirm the relation

(2.51)

J.H. Argvris & J. St. Doltsinis

10

connecting the different representations of the stress state,

2 . 6 Constitutive relations for incompressible viscous fluids In formulating the stress-strain relations appertaining to the fluid motion, i t i s convenient to split the stress state into hydrostatic and deviatoric contributions. We may ignore here an account of the standard Cartesian approach (see e.g , I 1 11) and apply instead the natural approach to this subject as developed in [7, 91. Considering first total stresses we write

=t =

%If

+

(2 .52)

=tb

and obtain the hydrosbtic part of the t o b l stress in the form

Qiy

=

- L'

=

c, e,t

I 3

b,

(2.53)

where the matrix

€6

(2 .54)

performs the summation of the component stresses in each row and yields the total hydrostatic stress i n each of the natural directions. The deviatoric part of the total stress follows then from (2.52) as

in which relation (2.51) between total and component definitions i s used. Partitioning next the component stress as

we may derive the hydrostatic and the deviatoric part by application of (2.51) to the total quantities of (2.53) and (2.55), respectively. A decomposition of the total natural rate of deformation

st =

4"

f

Sib

(2 .57)

into volumetric parts

(2 .58) and deviatoric parts

11


(2.59) proceeds along the same argument. Also the component natural rate of deformation

may be partitioned analogously. Consider next an incompressible fluid, i.e. a fluid undergoing only isochoric deformations. In this case the volumetric rate o f deformation must vanish. This yields,

(2.61) which i s equivalent to (2.32). In the absence of viscous effects the incompressible fluid i s described as an ideal one for which deviatoric stresses are absent. Then the stress field derives simply from a static pressure p Consequently, the total stresses reduce to

.

and by

(2.51)the component stresses become

Gc

ccy= - pff-e,

(2.63)

-

In a viscous incompressible fluid on the other hand, a rate of deformation which i s exclusively deviatoric because of (2.61) leads to deviatoric total stresses of the form

-

or to deviatoric component stresses,

where

p

denotes the viscosity coefficient o f the fluid.

For the viscous case the stress i s ultimately obiuined by a superposition of a hydrostatic contribution arising from (2.62), (2.63)and the deviatoric contribution of (2.64), (2.65). Thus, the total natural stress reads

bt

= 2 / 4 4- p e ,

(2.66)


12 and the component one

(2.67)

We observe that the constitutive relations i n (2.66) and (2.67) are expressed i n terms of corresponding stress and rate o f deformation measures.

I f standard computational procedures are to be applied to the analysis of the isochoric motion of an incompressible fluid, one may use the so-called penalty approach. The isochoric condition (2.61) can then be relaxed and the pressure p i s related to the volumetric rate of deformation as follows

where (2 .69)

k

represents the penalty parameter. In (2.68), may be interpreted as a modulus of viscous compressibility and i s expressed in (2.69) i n an analogous manner to the wellknown elastic bulk modulus. The strictly incompressible constitutive relations (2.66), (2.67) m y now be modified accordingly. For instance, (2.67) assumes i n the penalty approach the form

(2.70) in which

5 4 -1 2

(2.71)

may be used as an alternative penalty parameter.

2.7 Fluid motion coupled with thermal phenomena In this subsection we consider fluid motion coupled to thermal phenomena. To this purpose we assume the following unsteady temperature field

r *

= T(~,x)

(2 .72)

where denotes the position vector. In extension of the argument in subsection 2.2 the time mte of the temperature of a particle may be expressed in the alternative forms


13

(2 .73)

The different formulations of the tempemture gmdient in (2.73) may be compared to the definitions i n (2.10), ( 2 . l l ) a n d (2.12), respectively. In the present case the invariance condition of (2.14) becomes

(2 .74)

The time rate of the temperature i s associated with a mte of heat stored i n the fluid material. The latter may be expressed per unit material volumeas

(2 .75) where C denotes the specific heat capacity of the fluid. In accordance with (2.73) the rate of heat stored i n the material may be composed in the Eulerian approach of two parts. Thus, the rate o f heat stored in a unit volume when fixed i n space reads

a7

PJ = P a t

(2 .76)

and i s associated with the tempemture rate obiained a t a fixed location. The contribution

(2 .77) i s the heat convection term due to the motion o f the fluid and may be presented in one of the alternative formulations, natural or artesian, as shown in (2.77). We proceed next to the specification of the heat supply to a unit volume o f space due to a dire:ted heat flow, i.e. conduction. Following subsection 2.1 the heat flow vector with Cartesian entries

p

,


14

(2 .78)

may alternatively be represented by the component natural contributions

(2 -79) or by the total natural quantities

(2.80) The reader is reminded of the interrelations between the alternative representations of the heat flow vector in accordance with (2.4) and (2.6). When determining the heat flow through an infinitesimal tetrahedron element shown in fig. 2.5, asarising from component natural heat fluxes [3] we have to introduce the column matrix

,

Consider now in fig. 2.5 a component natural heat flux characterised by the intensity and the outprogressing through the tetrahedron. Noting the input f:.(* put f9+0'4):Ad of the heat rate emerging a t a distance 1% from the point o f input we deduce for the rate of heat supply to the element as contributed by the component natural heat flux

$'

s',"

(2.82) where

(2.83)

:i

.

denotes the rate of change of Genemlising for a l l natuml directions along we apply the column m t r i x Ct of (2.81) and obtain the rate of heat supply by summation of a l l individual contributions as expressed by (2.82). Hence, we find

(2 .84)

On the Natural Approach to Flow Problems i s the Cartesian counterpart of the column matrix

where

15

4,

and reads

We conclude this subsection by presenting a natural counterpart to the Fourier's law relating the heat flow to the temperature gradient [3]. Starting with the Cartesian form

(2.86) wh re tion

2

den0

3s

the thermal conductivity o f the fluid, we LJduce the natum rela-

(2.87) by an appropriate application o f (2.6)and (2.4) or (2.11). We note that

(2 .88) symbolises the natural thermal conductivity matrix connecting via (2.87) the total natwith the component temperature gradient We ural heat flow vector ft also observe that the connection between $t and the total temperature gradient q ( T ) i s simply given by the thermal conductivity 2 o f the material as i n

;P,(T)

.

(2.86).

3.

DISCRETISATION BY FINITE ELEMENTS

3.1 Weak form o f the equations governing fluid and thermal flow Bearing in mind our prospective application o f the finite element technique to the flow problem we write i n the following the basic equations i n their weak form assuming a Thus, a weak form o f the momentum finite volume Y bounded by the surface balance may be expressed i n natural terms as

s .

3

-

8

the associated rate of deformawhere symbolises a virtual velocity field and tion, Also, f denotes the body force vector acting per u n i t volume and n a normal


16

operator yielding the surfoce tractions. Alternative formulations of (3.1) i n natural or in Cartesian terms are possible as outlined i n section 2. The virtual rate o f kinetic energy, for instance, on the left-hand side i n (3.1) i s given i n terms of one of the expressions

(3.2)

2

offered in (2.8) for the scalar product of two vectors. Clearly, the accelerution consists, in the Eulerian apprcach adopted here of a I y a I part and a convective part and may be specified in the component natural form &$ of (2.32), the Cartesian form of (2.24), or the total natural form of (2.26). We also observe that the component natural stress Cc in (3.1) obeys the constitutive laws of subsection 2.6. For a weak formulation of the isochoric condition we rely on expression (2.61) and write i n na tura I terms

where

represents the virtual pressure field.

We next turn our attention to the heat flow as occurring concurrently with the fluid motion. The heat balance of the volume in question may be expressed i n natural terms as

Y

I/

r/

where ? denotes a virtual temperature field. In (3.4), the first integral on the lefthand side i s due to the rate of heat stored i n the material, i n accordance with (2.75). I t i s specified through the local term of (2.76) and the convective term of (2.77). The second integral reproduces the rate of heat supply (2.84) by heat conduction. I t balances the stored heat expression with due consideration o f the rate of dissipation in the material as given by the right-hand side of (3.4). The second integral i n (3.4) associated with the heat flux may be transformed as follows (cf. [ 31)

where due to (2.8) and (2.87)

17

On the Natural Approach to Flow Problems Furthermore, the boundary condition

5

under the temperature 7 expresses the local heat exchange between the surface and the surrounding medium under a temperature Ts ; the associated heat transfer coefficient i s denoted by o( Thus (3.5) m y be brought into the final form

.

By substitution of (3.8) i n (3.4) one obtains the fundamental expression for the derivation of the relevant finite element relations,

3.2 Natural finite element equations for fluids

To set up a finite element formulation o f the flow problem consider first the weak momentum equation (3.1) i n conjunction with an approximate representation of the velocity field within each finite element expressed by

The column matrix

comprises the component natural contributions to the velocity vector a t any one o f the r) nodes of an element

CorrespondingI y, the matrix

contains the diagonal matrices, (3.13)


18

o f dimensions 6 x 6 which interpolate the velocities depend only on the total natural coordinates Kt

.

gj . Note also that the

W.'J

J

The local part of the acceleration within the element may now be established immediately via (3.9) as

(3.14)

Before entering into the derivation o f the convective part o f the acceleration we observe that the velocity field (3.9) may alternatively be described by

where

and

3

13

i s here the super row matrix of the component nodal velocities,

the column matrix,

Hence, the velocity gradient may be written as

(3.18)

(3 .19)

The convective term of the acceleration (cf.

(2.21)) may now be expressed as,

(3.20) i n which the velocity gradient i s represented by (3.18). The total natural velocity appearing in (3.20) obeys the interpolation rule of (3.9) i n the form

19

On the Natural Approach to Flow Problems with the column matrix (cf. (3.10))

Here nodes

6

comprises the field of total natural velocitiesat each of the 0 element

J

App Iying next expression (2.22), we obtain the acceleration by a summation o f the local part (3.14) and the convective part (3.20) in the form,

(3.24)

We now proceed to the mte of deformation within the finite element. To this purpose we consider the total natural rate of deformation Jt of (2.42) and rewrite i t in the form

>= d,.-.1 ' where the operator

=

d,

(3.25)

i s the (6 x 6) diagonal matrix

J

and,

(3.27)

Btc

Here 6,9 symbolises the a - t h column of the matrix i n (2.6), respectively i n (2.7). Application o f the interpolation rule (3.21) furnishes the total natural rate o f deformation within the element as,


20

(3.28)

Turning our attention to the virtual velocity field

Gt

introduced in (3.1) we set,

(3.29)

@

The definition of the column matrix i s i n line with that of i n (3.22). As to & i t s formation i s that of W N of (3.12) but may be based on different interThe associated virtual rate of deformation reads # "'J' polation functions 6;' then in analogy to (3.28)

.

(3.30)

In finite element theory, forces are assumed to be transmitted exclusively through the element nodes. Let the column matrix

comprise the component natural element contribution to the force vector a t each of the n element nodes,

In accordance with the invariance rule (2.8), the component natural representation of the nodal force vector pu' of (3.32) corresponds to the total natural definition of the nodal velocity vector b$j of (3.23). Disregarding for simplicity the volume forces on the right-hand side of (3.1) and expressing the surfoce integral through the nodal quantities the virtual work expression (3.1) assumes for a finite element of volume c/ the form

(3.33)

1/ Introducing the kinematic relations (3.29), natural forces a t the element nodes as

(3.30) i n (3.33) we obtain the component

21


(3.34)

The acceleration term on the right-hand side of (3.34) may be transformed with the aid of (3.24) into,

(3.35) where

(3.36)

J corresponds to the Lagrangeun mass matrix while

(3.37)

accounts for the nonlinear convective contribution inherent to the present Eulerian approach. To specify the stress term on the right-hand side of (3.34) we call upon expression (2.67) for the component natural stresses and obtain

I/

J

Using the kinematics as prescribed in (3.28), the f i r s t integral on the right-hand side of (3.38) i s transformed into,

22


where

(3.40)

v represents the viscosity matrix of the element and reflects the deviatoric response of the isochoric fluid. With respect to the second integral on the right-hand side of (3.38) we introduce the approxima tion

P = V

(3.41)

to describe the pressure field within the element. In (3.41)

and contains the pivotal values of the pressure and 9C within the row matrix

p

i s the column matrix

the interpolation functions

Introducing (3.41) into (3.38) one obiuins,

(3.44)

(3.45)

J i s the hydrostatic element matrix. Using (3.44), (3.39) and (3.35), the component natural forces (3.34) of the element may ultimately be expressed as

23


3.3 Transition to artesian definitions; discretised Navier-Stokes equations Before proceeding to the assembly of the element contributions (3.46) within the region considered, we transform (3.46) into a global artesian system o f reference. Denoting the respective artesian element nodal forces by

and the corresponding velocities by

j=

I,

..., Y

(3.48)

we m y apply relation (2.4) connecting natural and artesian definitions of vectors to obtain on the element level,

(3.49) and

(3.50) We note also that in

v=wv

(3.51)

the interpolation matrix W corresponds to the definition of ,&# i n (3.12) but with (cf. (3.13)) of dimension 3 x 3, in order to maintain consistency with the entries artesian definition. One may now substitute (3.50) i n (3.51) to express Y i n terms of V, Relating on the other hand f l to via (2.4) and expressing the latter through the interpolation (3.9) we deduce a second expression for Y Thus,

c3J'

.

6

.

and hence

(3.53) Applying next the transformation to the velocities (2.6) we obtain for the toiul elementa l velocities

24


Jt = r B J J

(3.54)

An analogous argument to that used i n (3.52) yields in the present case

5

(3.55)

and hence

(3.56) Substituting in (3.49) nishes the Cartesian forces

I

as

given i n (3.46), and

as defined in (3.54) fur-

(3.57) Using (3.56) and (3.53) as well as (3.36) and (3.50) we may verify that the first term in the second expression of (3.57) reduces to

v =

[/pGWdlTB',,J

2

Y

- [/pGCdJ1 r/ Note the expression for the elemental mass matrix vv)

=IpGcld/ Y

= WI

C;

(3.58)

(3.59)

25

On the Natural Approach to Flow Problems Consider next the second term on the right-hand side o f (3.58). Application of (3.37) for the natural convectivity matrix yields the Cartesian counterpart

4

in which use i s made of the relation (2.18) connecting and denotes the super row matrix of the cartesian nodal velocities.

$,

, Also,

f

Finally, the Cartesian viscosity and hydrostatic elemental matri:

d = r&;&j dN rs,, I

(3.61)

and

represent standard transform tions and do not require further elaboration.

zi

, the weightWe observe i n the above finite element idealisation that identity of the interpolation functions, reduces the discretisation proing functions, with wj cedure to that of Galerkin. In most structural applications, this method leads to symmetric matrices and the associated solutions are known to possess the property of best approximation. In convection dominated flow problems, however, we adopt a suggestion o f [41 and prefer to apply the streamline upwind/Petrov-Galerkin technique. In this case zj and W j are taken to be different. Bearing i n mind the aforementioned publication i n which a detailed description o f the method i s given we restrict our present account to an elaboration o f the alternative natural formulation. Following [ 4 ] , the weighting functions 6j are formed as

,

(3.63) where w j i s the standard interpolation function a t the j - t h element node and S j a perturbation defined by


26

which induces an upwinding i n the streamline direction. The scalar coefficient (G i s specified in [4] asa function of the velocityand the element dimensions. The natural expression for Sj i n (3.64) may be seen to simply rely on the invariance of alternative expressions of scalar products as shown in (2.8). In (3.64) d j i s assumed to be a function of the total natural coordinates X t The associated gradient $< follows then the definition of (2.12) with dj i n place of the pressure. In conclusion we note that the upwind technique introduces an additional dependence on the velocity into the finite element characteristics. As outlined i n [41 under certain conditions the upwind scheme affects merely the weighting of the acceleration term in (3.34) but not that of the stress term, In this case the element viscosity matrix i s symmetric.

.

Turning next our attention to the entire flow domain, the element contributions to the nodal forces as given by (3.57) may be summed up and yield the global relation

R

which represents the discretised form of the Navier-Stokes equations. In (3.65) ,denotes the column matrix of the nodal forces applied to the flow domain, y and 1, are the corresponding velocities and accelerations, and the column matrix P defines the pressure field in the entire flow domain. The matrices fl D h / , 0 and N may and be deduced by a straightforward assembly procedure from the matrices m , b l h of the individual elements.

,d

3.4 lsochoric condition. Exact analysis and approximate penalty formulation We now proceed to the discretisation o f the isochoric condition using the natural methodology and consider to this end the last expression i n (3.3). Introducing a relation analogous to that of (3.41) for the variation of and expressing as i n (3.28) we deduce for a finite element h e condition

dt

(3.66)

v

J

where the matrix

(3.67)

I/

Y

h,

cincides with the matrix and i n (3.45) for the case when GJ'= * l j TO obtain the artesian form of (3.66) we refer to (3.54) and deduce ILj = %j

.

21


Hence the Cartesian counterpart of the natural matrix

f

is

(3.69)

The isochoric condition for the entire flow domain may now be symbolised by (cf. (3.68))

G'V

=

o

(3.70)

where the column matrix Y comprises the velocities a t the nodal points of the finite in (3.69). element mesh, and 6 i s composed by the individual element matrices

9

In the penalty approach the isochoric condition i s relaxed in accordance with (2.68). As a consequence the weak formulation in (3.3) i s correspondingly affected. Adopting the finite element approximation in (3.66) one obtains i n the penaltyopproach

(3.71)

The matrix

R"

may be seen to represent the integral expression,

(3.72)

Y Solution of (3.71) for the pressure yields

(3.73)

where use i s made of (3.54), (3.69) when forming the alternative Cartesian expression on the right-hand side of (3.73). Substitution of (3.73) in (3.57) determines a pure velocity formulation. Isolating the two last terms in the final expression in (3.57) we consequently have

The matrix

(3.75)

28


represents the elemental viscosity i n the penalty approach and i s a symmetric matrix i n an ordinary Galerkin approximation. The above procedure corresponds to the mixed finite element technique of [15] i n which velocity and pressure field are approximated independently, An alternative penalty formulation of the viscous incompressible problem may be obtained by substitution o f (2.68) i n (3.38). This leads to a pure velocity formulation i n (3.46) or (3.57) without the need o f a separate approximation for the pressure. On the other hand, this advantage involves necessarily a reduced integration [13, 141 Sumscheme for the volumetric part of theassociated viscosity matrix marisingthe discretised Navier-Stokes equations for the entire flow domain may be written in the penalty approach as

.

2

(3.76) where the relaxed isochoric constraint i n the viscosity matrix i s included in accordance with one or the other approximation technique.

-

4

in

3.5 Finite element equations for heat flow A s a final item we consider the finite element approximation of the heat balance in the fluid as governed by (3.4) and (3.8). To this end we write the tempemture field within the element as

(3.77) where the column matrix

comprises the temperatures a t the element nodes and the row matrix

the'interpolation functions. Analogously, we express the virtual tempemture field as

(3.80)

-

where the weighting functions ?j i n 2 ' may be constructed i n accordance with the streamline upwind/Petrov-Galerkin concept, as detailed i n (3.63) for Applying (3.77) we may obtain the local part o f the temperature rate as

aj

.

29


-==r at

ri

(3.81)

Correspondingly the convective part becomes

(3.82)

where use i s made of (3.21) for

5 .

With the aid of (3.80), (3.81) and (3.82) the first integral in the heat balance of (3.4) may be transformed into

(3.83) The matrix

(3.84)

represents the heat capacity matrix of the element in a Lagrangean approach and must be supplemented in the present Eulerian presentotion by the convective contribution associated in (3.83) with the coefficient matrix


30

k

The second integral expression in (3.85) refers to a cartesian specification, the transition from the first natural expression being a consequence o f (2.74). The second integral on the left-hand side of (3.4) may be put as a consequence of (3.8) into the finite element form

J

s

being the element surface. Application of (3.80) and (3.77) yields the equiwlent expression

(3.87) The element conductivity matrix i s thus given by

I t s transcription into the artesian form m y be established by substitution o f (2.88) for and application of the gradient relations (2.20), (2.17) and (2.18). We find

Arc

Furthermore, we observe in (3.86) that

(3.90)

S

J

represents a prescribed heat rate through the element surface.

On the Natural Approach t o Flvw Problems

31

Concerning the rate o f dissipation defined by the integral on the right-hand side o f (3.4), one may write,

and

(3.92)

J

Y

6,

Here t t B $ and Q may be deduced from the mechanical account o f the flow problem i n subsection 3.2. Collecting the contributions (3.83), (3.86) and (3.91) into the overall heat balance of the element as expressed by (3.4) we obtain

(3.93) where a

a

a

f = P,

fd

(3.94)

i s a generalised heat rate. The finite element equations for the entire flow domain assume then the form

(3.95) *

a

r,a

are column matrices comprising quantities a t the nodes o f the i n which 7, finite element mesh and , k L are the relevant global matrices deduced by assembly of the respective element matrices.

c

,


32

4.

NUMERICAL EXAMPLES

In this section we present some examples illustrating the application o f the preceding theory on the solution o f pure and thermally coupled flow problems. Details o f the numerical solution methods, omitted i n this paper, may be studied i n 1281. There, the numerical aspects are discussed taking account o f the pertinent literature on the subject [16 251, which include recent developments, We should stress here that the streamline upwind/Petrov-Galerkin scheme i s applied to a l l our examples. The capabili t y of this method i s demonstmted i n what follows for convection dominated flow i n two and three dimensions. The solution o f thermally coupled flows i s illustrated on the BBnard type instability. Cavity flows with free and forced convection including a change of phase are treated i n [ 2 8 ] .

-

4.1 Flow over a step The transient incompressible flow over a step demonstrates the applicability o f the independent p - Y formulation and a two stage solution strategy as described i n [4, 281 Due to the high Reynolds number a turbulent flow field develops necessitating the use of upwind techniques. The geometry of the flow domain and the boundary conditions used in the calculation are sketched i n fig. 4.1 together with the material data o f the medium (air). A t the inlet a constant velocity profile i s prescribed which yields a Reynolds number o f 14950 based on the step height. A zero velocity component i n cross flow direction i s assumed a t the upper side and zero pressure a t the outlet of the channel, The flow region i s discretised by a mesh o f 1700 bilinear plane elements QUAP4 as indicated i n fig. 4.1. Starting from a quiescent i n i t i a l condition the development o f the turbulent flow i s investigated up to a total duration o f = 67.5 ms using 900 time increments. In fig. 4.2 the onset of turbulent flow i s shown i n the upper two streamline plots while the other plots depict the fully turbulent flow field. The disturbances i n the flow field near the outlet may be caused by the somewhat unrealistic pressure boundary condition. Also, the zero cross flow condition a t the upper side o f the flow region seems to be not well adjusted to the process. Despite a l l these shortcomings the long-time exposure of the flow over a step ( w t e r , visualised b y aluminium powder) shown i n fig. 4 . 3 and extracted from 1261 compares quite well with the streamlines a t the instant = 56.25 ms of the numerical investigation.

t

t

4.2 Flow i n a quadratic duct with a step The efficiency of the upwind scheme and its three-dimensional generalisation involving the two stage solution algorithm i s demonstmted i n this example. The flow region, the boundary conditions and the data o f the fictitious material are depicted i n fig. 4.4. A t the inlet cross-section a constant flow velocity i s assumed, the Reynolds number of 200 A zero pressure condition i s adopted a t the being based on the duct dimension H outlet. The flow domain i s discretised by 1368 linear volumetric elements HEXE8 as shown i n fig. 4.5. Calculations were performed i n 60 time steps from the i n i t i a l canditions to a steady state a t dimensionless times = 6. A t the final stage, projections

.

t

.


33

o f the nodal point velocity vectors onto the xy- and yz-planes are shown i n fig, 4.6. The following example i s concerned with the solution o f a coupled fluid/thermaI problem.

4.3 BBnard convection i n a rectangular box In a fluid heated from below buoyancy driven convection rolls w i l l develop above a critical value o f the Rayleigh number (cf. fig. 4.7)

This process i s analysed for water enclosed i n a rectangular box, disregarding any threedimensional effects. The lower and upper plate o f the box are held a t a constant temperature, but the vertical side walls are assumed to be subject to an adiabatic state. The fluid i s i n i t i a l l y set a t the same temperature as the upper plate and i s then heated from below. As soon as the critical Rayleigh number Ra = 1708 i s exceeded convection rolls begin to develop. To avoid the difficulties associated with the bifurcation phenomenon a t the critical Rayleigh number a perturbation i n temperature i s applied which determines the rotational sense o f the first vortex. The analysis i s continued unt i l stationary conditions are attained. The mechanical and thermal data o f the fluid (water) are quoted i n fig. 4.7 together with the discretisation by QUAP4 plane elements. The Rayleigh number i s ewluated to be 20250 which exceeds by far the critical value. This fact facilitates the generation = 450 s and involves 60 time o f an unstable process. The calculation extends over steps, varying between 2.5 s and 20 s. The small increments prove necessary i n the i n i t i a l process of the formation o f the convection rolls within the time i n t e r w l between 100 sand 150 s. The temperature perturbation applied for the initiation o f the convective flow i s removed after 150 s when a l l vortices are formed. Fig. 4.9 exhibits isotherms and streamlines a t different stages o f the process. The development o f the convection rolls and the steady state condition i s i n good agreement with experimental and analytical investigations 1271. The time between the initiation o f convection up to the f u l l y developed flow corresponds to the predictions. The series o f differential interferograms reproduced in fig. 4.8 shows the formation o f convection rolls i n silicone o i l under similar conditions.

t

The calculation o f the coupled fluid and thermal problems was performed by an iterative sequential solution o f the two individual problems (cf. 1 3 2 ) . The thermal equation, dominant i n the BInard convection phenomenon, w a s solved first followed by the solution o f the flow problem. A l l coupling quantities were taken into account, i.e. the convective terms i n the thermal problem and the buoyancy forces i n the flow problem, the latter being calculated using the Boussinesq approximation. The iterative solution o f the discretised equations leads to linear equation systems with non-symmetric coefficient matrices due to the convection terms, The equation system o f the thermal problem was solved using the QR-factorisation for the non-symmetric coefficient matrix.


34

For the flow problem the penalty approach w a s applied with the convection terms on the right-hand side so that standard solution methods were eligible. Upwinding w a s used in both problems with an upwind parameter of 0.258. Convergence below the limit € = 10'' in the heat rates and velocities respectively was required to terminate the iteration of the individual problems. The sequential solutions were continued until both the velocity and the temperature increments were reduced below the convergence limit of t = to-' between consecutive iterations.

REFERENCES

-

Argyris, J.H. et al., Finite element method the natural approach, Fenomech Comput. Meths. Appl. Mech. Engrg. 17/18 (1979)1-106.

'78,

Argyris, J.H., Doltsinis, J.St., Pimenta, P.M. and Wtistenberg, H., Thermomechanical response of solidsat high strains natural approach, Fenomech '81, Comput. Meths. Appl. Mech. Engrg. 32 (1982)3-57.

-

Argyris, J.H. and Doltsinis, J.St., On the natural formulation and analysis of large deformation coupled thermomechanical problems, Comput. Meths. Appl Mech. Engrg. 25 (1981)195-253.

.

Brooks, A .N. and Hughes, T.J. R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Fenomech '81, Comput. Meths. Appl. Mech. Engrg. 32 (1982)199-259. Argyris, J.H. and Doltsinis, J.St., On the large strain inelastic analysis in natural formulation Part I . Quasistatic problems, Comput. Meths. Appl. Mech. Engrg. 20 (1979)213-252. Part II. Dynamic problems, Comput. Meths. Appl. Mech. Engrg. 21 (1980)91-128.

-

-

Argyris, J.H. e t al., Eulerian and Lagmngean techniques for elastic and inelastic deformation processes, TICOM 2nd Int. Conf., Austin, Texas, 1979. In: Compututional Methods in Nonlinear Mechanics (J .T. Oden, Editor), NorthHolland Publishing Company (1980)13-66. Argyris, J.H., Doltsinis, J.St. and Wtistenberg, H., Analysis of thermo-plastic forming processes natural approach, Computers and Structures, to appear.

-

Prager, W., Introduction to mechanics of continua, Ginn and Co.,

Boston

(1961).

Argyris, J, H,, Three-dimensional anisotropic and inhomogeneous elastic media, matrix analysis for small and large displacements, Ing.-Archiv 34 (1965)33-55. Argyris, J.H. and Doltsinis, J.St., A prime on superplasticity in natural formulation, Comput. Meths. Appl Mech. Engrg to appear.

.

.,

Hohenernser, K., Prager, W., Uber die Ansdtze der Mechanik isotroper Kontinw, ZAMM 12 (1932)21 6-226.

35


P2J Argyris, J. H. and Mareczek, G. , Finite element analysis of slow incompressible viscous fluid motion, Ing. Archiv 43 (1974) 92-109.

-

31 Malkus, D.S. and Hughes, T.J.R.,

Mixed finite element methods reduced and selective integration technique: a unification of concepts, Comput. Meths. Appl. Mech. Engrg. 15 (1975) 63-81. Oden, J.T.,

RIP-methods for Stokesian flows, In: Finite Elements i n Fluids, Vol.

4 (R.H. Gallagher e t a l . , Editors), John Wileyand Sons Ltd., 1982. Taylor, R.L. and Zienkiewicz, O.C., Mixed finite element solution of fluid flow problems, In: Finite Elements in Fluids, Vol. 4 (R.H. Gallagher et al., Editors), John Wileyand Sons Ltd., 1982. Felippa, C.A. and Park, K.C., Direct time integration methods i n nonlinear structural dynamics, Comput. Meths. Appl, Mech. Engrg. 17/18 (1979) 277-313.

71 Glowinski, R., Dinh, Q.V. and Periaux, J., Domain decomposition methods for nonlinear problems in fluid dynamics, Fenomech ‘81, Comput. Meths. Appl. Mech

. Engrg., to appear.

Hestenes, M.R., Stiefel, E., Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Stand. 49 (1952) 409-436. Jennings, A . and Malik, G.M., The solution o f sparse linear equations by the conjugate gradient method, Int. J. Num. Meths. Engrg. 12 (1978) 141-158. Dennis, J.E. and More, J .J., Quasi-Newton methods - Motivation and theory, SlAM Review 19 (1977) 46-89. Matthies, H. and Strang, G., The solution o f nonlinear finite element equations, Int. J. Num. Meths. Engrg. 14 (1974) 1613-1626.

,

Thomasset, F. Implementation of finite element methods for Navier-Stokes equations, Springer New-York, 1981. Hughes, T.J.R., Winget, J., Levit, I. and Tesduyer, T.E., New alternating direction procedures in finite element analysis based upon EBE approximate fhctorimtion, Recent Developments i n Computer Methods for Nonlinear Solid and Structural Mechanics (eds. S.N. Atluri and N. Perrone), ASME Applied Mechanics Symposium Series, New York, 1983.

G.J., Methods of Numerical Mathematics, Springer-Verlag, PI Marchuk, York - Heidelberg - Berlin, 1975.

,

New

Hughes, T. J .R., Levit, I. and Winget, J. An element by element solution algorithm for problems of structural and solid mechanics, Comput. Meths. Appl. Mech , Engrg , 36 (1983) 241 -254.

36


[26]

Toni, I., Experimental investigation o f flow separation over a step, IUTAM Symposium Freiburg 1957, Grenzschichtforschung/Boundary layer research, H. Gartler ed., Springer-Verlag, 1958, 377-386.

[27]

Kirchartz, K.R. Oertel, H., Zeitabhtingige Zellularkonvektion, (1982), T 211 T 213.

[28]

Argyris, J., Doltsinis, J.St., Pimenta, P.M. and Wustenberg, H., Natural finite element techniques for fluid motion, Comput. Meths. Appl. Mech. Engrg., to appear,

-

,

ZAMM 62


Paral lelopiped

Carterion directions (a)

Cartesian apprmch

Tetrahedrm

Natuml directions

(b)

Fig. 2.1

Natural apprmch

Cartesian and natural system of reference

37

w

00

4

t

Natural and Cartesian directions

(a)

.s

Reference system

2

Bl

Component definition

Cartesian definitions

Total definition

(Non unique cornpition of a vector)

(Unique decomposition of o vector)

(b)

Fig. 2.2

Alternative representations of vector

r

Natural and cartesian specifications of a vector for the two-dimensional case


Fig. 2.3

M o s s supply to a natural element due to a component velocity vector ycd

Fig. 2.4

Corresponding definitions of natural stresses and rates of deform tion

Fig. 2.5

Heat supply to a natural element due to a component heat flux vector

4:

39

P

0

-

v, = 10 m/s

vr = 0

yI

'vy= 0

-2s--l

Ov,

= "vv = 0

p=o

I

X

= 0360m H = a056 m

Material data (air1

S = 0.020 m

p = 1.293

L

Fig. 4.1

I(

= 17.3 10-6Pas kg/m3

Flow over a step. Description and finite element discretisation

,f H

I


Fig. 4.2

Fig. 4.3

Flow over a step. Streamlines during development o f turbulent flow

Visualisation of flow over a step by aluminium powder i n water [26]

41


42

v,.vy.v,.O

/t

v,. 1.0

T vy :v, :0

H

p.0

I

) .

L = 7.0 H = 2.5

s Fig. 4.4

i

g = 2.5

p

t

200.0

1.0

Flow in a quadratic duct with a step

A 1368 HEXEB -Elements

3306 Unknown velocities 1368 Unknown pressures

Fig. 4.5

Flow in a duct, Discretisation


I

I

I

43

A E C b

I

--_____________

$+ .T. . . . . . . ......... . . . . . . . . .

. . . . . . . . . . . . . . . . . . I

9

I

.

.

*

,

.

.

-

a

I T

. I

. . . .

. .

.

. . . .

,

. . . . . . . . . I

, , , , , , .

,

, ,

,

I

I

,

,

,

,

,

,

1

I

,

,

. . I

,

, ,

I

, ,

* 1

,

,

. -

.

, .

.

_ - - . . - .

, - - - ..

*""

100

S t r e s s - I n t e n s i t y F a c t o r Varia t i o n w i t h T i m e f o r a Propag a t i n g Crack [ A c t u a l Loading Condition]

88

S.N. Atluri & T. Nishioka

F i g . 10. Schematic R e p r e s e n t a t i o n of Domains Modeled by D i f f e r e n t D i s c r e t i z a t i o n Methods

Hybrid Methods of Analysis

with similar definitions for coupling are considered.

the

89

+

-

tractions ti and ti.

Three cases of

111.1 Coupling of FEM with Direct BIE Method

The notation is given in Fig. 10. technique yields the equations for V1:

Application of the Direct BIE

(111.2)

Application of the FEM to V1 yields the equation

K_s=_a III.l.A

Direct Coupling.

(111.3)

Equations (11.2,.3)

may be writen as follows:

(111.4)

where _q,* is the vector of nodal displacements at p for the BIE modeled is the vector nodal displacements at p for the FEM region and where CJ modeled region. The vector _a,* is that of nodal tractions at p for the BIE region, and (la is the vector of equivalent nodal forces for the FEM region. Two possibilities arise: (i) lump the tractions at BIE nodes, (ii) distribute the forces for the FEM region. + By satisfying the equations ga = q* and the condition t = -ti at p, -a i using either (i) or (ii), direct coupling is achieved as in a substracturing procedure. However, the assembled equations for V1 U V2 are unsymmetric. Thus, in the direct-coupling procedure, a significant advantage of the FEM (viz: symmetric banded matrices) is lost without appearing to gain much. III.1.B Coupling Through the Variational Method. The function ui(p), where p is a point in V1, generated from the solution of (111.2), using the Direct BIE method, satisfies the Navier equations exactly. Let the Let this be FEM interpolation for the displacement field at p be -UFO. written in the form


90

where is the vector of FEM nodal displacements at p. Let the known 9-Fp and Su be not yet substituted into boundary values of ui and ti on S ui i (111.2). Then the solution u (p) in V1, that satisfies the inter-region i continuity condition:

at all points of p, and the relevant boundary conditions at S and S 4 u1 01 can be obtained from the stationary condition of the functional

The displacements

and tractions may be interpolated over S1 =

U p, as:

_u(Q)

=

!j*(Q)g*

sul

sul

(III.7a)

where *! and N* are functions defined appropriately over S1, Q is a point on S1, and q* and Q* are the master-vectors of nodal displacements and tractions over S1. From (111.2) it follows that

-

9*

= B-l

A9

*

(requiring the inversion of an unsymmetric, densely populated matrix). Equation (III.7b) then yields

L(Q)

=

k!

*,B-1,A q *

Suhstitution of (111.8) Into (111.6) gives:

(111.8)


"

91

I

I

(111.9)

It should be noted that ['plBIE

where q

'

'p(g*

gFp)

are, as yet, unknown.

-FP

87T

=

P

The stationary condition

(6cJ*) = 0

leads to algebraic equations for -q* in terms of -Fp' q On expressing -q* thus in terms of q the functional T can be expressed in the form -Fp P

[np IBIE

T = f !FP

T

[-KlBIE 4Fp

+

(111.10)

gBIE gFp

where -%IE is now symmetric. Thus, a symmetric equivalent stiffness matrix has been obtained for the BIE modeled region, which can be added to that of the FEM modeled region. Thus, a symmetric system matrix is obtained, at the expense, however, of inverting the unsymmetric matrix B_. The procedure given in Eq. (111.9) is general and can be used to link several BIE and FEM modeled regions. A simplification occurs if the condition u

= u

-Fo

-B

where served

onp

(111.11)

sB

is the B I E interpolation for 2. The integral over p in (111.6) the purpose of enforcing this condition. This integral now

reduces to

if (111.11) is satisfied a priori.

!,(Q)

= _N*(Q) g*

on

s1

Thus, for the BIE region

- P (111.12)

This

results in

some simplification to

the

algebra leading t o the


92

equivalent stiffness matrix [,KIBIE defined in (111.10). However, the inversion of ,B still remains. Further simplifications arise if the BIE region is completely surrounded by FEM regions. In this case Sl = P and S = = 0. If u1 '51 the assumed displacement field at P for the BIE is identical to that for the FEM assumed displacement field at P, then an equivalent symmetric stiffness matrix for the BIE region can, a priori, be obtained as: *-1

TJr

4) ,N + ,N*

[(,M ,B

T

~ r - 1

(,M ,B ,A)

I

dp

(111.13)

The inversion of B still remains. 111.2 Coupling of FEM with Indirect Boundary Solution Method Consider the mixed boundary value problem for the BEM region V1:

u

i

=

-

u

at

i

(III.14a)

So

1 (III.14b) (111.144

s l = s uso.up u1

(I11.14d)

1

where Sl is the boundary of V1. single-layer potential:

The solution may be represented by a

(111.15) the unknown single where p is a point in V1, Q is a point on S1, S,(Q) layer potential, and U is the known Kelvin solution [ 1 6 ] . The stress ji field corresponding to (111.15) may be written as: /-

t (P) = j

- f Sj(P) +

where P is also on S1.

Si(Q)Tji(P,Q)

dSQ

In vector form, (III.15,.16)

(111.16)

may be written as: (111.17)

93


(111.18)

since

is continuous at the boundary [16], it follows that: (111.19)

Now, S ( Q ) may be interpolated over S1 as: =

_M(Q)s

(111.20)

where a is a vector of unknown parameters; or the boundary S1may be partitioned into elements S 1 , S2, ..,,SM; and the potenial 5 may be locally interpolated over each boundary element. The resulting interpolation could still be written in the form of (111.20) where, now, CL is a vector of nodal values of 5, On substituting (111.20) into (III.l7,.18,and . 1 9 ) , we obtain:

t ( P ) = “P)?

(111.23)

Since ~ ( p ) in (111.17) satisfies the Navier equations of elasticity identically, the one that satisfies the boundary conditions (111.14a,b,c) may be determined from the simplified potential energy functional [16], as the condition:

T

- b & y - yFP]. ds

(111.24)

is minimum

(111.25) where

q -FP

functions. varying (

are On T

~

nodal displacements at p, and M substituting (111.21,.22,.23, with ) ~respect ~ ~ to both and g

a

FP’

TP

are the interpolation

and . 2 5 ) into (111.24) and one obtains:

94

S.N. Atluri & T, Nishioka

(111.26)

where (_P* + P*T ) / 2 is the symmetric stiffness matrix of the region V 1 modeled by indirect boundary solution (IBS) method. Eqs. (111.26) may now be added to other eymmetric equations of the FEM modeled region V2. Thus, unlike the direct boundary integral method, no unsymmetric-matrix inversions arrive in the case of coupling of IBS method with FEM. However, a close examination of Eqs. (11.17,.18,.19, and $ 2 4 ) reveals that surface integrations must be performed twice. ACKNOWLEDGEMENTS The results presented herein were obtained during the course of investigations supported by the U.S. AFOSR under grant 81-0057C to Georgia Institute of Technology. The authors gratefully acknowledge this support as well as the encouragement of Dr. A. Amos. It is a pleasure to sincerely thank Ms. J. Webb for her assistance in the preparation of this manuscript. FOOTNOTES 1. Regents' Professor of Mechanics 2. Visiting Assistant Professor 3. Note, however, that in the presently considered symmetric 'mode I' problem only C J and ;~ a(') are nonzero. 33 4. This can be derived [16] from the usual potential energy functional, when the displacement field, in addition to satisfying the compatibility condition, also satisfies the equilibrium equations. REFERENCES Atluri, S. N., "Higher-Order, Special, and Singular Finite Elements", Chapter 4 in: State-of-the-Art Surveys 2 Finite Element Technology (Eds.: A. K. Noor and W. D. Pilkey), ASME, New York, NY, (19831, pp. 37-126. Atluri, S . N. and Kathiresan, K., "3-D Analyses of Surface Flaws in Thick-Walled Reactor Pressure Vessels Using - a Displacement-Hybrid Finite Element Method", Nuclear Engineering and Design, Vol. 51, No. 2, (1979), pp. 163-176. S., Eaetanya, A. N., and Shah, R. C., Kobayashi, A. "Stress-Intensity Factors for Elliptical Cracks" in: Prospects of Fracture Mechanics (Eds.: G. C. Sih, H. C. van Elst, and D. Brock), Noordhoff Int. Pub., (19751, pp. 525-544.

Sorensen, D. R. and Smith, F. W., "Semielliptical Surface Cracks Subjected to Shear Loading" in: Pressure Vessel Technology, Part I1 (Materials and Fabrication) Proceedings, Vol. 3. ICPVT, Tokyo,


95

ASME, NY, (1977), pp. 545-551. Vijayakumar, K. and Atluri, S . N., "An Embedded Elliptical Crack, in an Infinite Solid, Subject to Arbitrary Crack-Face Tractions", -Journal of Applied Mechanics, Vol. 48, (March 1981), pp. 88-96. Nishioka, T. and Atluri, S . N., "Analytical Solution for Embedded Elliptical Cracks, and Finite Element Alternating Method for Elliptical Surface Cracks, Subjected to Arbitrary Loadings", Engineering Fracture Mechanics, Vol. 17, No. 3, (19831, pp. 247-268. Nishioka, T. and Atluri, S . N., "Analysis of Surface Flaws in Pressure Vessels by a New 3-Dimensional Alternating Method" in: Aspects of Fracture Mechanics in Pressure Vessels and Piping, ASME PVP, Vol.58, (19821, pp. 17-35: Nishioka, T. and Atluri, S. N., "Integrity Analyses of Surface-Flawed Aircraft Attachment Lugs: A New, Inexpensive, 3-D Alternating Method," AIAA Paper No. 82-0742, 23rd SDM Conference, AIAA/ASCE/ASME/AHS, (10-12 May 1982), New Orleans, pp. 287-300. O'Donoghue, P., Nishioka, T., and Atluri, S . N., "Multiple Surface Cracks in Pressure Vessels", Engineering Fracture Mechanics (In Press), Georgia Tech Report (1983). Mondkar, D. P. and Powell, G. H., "Large Capacity Eqn. Solver for Structural Analysis", Computers & Structures, Vol. 4, (1974), pp. 699-728. Kobayashi, A. S . , "Hybrid Experimental-Numerical Stress Analysis", Experimental Mechanics, Vol. 23, No. 3, (19831, pp. 338-347. Kalthoff, J. F., Beinert, J., and Winkler, S . , "Measurements of Dynamic Stress Intensity Factors for Fast Running and Arresting Cracks in Double Cantilever Beam Specimens" in Fast Fracture and Crack Arrest (Eds.: G. T. Hahn, and M. F. Kanninen), ASTM STP 627, (1977), pp. 161-176.

--

Atluri, s. N., Nishioka, T., and Nakagaki, M., "Numerical Modeling of Dynamic and Nonlinear Crack Propagation in Finite Bodies by Moving Singular Elements'' in Nonlinear and Dynamic Fracture Mechanics (Eds.: N. Perrone and S . N. Atluri), ASME-AMD Vol. 35, (1979), pp. 37-67. Nishioka, T. and Atluri, S . N., "Numerical Modeling of Dynamic Crack Propagation in Finite Bodies, by Moving Singular Elements, Part 1 - Formulation, Part 11-Results", Journal of Applied Mechanics, Vo. 47, (1980), pp. 570-583. Nishioka, T. and Atluri, S . N., "Finite Element Simulation of Fast Fracture in Steel DCB Specimen", Engineering Fracture Mechanics, Vol. 16, No. 2, (1982), pp. 157-175. Atluri, S .

N. and Grannell, J. J., Boundary Element Methods (BEM) GIT-ESM-SA-78-16, Georgia

and Combination BEM-FEM, Report Institute of Technology, (19781, 84 pp.


Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

97

CHAPTER 4 THE POSTPROCESSING TECHNIQUE IN THE FINITE ELEMENT METHOD. THE THEORY & EXPERIENCE I. BabuKka, K. Izadpanah, & B. Szabo

The p a p e r a d d r e s s e s t h e h , p , and h-p versions of t h e f i n i t e e l e m e n t method i n c o n n e c t i o n w i t h a postprocessing technique f o r e x t r a c t i n g t h e values of a f u n c t i o n a l . T h i s t e c h n i q u e combines t h e f i n i t e e l e m e n t method w i t h t h e a n a l y t i c a l i d e a s of t h e t h e o r y of p a r t i a l d i f f e r e n t i a l e q u a t i o n s of e l l i p t i c type.

1.

INTRODUCTION

F i n i t e e l e m e n t c o m p u t a t i o n s i n s t r u c t u r a l mechanics u s u a l l y have two p u r p o s e s : ( 1 ) t o d e t e r m i n e t h e s t r e s s and d i s p l a c e m e n t f i e l d s and ( 2 ) t o d e t e r m i n e t h e v a l u e s of c e r t a i n f u n c t i o n a l s d e f i n e d on d i s p l a c e m e n t f i e l d s as, f o r example, t h e s t r e s s i n t e n s i t y f a c t o r s , stresses at s p e c i f i c p o i n t s , r e a c t i o n s , e t c . C o m p u t a t i o n s of t h e s e v a l u e s i n v o l v e t h e f i n i t e e l e m e n t s o l u t i o n . F o r example, t h e s t r e s s components are o f t e n computed a t t h e Gauss p o i n t s of t h e e l e m e n t s and t h e s t r e s s e s a t any o t h e r p o i n t s are t h e n computed by t h e i n t e r p o l a t i o n t e c h n i q u e , t h e s t r e s s i n t e n s i t y f a c t o r s is d e t e r m i n e d by t h e J - i n t e g r a l o r c u r v e f i t t i n g t e c h n i q u e , e t c . We s h a l l r e f e r t o t h e s e o p e r a t i o n s as postprocessing. U s u a l l y t h e v a l u e s of t h e s e f u n c t i o n a l s a r e needed t o be known w i t h h i g h e r a c c u r a c y and r e l i a b i l i t y t h a n t h e d i s p l a c e m e n t o r stress field itself.

I P a r t i a l l y s u p p o r t e d by t h e O f f i c e of Naval R e s e a r c h u n d e r g r a n t number N O 0 0 1 4-77-C-0623. 2 P a r t i a l l y s u p p o r t e d by t h e O f f i c e -of Naval R e s e a r c h u n d e r g r a n t number NO001 4-81 -K-0625.

98

1. Babuika et al.

Assuming t h a t we h a v e t h e f i n i t e e l e m e n t s o l u t i o n and w i s h t o determine c e r t a i n f u n c t i o n a l values t h e following questions arise: 1 ) What s h o u l d t h e r e l a t i o n s h i p be b e t w e e n t h e computat i o n a l e f f o r t s p e n t on t h e f i n i t e e l e m e n t s o l u t i o n and t h e e f f o r t s p e n t on p o s t p r o c e s s i n g : Is it b e t t e r t o u s e a v e r y s i m p l e and i n e x p e n s i v e p o s t p r o c e s s i n g t e c h n i q u e as f o r example d i r e c t e v a l u a t i o n of t h e s t r e s s e s from t h e f i n i t e e l e m e n t s o l u t i o n i n t h e d e s i r e d p o i n t s o r s h o u l d one s e l e c t a more O f c o u r s e we h a v e t o r e l a t e t h e a n s w e r t o expensive technique. t h e a c h i e v e d a c c u r a c y and t o t h e r e l i a b i l i t y and r o b u s t n e s s of t h e postprocessing procedures under consideration. 2 ) G i v e n a f i n i t e e l e m e n t s o l u t i o n , what is t h e l a r g e s t a c c u r a c y of t h e f u n c t i o n a l v a l u e s one c a n a c h i e v e by t h e postprocessing technique. I n o t h e r w o r d s , what is t h e maximal i n f o r m a t i o n c o n t a i n e d i n t h e f i n i t e e l e m e n t s o l u t i o n which c o u l d be u s e d f o r t h e e x t r a c t i o n of t h e d e s i r e d v a l u e . 3 ) How do t h e v a r i o u s v e r s i o n s of t h e f i n i t e e l e m e n t method, i . e . , t h e h - v e r s i o n , t h e p - v e r s i o n and t h e h-p v e r s i o n b e a r on t h e i m p o r t a n c e of p r o p e r s e l e c t i o n of t h e postprocessing techniques. T h e s e q u e s t i o n s a r e d i s c u s s e d i n some d e t a i l s . 2.

THE EXTENSION OPERATORS. THE h , p AND h-p T H E FINITE ELEMENT METHOD

VERSIONS OF

T h e r e a r e t h r e e v e r s i o n s of t h e f i n i t e e l e m e n t methods b a s e d on t h e common v a r i a t i o n a l ( e n e r g y ) p r i n c i p l e . They are c h a r a c t e r i z e d by t h e s y s t e m a t i c s e l e c t i o n ( e x t e n s i o n ) of t h e f i n i t e e l e m e n t s p a c e s l e a d i n g t o t h e c o n v e r g e n c e of t h e f i n i t e e l e m e n t s o l u t i o n s t o t h e e x a c t one. The h-version is t h e c l a s s i c a l and most commonly u s e d method of e x t e n s i o n : t h e p o l y n o m i a l d e g r e e of e l e m e n t s p is f i x e d and mesh r e f i n e m e n t is u s e d f o r c o n t r o l l i n g t h e e r r o r of a p p r o x i m a t i o n ( h r e f e r s t o t h e s i z e of t h e e l e m e n t ) . T y p i c a l l y t h e p o l y n o m i a l d e g r e e of e l e m e n t s is l o w , u s u a l l y p = 1 o r 2 . P r o p e r s e l e c t i o n of t h e mesh and i t s r e f i n e m e n t s t r o n g l y i n f l u e n c e s t h e e r r o r and i t s d e p e n d e n c e on t h e c o m p u t a t i o n a l effort. I n t h e p v e r s i o n t h e mesh is f i x e d and t h e p o l y n o m i a l d e g r e e of e l e m e n t s is i n c r e a s e d e i t h e r u n i f o r m l y o r s e l e c t i v e l y o v e r t h e mesh. The h-p version combines t h e h and p - v e r s i o n s , i . e . , e r r o r r e d u c t i o n is a c h i e v e d by a p r o p e r mesh r e f i n e m e n t and conc u r r e n t c h a n g e s i n t h e d i s t r i b u t i o n of t h e p o l y n o m i a l d e g r e e of elements. The p e r f o r m a n c e of t h e v a r i o u s e x t e n s i o n s o p e r a t o r s c a n b e compared f r o m v a r i o u s p o i n t s of v i e w , t h e most i m p o r t a n t of which a r e human and c o m p u t e r - r e s o u r c e s r e q u i r e m e n t i n r e l a t i o n

The Postprocessing Technique in the Finite Element Method

99

t o t h e d e s i r e d l e v e l of p r e c i s i o n . Such r e l a t i o n s h i p s a r e d i f f i c u l t t o q u a n t i f y and a r e s u b j e c t d u e t o v a r i o u s f a c t o r s , t h e r e f o r e t h e p e r f o r m a n c e o f t h e e x t e n s i o n o p e r a t o r s is u s u a l l y O f course r e l a t e d t o t h e number of d e g r e e s of f r e e d o m N . e v a l u a t i o n of an e x t e n s i o n o p e r a t o r would n o t b e m e a n i n g f u l w i t h o u t c o n s i d e r i n g t h e g o a l s of c o m p u t a t i o n . F o r e x a m p l e , i f o n l y s t r e s s i n t e n s i t y f a c t o r s a r e d e s i r e d , t h e n t h e a c c u r a c y of t h e computed d i s p l a c e m e n t s , r e a c t i o n s o r s t r e s s e s are n o t of importance. I n many c a s e s t h e c o m p u t a t i o n h a s m u l t i p l e g o a l s .

3.

rix/

THE MODEL PROBLEM

I n o r d e r t o i l l u s t r a t e t h e e s s e n t i a l p r o p e r t i e s of f i n i t e e l e m e n t s o l u t i o n and e x t r a c t i o n t e c h n i q u e s , we h a v e s e l e c t e d a model p r o b l e m which r e p r e s e n t s some o f k e y f e a t u r e s of a l a r g e c l a s s of e n g i n e e r i n g problems. S p e c i f i c a l l y l e t u s consider t h e plane s t r a i n p r o b l e m of twodimensional e l a s t i c i t y (homogeneous i s o t r o p i c m a t e r i a l ) w i t h E and u r e p r e s e n t i n g t h e modulus of e l a s t i c i t y and P o i s s o n k-l+l+ r a t i o r e s p e c t i v e l y (E > 0 , 0 < u < . 5 ) . The domain Figure 1 D, a square panel with a The model p r o b l e m c r a c k is shown i n F i g . 1 .

i

We s h a l l be c o n c e r n e d h e r e w i t h p r o b l e m s i n which o n l y t r a c t i o n s are p r e s c r i b e d a t t h e b o u n d a r y ( i . e . , f i r s t b o u n d a r y v a l u e p r o b l e m of e l a s t i c i t y ) .

We d e n o t e t h e d i s p l a c e m e n t v e c t o r f u n c t i o n by and t h e c o r r e s p o n d i n g s t r e s s t e n s o r by

_u = { u l , u 2 )

The s t r a i n e n e r g y f u n c t i o n a l is

+

( l - v ) ( - au2)2 a x2

+ 1-2v 2

au

(-Jaxl

+

-)a u 2 ax2

]dx1dx2

.

(3-1)

I. Babufka et al.

100

The s o l u t i o n u s a t i s f i e s t h e Navier-Lam; e q u a t i o n s . I t is p o s s i b l e t o e x - r e s s t h e s o l u t i o n t h r o u g h two holomorphic f u n c t i o n s $ ( z p , $ ( z ) u s i n g t h e t h e o r y of M u s k h e l i s h v i l i [ I ] .

where z

and and

z

=

= x1 +'(z).

x1

+

-

ix2,

ix2,

=

p

E 2vj, = 3-4u K

(3.7)

mean c o n j u g a t e v a l u e s t o

reap.

z

The components of t h e s t r e s s t e n s o r a r e e x p r e s s e d by KolosovMuskhe 1i s h v il i f o r m u l a e

and

Re + ' ( z )

i s t h e real p a r t of

$' (z)

.

The c o r r e s p o n d e n c e between t h e d i s p l a c e m e n t s (and t h e s t r e s s ) f i e l d and t h e f u n c t i o n s and $ is one t o one up t o t h e c o n s t a n t s y and y ' i n + and $ , r e s p e c t i v e l y , s a t i s f y i n g y 7' = 0 . the relation

+

-

I n our model problem we c o n s i d e r t h e f o l l o w i n g ( e x a c t ) s o l u t i o n

101


T(z)

where

=

m.

B ( z ) is a h o l o m o r p h i c f u n c t i o n on D. F u n c t i o n ze1/2 is t o on D . b e u n d e r s t o o d as t h e p r i n c i p a l b r a n c h of z-’/2 F u n c t i o n ~ ( z ) is u n i q u e l y d e f i n e d b y ( 3 . 9 ) and ( 7 . 7 ) ( 3 . 8 ) . The t r a c t i o n s on t h e b o u n d a r y o f D a r e d e f i n e d by ( 3 . 4 ) ( 3 . 5 ) . I t c a n b e r e a d i l y v e r i f i e d t h a t t h e two e d g e s of t h e crack a r e t r a c t i o n f r e e . We w i l l now d i s c u s s t h e f i n i t e e l e m e n t s o l u t i o n and t h e p o s t p r o c e s s i n g t e c h n i q u e i f t h e t r a c t i o n s a r e p r e s c r i b e d on t h e b o u n d a r y of D s o t h a t t h e e x a c t s o l u t i o n t o t h e p r o b l e m is g i v e n by ( 7 . 7 ) - ( 7 . 9 ) . S p e c i f i c a l y we now c o n s i d e r t h e c a s e E = 1 , v = 7. The s t r a i n e n e r g y of t h e exact s o l u t i o n is: W = 42.16491 240.

4.

THE FINITE ELEMENT SOLUTION

We h a v e s o l v e d t h e model p r o b l e m by t h e t h e f i n i t e e l e m e n t method.

The

h and p - v e r s i o n s p - v e r s i o n of t h e f i n i t e

6

A

The meshes f o r t h e

of

Figure 2 p-version,

A:

Mesh 1 , B: Mesh 2

e l e m e n t method was implemented i n t h e e x p e r i m e n t a l c o m p u t e r program COMET-X d e v e l o p e d a t t h e C e n t e r f o r C o m p u t a t i o n a l Mechanics of Washington U n i v e r s i t y i n S t . Louis [2]. The two meshes shown i n F i g . 2A,B were u s e d . The p o l y n o m i a l d e g r e e s w e r e t h e same f o r a l l elements and r a n g e d f r o m 1 t o 8 . The s h a p e f u n c t i o n s on t r a p e z o i d a l e l e m e n t s of mesh 2 w e r e c o n s t r u c t e d by b l e n d i n g f u n c t i o n t e c h n i q u e . The h - v e r s i o n s o l u t i o n was o b t a i n e d b y means of t h e c o m p u t e r p r o g r a m FEARS d e v e l o p e d a t t h e U n i v e r s i t y of Maryland [ 7 , 41.

102

I. BabuSka et al.

FEARS u s e s q u a d r i l a t e r a l e l e m e n t s of d e g r e e one. The program is a d a t i v e and r o d u c e s a s e q u e n c e of n e a r l y o p t i m a l meshes. The mesh from t h i s s e q u e n c e w i t h 319 S e e [3P [41 [51 f 6 1 “71 e l e m e n t s and number of d e g r e e s of freedom N = 617 is shown i n F i g . 3.

.

Figure 3 The mesh c o n s t r u c t e d by t h e a d a p t i v e program FEARS

5.

ERROR OF THE FINITE ELEMENT SOLUTION MEASlJRED I N ENERGY NORM

We d e n o t e t h e e x a c t s o l u t i o n by uo and t h e f i n i t e e l e m e n t s o l u t i o n by ypE. The e r r o r o f t h e f i n i t e element s o l u t i o n is d e n o t e d by

2,

-e

=

so - gFE.

We measure t h e magnitude of t h e e r r o r by t h e energy norm

n*IIE,


103

T h i s measure is e q u i v a l e n t t o m e a s u r i n g t h e e r r o r i n t h e s t r e s s components by i n t e g r a l s of i t s s q u a r e s ( t h e L2 norm). I n o u r c a s e when t r a c t i o n s are s p e c i f i e d at t h e boundary

and

(5.7) The e x t e n s i o n s o p e r a t o r s u n d e r c o n s i d e r a t i o n m o n o t o n i c a l l y i n c r e a s e t h e f i n i t e e l e m e n t s p a c e s e i t h e r by i n c r e a s i n g t h e d e g r e e of e l e m e n t s o r r e f i n i n g t h e mesh. Therefore t h e energy norm of t h e e r r o r m o n o t o n i c a l l y d e c r e a s e s . We c a n w r i t e

and e x p e c t t h a t f o r p r o p e r l y c h o s e n p t h e f u n c t i o n C ( N ) is n e a r l y c o n s t a n t e s p e c i a l l y f o r l a r g e r N. The number p > 0 i s t h e r a t e of c o n v e r g e n c e of t h e e r r o r measured i n t h e e n e r g y norm.

I t is p o s s i b l e t o e s t i m a t e t h e v a l u e of p . I n o u r c a s e t h e r a t e p is governed by t h e s t r e n g t h of t h e s i n g u l a r i t y of t h e s o l u t i o n . I t can be shown t h a t f o r t h e p - v e r s i o n [ 8 ] , [ 9 ]

w i t h E > 0 a r b i t r a r i l y small and C i n d e p e n d e n t of N. The h - v e r s i o n u s i n g t h e u n i f o r m mesh y i e l d s t h e e s t i m a t e

r

SXds

r

=

(3.11’)

0.

We introduce a notation analogous to ( 3 . 7 ) . 0 , u = (u,x), v = (V,C)

We put, for

(3.12) L

Then ( 3 . 1 0 ) and ( 3 . 1 1 ) are, 0

G (U,V)

0

= 3; ( V )

( V P ) O.

We want to study the form of the variational problems a little more closely. For simplicity let us assume that y = p. In the applications this is usually true or else it can be achieved by changing variables. Let us write ( 3 . 7 ) and ( 3 . 8 ) in the obvious notations,

G0 ( U , V )

= G0 (

(u,X),(v,$))

= All(u,v)

0 (X,V)+ A21 0 (u,$ 1 + A 02 & X , $1 + A12

(3.13)

We want to demonstrate the symmetries here, using ( 2 . 5 ) . we have, by ( 2 . 5 ) 1 ,

First

Mixed Variational Methods for Interface Problems

(3.14)

4 2

Next ,

155

2.512

yields,

Clearly one has the choice of using (VP)l or (VP)O. The advantage of (VP)l is that it yields ui directly as part of the solution. Its disadvantage is that it makes the computation of the external field a little complicated. One must determine from the interior solution, compute u+ and - from theu-transition onditions and then do the two inteun grations in ( 2 . 8 ) . (VP)s yields the external field more readily with the single integration ( 2 . 1 2 ) but requires another integration, ( 2 . 1 5 ) I. to obtain u;. We observe that there is really a whole family of variational problems (VP)6 f 0 < 6 < 1. We simply multiply (VP) by 6 and (VP) by (1 - 6) and add. Then if we put Ir = (u,rP,x), Ir = (v,l,C) and (3.18)

we have the variational problems, find 6 6 G ( b , b ) = 3 (k)

Ir

such that for any b, (VP)a.

One can check that for

6 = 1 / 2 we have the symmetry relation, G 1 / 2 (b, -Ir) = G 1 ' 2 ( b , k ) . (3.19)

We return to this relation in the next sections. An analysis of the problems P1 and Po, as well as the variational problems (VP) and (VP)O, is presented in [ 1 1 , for the case $ 0 > 0. (The case $0 = 0 can be treated similarly.) We review the results briefly. There are some technical conditions. We have indicated in section two that a countable infinity of $,'s must be avoided. Further, if q in ( 1 . 2 ) 1 is real and positive in ri then it could happen that the problem Lu + qu = 0 in n, pun = 0 on r could have non-zero solutions. We assume that n is such that this cannot happen. Then the following facts have been established.

156

J. Bieluk & R.C. MucCumy

1. Suppose f f Hr(r) and g f Hr- (r) for some r 1/2. Then (6")has a unique (generalizeh.) solution (u,cp) with U f Hr+1/2(W u f ~ ~ ( r U; ) ,E H,- (r) If one computes 'u 'a%-'Li mo;:'( the transition conaitions then (2.8) yields a (classical) solution of (1.212 in n,' satisfying the radiation condition and with u f H, loc(n+). The combined function is a (generalized) solution of ( P F ~ ) . 2. Under the same conditions (Po) has a unique generalized solution ( u , ~ ) ,with same regularity; (2.13) yields a solution; of (1.2)2 with the same regularity and the combined function yields a (generalized) solution of (Pp0). 3. For f E , H ~ / ~ ( T and ) g E H- 2 ( r )( v P ) ~( ( ~ ~ 1 0have ) u f H1 (l4 and unique solutions ( u , ~ )( ( u , ~) ) wi[h V(X) E H-1/2(r). Results 1 and 2 are established by using known facts about boundary value problems in to reduce P1 ( P o ) to an equation of Riesz-Schauder type for ~ ( x on ) the space H-1/2(r). Then one can use the uniqueness of solutions of (P o ) to show the homogeneous Riesz-Schauder equations have onfy the trivial solution. In order to prove result 3 one has to establishcoercivity results of the form

.

-

i ) ~+~ I I C P I I ~ (r) The estimates (3.20) can where I I U I=I ~I I U I( t I be established by c nsidering'48e ad joint variational problems for (VP)l and (VP)8, respect'vely. It turns out that because of (3.17) the adjoint of (VP)I ((VP)O) is essentially (VP)O ((VP)l); hence one has a symmetric argument.

4.

Approximate variational problems.

In order to implement the variational problems numerically one introduces finite dimensional approximate spaces. We illustrate with (VP)l; the 0th rs are analogous. According to result 3 in section 3 , (VP)' has a solution (u,cp) f H1 (12) x H - 1 / 2 (r). We introduce families of subspaces, (4.1)

These are to be finite dimen ional and to depend on parameters hfl and hy. We put s h = S'fi x Shy. Then our approximate variational problem is: Find Uh = (uh ,cp h ) E Sh such that for any Vh = (vh , $ h E Sh , (AVP) h h h G ( U ,V ) = 3(V ) . (AVP)' is equivalent to sets of algebraic equations. Let

157


h

(W~,...,W

h

h

I

h

(41,...,QN ) be bases for

uh

hr.

and

S

N

Nhii =

z

h h

h

UiWiI q

=

i=l

kllii

L1

hr

LA Nh,. Then we have

and (AVP)

S

hr h h z rp.4. i=l 1 1

(4.2)

is equivalent to the algebraic equations h h h h +

~~~2~ = 2; &21ii

sh E

N

1R hL' I

(4 h )i

&222 =

+

a

(4.3)

(4.4) =

SFqi hds P

L

and the matrices are determined by

We shall say a little more about numerical implementations in the next section. Here we want to review some further theoretical results from [l]. The results requ're the fo lowing approximation properties of the spaces s'rl and SAT: (A.1) There exists a constant 1 , and an inteaer k such'that for any w E H 4 ( l l ) I 1 0 and a & I

kl

k1

>

>

1

1/2

there is a

r'$

The following results are established in [l]. Put h = hn + hr- Then if h is sufficientxy small: (1) Equations (4.5) have a unique solution. (2) Suppose U = {u,rp] is the solution of (VP)l and

J. Bielak & R.C.MacCamy

158

1

E H - 1 1 +€(TI with E < min(k,k 1 in (A.l) and 7A.2). $hen there exisrs a constant dent of h such that,

u E H1+~(12),

c, indepen-

Ashan example of the meaning of the above result one can take s l2 to consist of piecewise linear functions in R (k = 2) land Shy to consist of piecewise constant functions on L' (k = 1). Suppose then that the solution of (VP) has u E H2(n) and cp E H1/2(r) ( € = 1). Then take E = 1 in (4.6) and get (4.7)

Thus we obtain order h convergence in the natural n rm for (VP)l. One can also show that there is a constant c independent of the choice of h such that

P

The proofs of the above results proceed in several stages. One shows f'rst that the coercivity result{ (3.20) hold for any Uh E S' when V is restricted to V This is done by first using (3.20) to get a V E S which makes the inequalities valid and then using regul rity results to show that V can be approximated with a V' E Sh. These coercivity results on Sh enable one to establ'sh optimality; that is, to show that U is approximated by ,'U in the natural norm, as well as it is ossible to approximate U, in that norm, by elements of S Then one invokes (A.l) and (A.2). The L estimate (4.8) is obtained via the Aubin-Nitsche tricz.

.

R.

5.

Implementation of the numerical procedure.

For purposes of illustration we now discuss the actual implementation of the finite element method described in the preceding section for the case in which n is the unit circle and f and g are given by u$ and U6,nr respectively, where uo represents a harmonic incident plane wave field. The material in ,Li is homogeneous and uo is symmetric with respect to a diameter. The problem has been solved exactly in [ 7 1 . We divide the region R into circular sectors, with wedges around the origin, and consider a piecewise linear approximation for u in the polar coordinates r and 8 ; cp is taken to be a piecewise constant function on r. With this approximation the elements of the matrices 8 1 1 and Al2 and the load vector can be evaluated explicitly by direct integration. For WZ1 we integrate numerically with

sh


159

standard Gauss-Legendre formulas since the kernel in (2.4) that enters into the bilinear form Azl in ) Q ! : ( is continuous. Due to the logarithmic singularity of gPo which appears in the bilinear form A22 we use a modified Gauss-Legendre formula [ 8 1 that accommodates this singularity explicitly in evaluating the elements of &2. Nhi, X Nhn matrix with the elements indicated inA1h. iil;n and $21 are Nhn Nhy and Nhy Nhid, respectively, and $22 Nh The last three matrices will be full. will be NhT However, since tf;e diameters of the elements inside l2 and the lengths of the intervals on l' are about the same and of size h, then we see that Nh = N2 Thus, although the matrices $12, $21 and 822 fg11 their size is much smaller than that of $11 and an effective numerical procedure is still possible. Remarks: (i) Note that while $11 and $22 are symmetric matrices, $12 and f?21 are not generally the transpose of each other. Therefore, the system (4.3) is, in general, asymmetric. (ii) The form of equations (4.3) permits condensation. Suppose we are primarily concerned with the interior region. Then we may eliminate gh and consider the system,

.

.

sh

where BJ~ = -1/23?123322$321 and = -A12Aj$fh. The matrix B has nonzero elements only for nodes on t e boundary r. Fkus it represents the impedance of the exterior region n+, and constitutes, in effect, a discretized nonlocal absorbing represents the corresponding effective forcing boundary. function. Although 211 is in general an asymmetric matrix it turned out to be symmetric for the present problem. (iii) The condensation procedure requires that the matrix $22 be inverted; thus, it is not valid for values of Po for which the operator S[cpl in (2.4) cannot be inverted. Direct solution of the complete system (4.3)1, however, was possible for values of PO approaching these critical values. (iv) Other condensation schemes are clearly possible; see [ l ] for a scheme that is applicable if one is mainly concerned with the exterior region. (v) A symmetric discretized formulation of the general problem P is always possible by using the variational formulation VP1i2 in view of the symmetry relationship (3.19) provided one chooses real basis functions. The price we pay for this symmetry is that the system (4.3) is replaced by a set of similar structure of Nhn + 2Nhy equations instead of the Nhn + Nhy in (4.3). See [l] for details. After condensation, however, the corresponding system leads to equations of the form,

sh

wnich is similar to ( 5 . 1 ) and is clearly symmetric. the corresponding effective forcing function.

Lh

is

J. Bielak & R.C MacCamy

160

A comparison between the exact and approximate values of u at the center of the circle is shown in Table 1 for different values of Nhy (Nhn = N i ) for several combinations of the system parameters. The rresults tend to confirm our theoretical estimates that the convergence is of order h2 for the elements used.

6.

Two dimensional electromagnetic problems.

Many of the problems of electromagnetic problems can be idealized in the following way. One has a field everywhere in space, which we think of as filled with air. One introduces dielectric or metallic obstacles and seeks to determine both the fields induced in the obstacles and the distortion of the original field outside. This is an interface problem: one has different sets of Maxwell's equations in air and in the obstacles and transition conditions across the boundaries. The above problems are usually considered for the case of time-periodic fields of a single frequency and this is the case we consider. (By taking inverse Fourier transforms one can, in principle, solve time dependent problems from the periodic case.) The variables to be determined are the electric and magnetic fields t?. and 3. These satisfy Maxwell's equations. We write down theze equations when the material in question is either a dielectric or non-ferromagnetic metal in the time periodic case with frequency w. Further, we render the position variables x non-dimensional by dividing by a representative lengtf; a. The equations are: curl

2=

iwpa2,

curl

3

= KE,

where

K

-iwEa

=

for dielectrics,K

=

Ua

for metal.

(6.2)

Let us first do some scaling in the problem. Air is a dielectric with permeabilities p ~ €0. , We introduce dimensionless fields ;and 2 by writing, 8

H

=

hog,

$

=

iwgOahOg.

(6.31

Then (6.1) becomes ,

where k

2

= w l o€ a 2 =

P2

for dielectrics, k

=

icupoUa2

=

ia2

for metal.

161


The parameters k, a and $ are dimensionless. We allow for the possibility that the obstacles are inhomogeneous so that k , a and $ can depend on position. We can now describe the interface problem. Let n denote the obstacle regi n and d its exterior. Then in ll+ we have p = p o l k = = ui2a2C(OE0. Thus we have, curl

3

=

8,

curl

;=

P 20-E in

ri+

curl

3

=

c”)#,

curl

#

kg

fA

=

in

MO

.

r

The transition conditions across = an are that the tangential components of g and 2 are continuous, that is, if 2 is the normal to r:

-

n x g + = s x g ;

N

n x z + = n x g -

on

rl

(6.7)

n+ and

where the plus and minus denote limits from

n.

The problem is to be drive? by an incident field E 0 ,# 0 The differences 5 50- and satisfying (6.6)1 in all space. H -Zo are to satisfy a radiation condition. If we let E N and H represent the scattered fields in n+ then we sty11 have gquations (6.6) but (6.7) is replaced by,

-

n x g - = g x g+ + 2 x g 0;

n x g + = n x x - + n x g0 on

rl

N

with

3

and

2

(6.8)

satisfying a radiation condition.

We now specialize the geometry. We suppose that the obstacles consist of cylinders of uniform cross section n parallel to the z-axis. Then we limit ourselves to fields E, # (and which depend only on x and y, not z. N It can be shown that all such fields are combinations of fields of the following type: A 1 A 2 A Transverse magnetic (TM): 2 = E(x,y)k; 5 = H (x,y)i + H (x,y)j Transverse electric (TE):

5

=

H(x,j)t;

3

1

A

= E (X,j)l

h + E2 (x,j)].

(6.9)

Let us determine the structure of such fields. (TM) We have, A A p IA 2h H2 - H1 = kE. i+H J), E 1 - Ex] = -(H PO X Y Y

(*)

(6.10)

One can allow E o and go to have singularities in In fact, when = 0 one must have singularities in non-trivial problem. order to obtain

T2

n+.

162

J. Bielak & R.C. MacCamy

We introduce a function u

u

by the formulas, 2 u =

HIr

=

X

Po

(6.11)

PO

Then (6.1011 is satisfied if E = u if PO u ~ + )(-PO~ U (T

cc

and (6.lOI2 is satisfied )

Y Y

=

-ku.

(6.12)

Converse1 if u satisfies (6.12) and we put E = u and define Hy: H2 by (6.11) we have a solution of (6.10). Observe that onthe surface of our cylinder we have, for fields

TM

(6.13) where T and v are the unit tangent and normal to the boundary of n -in the x-y plane. ( T E ) We have, E:

-

E1 =

h!- h, Po

This time we define

u

uY = k E 1,

A A 10 20 H i - Hxj = k ( E i + E 1 ) . Y

(6.14)

by, LI

X

=

-kE

2,

H=u,

(6.15)

so that (6.16) Instead of (6.13) we have, (6.17) We can now obtain four different problems, all fitting the framework discussed in the preceding sections. We again let n+ denote the exterior of n in the x-y plane. The exterior region is air.

-

TM fields.

(I)

Dielectric cylinder

(11)

Dielectric cylinder - TE fields.

Mixed Variational Methods f o r Interface Problems

-

u

163

+ + uo/ +

u

=

(111) Metallic cylinder

uxx

+

u YY

-

u (IV)

=

=

-8,2 in

+ + uo; +

u

Metallic cylinder 3

-

u

=

+ + uo, +

u

Remarks: 1. For any materials except ferromagnetic ones there is only a small variation in p. Hence it is not a bad approximation to assume p/po = 1. 2. Although the theory can be carried through for any choice of the parameters there are really only two important cases. At low (say, 60 cycle) frequencies the parameter $ for a dielectric is very small while the parameter a for metals is O(1). At higher frequencies, say w = 0 ( 1 0 1 0 ) the parameter B is O ( 1 ) but the parameter a is very large. Thus the dielectric problems are both meaningful at higher frequencies. For the metallic cylinder problem at low frequencies one can, with small error, put $ - 0. This is what is usually done with a statement that ong 'neglects displacement current in air". This is the origin of our problem (PO). At higher frequencies the usual approximation is that the metal has "infinite conductivity" in which case one simply solves an exterior boundary value problem with equal to zero on the obstacle. Thus problems I11 and -tang e IV are really meaningful only if Po = 0.

Acknowledgement. This work was supported by the National Science Foundation under Grants CEE-8210859 (J.B.) and MCS-8219675 (R.C. MacC.).

T-able 1.

Relative Displacement at Origin

N

Re 1/2

.25n .50n n

2

.25n -50s n

hr

Im

Re

Exact

20

10

C

Im

Re

Im

Re

Im

0.8807 0.5327 0.1343

0.1421 0.0249 -0.2314

0.8802 0.5335 0.1367

0.1401 0.0225 -0.2301

0.8800 0.5334 0.1369

0.1398 0.0222 -0.2295

0.8800 0.5333 0.1368

0.1398 0.0222 -0.2293

1.8209 -0.5755 -2.3700

-0.0394 1.7429 -0.5224

1.8051 -0.5650 -1.8809

-0.0419 1.7012 -0.6694

1.8003 -0.5619 -1.7531

-0.0421 1.6856 -0.6850

1.7983 -0.5606 -1.7050

-0.0420 1.6785 -0.6859


165

References [l]

Bielak, J. and MacCamy, R.C., An exterior interface problem in two-dimensional elastodynamics, Quart. of Appl. Math. 41 ( 1 9 8 3 ) 1 4 3 - 1 6 0 .

[2]

Fix, G.J., Hybrid finite element methods, in: Noye, John (ed.), Numerical Simulation of Fluid Motion (North-Holland, Amsterdam, 1 9 7 8 ) .

[3]

MacCamy, R.C. and Marin, S.P., A finite element method for exterior interface problems, lnt. Jrnl. Math.and Math.Ana1. 3 ( 1 9 8 0 ) 3 1 1 - 3 5 0 .

[4] Aziz, A.K. and Kellogg, R.B., Finite element analysis of a scattering problem, Math. of Comp. 3 7 ( 1 9 8 1 ) 261-272. [5]

Johnson, C. and Nedelec, J.C., On the coupling of boundary integral and finite element methods, Math. of Comp. 3 5 ( 1 9 8 0 ) 1 0 6 3 - 1 0 7 9 .

[6]

Hsiao, G. and MacCamy, R.C., Solutions of boundary value problems by integral equations of the first kind, SIAM Review 1 5 ( 1 9 7 3 ) 6 8 7 - 7 0 5 .

[7]

Trifunac, M.D., Surface motion of a semi-cylindrical alluvial valley for incident plane SH waves, Bull. Seism. SOC. Am 6 1 ( 1 9 7 1 ) 1 7 5 5 - 1 7 7 0 .

[8]

Harris, C.G. and Evans, W.A.B., Extension of numerical quadrature formulae to cater for end point singular behavior over finite intervals, Int. J. Comp. Maths. 6 B (1977)

219-227.



167

CHAPTER 7 PRECONDITIONED ITERATIVE METHODS FOR NONSELFADJOINT OR INDEFINITE ELLIPTIC BOUNDARY VALUE PROBLEMS J.H. Bramble & J.E. Pasciak

We c o n s i d e r a G a l e r k i n - F i n i t e Element a p p r o x i m a t i o n t o a general l i n e a r e l l i p t i c boundary v a l u e Droblem which may be n o n s e l f a d j o i n t o r i n d e f i n i t e .

\Je

show how t o p r e c o n d i t i o n t h e equations s o t h a t t h e r e s u l t i n g systems o f l ' i n e a r a l g e b r a i c equations l e a d t o i t e r a t i o n procedures whose i t e r a t i v e convergence r a t e s a r e independent o f t h e number o f unknowns i n t h e s o l u t i o n .

1.

INTRODUCTION. I n r e c e n t y e a r s , t h e a p p l i c a t i o n o f i t e r a t i v e methods t o

p r e c o n d i t i o n e d l i n e a r systems has been e x t r e m e l y s u c c e s s f u l i n a v a r i e t y of complex p h y s i c a l a p p l i c a t i o n s [3,16].

Many a r t i c l e s a r e a v a i l a b l e

i n t h e l i t e r a t u r e which r e p o r t on t h e f a v o r a b l e performance o f such methods [3,6,10,12]. The two aspects o f a r e s u l t i n g a l g o r i t h m c o n s i s t o f t h e p r e c o n d i t i o n e r and t h e u n d e r l y i n g i t e r a t i v e method [l ,8,12].

Various

i t e r a t i v e methods, t h e most p o o u l a r b e i n g t h e c o n j u g a t e g r a d i e n t (CG) and c e r t a i n normal forms o f t h e CG method, have been c o n s i d e r e d e x t e n s i v e l y b o t h f r o m a t h e o r e t i c a l and an experimental v i e w p o i n t (see references t h e r e i n ) .

[lo]

I t has been demonstrated t h a t , i n general

and t h e

,

i t e r a t i v e a1 g o r i thms w i t h t h e same t h e o r e t i c a l convergence r a t e s

J.H. Bramble & J.E. Pasciak

168

converge, i n p r a c t i c e , a t about t h e same r a t e ' .

The q u e s t i o n o f choosing

an a p p r o p r i a t e p r e c o n d i t i o n e r i s much more d i f f i c u l t .

The

p r e c o n d i t i o n e r must i n some way be s i m i l a r t o t h e i n v e r s e o f t h e system which i s b e i n g solved.

Consequently, t h e e v a l u a t i o n o f t h e

p r e c o n d i t i o n e r u s u a l l y r e q u i r e s t h e s o l u t i o n o f a system o f equations and

s o i f t h e method i s t o r e s u l t i n an improvement o f computational e f f i c i e n c y , t h e p r e c o n d i t i o n e r must have some p r o p e r t y which makes i t e a s i e r t o s o l v e than t h e o r i g i n a l system.

The i t e r a t i v e convergence

r a t e o f t h e a l g o r i t h m i s extremely s e n s i t i v e t o t h e choice o f Indeed, t h e c h o i c e o f a more a n n r o o r i a t e

preconditioner.

p r e c o n d i t i o n e r may reduce t h e number o f i t e r a t i o n s by an o r d e r o f magnitude o r more i n a g i v e n problem. I n t h i s paper we i l l u s t r a t e some techniques f o r a n a l y s i n g p r e c o n d i t i o n e d i t e r a t i v e methods f o r nonsymmetric problems.

We w i 11

discuss t h e problem o f choosing an a p p r o p r i a t e p r e c o n d i t i o n e r and study two d i f f e r e n t i t e r a t i v e a l g o r i t h m s .

T y p i c a l f i n i t e element

d i s c r e t i z a t i o n o f an e l l i p t i c boundary value DrOblem leads t o a m a t r i x problem (1.1) where

Mc = d. M

i s the " s t i f f n e s s " matrix associated w i t h the d i s c r e t i z a t i o n

and i s n o n s i n g u l a r and

d

and

Mi;' such t h a t

preconditioner (M1)-l

c

are vectors.

M1

i s symmetric p o s i t i v e d e f i n i t e ,

i s e a s i e r t o compute than

i n some sense"

(M)-'.

We seek a

(M)-',

and

(M1)-'

"approximates

System ( 1 . 1 ) can o f course be r e p l a c e d by t h e

e q u i v a l e n t system (1.2) The m a t r i x

Mt M i ' M i ' Mc = Mt M i '

M'

f

M

t

M i ' 'M;

M

M-i

d

.

i s symmetric p o s i t i v e d e f i n i t e and t h e

f i r s t a l g o r i t h m i s d e f i n e d by a o p l y i n g t h e conjugate g r a d i e n t method t o (1.2). (1.3)

A l t e r n a t i v e l y , (1.1) i s e q u i v a l e n t t o t h e problem -1

t

-1 d.

Mi;' Mt M i ' Mc = M1 M M1

The number o f i t e r a t i o n s t o reach a d e s i r e d accuracy may vary by a t most a f a c t o r of f i v e [6,10].

Nonselfadjoint or Indefinite Elliptic Boundary Value Problems

169

M" : M - l Mt MY;' M a l t h o u g h n o t u s u a l l y symmetric, i s a 1 symmetric o p e r a t o r w i t h r e s p e c t t o t h e i n n e r p r o d u c t defined by

The m a t r i x

I n e CG method can be a p p l i e d t o ( 1 . 3 ) i n t h e and leads t o A l g o r i t h m I 1 o f S e c t i o n 2.

,.>>

i n n e r product

Our a n a l y s i s suggests t h a t t h e

p r e c o n d i t i o n e d i t e r a t i v e method based on (1.3) i s more r o b u s t t h a n t h a t based on ( 1 . 2 ) s i n c e r e s u l t s f o r (1.2) r e q u i r e a d d i t i o n a l hypotheses.

In

f a c t , we have n o t been a b l e t o o b t a i n r e s u l t s f o r t h e scheme based on (1.2) unless t h e elements used i n t h e methods a r e o f " q u a s i - u n i f o r m " size. We s h a l l p r e s e n t two general theorems which can be used t o d e r i v e c e r t a i n d i s c r e t e s t a b i l i t y estimates.

Such e s t i m a t e s l e a d t o bounds on

t h e i t e r a t i v e convergence r a t e s o f a l g o r i t h m s f o r f i n d i n g t h e s o l u t i o n o f m a t r i x equations r e s u l t i n g from t h e f i n i t e element d i s c r e t i z a t i o n o f e l l i p t i c boundary v a l u e problems which may be nonsymmetric and/or indefinite.

We show how these general r e s u l t s can be a p p l i e d i n a

f i n i t e element a p p r o x i m a t i o n t o t h e Poincar;

problem.

Both s t r a t e g i e s

depend upon a p r i o r i s t a b i l i t y e s t i m a t e s f o r t h e continuous problem and use t h e a p p r o x i m a t i o n p r o p e r t i e s o f t h e d i s c r e t i z a t i o n t o d e r i v e t h e s t a b i l i t y e s t i m a t e f o r t h e matr-ix problems. The f i r s t theorem leads t o a s t r a t e g y which uses a p o s i t i v e d e f i n i t e symmetric problem as a p r e c o n d i t i o n e r f o r a more c o m p l i c a t e d nonsymmetric and/or i n d e f i n i t e problem.

The problem o f t h e e f f i c i e n t

s o l u t i o n o f p o s i t i v e d e f i n i t e problems, a l t h o u g h n o t c o m p l e t e l y s o l v e d , has been e x t e n s i v e l y researched.

For example, m a t r i c e s corresponding

t o p o s i t i v e d e f i n i t e symmetric problems o f t e n have c e r t a i n diagonal dominance p r o p e r t i e s which i m p l y t h a t v a r i o u s sparse m a t r i x packages [9,11]

can be used f o r t h e i r s o l u t i o n .

Also, t h e r e a r e " f a s t s o l v e r "

a l g o r i t h m s a v a i l a b l e f o r c e r t a i n e l l i p t i c oroblems on a v a r i e t y o f domains [5,14,15].

Our a n a l y t i c a l r e s u l t s guarantee t h a t t h e i t e r a t i v e

convergence r a t e f o r o u r a l g o r i t h m s i s independent o f t h e number o f unknowns i n t h e system.

Thus t h e c o s t o f convergence t o a g i v e n

accuracy grows l i n e a r l y w i t h t h e s i z e o f t h e problem. The f i r s t s t r a t e g y i s a p p l i c a b l e t o , f o r example, problems where the d i f f e r e n t i a l operator d e f i n i t e operator

L

A

can be decomposed i n t o a symmetric p o s i t i v e

and a compact ( b u t n o t s m a l l ) p e r t u r b a t i o n

B.

The

J. H. Bramble & J. E. Pasciak

170 operators

A, L, and B

a r e approximated by d i s c r e t e o p e r a t o r s

Ah,

and Bh d e r i v e d by f i n i t e elements. The d i s c r e t e approximation Lhy t o t h e s o l u t i o n u o f t h e o r i g i n a l problem i s d e f i n e d as t h e s o l u t i o n of (Lh

(1.4)

-+ B h ) U

= F.

Problem ( 1 . 4 ) can be r e p l a c e d by t h e e q u i v a l e n t problem Lhl (Lh t Bh)U = Lh’ F

(1.5)

,

We d e r i v e t h e a p p r o p r i a t e s t a b i l i t y e s t i m a t e s f o r ( 1 . 5 ) which guarantee t h a t t h e CG method a p p l i e d , w i t h r e s p e c t t o

,

t o (1.3) converges

a t a r a t e independent o f t h e number o f unknowns i n t h e d i s c r e t i z a t i o n . a d d i t i o n , t h e s t a b i l i t y r e s u l t s y i e l d immediately e s t i m a t e s f o r t h e discretization error

In

u-U.

We g i ve a second theorem which , under a d d i t i o n a l hypotheses, provides another s t a b i l i t y estimate.

T h i s e s t i m a t e , under a f u r t h e r

r e s t r i c t i o n , can be used t o show t h a t t h e CG method a p p l i e d t o (1.2) converges t o t h e s o l u t i o n o f (1.2) a t a r a t e which i s independent o f t h e number o f unknowns i n t h e d i s c r e t i z a t i o n . An o u t l i n e o f t h e remainder o f t h e paper i s as f o l l o w s .

I n Section 4

we d e s c r i b e two conjugate g r a d i e n t a l g o r i t h m s f o r m a t r i x problems. S e c t i o n 3 g i v e s some p r e l i m i n a r i e s and n o t a t i o n t o be used i n t h e paper. I n S e c t i o n 4 we s t a t e t h e t y p e o f e s t i m a t e s needed t o guarantee r a p i d convergence f o r some it e r a t i ve methods f o r s o l v i ng nonsymmetri c and/or i n d e f i n i t e problems.

Two theorems used t o d e r i v e t h e s t a b i 1 ity estimates

a r e given i n S e c t i o n 5.

I n S e c t i o n 6 we apply t h e theorems t o a f i n i t e

element approximation o f a general e l l i p t i c boundary value problem. F i n a l l y i n S e c t i o n 7 we a p p l y a s t a b i l i t y e s t i m a t e t o bound t h e discretization error.

2.

CONJUGATE GRADIENT ALGORITHMS.

We d e s c r i b e t h e a l g o r i t h m s which r e s u l t from a p p l y i n g t h e conjugate g r a d i e n t method t o t h e p r e c o n d i t i o n e d sqstems (1.2) and ( 1 . 3 ) . I n e i t h e r t o the co o f (1.1) and t h e i t e r a t i v e a l g o r i t h m produces a sequence o f

case we assume t h a t we a r e g i v e n an i n i t i a l a p p r o x i m a t i o n solution

c


iterates

ci

for

residual error

i > 0.

d-Mc

We s t o p t h e i t e r a t i v e procedure when t h e

becomes s u f f i c i e n t l y s m a l l .

We n o t e t h a t a p p l y i n g

t h e c o n j u g a t e g r a d i e n t method t o p r e c o n d i t i o n e d systems as

i11 u s t r a t e d in t h e f o l 1owing a1 g o r i thms i s n o t novel however we in c l ude t h e d e t a i l s f o r completeness. A p p l y i n g t h e c o n j u g a t e g r a d i e n t method t o (1.2) g i v e s t h e f o l l o w i n g algorithm: M I = Mt Mi;' M i ' M

ALGORITHM I . (1)

Define

(2)

For

t -1 M-l ( d-Mco). ro = po = M M1 1

i> 0

define

ri o pi a i = (MI p i l o p i = c:

Cit1

1

+ a. p

i i

A p p l y i n g t h e c o n j u g a t e g r a d i e n t method i n t h e p r o d u c t t o (1.3) g i v e s t h e f o l l o w i n g a l g o r i t h m : M" = M i ' Mt MY1 M

ALGORITHM 11. (1)

Define

(2)

For

.

ro = po = M i ' Mt Mi1(d-Mco).

i> 0

define

ci+l

= ci

+ a. 1 pi

ri+l

= ri

- a i M"

t~~

inner

171

J.H. Bramble & J. E. Pasciak

172

3.

PRELIMINARIES AND NOTATION. Throughout t h i s paper we s h a l l be concerned w i t h s o l v i n g boundary

value problems on a bounded domain

.

r

boundary

n

contained i n

R2

with

To s t a t e o u r s t a b i l i t y estimates, we s h a l l make use o f

v a r i o u s spaces o f f u n c t i o n s d e f i n e d on

R

.

The space

L2(R)

i s the

c o l l e c t i o n o f square i n t e g r a b l e f u n c t i o n s on R ; t h a t is,a f u n c t i o n defined f o r (x,y) i n R i s i n L2(R) i f

The

LL(R)

f(x)

i n n e r p r o d u c t i s d e f i n e d by

(f,g)

: f(x,y)

g(x,y)dxdy

for

f, g E

2 L (Q).

R 1 We s h a l l a l s o use t h e Sobolev space H (n). Loosely, a f u n c t i o n af af 2 f, and - a r e a l l i n L (R). Thus f o r i n H1(R) i f aY 1 f u n c t i o n s i n H (n), we can d e f i n e t h e D i r i c h l e t f o r m by

We s h a l l a l s o denote t h e E

r

L

2

(r)

f g ds

f

is

i n n e r p r o d u c t by

. 2

r, t h e Sobolev space o f L ( a ) - f u n c t i o n s 2 rth o r d e r p a r t i a l d e r i v a t i v e s belong t o L ( Q ) w i l l be denoted by

For any p o s i t i v e i n t e g e r whose Hr( 0). We a l s o l e t values o f and

Ci

C

and

and

C Ci

Ci

for

i> 0

denote p o s i t i v e c o n s t a n t s .

may be d i f f e r e n t i n d i f f e r e n t places however

s h a l l always be independent o f t h e mesh parameter

h

The C

defining

173

Nonseljadjoint or Indefinite Elliptic Boundary Value Problems t h e a p p r o x i m a t i o n method.

Thus

C

and

Ci

w i l l always be independent

of t h e number o f unknowns i n t h e d i s c r e t i z a t i o n . To d e f i n e t h e a p p r o x i m a t i o n o f l a t e r s e c t i o n s we s h a l l need a c o l l e c t i o n o f f i n i t e element a p p r o x i m a t i o n subspaces { S h l , 0 < h( 1, 1 c o n t a i n e d i n H (R). T y p i c a l l y , f i n i t e element a p p r o x i m a t i o n subspaces

Q i n t o subregions o f s i z e h and t o be t h e s e t of f u n c t i o n s which a r e continuous on R and

a r e d e f i n e d b y p a r t i t i o n i n g t h e domain defining

Sh

piecewise p o l y n o m i a l when r e s t r i c t e d t o t h e subregions (see [4,7,17] for details). o f size on

h

and d e f i n e

Sh

t o be t h e f u n c t i o n s which a r e continuous

and l i n e a r on each o f t h e t r i a n g l e s .

52

R i n t o triangles

For example, one c o u l d p a r t i t i o n

be p a r t i t i o n e d i n t o r e c t a n g l e s and f u n c t i o n s which a r e continuous on

Sh

R could

Alternatively,

c o u l d be defined t o be t h e

R and b i l i n e a r on each o f t h e

rectangles.

4.

ESTIMATES FOR THE CONJUGATE. GRADIENT METHOD. Our a n a l y s i s o f i t e r a t i v e a l g o r i t h m s f o r p r e c o n d i t i o n e d systems i s

based on s t a b i l i t y e s t i m a t e s f o r t h e continuous o r n o n d i s c r e t e problem and t h e e r r o r e s t i m a t e s between t h e continuous s o l u t i o n s and t h e i r d i s c r e t e approximations.

To s t u d y t h e p r o p e r t i e s o f t h e s o l u t i o n s o f

boundary v a l u e problems i n p a r t i a l d i f f e r e n t i a l equations, i t i s natural t o consider operators i n t h e i r basis f r e e representations since complete s e t s o f b a s i s f u n c t i o n s a r e u s u a l l y t o o complex t o be o f much p r a c t i c a l value.

Consequently, i t i s n a t u r a l t o t h i n k o f t h e process

o f s o l v i n g f o r t h e d i s c r e t e s o l u t i o n o f t h e f i n i t e element equations as a b a s i s f r e e o p e r a t o r on t h e f i n i t e element subspace

Sh

of

H'(R)

r e p r e s e n t d i f f e r e n t i a l and s o l u t i o n o p e r a t o r s by t h e n o t a t i o n L,

or

T

denoted

.

We

A, B,

whereas t h e i r d i s c r e t e c o u n t e r p a r t s s h a l l be r e s p e c t i v e l y

Ah, Bh, Lh

and

Th.

The CG method can be a p p l i e d t o f i n d t h e s o l u t i o n

X

o f t h e problem

Lhx=Y where

L,,

i s a symmetric p o s i t i v e d e f i n i t e o p e r a t o r w i t h r e s p e c t t o some

inner product (cf.

L13J).

produces an a p p r o x i m a t i o n

The CG a l g o r i t h m r e q u i r e s an i n i t i a l guess X

n

to

X

after

n

i t e r a t i v e steps.

X,

It i s

and

114

J. H. Bramble & J. E. Pasciak

w e l l known t h a t

where

i s t h e c o n d i t i o n number f o r

y

r a t i o o f t h e l a r g e s t eigenvalue o f i f Lh

where

L

Lh h

and i s d e f i n e d t o be t h e

t o the smallest.

We n o t e t h a t

s a t i s f i e s the i n e q u a l i t y

(*,-),

number y

denotes t h e H - i n n e r product, then t h e c o n d i t i o n

i s bounded by

Thus estimates o f t h e t y p e (4.3)

Cl/Co.

i n c o n j u n c t i o n w i t h (4.2) l e a d t o convergence e s t i m a t e s f o r t h e CG method a p p l i e d t o (4.1). The problem o f f i n d i n g t h e f i n i t e element s o l u t i o n i n t h e examples X

of a

Sh.

We

o f l a t e r s e c t i o n s can be reduced t o s o l v i n g f o r t h e s o l u t i o n nonsingular operator equation (4.4)

AhX=Y Ah

where

i s a nonsymmetric and/or n o n p o s i t i v e o p e r a t o r on

s h a l l f i r s t p r e c o n d i t i o n t h e system, m u l t i p l y by t h e a d j o i n t and t h e n a p p l y t h e CG method i n t h e a p p r o p r i a t e i n n e r p r o d u c t . We assume t h a t we have a symmetric p o s i t i v e d e f i n i t e o p e r a t o r Th

d e f i n e d on

Sh

f o r a preconditioner.

The types o f p r e c o n d i t i o n e r s

f o r which we can g e t a n a l y t i c r e s u l t s w i l l be d e s c r i b e d i n l a t e r s e c t i o n s We note t h a t problem (4.4) can be r e p l a c e d by t h e problem o f finding X i n S satisfying

A; Th Th Ah X =

(4.5) where t o the

A;

i s the L

2

(a)

" LL(a)

At

Th Th

- adjoint

of

Ah.

The CG method w i t h r e s p e c t

i n n e r p r o d u c t can be used t o s o l v e (4.5).

The

convergence r a t e o f t h e r e s u l t i n g a l g o r i t h m i s bounded by (4.2) i n 2 t h e L (a) norm where Y i s bounded by Cl/Co f o r any Co and C1

satisfying

175


I n c e r t a i n a p p l i c a t i o n s , e s t i m a t e ( 4 . 6 ) can be used t o d e r i v e bounds on t h e i t e r a t i v e convergence r a t e o f A1 g o r i thm I. A l t e r n a t i v e l y , problem ( 4 . 4 ) i s a l s o e q u i v a l e n t t o t h e problem o f finding

i n Sh

X

satisfying

T A*T A X = T A*T Y . h h h h h h h

(4.7) The o p e r a t o r inner product

B

?

T A* T A h h h h

( T i ' W, V ) .

i s symmetric p o s i t i v e d e f i n i t e i n t h e

A p p l y i n g t h e CG method t o t h e s o l u t i o n o f

(4.7) i n t h i s i n n e r p r o d u c t g i v e s an a l g o r i t h m which converges a t a r a t e d e s c r i b e d by (4.2) where CO(Til W,W)

(4.8)

f o r any

y < C1/Co

f (Th AhW, AhW)

5

Cl(Thl

Co W,W)

and

C1

for all

satisfying W

E

Sh

.

I n a p p l i c a t i o n s , e s t i m a t e (4.8) i s used t o d e r i v e i t e r a t i v e convergence r a t e s f o r A1 g o r i thm II.

5.

STABILITY THEOREM.

In t h i s s e c t i o n we g i v e general r e s u l t s which can be used t o d e r i v e e s t i m a t e s o f t h e form (4.6) and ( 4 . 8 ) . L e t R be a continuous o p e r a t o r and Rh be i t s d i s c r e t e Theorem 1. approximation. Assume t h a t t h e f o l l o w i n g s t a b i l i t y and e r r o r e s t i m a t e s h o l d:

For any

E

> 0

there e x i s t s

CE

such t h a t

176


Then t h e r e e x i s t s ho

0

such t h a t f o r

h < ho

(5.4) Remark 1.

E s t i m a t e ( 5 . 4 ) combined w i t h

guarantees a u n i f o r m ( i n d e p e n d e n t of

h ) i t e r a t i v e convergence r a t e f o r

t h e CG i t e r a t i o n f o r t h e s o l u t i o n o f (I+Rh) where

*

oroduct.

*

(I+Rn)U = F 1 H (R) i n n e r I + R h = ThAh and

denotes t h e a d j o i n t w i t h r e s D e c t t o t h e I n o u r f i n i t e element a p p l i c a t i o n s ,

Thus ( 5 . 4 ) and ( 5 . 5 ) w i l l i m p l y ( 4 . 8 ) f o r t h e p a r t i c u l a r examples o f the next section. Theorem 2.

Let

T1

and

T2

be c o n t i n u o u s o p e r a t o r s and

be t h e i r c o r r e s p o n d i n g d i s c r e t e a p p r o x i m a t i o n s . three estimates hold:

for

i = 1,2.

Then

TA

and

Th2

Assume t h a t t h e f o l l o w i n g

177

Nonselfadjoirit or Indefinite Elliptic Boundary Vulue Problems Remark 2.

Estimate

(5.8) i s an i n v e r s e p r o p e r t y f o r t h e o p e r a t o r

Th1

and i n a p p l i c a t i o n s i s d e r i v e d from t h e h y p o t h e s i s t h a t t h e mesh Estimate (5.9) coincides w i t h

elements a r e of "quasi uniform" s i z e .

.

Ah = (T;)-l

( 4 . 6 ) when Remark 3.

The proofs o f t h e above two theorems a r e s i m p l e and

consequently w i l l n o t be i n c l u d e d .

6.

THE P O I N C A R i PROBLEM. To i l l u s t r a t e o u r approach we c o n s i d e r a f i n i t e element

a p p r o x i m a t i o n o f t h e Poincare' problem i n t h i s s e c t i o n .

We c o n s i d e r t h e

f o l l o w i n g model problem: -Au t

au f KU ax

= f

in

R

(6.1) au -arl + a % au where

A

=

a2 ax

f

fYu=O

,a2 ~ ,2n

and

tangential d i r e c t i o n s along

r'.

on

r

a r e r e s p e c t i v e l y t h e normal and

T

For s i m p l i c i t y we have c o n s i d e r e d

c o n s t a n t c o e f f i c i e n t s i n d e f i n i n g t h e d i f f e r e n t i a l e q u a t i o n as w e l l as t h e boundary c o n d i t i o n .

Our r e s u l t s and i t e r a t i v e a1 g o r i thms e x t e n d t o

v a r i a b l e c o e f f i c i e n t problems w i t h o u t any c o m p l i c a t i o n s .

We a l s o assume

t h a t t h e s o l u t i o n o f (6.1) e x i s t s and i s unique. The f i n i t e element a p p r o x i m a t i o n t o (6.1) can t h e n be d e f i n e d by t h e G a l e r k i n technique.

M u l t i p l y i n g (6.1) by an a r b i t r a r y f u n c t i o n

i n t e g r a t i n g by p a r t s shows t h a t t h e s o l u t i o n

The f i n i t e element a p p r o x i m a t i o n function

U

in

Sh

U

which s a t i s f i e s

to

u

u

41 and

satisfies

i s t h e n d e f i n e d t o be t h e


I78

Equation (6.3) can be used t o d e r i v e a system o f equations o f t h e form

(1.1) d e f i n i n g t h e d i s c r e t e s o l u t i o n U, i . e . , u s i n g a b a s i s f o r S h y (6.3) g i v e s N equations f o r t h e N unknowns d e f i n i n g U i n t h a t basis. To d e s c r i b e i t e r a t i v e methods f o r t h e s o l u t i o n o f (6.3) and/or t h e corresponding m a t r i x system, we s h a l l need t o use some o p e r a t o r notation.

F i r s t we consi der t h e Neumann problem w - A w = f

i n G.

aw

r

on

- = 0

au

Given a f u n c t i o n

f

D(w,e)

(a),t h e

f

to

w

o f (6.4) i s i n

i s s u f f i c i e n t l y smooth.

H2(G.)

We denote t h e

=

w. T i s a as t h e map which takes f t o T f 2 2 L (G.) i n t o H ( Q ) . The f i n i t e element approximation t o

+ (w,e)

W

in

Th f 5 W .

satisfying

Sh

= (f,e)

The d i s c r e t e s o l u t i o n o p e r a t o r takes

solution

T

(6.4) i s t h e f u n c t i o n (6.5)

2

r

i f as we s h a l l assume,

solution operator bounded map o f

L

in

Th

for all Th

e

E

Sh

.

can t h e n be defined as t h e map which

i s a map from

2

L (Q)

onto

Sh

and t h e

f o l l o w i n g convergence e s t i m a t e i s w e l l known ( c f . [ 2 ] ) :

I n a s i m i l a r manner, we can d e f i n e s o l u t i o n o p e r a t o r s f o r t h e f o l l o w i n g v a r i a t i o n a l problems:

and

We d e f i n e t h e s o l u t i o n o p e r a t o r s

R1z:

2

X and R w

I

$.

The corresponding


X

f i n i t e element approximations a r e g i v e n by t h e s o l u t i o n s

and

179

Y

in

satisfying

Sh

az

D(X,e)

+ (X,e)

= (5’8)

D(Y,e)

+ (Y,e)

=

for all

e

E

Sh

,

respectively. R

1 h

z :X

The d i s c r e t e s o l u t i o n o p e r a t o r s a r e t h e n d e f i n e d by 2 and Rh w Y and t h e f o l l o w i n g convergence e s t i m a t e s h o l d :

=

and

(6.8)

I n terms o f o p e r a t o r s , problem (6.1) i s e q u i v a l e n t t o ( I + R 1 + R2 ) u =- T A u = T f . The e x i s t e n c e and uniqueness p r o p e r t i e s o f s o l u t i o n s o f (6.1) can be used t o show t h a t f o r any

E

> 0

there i s a constant

CE

such t h a t

The d i s c r e t e e s t i m a t e

i s immediate f r o m t h e d e f i n i t i o n o f

i Rh

i n terms o f o p e r a t o r s as 2 ( I + Rh1 + Rh)U z T A U = Th f h h

.

Problem (6.3) can be s t a t e d

.

180


A p p l y i n g Theorem 1 we g e t t h e f o l l o w i n g s t a b i l i t y e s t i m a t e :

The second i n e q u a l i t y i n (6.11) can be e a s i l y d e r i v e d from t h e d e f i n i t i o n s

Co

The constants size (6.12)

and

i n (6.10) a r e independent o f t h e mesh

C1

Now i t i s easy t o check t h a t

h.

W,V) = D(W,V) t (W,V)

(Ti

Comparing ( 6 12), (6.11),

for all

W,V€Sh

(4.7) and (4.8) i m p l i e s t h a t t h e CG method

applied t o

T h A t ThAh U = ThA;

(6.13)

Th f

converges w i t h a r e d u c t i o n p e r i t e r a t i o n which can be bounded independently o f t h e number of unknowns. r e s p e c t i v e l y denote t h e " s t i f f n e s s " m a t r i c e s N corresponding t o ( 6 . 3 ) and (6.5) i n a g i v e n b a s i s 8 = E8ili,l Let

for

Sh.

basis

8

M

and

M1

I f the coefficients o f a function a r e r e p r e s e n t e d by t h e v e c t o r d

in

Sh

i n terms o f t h e

then

-1 t M1 M MY;' Mc Th A;

gives the c o e f f i c i e n t s o f t h e sequen'ce o f v e c t o r s

W

c

ci

ThAh W

i n terms o f

8.

Consequently,

generated by A l g o r i t h m I 1 gives t h e

c o e f f i c i e n t s o f t h e sequence o f f u n c t i o n s generated by t h e CG method appl ied t o (6.13).

Thus t h e it e r a t i ve convergence estimates f o r

t h e CG method a p p l i e d t o (6.13) i m p l y i t e r a t i v e convergence r a t e s f o r A1 g o r i thm

II .

The above procedure i s an example o f an i t e r a t i v e convergence analysis i n

H1(,).

o p e r a t o r on

Sh

(6.14)

We a l s o n o t e t h a t i f

T,,l

i s another d i s c r e t e

which i s s p e c t r a l l y e q u i v a l e n t t o

Co(Th W,W) < (TA W,W)

5

C1(Th W,W)

Th

for all

i n t h e sense t h a t W E Sh

181

Nonselfadjoint or Indejinite Elliptic Boundary Value Problems

then

can be r e p l a c e d by

Th

1 Th

i n (6.11).

2 We n e x t c o n s i d e r an i t e r a t i v e a n a l y s i s i n L ( Q ) based on 1 2 2 H ( Q ) denote t h e s o l u t i o n o p e r a t o r Theorem 2. L e t T : L ( Q ) -f

f o r problem ( 6 . 1 )

with

B = 0, i . e . ,

T1 f

The s o l u t i o n o p e r a t o r

Eu.

T1

s a t i s f i e s an e s t i m a t e o f t h e form

We have r e s t r i c t e d t o t h e case o f

13

t h a t case.

and

Assume t h a t b o t h

T1

f i n i t e element subspaces and l e t discrete s o l u t i o n operators.

TA

0

s i n c e (6.15) i s w e l l known i n T

and

can be approximated i n t h e same Th

denote t h e c o r r e s p o n d i n g

The f o l l o w i n g convergence e s t i m a t e s a r e we1 1

known f o r a wide c l a s s o f f i n i t e element a p p l i c a t i o n s [271:

We f i n a l l y assume t h a t t h e i n v e r s e p r o p e r t i e s

are a l s o s a t i s f i e d .

Estimates o f t h e t y p e (6.17) can u s u a l l y be A p p l y i n g Theorem 2

d e r i v e d f r o m i n v e r s e assumptions f o r t h e subspaces. gives t h a t

for all E s t i m a t e (6.18) guarantees t h a t t h e CG method a p p l i e d i n

W

LL(Q)

E

Sh

.

f o r the

solution o f

A[ ThThAh X = A;

(6.19) where

An

(T:)-'

ThTh f

w i l l converge t o t h e s o l u t i o n

X

a t a rate

which

The r e s u l t i n g a l g o r i t h m

i s independent o f t h e number o f unknowns i n

Sh.

does n o t however correspond t o A l g o r i t h m I .

To guarantee r a p i d i t e r a t i v e

182

J.H. Bramble & J. E. Pasciak

convergence r a t e s f o r A l g o r i t h m I we must make a d d i t i o n a l assumptions. Again we use t h e b a s i s coefficients o f

(6.20)

W

for

63

Sh

i n t h e basis

cw )

co(cw

63.

If W

L

E

we denote by

Sh

the

Cw

We r e q u i r e t h a t

5

< (W,W)

-

,

(a

C1(Cw* Cw)

for a l l

W

E

Sh

.

Estimate (6.20) s t a t e s t h a t t h e Gram o r mass m a t r i x i s " e q u i v a l e n t " t o the coordinate i n n e r product.

for all

N

dimensional v e c t o r s

Combining (6.19) and (6.20) i m p l i e s

c.

E s t i m a t e (6.21) i s f i n a l l y an

e s t i m a t e which can be a p p l i e d t o guarantee uniform i t e r a t i v e convergence r a t e s f o r A1 g o r i thm I.

7.

AN ESTIMATE FOR THE DISCRETIZATION ERROR. I n order t o estimate the d i s c r e t i z a t i o n e r r o r

defined by (6.2) and (6.3) projection

Ph

onto

I t i s w e l l known t h a t

for

v

E

Hr(Q)

Sh(cf. [2,7]).

Sh.

Ph

and some

.

with

1 2 Rh = Rh + Rh

with

u

and

r e s p e c t i v e l y , we i n t r o d u c e t h e H (Q)v c H 1 ( n ) by

It i s defined f o r

satisfies

r > 1 which depends on t h e c h o i c e o f u-U

we need o n l y c o n s i d e r

Hence we apply (5.4) t o o b t a i n

.

U

1

I n view o f (7.2), t o e s t i m a t e

Ph u-U

u-U

From t h e d e f i n i t i o n s o f

R

1

, Rh,1

see t h a t (I+Rh)(Ph u-U) = Ph(R 1+R 2 ) ( P h - I ) u

.

R

2

and

2 Rh

we


183

Hence

from which i t follows immediately t h a t (7.3) Thus u s i n g ( 7 . 2 ) we obtain the estimate f o r the d i s c r e t i z a t i o n e r r o r ,

REFERENCES. 0. Axelsson; A c l a s s of i t e r a t i v e methods f o r f i n i t e element equations , Comp. Methods Appl. Mech. Engng., V. 9 , p p . 123-137.

I . Babuika and A . K . Aziz; Part I . Survey l e c t u r e s on the mathematical foundations of the f i n i t e element method , The Mathematical Foundations of the Finite Element Method w i t h Applications t o P a r t i a l D i f f e r e n t i a l Equations, A . K . Aziz, ed. Academic Press, New York, 1972. J.H. Bramble and J.E. Pasciak; An e f f i c i e n t numerical procedure f o r the computation of steady s t a t e harmonic c u r r e n t s i n f l a t p l a t e s , COMPUMAG conf., Genoa, 1983. J.H. Bramble, J.E. Pasciak, and A . H . Schatz; Preconditioners f o r i n t e r f a c e problems on mesh domains, p r e p r i n t .

B . L . Buzbee, F.W. Dorr, J.A. George, and G.H. Golub; The d i r e c t s o l u t i o n of t h e d i s c r e t e Poisson equation on i r r e g u l a r regions , SIAM J . Numer. Anal., V. 8 , 1971, pp. 722-736. R. Chandra; Conjugate gradient methods f o r p a r t i a l d i f f e r e n t i a l equations, Yale University, Dept. of Comp. S c i . Report No. 129, 1978. P.G. C i a r l e t ; The f i n i t e element method f o r e l l i p t i c problems, North-Holland, Amsterdam, 1978.

P . Concus, G. Golub, and D. O'Leary , A generalized conjugate gradient method f o r the numerical s o l u t i o n of e l l i p t i c p a r t i a l d i f f e r e n t i a l equations , i n Sparse Matrix Computation, J . Bunch and D. Rose, e d s . , Academic Press, New York, 1976, pp. 309-322.

J.H. Bramble & J.E. Pusciuk

184

[9]

S.C. E i s e n s t a t , M.C. Gursky, M.H. S c h u l t z , A.H. Sherman; Yale sparse m a t r i x package, I . t h e symmetric codes, Yale Univ. Dept. o f Comp. Sci . Report No. 112.

[lo]

H. Elman; I t e r a t i v e methods f o r l a r g e , sparse, nonsymmetric systems o f l i n e a r equations, Yale Univ. Dept. o f Comp. S c i . Report No. 229, 1978.

[ll]A. George and J.W.H.

L i u ; User Guide f o r SPARSPAK, Waterloo Oept. o f Comp. S c i . Report No. CS-78-30.

[12] J.A. M e i j e r i n k and H.A. Van d e r Vorst; An i t e r a t i v e s o l u t i o n method f o r l i n e a r systems o f which t h e c o e f f i c i e n t m a t r i x i s a symmetric M-matrix , Math. Comp. 1973, V. 31, pp. 148-162. [13] W.M. P a t t e r s o n ; I t e r a t i v e methods f o r t h e s o l u t i o n o f a l i n e a r o p e r a t o r e q u a t i o n i n H i l b e r t space - A survey, l e c t u r e notes i n mathematics, S p r i n g e r - V e r l a g , No. 394, 1974. [14] W . Proskurowski and 0. Widlund; On t h e numerical s o l u t i o n o f H e l m h o l t z ' s e q u a t i o n by t h e capacitance m a t r i x method , Math. Comp., V. 20, 1976, pp. 433-468. [15] A.H. Schatz; E f f i c i e n t f i n i t e element methods f o r t h e s o l u t i o n o f second o r d e r e l 1i p t i c boundary v a l u e problems on piecewise smooth domains , Proceedings o f t h e conference Construct! ve methods f o r s i n g u l a r problems , November 1983, Oberwolfach, West Germany, P . G r i s v a r d , W. Wendland and J. Whi teman, e d i t o r s , S p r i nger-Verl ag l e c t u r e notes i n mathematics, t o appear. [16] J . Simkin and C.W. Trowbridge; On t h e use o f t h e t o t a l s c a l a r p o t e n t i a l i n t h e numerical s o l u t i o n o f f i e l d problems i n e l e c t r o m a g n e t i c s , I n t e r . J . Numer. Math. Eng., 1979, V . 14, pp. 423-440. [17] O.C. Z i e n k i e w i c z ; The f i n i t e element method McGraw-Hill , 1977.

, 3rd e d i t i o n ,


185

CHAPTER 8 ON THE UNIFICATION OF FINITE ELEMENTS & BOUNDARY ELEMENTS

Cd.Brebbia

T h i s p a p e r r e v i e w s some of t h e a p p l i c a t i o n s of b o u n d a r y e l e m e n t methods f o r t h e s o l u t i o n of e n g i n e e r i n g p r o b l e m s . The p a p e r c o n s i d e r s how t h e new t e c h n i q u e r e l a t e s t o c l a s s i c a l f i n i t e e l e m e n t s , by r e v i e w i n g t h e f u n d a m e n t a l s of m e c h a n i c s , i n p a r t i c u l a r v i r t u a l T h i s a p p r o a c h g i v e s a common b a s i s work a n d a s s o c i a t e d p r i n c i p l e s . f o r a l l approximate techniques and h e l p s t o understand t h e r e l a t i o n s h i p between f i n i t e a n d boundary e l e m e n t method. The p a p e r s t r e s s e s t h e r a n g e of a p p l i c a t i o n s f o r which t h e boundary e l e m e n t method c a n g i v e a c c u r a t e r e s u l t s and be computat i o n a l l y e f f i c i e n t .

1.

INTRODUCTION

I n t h e l a s t few y e a r s t h e a p p l i c a t i o n s of b o u n d a r y i n t e g r a l e q u a t i o n s i n e n g i n e e r i n g have undergone i m p o r t a n t c h a n g e s . The b r a v e a t t e m p t s d u r i n g t h e s i x t i e s and e a r l y s e v e n t i e s p i o n e e r s s u c h a s Jawson [ I ] , Symm [ 2 ] , Massonet [ 3 ] , Hess [ 4 ] , C r u s e [ 5 ] and few o t h e r s , h a v e now b o r n e f r u i t i n t h e newly d e v e l o p e d boundary e l e m e n t method. I n t h i s way boundary i n t e g r a l e q u a t i o n s h a v e become a n e n g i n e e r i n g t o o l r a t h e r t h a n a mathem a t i c a l method w i t h i m p o r t a n t b u t r a t h e r r e s t r i c t i v e a p p l i c a t i o n s . S i n c e t h e e a r l y 1 9 6 0 ’ s a small g r o u p a t Southampton U n i v e r s i t y i n England s t a r t e d working on t h e a p p l i c a t i o n s of i n t e g r a l e q u a t i o n s t o s o l v e s t r e s s a n a l y s i s p r o b l e m s . Some of t h i s work h a s b e e n r e p o r t e d a t t h e f i r s t i n t e r n a t i o n a l C o n f e r e n c e on V a r i a t i o n a l Methods i n E n g i n e e r i n g , h e l d t h e r e i n 1972 [ 6 ] . More i s e x p e c t e d t o b e p r e s e n t e d d u r i n g t h e s e c o n d C o n f e r e n c e reconvened f o r 1985. These C o n f e r e n c e s a r e h e l d t o d i s c u s s t h e d i f f e r e n t t e c h n i q u e s of e n g i n e e r i n g a n a l y s i s and how t h e y a r e i n t e r r e l a t e d . The i m p o r t a n c e of t h e B I E p r e s e n t a t i o n s d u r i n g t h e 1st C o n f e r e n c e i s t h a t t h i s was t h e f i r s t t i m e t h a t boundary i n t e g r a l e q u a t i o n s w e r e i n t e r p r e t e d a s a v a r i a t i o n a l t e c h n i q u e . The work a t Southampton was c o n t i n u e d t h r o u g h o u t t h e s e v e n t i e s t h r o u g h a s e r i e s of t h e s e s m a i n l y c o n c e r n e d w i t h b o u n d a r y i n t e g r a l A t t h e Same t i n e new d e v e l o p m e n t s s o l u t i o n s of e l a s t o s t a t i c p r o b l e m s . i n f i n i t e e l e m e n t s s t a r t e d t o f i n d t h e i r way i n t o b o u n d a r y i n t e g r a l equat i o n s and t h e p r o b l e m of how t o r e l a t e t h e t e c h n i q u e t o o t h e r a p p r o x i m a t e s o l u t i o n s was s o l v e d u s i n g w e i g h t e d r e s i d u a l s [ 7 ] . T h i s work a t Southampton U n i v e r s i t y c u l m i n a t e d a r o u n d 1978 when t h e f i r s t book was p u b l i s h e d w i t h t h e t i t l e “Boundary E l e m e n t s ” [ 8 ] . The work was expanded t o encompass t i m e d e p e n d e n t and n o n - l i n e a r p r o b l e m s i n two s u b s e q u e n t books [ 9 ] ,[ l o ] , o n e of them v e r y r e c e n t l y p u b l i s h e d [ l o ] . The i m p o r t a n c e of t h i s work i s t h a t i t s t r e s s e s t h e common p r i n c i p l e s and f u n d a m e n t a l s r e l a t i o n s h i p s g o v e r n i n g

186

C A . Brebbia

t h e d i f f e r e n t t e c h n i q u e s , r a t h e r than t r y i n g t o set t h e boundary element method a s a completely s e p a r a t e c o m p u t a t i o n a l t e c h n i q u e . F i v e i m p o r t a n t i n t e r n a t i o n a l c o n f e r e n c e s have a l r e a d y been h e l d on t h e t o p i c of boundary elements i n 1978 (Southampton) [ l l ] , 1980 (Southampton) [ 121 , 1981 ( C a l i f o r n i a ) [ 131 , 1982 (Southampton) [ 141 , 1983 (Hiroshima) [ 1 5 ] and t h e n e x t one i s t o b e h e l d i n J u l y 1984 on board t h e Queen E l i z a b e t h I1 c r u i s e r . The frequency of t h e meetings and t h e i n c r e a s i n g number of p a p e r s p r e s e n t e d a t each of them i s e v i d e n c e of t h e h e a l t h y growth of t h e new method. I n a d d i t i o n , a s e r i e s of s t a t e of t h e a r t books a r e r e g u l a r l y p u b l i s h e d t o h i g h l i g h t t h e main developments of t h e t e c h n i q u e [ 161 [ 1 7 1 [ 181. The s u c c e s s and r a p i d a c c e p t a n c e of t h e new t e c h n i q u e i s due t o some important a d v a n t a g e s o v e r c l a s s i c a l f i n i t e e l e m e n t s , which a r e b e t t e r understood by reviewing t h e main c h a r a c t e r i s t i c s of t h e method. The boundary element method a s understood nowadays i s a r e d u c t i o n t e c h n i q u e based on boundary i n t e g r a l e q u a t i o n f o r m u l a t i o n s and i n t e r p o l a t i o n f u n c t i o n of t h e t y p e used i n f i n i t e e l e m e n t s , The main c h a r a c t e r i s t i c of t h e method i s t h a t i t reduces t h e d i m e n s i o n a l i t y of t h e problem by one and hence produces a much smaller system of e q u a t i o n s and more i m p o r t a n t f o r the practicing engineer, considerable reductions i n the data required t o run a problem. The l a t t e r advantage i s making boundary elements a f a v o u r i t e f o r many mechanical e n g i n e e r i n g problems when t h e n u m e r i c a l model h a s t o be i n t e r f a c e d w i t h mesh g e n e r a t o r s and o t h e r CAD f a c i l i t i e s . I n a d d i t i o n t h e numerical accuracy of t h e method i s g e n e r a l l y g r e a t e r t h a n t h a t of f i n i t e e l e m e n t s , which have l e d many e n g i n e e r s t o u s e BEM f o r problems such a s f r a c t u r e mechanics and o t h e r s where s t r e s s c o n c e n t r a t i o n can o c c u r . T h i s accuracy i s due t o u s i n g a mixed f o r m u l a t i o n . t y p e of approach f o r which a l l boundary v a l u e s a r e o b t a i n e d w i t h s i m i l a r d e g r e e of a c c u r a c y . I n t h i s r e s p e c t BEM i s c l o s e l y r e l a t e d t o t h e mixed f o r m u l a t i o n s p i o n e e r e d by R e i s s n e r [ 1 9 ] and e x c e l l e n t l y e x p l a i n e d and g e n e r a l i z e d by Washizu [ZO] and Pian and Tong [ 2 1 ] . The method i s a l s o w e l l s u i t e d t o problem s o l v i n g w i t h i n f i n i t e domains such a s t h o s e f r e q u e n t l y o c c u r r i n g i n s o i l mechanics and hydrodynamics, and f o r which t h e c l a s s i c a l domain methods a r e u n s u i t a b l e . A boundary s o l u t i o n i s f o r m u l a t e d i n terms of i n f l u e n c e f u n c t i o n s o b t a i n e d by a p p l y i n g a fundamental s o l u t i o n . I f t h e s o l u t i o n i s s u i t a b l e f o r a n i n f i n i t e domain no o u t e r b o u n d a r i e s need t o be defined. I t i s now g e n e r a l l y a c c e p t e d t h a t t h e b e s t way of f o r m u l a t i n g boundary elements f o r g e n e r a l e n g i n e e r i n g problems i s by u s i n g weighted r e s i d u a l t e c h n i q u e s , as shown i n r e f e r e n c e s [ 7 ] , [ a ] and [ 101. T h i s f o r m u l a t i o n c l o s e l y r e l a t e s t h e BEM t o t h e v a r i a t i o n a l methods and t o t h e o r i g i n a l It a l s o allows i n t e r p r e t a t i o n of v i r t u a l work proposed by B e r n o u l l i . f o r c o m p l i c a t e d n o n - l i n e a r and t i m e dependent problems t o be p r o p e r l y f o r m u l a t e d , w i t h o u t need t o f i n d a n i n t e g r a l expansion beforehand.

The term boundary element now a l s o i m p l i e s t h a t t h e s u r f a c e of t h e domain i s d i v i d e d i n t o a s e r i e s of elements o v e r which t h e f u n c t i o n s under c o n s i d e r a t i o n v a r y i n a c c o r d a n c e w i t h some i n t e r p o l a t i o n f u n c t i o n s , i n much t h e same way a s i n f i n i t e elements. By c o n t r a s t w i t h p a s t i n t e g r a l e q u a t i o n s f o r m u l a t i o n s - which were r e s t r i c t e d t o c o n c e n t r a t e d s o u r c e s t h e s e v a r i a t i o n s p e r m i t t h e p r o p e r d e s c r i p t i o n of curved s u r f a c e s i n a d d i t i o n t o working w i t h more a c c u r a t e h i g h e r o r d e r i n t e r p o l a t i o n functions.

The Unification o f Finite Elements & Boundary Elements

187

Summarizing, a f t e r y e a r s of r e s e a r c h and development t h e b o u n d a r y e l e m e n t method h a s emerged a s a p o w e r f u l m a t h e m a t i c a l t o o l f o r t h e s o l u t i o n of a l a r g e v a r i e t y of e n g i n e e r i n g p r o b l e m s . The a c c e p t a n c e of t h e t e c h n i q u e amongst p r a c t i c i n g e n g i n e e r s i s m a i n l y due t o t h e f o l l o w i n g a d v a n t a g e s : i) Simple d a t a p r e p a r a t i o n , which c o n s i d e r a b l y r e d u c e s t h e amount of manpower r e q u i r e d t o r u n a problem i i ) More a c c u r a t e r e s u l t s , which makes t h e t e c h n i q u e e s p e c i a l l y a t t r a c t i v e f o r s t r e s s c o n c e n t r a t i o n problems, f r a c t u r e mechanics a p p l i c a t i o n and o t h e r s . T h i s i n c r e a s e d a c c u r a c y a l s o a l l o w s t h e d e s i g n e r t o work w i t h c o a r s e r meshes t h a n i n f i n i t e e l e m e n t s w i t h f u r t h e r r e d u c t i o n i n manpower. i i i ) D e f i n i t i o n of s y s t e m and i n t e r p r e t a t i o n of r e s u l t s become e a s i e r which p e r m i t s a b e t t e r i n t e r f a c i n g t o s u r f a c e m o d e l l i n g and o t h e r CAD systems. i v ) Problems w i t h i n f i n i t e domains c a n b e s o l v e d a c c u r a t e l y , which makes t h e method w e l l s u i t e d f o r a p p l i c a t i o n s s u c h a s s o i l m e c h a n i c s and hydrodynamics.

2.

FUNDAMENTAL PRINCIPLES

I n what f o l l o w s we w i l l c o n s i d e r p r o b l e m s i n l i n e a r e l a s t i c i t y f o r which t h e problem c a n b e e x p r e s s e d i n f u n c t i o n of a s e t o f e q u i l i b r i u m e q u a t i o n s and a n o t h e r s e t of c o m p a t i b i l i t y r e l a t i o n s , r e l a t e d t o g e t h e r by c o n s t i t u t ive laws. These e q u a t i o n s w i l l b e w r i t t e n u s i n g t h e i n d i c i a 1 n o t a t i o n . Dynamic l o a d i n g w i l l n o t b e c o n s i d e r e d e x p l i c i t l y b u t i t c a n b e e a s i l y i n c l u d e d u s i n g D ' A l e m b e r t ' s h y p o t h e s i s , i . e . by c o n s i d e r i n g t h a t a t a g i v e n time t h e dynamic and s t a t i c f o r c e s a r e i n e q u i l i b r i u m . T h i s s i m p l e b u t b r i l l i a n t i d e a f a c i l i t a t e s t h e dynamic a n a l y s i s .

The a p p r o x i m a t e methods of s o l u t i o n u s e d i n e n g i n e e r i n g a n a l y s i s h a v e a l l a common b a s i s n o t o n l y g i v e n by t h e f u n d a m e n t a l e q u a t i o n s of p h y s i c s b u t a l s o by t h e f a c t t h a t t h e a c t u a l a p p r o x i m a t i o n s c a n b e i n t e r p r e t e d u s i n g t h e p r i n c i p l e of v i r t u a l work. The a p p l i c a t i o n of t h i s p r i n c i p l e i n d i f f e r e n t ways g i v e s r i s e t o t h e d i v e r s e t e c h n i q u e s o f e n g i n e e r i n g a n a l y s i s . I t is important t o p o i n t o u t t h a t t h e p r i n c i p l e i t s e l f i s a fundamental i d e a b a s e d on p h i l o s o p h i c a l and p h y s i c a l i n t u i t i o n r a t h e r t h a n h i g h e r mathematics. I n t h i s r e s p e c t i t i s i n t e r e s t i n g t o remark t h a t t h e p r i n c i p l e h a s b e e n d i s c u s s e d s i n c e t h e b e g i n n i n g of w e s t e r n c i v i l i z a t i o n and i s r e l a t e d t o t h e ' p o t e n t i a l i t i e s ' o f p h y s i c a l s y s t e m s a s d i s c u s s e d by A r i s t o t l e [ 2 2 ] . From c l a s s i c a l a n t i q u i t y onward t h e p r i n c i p l e h a s b e e n f r e q u e n t l y a p p l i e d and s e v e r a l w e l l known f i e l d s of m a t h e m a t i c s r e l a t e d t o i t , s u c h a s t h e C a l c u l u s of V a r i a t i o n s , F u n c t i o n a l A n a l y s i s , D i s t r i b u t i o n T h e o r y , e t c . These m a t h e m a t i c s , a l t h o u g h i m p r e s s i v e , s h o u l d n o t d i s t r a c t u s f r o m t h e e l e g a n c e , s i m p l i c i t y and g e n e r a l i t y o f t h e o r i g i n a l v i r t u a l work s t a t e m e n t . I n t h i s s e c t i o n we w i l l t r y t o p o i n t o u t how t h e p r i n c i p l e of v i r t u a l work can b e u s e d t o g e n e r a t e models i n s o l i d m e c h a n i c s . T h i s i s f i r s t done by assuming t h a t t h e same p h y s i c a l e q u a t i o n s a p p l y t o two d i f f e r e n t s t a t e s , o n e i s t h e ' a c t u a l ' and t h e o t h e r i s t h e ' v i r t u a l ' s t a t e . The a c t u a l s t a t e i s u s u a l l y d e f i n e d i n t e r m s of a n a p p r o x i m a t i o n i n t h e p r a c t i c e . The p r o d u c t s of t h e s e two s t a t e s g i v e r i s e t o v i r t u a l work s t a t e m e n t s . T h i s s e c t i o n w i l l a t t e m p t t o c l a s s i f y t h e s e s t a t e m e n t s d e p e n d i n g on which t y p e of r e l a t i o n s h i p s a r e i d e n t i c a l l y s a t i s f i e d a n d which a r e t o b e imposed on t h e a p p r o x i m a t e f u n c t i o n s . T h e s e f o r m u l a t i o n s a r e a s w e l l known a s v i r t u a l d i s p l a c e m e n t s and v i r t u a l f o r c e s , b u t c a n a l s o b e some t y p e o f mixed o r h y b r i d a p p r o a c h . We w i l l p a r t i c u l a r l y c o n s i d e r t h e p o s s i b i l i t y of p r o d u c i n g g e n e r a l i z e d f o r m u l a t i o n s and t a k i n g them t o t h e b o u n d a r y , a s i t

C.A. Brebbia

188

i s due i n boundary elements. The s i m p l i c i t y of t h e v i r t u a l work approach allows f o r t h e f o r m u l a t i o n of very g e n e r a l approximate models, v a l i d even f o r non l i n e a r and time dependent problems. The formulation of d i f f e r e n t techniques - i n c l u d i n g boundary elements - becomes then independent of t h e e x i s t e n c e o r , o t h e r w i s e of a f u n c t i o n a l o r i n t e g r a l s t a t e m e n t . These f o r m u l a t i o n s w i l l not b e d i s c u s s e d h e r e , but t h e i n t e r e s t e d r e a d e r i s r e f e r r e d t o [ 2 3 ] . VIRTUAL WORK

The V i r t u a l Work p r i n c i p l e can be i n t e r p r e t e d a s t h e work done by one s t a t e ( ' a c t u a l ' ) over a n o t h e r ( ' v i r t u a l ' ) . This work can be expressed i n d i f f e r e n t ways, depending on t h e v a r i a b l e s under c o n s i d e r a t i o n . For i n s t a n c e i f one is d e a l i n g with displacements and body and t r a c t i o n f o r c e s one can w r i t e t h e following v i r t u a l work s t a t e m e n t

Notice t h a t t h e work h a s been d e f i n e d i n terms of t h e usual i n n e r p r o d u c t , 5.e. t h e m u l t i p l i c a t . i o n of t h e v a r i a b l e s i n t e g r a t e d over t h e domain and e x t e r n a l s u r f a c e . bk a r e t h e body f o r c e s , t k t h e s u r f a c e t r a c t i o n s and u k t h e displacement components. The v i r t u a l f i e l d i s i n d i c a t e d by an asterisk

.

The same p r i n c i p l e can a l s o be expressed i n terms of t h e i n t e r n a l work, which g i v e s ,

I.

jk

€Jk

dR =

I

ufk

E~~

dR

a j k and c j k a r e t h e s t r e s s and s t r a i n components r e s p e c t i v e l y . S t i l l more i n t e r e s t i n g l y , v i r t u a l work could be given a s a r e l a t i o n s h i p between c o m p a t i b i l i t y e q u a t i o n s and s t r e s s f u n c t i o n s . I f t h e c o m p a t i b i l i t y r e l a t i o n s h i p s a r e expressed by t h e Rk components of a c o m p a t i b i l i t y v e c t o r and t h e a s s o c i a t e d s t r e s s f u n c t i o n xk one can w r i t e ,

These t h r e e s t a t e m e n t s a r e e q u a l l y v a l i d and they can even be added t o f i n d an extended v e r s i o n of v i r t u a l work a s w e w i l l s e e soon. This p r e s e n t a t i o n of v i r t u a l work h a s some advantages over t h e more c l a s s i c a l v a r i a t i o n a l t y p e of approach a s we w i l l s e e s h o r t l y . The c l a s s i c a l approach o r i g i n a t e d w i t h B e r n o u l l i c o n s t r a i n t e q u a t i o n s , u s u a l l y r e q u i r e s t h e d e f i n i t i o n of some Lagrangian m u l t i p l i e r s t o g e n e r a l i z e t h e p r i n c i p l e s . Our approach i n s t e a d i s much s i m p l e r . VIRTUAL DISPLACEMENTS I t i s now easy t o deduce d i f f e r e n t v e r s i o n s of t h e v i r t u a l work p r i n c i p l e by applying t h e above e q u a t i o n s . Let u s s t a r t with i d e n t i t y ( 1 ) i n t e g r a t i n g by p a r t s t h e s u r f a c e i n t e g r a l on t h e r i g h t hand s i d e . I n o r d e r t o do

The Unification of Finite Elements & Boundary Elements t h i s we can use t h e w e l l known Gauss theorem and f o r l i n e a r s t r a i n displacement r e l a t i o n s - which we a c c e p t a r e i d e n t i c a l l y s a t i s f i e d ob t a i n ,

189

-

where t k . = nj ujk ; n . a r e t h e d i r e c t i o n c o s i n e s of t h e normal w i t h r e s p e c t 3 t o x j , a x i s . I f f u r t h e r m o r e we a c c e p t i ) r e c i p r o c i t y a s given by e q u a t i o n (21, i i ) t h a t t h e v i r t u a l displgcements u t are-such t h a t t h e d i s p l a c e m e n t s boundary c o n d i t i o n on r l (uk = uk on r l where Uk a r e known v a l u e s ) a r e -0 on r l and i i j . ) t h a t t h e v i r t u a l f i e l d identically satisfied, i.e. s a t i s f i e s e q u i l i b r i u m , one f i n d s ,

$

which i s t h e u s u a l e x p r e s s i o n f o r v i r t u a l work. Notice t h a t t h e o t h e r p a r t of t h e boundary r 2 i s - t h a t on which t h e t r a c t i o n boundary c o n d i t i o n s a r e p r e s c r i b e d , i . e . tk = t on r

2'

k

Another form of v i r t u a l disolacements can b e o b t a i n e d by i n t e g r a t i n g by p a r t s t h e l e f t hand s i d e i n t e g r a l i n ( 5 ) . This g i v e s

1

dQ =

(tk

-

r2 The above s t a t e m e n t i s e q u i v a l e n t t o ( 5 ) provided t h a t we a c c e p t t h a t t h e s t r a i n - d i s p l a c e m e n t e q u a t i o n s and c o n s t i t u t i v e r e l a t i o n s h i p s a r e i d e n t i c ally satisfied. The above r e s t r i c t i o n s t o v i r t u a l work g i v e r i s e t o t h e p o s s i b i l i t y of d e f i n i n g a f u n c t i o n a l c a l l e d t o t a l p o t e n t i a l energy, composed of two p a r t s , i . e . t h e i n t e r n a l s t r a i n energy f u n c t i o n ,

and t h e p o t e n t i a l of t h e l o a d s (assuming they a r e c o n s e r v a t i v e

"-j

-tk uk

dr

-

1

bk uk dR

I-2 The t o t a l p o t e n t i a l energy i s then

Equilibrium s t a t e m e n t s ( 5 ) o r ( 6 ) f o r i n s t a n c e a r e now d e f i n e d by t h e ' v a r i a t i o n ' of n , i . e .

*

*

*

n = u +n = o

(10)

190

C A . Brebbia

Notice that Potential energy is function of the displacements and strains. As it is well known this principle is the basis of the stiffness finite element formulations. Principle of Virtual Forces The converse of the Principle of Virtual Displacements is the Principle of Virtual Forces which can be described in several different ways. In this paper we will start by using the virtual work relationship (equation ( 2 ) )

Accepting that

* E

ij

9:

=

l(uiyj+u

,t

j

. ) we can transform the right hand side term

of ( 1 1 ) into,

Furthermore accepting that the

G

state satisfies the equilibrium equations

one can write ( 1 2 ) as,

9

0, then

if

8

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