cover
cover
next page >
Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation title: author: publisher: isbn10 | asin: print isbn13: ebook isbn13: language: subject publication date: lcc: ddc: subject:
cover
file:///C|/Download/_17910______/files/cover.html[2/21/2009 2:21:53 PM]
next page >
page_iii
< previous page
page_iii
next page > Page iii
Parameter Sensitivity in Nonlinear Mechanics Theory and Finite Element Computations Michal Kleiber * Polish Academy of Sciences, Warsaw, Poland in collaboration with Horatio Antúnez Polish Academy of Sciences, Warsaw, Poland Tran Duong Hien Technical University of Szczecin, Poland Piotr Kowalczyk Polish Academy of Sciences, Warsaw, Poland
Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation
< previous page
page_iii
file:///C|/Download/_17910______/files/page_iii.html[2/21/2009 2:21:53 PM]
next page >
page_iv
< previous page
page_iv
next page > Page iv
Published in 1997 by John Wiley & Sons Ltd. Baffins Lane, Chichester, West Sussex P019 1UD, England National 01243 779777 International (+44) 1243 779777 e-mail (for orders and customer service enquiries):
[email protected] Visit our Home Page on http://www.wiley.co.uk or http://www.wiley.com The copyright for Chapter 7 has been retained by Springer-Verlag The copyright for Section 9.5 has been retained by Elsevier Science S.A. All Rights Reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London, UK W IP 9HE without the permission in writing of the publisher. Other Wiley Editorial Offices John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA VCH Verlagsgesellschaft mbH, Pappelallee 3, D-69469 Weinheim, Germany Jacaranda Wiley Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W 1L1, Canada John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 Library of Congress Cataloging-in-Publication Data Kleiber, Michal. * Parameter sensitivity in nonlinear mechanics : theory and finite element computations / M. Kleiber, in collaboration with H. Antúnez, T.D. Hien, P. Kowalczyk. p. cm. Includes bibliographical references and index ISBN 0-471-96854-4 1. Mechanics, Applied. 2. Nonlinear mechanics. 3. Finite element method. 4. Sensitivity theory (Mathematics). I. Antúnez, H. II. Hien, Tran Duong. III. Kowalczyk, P. IV. Title. TA350.K654 1997 96-40503 620.1'054'01515355dc21 CIP British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-471-96854-4 file:///C|/Download/_17910______/files/page_iv.html[2/21/2009 2:21:54 PM]
page_iv
Produced from camera-ready copy supplied by the authors Printed and bound in Great Britain by Bookcraft Ltd, Midsomer Norton This book is printed on acid-free paper responsibly manufactured from sustainable forestation, for which at least two trees are planted for each one used for paper production. Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation
< previous page
page_iv
file:///C|/Download/_17910______/files/page_iv.html[2/21/2009 2:21:54 PM]
next page >
page_v
< previous page
page_v
next page > Page v
CONTENTS Preface
xi
Part I Preliminaries
1
1 Motivation: Sensitivity and Large-Scale Systems
3 3
1.1 Aims and Scope of the Chapter 3 1.2 On System Optimization 6 1.3 On Reliability-Based System Design 11 1.4 On System Identification 13 1.5 On the Stochastic Finite Element Method 2 Nonlinear Solid Mechanics: Continuous and Semi-Discretized Formulation
17 17
2.1 Introductory Comments 18 2.2 Equations of Nonlinear Solid Mechanics 18 2.2.1 Problem Description 20 2.2.2 Strain-Displacement Relations 20 2.2.3 Constitutive Equations 23 2.2.4 Equations of Motion
file:///C|/Download/_17910______/files/page_v.html[2/21/2009 2:21:55 PM]
page_v
24 2.2.5 Boundary Conditions 24 2.2.6 Variational Formulation 32 2.3 Constitutive Models for Inelastic Materials 32 2.3.1 Elasto-Plasticity and Elasto-Viscoplasticity 32 2.3.1.1 Linear Elasticity 32 2.3.1.2 Elasto-Plasticity 38 2.3.1.3 Elasto-Viscoplasticity 43 2.3.2 Rigid Plasticity and Rigid Viscoplasticity 45 2.4 The Semi-Discretized Formulation 54 2.5 Remarks on Tensor and Matrix Notation 3 Concepts of Sensitivity Analysis for Linear Systems
59 59
3.1 Introductory Remarks and Notation 64 3.2 Discretized Systems: Statics 72 3.3 Discretized Systems: Dynamics 77 3.4 Distributed Parameter Systems 77 3.4.1 General Comments 79
file:///C|/Download/_17910______/files/page_v.html[2/21/2009 2:21:55 PM]
page_v
3.4.2 The Direct Differentiation Method 81 3.4.3 The Adjoint System Method 84 3.5 Shape Sensitivity 89 3.6 Summary of the Chapter Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation
< previous page
page_v
file:///C|/Download/_17910______/files/page_v.html[2/21/2009 2:21:55 PM]
next page >
page_vi
< previous page
page_vi
next page > Page vi
Part II The Sensitivity of Nonlinear Systems
95
4 The Basic Concepts of Nonlinear Quasi-Static Problems at Regular States
97 97
4.1 The Direct Differentiation Method 97 4.1.1 The Discretized Formulation 104 4.1.2 The Distributed Parameter Formulation 106 4.1.3 On Time Integration of Sensitivity Equations 107 4.1.4 The Total versus Incremental Approach to Nonlinear Elastic Problems 110 4.1.5 Configuration-Dependent Forces 112 4.1.6 The 'Semi-Analytical' Approach 115 4.2 The Adjoint System Method 115 4.2.1 'he Adjoint Sensitivity Formulation 118 4.2.2 An Exercise in Using the Variational ASM 119 4.3 Some Illustrative Examples 5 Inelastic Systems
127 127
5.1 Small Deformations: The General 3D Case 127
file:///C|/Download/_17910______/files/page_vi.html[2/21/2009 2:21:55 PM]
page_vi
5.1.1 Theoretical Background 130 5.1.1.1 Linear Elasticity 131 5.1.1.2 Rate-Independent Elasto-Plasticity 148 5.1.1.3 Elasto-Viscoplasticity 151 5.1.2 Some Illustrative Examples 162 5.2 Small Deformations: The Plane Stress Case 162 5.2.1 Theoretical Background 163 5.2.1.1 Linear Elasticity 163 5.2.1.2 Rate-Independent Elasto-Plasticity 170 5.2.1.3 Elasto-Viscoplasticity 172 5.2.2 Some Illustrative Examples 177 5.3 Large Deformations 177 5.3.1 Theoretical Background 184 5.3.2 An Illustrative Example 189 5.4 Conclusions and Remarks on Computer Implementation 6 Shape Sensitivity
193 193
6.1 The Domain Parameterization ApproachBasic Concepts and Notation
file:///C|/Download/_17910______/files/page_vi.html[2/21/2009 2:21:55 PM]
page_vi
202 6.2 The Direct Differentiation MethodDPA/DDM 209 6.3 On Design Parameters Entering the Kinematic Boundary Conditions 212 6.4 The Adjoint System MethodDPA/ASM 218 6.5 Summary of the Chapter 7 Buckling and Post-Buckling
219 219
7.1 Introductory Remarks 221 7.2 On Solving Buckling Problems Using FEM 227 7.3 Sensitivity in Buckling and Post-Buckling Problems 227 7.3.1 Pre-Critical Analysis 229 7.3.2 Critical-Point Analysis 233 7.3.3 Post-Critical Analysis Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation
< previous page
page_vi
file:///C|/Download/_17910______/files/page_vi.html[2/21/2009 2:21:55 PM]
next page >
page_vii
< previous page
page_vii
next page > Page vii 236
7.4 The Continuum-Based Approach 238 7.5 Numerical Illustrations 242 7.6 Summary of the Chapter 8 Nonlinear Dynamics
245 245
8.1 Problem Statement 248 8.2 The Direct Differentiation Method 251 8.3 The Adjoint System Method 254 8.4 Computational Issues 256 8.5 Numerical Illustrations 260 8.6 Summary of the Chapter 9 Metal Forming Using the Flow Approach
263 263
9.1 Introductory Comments 264 9.2 Discretized Formulation 264 9.2.1 Equations of Flow Approach 270 9.2.2 Boundary Friction 271
file:///C|/Download/_17910______/files/page_vii.html[2/21/2009 2:21:56 PM]
page_vii
9.2.3 Contact and Friction 273 9.2.4 Pressure Stabilization 275 9.2.5 Free Surfaces 276 9.3 Material Parameter Sensitivity 276 9.3.1 Formulation 282 9.3.2 Material Model at Low Effective Strain Rates 283 9.3.3 Analytical Illustrations 283 9.3.3.1 Pure Shear Strain 284 9.3.3.2 Free Forging 287 9.3.4 Computational Illustrations 287 9.3.4.1 Direct Extrusion 293 9.3.4.2 Cutting 296 9.4 Shape Sensitivity for Steady-State Problems 296 9.4.1 General Comments 296 9.4.2 The Domain Parameterization Approach 297 9.4.2.1 Transformation to Reference Configuration 298 9.4.2.2 Design Variation of Problem Variables
file:///C|/Download/_17910______/files/page_vii.html[2/21/2009 2:21:56 PM]
page_vii
299 9.4.2.3 Sensitivity Expressions 300 9.4.3 Discretization 303 9.4.4 Analytical Illustrations 305 9.4.4 Computational Illustrations 305 9.4.5.1 Extrusion 309 9.4.5.2 Rolling 313 9.5 Deep Drawing Problems 313 9.5.1 Formulation 320 9.5.2 Parameter Sensitivity Analysis 324 9.5.3 Computational Illustrations 328 9.6 Summary of the Chapter 10 Nonlinear Thermal Systems
331 331
10.1 Introductory Remarks 332 10.2 Continuum-Based Formulation for Transient Problems 332 10.2.1 On Solving Nonlinear Heat Transfer Problems 336 10.2.2 The Direct Differentiation Method 338
file:///C|/Download/_17910______/files/page_vii.html[2/21/2009 2:21:56 PM]
page_vii
10.2.3 The Adjoint System Method Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation
< previous page
page_vii
file:///C|/Download/_17910______/files/page_vii.html[2/21/2009 2:21:56 PM]
next page >
page_viii
< previous page
page_viii
next page > Page viii 341
10.3 Semi-Discretized Formulation 341 10.3.1 Finite Element Equations 350 10.3.2 Computational Illustration 351 10.4 Shape Sensitivity for Transient Problems 351 10.4.1 Introduction 352 10.4.2 The Material Derivative ApproachMDA/DDM 359 10.4.3 The Domain Parameterization ApproachDPA/DDM 361 10.4.4 Comparison of Methods 362 10.4.5 The Adjoint System Method 364 10.5 Kirchhoff Transformation in Shape Sensitivity Steady-State Problems 364 10.5.1 Formulation 366 10.5.2 Computational Illustrations 368 10.6 Summary of the Chapter Appendix A Some Mathematical Background
369
A.1 Concepts from Variational Calculus
369 369
A.1.1 Fundamental Problem file:///C|/Download/_17910______/files/page_viii.html[2/21/2009 2:21:57 PM]
page_viii
373 A.1.2 Integral Form of the Euler Equation 375 A.1.3 Variable End-Point Problem A.2 Directional Derivatives
379 379
A.2.1 The Gâteaux Derivative 381 A.2.2 The Fréchet Derivative Appendix B Stress and Stress-Rate Measures
383
B.1 Stress Measures
383
B.2 Stress-Rate Measures
386
References
389
Index
401
Glossary of Symbols
403
Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation
< previous page
page_viii
file:///C|/Download/_17910______/files/page_viii.html[2/21/2009 2:21:57 PM]
next page >
page_ix
< previous page
page_ix
next page > Page ix
ACKNOWLEDGEMENTS
This book was written at the Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, during 19931996, in the combined framework of the Institute's statutory research programme and the research grant No. 7T07A01810 from the Polish Committee for Scientific Research.
The support of the above institutions is gratefully acknowledged Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation
< previous page
page_ix
file:///C|/Download/_17910______/files/page_ix.html[2/21/2009 2:21:58 PM]
next page >
page_xi
< previous page
page_xi
next page > Page xi
PREFACE Until quite recently researchers and practitioners in the field of structural and solid mechanics were mainly interested in developing methods for the efficient response evaluation of bodies subject to static and dynamic loadings. Highly advanced methods have been developed in which the wide use of computers has turned out to be indispensable for any sophisticated assessment of realistic structural behaviour. Particular difficulties have been encountered and gradually surmounted in the case of problems with complex geometries, non-classical (unilateral) boundary conditions and complicated, highly nonlinear and path-dependent behaviour of the underlying material; additional complications arose when some uncertainty as to the problem characteristics had to be considered requiring solution techniques typical of random field theory. A solution methodology that is widely recognized as the most effective and universal tool in dealing with such a class of problems is known as the finite element method (FEM). Among many representative monographs and textbooks that address the aforementioned problems we shall just mention here those listed as [19, 37, 75, 85, 100, 102, 141, 156, 198] in the references at the end of the book. The books selected either represent recognized milestones in the development of the FEM, [19, 37, 75, 85, 141, 156, 198], or are particularly relevant to the subject matter of this book in terms of the material covered and/or notation employed. In the last several years a new fruitful area of engineering research has emerged in the form of the so-called sensitivity analysis (SA), cf. [2, 52, 61, 64, 71, 102, 106], for instance. SA is concerned with the relationship between parameters defining the system at hand and the system behaviour characterized by a response functional. What is most frequently sought is the gradient (or, sometimes, higher derivatives) of the response functional with respect to the system parameters. In the case where the parameters are at the design stage, subject to changes at the discretion of the analyst, we speak of design sensitivity analysis (DSA). In practice, we may also be interested in knowing changes in the system behaviour under usually unavoidable parameter imperfectionsimperfection sensitivity analysis (ISA) is the term used in this case. From a general mathematical point of view both DSA and ISA can be considered very similar and no particular attention need be paid to the above distinction within SA. Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation
< previous page
page_xi
file:///C|/Download/_17910______/files/page_xi.html[2/21/2009 2:21:58 PM]
next page >
page_xii
< previous page
page_xii
next page > Page xii
The reason for interest in SA results from the variety of uses for sensitivity coefficients; besides mechanics, extensive parallel developments in this field have taken place in modelling control systems, followed by applications in such diverse disciplines as thermodynamics and physical chemistry, cf. [2] for a brief review. Even more than in traditional response analysis, SA employed for realistic structural configurations calls, as a rule, for the use of sophisticated computer-assisted methods. SA for linear systems1 is now theoretically a wellestablished engineering field; for a thorough exposition of the subject the reader may consult excellent books [52, 64, 71], representative original contributions [13, 42, 81, 82, 83, 102, 136, 172] and some review articles [2, 61, 173]; useful conference proceedings are also available, [15, 48, 70, 92, 106, 134, 143]. Surprisingly, even in this relatively mature domain of research there is no special textbook addressing computational techniques typical of SA; this is a highly regrettable situation that hampers further development of the methodology and prevents its widespread use in the design environment. This book is primarily concerned with solid mechanics background and computational techniques relevant to SA problems with any material and/or kinematic nonlinearity. By its very nature, SA for linear systems is automatically included in the scope of the book as a special case and will therefore be discussed in some detail in many places throughout the book. However, no extensive separate treatment of these problems is offered and some of the relevant techniques are simply not covered on the grounds that they do not lend themselves to easy generalizations. Thus, the present book is by no means intended to fill the gap in the field of computer-assisted SA for linear systems. Instead, the book addresses the new and challenging problem of SA for structural systems with any nonlinearity, the linear structural systems being briefly treated in Chapter 3 to introduce the basic concepts and terminology. It should be emphasized that with this text we do not attempt a review, and even less so a unification, of the variety of concepts put forward in the existing literature. On the contrary, we shall intentionally restrict our presentation to a focused exposition of some selected aspects of the subject but then try to discuss them down to the level of effective computer implementation in a finite element program. The book consists of two major parts. Part I is sort of an extended introduction to the main subject studied in the book. It starts with a brief review of the role sensitivity plays in many branches of engineering in Chapter 1. To introduce notation and fix mathematical background a comprehensive account of nonlinear solid mechanics fundamentals is given in Chapter 2 using both the continuum-based and discretized setting. Chapter 3 completes the first part of the book by reviewing the basic concepts of sensitivity analysis for linear systems. 1A word about terminology: the term 'linear system' takes reference to its standard use in the field of solid and structural mechanics. Thus, for fixed system characteristics any change in the value of external agents acting upon the system results in the proportional change in the value of displacements, strains, stresses, etc. However, dependence of the problem equation on system parameters need not be linear! Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation
< previous page
page_xii
file:///C|/Download/_17910______/files/page_xii.html[2/21/2009 2:21:59 PM]
next page >
page_xiii
< previous page
page_xiii
next page > Page xiii
Part II of the book opens up with Chapter 4 in which fundamental concepts of sensitivity analysis for nonlinear quasi-static problems at regular states are presented. The sensitivity of inelastic systems is discussed in Chapter 5, followed in Chapters 6 and 7 by shape sensitivity, and buckling and post-buckling sensitivity formulations, respectively. The most complex problem of nonlinear dynamic sensitivity is undertaken in Chapter 8. The remainder of the book deals with some extensions of the sensitivity formulations derived earlier. It consists of applications of the sensitivity concept to metal forming formulations including contact/friction effects in Chapter 9 followed by discussion of the sensitivity framework for nonlinear transient thermal problems in Chapter 10. The appendices given at the end of the book are believed to be useful in that they treat in more detail some aspects of the formalism, presented in the main body of the text in a, of necessity, rather sketchy way. We should observe already at this point (as we will in many places later on) that in order to deal adequately with some of the problems discussed in the book, notably those with inequality constraints like inelasticity with elastic range or contact/friction problems, much more mathematically sophisticated formalisms than those used here may appear necessary. Our, in places, simplistic presentation derives from our general methodological attitude that selection of the best mathematical tool for a particular engineering task is influenced by factors other than the purely theoretical. For we believe that better results often come from a crude (but adequate!) method that the analyst thoroughly understands than from a sophisticated, but not fully comprehensive, method. The subject of the book belongs to a very fast-developing research area known as computational mechanics. Its very essence consists in an attempt to combine theoretical modelling techniques with effective computer implementation. These twofold objectives were also pursued in writing this book. Thus, besides the theory and results of numerical computations presented in the book many suitable finite element codes have been developed by the authors and used for computing the examples. The book is believed to prove useful to graduate students and active researchers in university departments of civil, mechanical and aerospace engineering as well as applied mathematics. It should also attract industry-based engineers interested in advanced aspects of the numerical analysis of nonlinear structural mechanics problems. The essential prerequisite is a good knowledge of basic solid and structural mechanics, the finite element method and variational calculus. Knowledge of design sensitivity analysis for linear structural systems will undoubtedly be an asset although the subject is reviewed in some detail in Chapter 3. For effective use of the technique presented a solid background in computer programming is also recommended. The book is a substantially extended version of lecture notes prepared for courses and series of lectures given by the main author at the Institute of Fundamental Technological Research in Warsaw, Poland, The University of Tokyo, Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation
< previous page
page_xiii
file:///C|/Download/_17910______/files/page_xiii.html[2/21/2009 2:21:59 PM]
next page >
page_xiv
< previous page
page_xiv
next page > Page xiv
Japan, The University of Connecticut in Stoors, Conn., USA, The University of Cape Town, South Africa, The Korean Advanced Institute of Science and Technology in Taejon, Korea and The University of Hong Kong. Regardless of its advanced character, the presentation is such as to enable the teacher to use it as a supplementary textbook for a one-semester graduate or post-graduate course. The whole content can certainly not be covered during such a course and selection of the material must be based on students' curriculum. For students who have taken courses in structural analysis, basic continuum mechanics and the finite element method, which is here assumed to be a fairly typical situation, the organization of various courses could, to a large extent, be based on the following parts of the book: - Course on Computer-Oriented Nonlinear Solid Mechanics: Chapter 2, Chapter 4, Chapter 5, Chapter 7 (Section 7.2), Chapter 8. - Course on Sensitivity Analysis for Linear Systems: Chapter 1, Chapter 3. - Course on Sensitivity Analysis for Nonlinear Static Systems: Chapter 3, Chapter 4, Chapter 5, Chapter 6. -Course on Parameter Sensitivity in Buckling and Dynamics: Chapter 3 (Section 3.3), Chapter 7, Chapter 8. - Course on Shape Sensitivity for Metal Forming Problems: Chapter 3. Chapter 9. - Course on Computer-Oriented Nonlinear Stationary and Transient Thermal Analysis: Chapter 10. The book in its present form is an outcome of a collaborative effort of a research group at the Institute of Fundamental Technological Research, Polish Academy of Sciences. Among the main author's collaborators, I)r H. Antúnez has written major parts of Chapter 6 on shape sensitivity formulations and Chapter 9 on sensitivity in metal forming. Dr T.D. Hien, now Professor at the Technical University of Szczecin, Poland, has collaborated with the main author on the subject of sensitivity analysis from the very beginning of the project over the years he has significantly influenced the final shape of the general theory as presented in the book and has been the main contributor in Chapter 8 on parameter sensitivity in nonlinear dynamics. He has also solved all the numerical problems described in Chapter 7. Dr P. Kowalczyk has cooperated with the main author on sensitivity analysis of inelastic systemsSections 5.2 and 5.3 of Chapter 5 are largely his contribution. Besides the theoretical aspects, the three co-authors have developed computer programs used for solving all the numerical examples discussed in the book. The main author gratefully acknowledges creative research contacts with his other colleagues not listed on the front page who collaborated with him on the subject matter of the book at different times and in different locations. Among them, the author is particularly indebted to Professor T. Ilisada and Dr Xian Chen at the University of'Tokkyo, Japan, Mr E. Postek and Dr W. Sosnowski Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation
< previous page
page_xiv
file:///C|/Download/_17910______/files/page_xiv.html[2/21/2009 2:22:00 PM]
next page >
page_xv
< previous page
page_xv
next page > Page xv
at the author's home institution and Dr A. Sluzalec *, Professor at the Technical University in Czestochowa*, Poland. Special thanks are directed to Professor Z. Mróz, the author's colleague at the Institute, for his (even if only indirect) influence on the author's understanding of many crucial aspects of the theory developed in the book. Dr T.D. Hien's typesetting work went far beyond the standard technical tasks-on the way he contributed a lot towards a better transparency of the derivations, improved overall composition of the text and influenced many other details crucial to the final appearance of the book. MICHAL* KLEIBER WARSAW NOVEMBER 1996 Start of Citation[PU]John Wiley & Sons, Ltd. (UK)[/PU][DP]1997[/DP]End of Citation
< previous page
page_xv
file:///C|/Download/_17910______/files/page_xv.html[2/21/2009 2:22:00 PM]
next page >
Part I PRELIMINARIES
Chapter 1 Motivation: Sensitivity and Large-Scale Systems 1.1
A i m s and Scope of the Chapter
S e t t i n g out to read a book on p a r a m e t e r sensitivity of s t r u c t u r a l response one m a y wonder what are t h e p o t e n t i a l gains from s t u d y i n g t h e hook. T h e s e possible gains, being at t h e s a m e t i m e driving forces behind t h e current sensitivity d e v e l o p m e n t s , can be phrased as follows: • G r a d i e n t s of functions describing s y s t e m b e h a v i o u r with respect to p a r a m e ters e n t e r i n g any specific theory employed are indispensable in t h e m a j o r i t y of a l g o r i t h m s used for such f u n d a m e n t a l problems of engineering as system o p t i m i z a t i o n and reliability assessment. As a m a t t e r of fact, t h e overall c o m p u t a t i o n a l cost required by such a l g o r i t h m s d e p e n d s strongly on t h e efficiency of gradient evaluation. • It is now broadly accepted t h a t any realistic large-scale engineering simulation has t o b e c o m p l e m e n t e d by an extensive s t u d y on response sensitivity to s y s t e m p a r a m e t e r s j u s t to broaden our u n d e r s t a n d i n g of t h e s y s t e m behaviour. To m a k e t h e first of t h e above reasons for s t u d y i n g p a r a m e t e r sensitivity clearer we shall briefly review in this c h a p t e r formulations typical of s y s t e m o p t i m i z a t i o n (Section 1.2), reliability (Section 1.3), identification (Section 1.4) and analysis by t h e so-called stochastic finite element m e t h o d (Section 1.5).
1.2
On S y s t e m Optimization
Only a very sketchy discussion of a formulation lying at t h e core of t h e c o n t e m p o r a r y o p t i m u m design theory of s t r u c t u r a l s y s t e m s will be given. For a fuller account of t h e subject t h e reader is referred to r e p r e s e n t a t i v e books [65, 69, 98, 147, 154, 158, 179], review articles [10, 11] a n d conference proceedings [15, 70, 143].
4 • Motivation: Sensitivity and Large-Scale Systems Any formulation for t h e o p t i m u m selection of p a r a m e t e r s requires p r o p e r identification of: (i) design variables ( p a r a m e t e r s ) describing t h e s y s t e m , (ii) a cost (objective) function to be o p t i m i z e d , a n d (iii) c o n s t r a i n t s assuring safe s y s t e m performance. Design variables are d e n o t e d here by a vector h = {h }, g — 1,2,. . . , D: by changing t h e values of t h e c o m p o n e n t s in this vector we define different s y s t e m s a m o n g which an o p t i m a l one is t o be sought. Design variables m a y have a continuous or discrete n a t u r e ; t h e former case is a s s u m e d here for simplicity with each variable having a prescribed range of variation. T h e very notion of system o p t i m i z a t i o n implies existence of a cost function / ( h ) t h a t can be used as a m e a s u r e of goodness of t h e design h. T h e cost function / can be a scalar- or vector-valued function; t h e l a t t e r case corresponds t o t h e so-called m u l t i c r i t e r i a o p t i m i z a t i o n t h a t goes beyond t h e scope of this brief discussion. U p p e r a n d lower limits i m p o s e d on design variables are typical r e p r e s e n t a t i v e s of t h e so-called inequality constraints. Very often we also need equality constraints -equations of e q u i l i b r i u m imposed on t h e solution are just one e x a m p l e of such c o n s t r a i n t s . e
Having defined a set of design variables h, a cost function / ( h ) , a n u m b e r of equality c o n s t r a i n t s g {h) = 0, c = l , 2 , . . . , C " , a n d a n u m b e r of inequality c o n s t r a i n t s (Ah) = r - r > = £ (0)
( E ^ A ^
(0
a=l
% = 1
+ ^Y
side of
(1.39) '
for which t h e necessary conditions a r e do
'
N
'
D
40)
T h e y form t h e linear s y s t e m of D algebraic e q u a t i o n s for A / ^ , g = 1,2 I). which can be solved by using t h e effective techniques of linear algebra. O n c e this has been d o n e , A h serves t o u p d a t e t h e p a r a m e t e r vector as 0)
( 0 )
= h
( 0 )
+ Ah
(1.41)
( 0 )
and t h e iteration step is r e p e a t e d by e x p a n d i n g Eq. (1.34) a r o u n d t h e new values of h. T h e general i t e r a t i v e formulae for t h e (UJ + 1) correction to h read £
(fX">
a= \
\
= 1
+ d
