Introduction to Shape Optimization: Theory, Approximation, and Computation (Advances in Design and Control)

292 Pages·2003·12.76 MB·English

by  J. Haslinger

Introduction to
Shape Optimization
Advances in Design and Control
SIAM's Advances in Design and Control series consists of texts and monographs dealing with
all areas of design and control and their applications. Topics of interest include shape
optimization, multidisciplinary design, trajectory optimization, feedback, and optimal control.
The series focuses on the mathematical and computational aspects of engineering design and
control that are usable in a wide variety of scientific and engineering disciplines.
Editor-in-Chief
John A. Burns, Virginia Polytechnic Institute and State University
Editorial Board
H. Thomas Banks, North Carolina State University
Stephen L. Campbell, North Carolina State University
Eugene M. Cliff, Virginia Polytechnic Institute and State University
Ruth Curtain, University of Groningen
Michel C. Delfour, University of Montreal
John Doyle, California Institute of Technology
Max D. Gunzburger, Iowa State University
Rafael Haftka, University of Florida
Jaroslav Haslinger, Charles University
J. William Helton, University of California at San Diego
Art Krener, University of California at Davis
Alan Laub, University of California at Davis
Steven I. Marcus, University of Maryland
Harris McClamroch, University of Michigan
Richard Murray, California Institute of Technology
Anthony Patera, Massachusetts Institute of Technology
H. Mete Soner, Koc University
Jason Speyer, University of California at Los Angeles
Hector Sussmann, Rutgers University
Allen Tannenbaum, University of Minnesota
Virginia Torczon, William and Mary University
Series Volumes
Haslinger, J. and Makinen, R. A. E., Introduction to Shape Optimization: Theory,
Approximation, and Computation
Antoulas, A. C., Lectures on the Approximation of Linear Dynamical Systems
Gunzburger, Max D., Perspectives in Flow Control and Optimization
Delfour, M. C. and Zolesio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and
Optimization
Betts, John T., Practical Methods for Optimal Control Using Nonlinear Programming
El Ghaoui, Laurent and Niculescu, Silviu-lulian, eds., Advances in Linear Matrix Inequality
Methods in Control
Helton, J. William and James, Matthew R., Extending H°°Control to Nonlinear Systems:
Control of Nonlinear Systems to Achieve Performance Objectives
Introduction to
Shape Optimization
Theory, Approximation,
and Computation
J. Haslinger
Charles University
Prague, Czech Republic
R.A. E. Makinen
University of Jyvaskyla
Jyvaskyla, Finland
Society for Industrial and Applied Mathematics
Philadelphia
Copyright © 2003 by the Society for Industrial and Applied Mathematics.
1098765432 1
All rights reserved. Printed in the United States of America. No part of this book may
be reproduced, stored, or transmitted in any manner without the written permission
of the publisher. For information, write to the Society for Industrial and Applied
Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688.
Library of Congress Cataloging-in-Publication Data
Haslinger, J.
Introduction to shape optimization : theory, approximation, and computation / J.
Haslinger, R.A.E. Makinen.
p. cm. - (Advances in design and control)
Includes bibliographical references and index.
ISBN 0-89871-536-9
1. Structural optimization-Mathematics. I. Makinen, R.A.E. II. Title. III. Series.
TA658.8.H35 2003
624.17713-dc21 2003041589
NAG is a registered trademark of The Numerical Algorithms Group Ltd., Oxford, U.K.
IMSL is a registered trademark of Visual Numerics, Inc., Houston, Texas.
is a registered trademark.
Contents
Preface ix
Notation xiii
Introduction xvii
Part I Mathematical Aspects of Sizing and Shape Optimization 1
1 Why the Mathematical Analysis Is Important 3
Problems 12
2 A Mathematical Introduction to Sizing and Shape Optimization 13
2.1 Thickness optimization of an elastic beam: Existence and
convergence analysis 13
2.2 A model optimal shape design problem 24
2.3 Abstract setting of sizing optimization problems: Existence and
convergence results 38
2.4 Abstract setting of optimal shape design problems and their
approximations 45
2.5 Applications of the abstract results 50
2.5.1 Thickness optimization of an elastic unilaterally
supported beam 50
2.5.2 Shape optimization with Neumann boundary value
state problems 53
2.5.3 Shape optimization for state problems involving mixed
boundary conditions 61
2.5.4 Shape optimization of systems governed by
variational inequalities 64
2.5.5 Shape optimization in linear elasticity and contact
problems 71
2.5.6 Shape optimization in fluid mechanics 83
Problems 93
v
vi Contents
Part II Computational Aspects of Sizing and Shape Optimization 97
3 Sensitivity Analysis 99
3.1 Algebraic sensitivity analysis 99
3.2 Sensitivity analysis in thickness optimization 107
3.3 Sensitivity analysis in shape optimization 109
3.3.1 The material derivative approach in the continuous setting 109
3.3.2 Isoparametric approach for discrete problems 119
Problems 125
4 Numerical Minimization Methods 129
4.1 Gradient methods for unconstrained optimization 129
4.1.1 Newton's method 131
4.1.2 Quasi-Newton methods 131
4.1.3 Ensuring convergence 133
4.2 Methods for constrained optimization 134
4.2.1 Sequential quadratic programming methods 136
4.2.2 Available sequential quadratic programming software . . 138
4.3 On optimization methods using function values only 139
4.3.1 Modified controlled random search algorithm 140
4.3.2 Genetic algorithms 141
4.4 On multiobjective optimization methods 144
4.4.1 Setting of the problem 145
4.4.2 Solving multiobjective optimization problems by
scalarization 146
4.4.3 On interactive methods for multiobjective optimization . 147
4.4.4 Genetic algorithms for multiobjective optimization
problems 148
Problems 150
5 On Automatic Differentiation of Computer Programs 153
5.1 Introduction to automatic differentiation of programs 153
5.1.1 Evaluation of the gradient using the forward and
reverse methods 156
5.2 Implementation of automatic differentiation 160
5.3 Application to sizing and shape optimization 165
Problems 167
6 Fictitious Domain Methods in Shape Optimization 169
6.1 Fictitious domain formulations based on boundary and distributed
Lagrange multipliers 170
6.2 Fictitious domain formulations of state problems in shape optimization 181
Problems 197
Contents vii
Part III Applications 199
7 Applications in Elasticity 201
7.1 Multicriteria optimization of a beam 201
7.1.1 Setting of the problem 201
7.1.2 Approximation and numerical realization of (P ) 204
w
7.2 Shape optimization of elasto-plastic bodies in contact 211
7.2.1 Setting of the problem 211
7.2.2 Approximation and numerical realization of (P) 213
Problems 219
8 Fluid Mechanical and Multidisciplinary Applications 223
8.1 Shape optimization of a dividing tube 223
8.1.1 Introduction 223
8.1.2 Setting of the problem 224
8.1.3 Approximation and numerical realization of (P) 227
e
8.1.4 Numerical example 230
8.2 Multidisciplinary optimization of an airfoil profile using genetic
algorithms 230
8.2.1 Setting of the problem 232
8.2.2 Approximation and numerical realization 236
8.2.3 Numerical example 239
Problems 243
Appendix A Weak Formulations and Approximations of Elliptic
Equations and Inequalities 245
Appendix B On Parametrizations of Shapes and Mesh Generation 257
B.1 Parametrization of shapes 257
B.2 Mesh generation in shape optimization 260
Bibliography 263
Index 271
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Preface
Before we explain our motivation for writing this book, let us place its subject in a more
general context. Shape optimization can be viewed as a part of the important branch of
computational mechanics called structural optimization. In structural optimization prob-
lems one tries to set up some data of the mathematical model that describe the behavior of a
structure in order to find a situation in which the structure exhibits a priori given properties.
In other words, some of the data are considered to be parameters (control variables) by
means of which one fine tunes the structure until optimal (desired) properties are achieved.
The nature of these parameters can vary. They may reflect material properties of the struc-
ture. In this case, the control variables enter into coefficients of differential equations. If one
optimizes a distribution of loads applied to the structure, then the control variables appear on
the right-hand side of equations. In shape optimization, as the term indicates, optimization
of the geometry is of primary interest.
From our daily experience we know that the efficiency and reliability of manufactured
products depend on geometrical aspects, among others. Therefore, it is not surprising that
optimal shape design problems have attracted the interest of many applied mathematicians
and engineers. Nowadays shape optimization represents a vast scientific discipline involving
all problems in which the geometry (in a broad sense) is subject to optimization. For a finer
classification, we distinguish the following three branches of shape optimization:
(i) sizing optimization: a typical size of a structure is optimized (for example, a thickness
distribution of a beam or a plate);
(ii) shape optimization itself: the shape of a structure is optimized without changing the
topology;
(iii) topology optimization: the topology of a structure, as well as the shape, is optimized
by, for example, creating holes.
To keep the book self-contained we focus on (i) and (ii). Topology optimization needs
deeper mathematical tools, which are beyond the scope of basic courses in mathematics, to
be presented rigorously.
One important feature of shape optimization is its interdisciplinary character. First,
the problem has to be well posed from the mechanical point of view, requiring a good
understanding of the physical background. Then one has to find an appropriate mathe-
matical model that can be used for the numerical realization. In this stage no less than
three mathematical disciplines interfere: the theory of partial differential equations (PDEs),
approximation of PDEs (usually by finite element methods), and the theory of nonlinear
ix