[Michael C. Delfour, Jean-Paul Zolésio] Shapes an(BookFi.org)

Advances in Design and ControlSIAMs 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-ChiefRalph C. Smith, North Carolina State University

Editorial BoardAthanasios C. Antoulas, Rice UniversitySiva Banda, Air Force Research LaboratoryBelinda A. Batten, Oregon State UniversityJohn Betts, The Boeing Company (retired)Stephen L. Campbell, North Carolina State University Michel C. Delfour, University of Montreal Max D. Gunzburger, Florida State University J. William Helton, University of California, San Diego Arthur J. Krener, University of California, DavisKirsten Morris, University of WaterlooRichard Murray, California Institute of TechnologyEkkehard Sachs, University of Trier

Series VolumesDelfour, M. C. and Zolsio, J.-P., Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second EditionHovakimyan, Naira and Cao, Chengyu, L1 Adaptive Control Theory: Guaranteed Robustness with Fast AdaptationSpeyer, Jason L. and Jacobson, David H., Primer on Optimal Control TheoryBetts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second EditionShima, Tal and Rasmussen, Steven, eds., UAV Cooperative Decision and Control: Challenges and Practical ApproachesSpeyer, Jason L. and Chung, Walter H., Stochastic Processes, Estimation, and ControlKrstic, Miroslav and Smyshlyaev, Andrey, Boundary Control of PDEs: A Course on Backstepping DesignsIto, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and ApplicationsXue, Dingy, Chen, YangQuan, and Atherton, Derek P., Linear Feedback Control: Analysis and Design with MATLABHanson, Floyd B., Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation Michiels, Wim and Niculescu, Silviu-Iulian, Stability and Stabilization of Time-Delay Systems: An Eigenvalue- Based ApproachIoannou, Petros and Fidan, Bars, Adaptive Control TutorialBhaya, Amit and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix ProblemsRobinett III, Rush D., Wilson, David G., Eisler, G. Richard, and Hurtado, John E., Applied Dynamic Programming for Optimization of Dynamical SystemsHuang, J., Nonlinear Output Regulation: Theory and ApplicationsHaslinger, J. and Mkinen, R. A. E., Introduction to Shape Optimization: Theory, Approximation, and ComputationAntoulas, Athanasios C., Approximation of Large-Scale Dynamical SystemsGunzburger, Max D., Perspectives in Flow Control and OptimizationDelfour, M. C. and Zolsio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and OptimizationBetts, John T., Practical Methods for Optimal Control Using Nonlinear ProgrammingEl Ghaoui, Laurent and Niculescu, Silviu-Iulian, eds., Advances in Linear Matrix Inequality Methods in ControlHelton, J. William and James, Matthew R., Extending H1 Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives

Society for Industrial and Applied MathematicsPhiladelphia

Shapes andGeometriesMetrics, Analysis, Differential Calculus, and OptimizationSecond edition

M. c. delfourUniversit de MontralMontral, QubecCanada

J.-P. ZolsioNational Center for Scientific Research (CNRS) and National Institute for Research in Computer Science and Control (INRIA)Sophia Antipolis France

is a registered trademark.

Copyright 2011 by the Society for Industrial and Applied Mathematics

10 9 8 7 6 5 4 3 2 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 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA.

Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended.

The research of the first author was supported by the Canada Council, which initiated the work presented in this book through a Killam Fellowship; the National Sciences and Engineering Research Council of Canada; and the FQRNT program of the Ministre de lducation du Qubec.

Library of Congress Cataloging-in-Publication Data Delfour, Michel C., 1943- Shapes and geometries : metrics, analysis, differential calculus, and optimization / M. C. Delfour, J.-P. Zolsio. -- 2nd ed. p. cm. Includes bibliographical references and index. ISBN 978-0-898719-36-9 (hardcover : alk. paper) 1. Shape theory (Topology) I. Zolsio, J.-P. II. Title. QA612.7.D45 2011 514.24--dc22 2010028846

This book is dedicated to

Alice, Jeanne, Jean, and Roger

j

Contents

List of Figures xvii

Preface xix1 Objectives and Scope of the Book . . . . . . . . . . . . . . . . . . . . xix2 Overview of the Second Edition . . . . . . . . . . . . . . . . . . . . . xx3 Intended Audience . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii4 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

1 Introduction: Examples, Background, and Perspectives 11 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Geometry as a Variable . . . . . . . . . . . . . . . . . . . . . 11.2 Outline of the Introductory Chapter . . . . . . . . . . . . . . 3

2 A Simple One-Dimensional Example . . . . . . . . . . . . . . . . . . 33 Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Optimal Triangular Meshing . . . . . . . . . . . . . . . . . . . . . . . 76 Modeling Free Boundary Problems . . . . . . . . . . . . . . . . . . . 10

6.1 Free Interface between Two Materials . . . . . . . . . . . . . 116.2 Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 12

7 Design of a Thermal Diuser . . . . . . . . . . . . . . . . . . . . . . 137.1 Description of the Physical Problem . . . . . . . . . . . . . . 137.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 147.3 Reformulation of the Problem . . . . . . . . . . . . . . . . . . 167.4 Scaling of the Problem . . . . . . . . . . . . . . . . . . . . . . 167.5 Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8 Design of a Thermal Radiator . . . . . . . . . . . . . . . . . . . . . . 188.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 188.2 Scaling of the Problem . . . . . . . . . . . . . . . . . . . . . . 20

9 A Glimpse into Segmentation of Images . . . . . . . . . . . . . . . . 219.1 Automatic Image Processing . . . . . . . . . . . . . . . . . . 219.2 Image Smoothing/Filtering by Convolution and Edge Detectors 22

9.2.1 Construction of the Convolution of I . . . . . . . . 239.2.2 Space-Frequency Uncertainty Relationship . . . . . 239.2.3 Laplacian Detector . . . . . . . . . . . . . . . . . . . 25

vii

viii Contents

9.3 Objective Functions Dened on the Whole Edge . . . . . . . 269.3.1 Eulerian Shape Semiderivative . . . . . . . . . . . . 269.3.2 From Local to Global Conditions on the Edge . . . 27

9.4 Snakes, Geodesic Active Contours, and Level Sets . . . . . . 289.4.1 Objective Functions Dened on the Contours . . . . 289.4.2 Snakes and Geodesic Active Contours . . . . . . . . 289.4.3 Level Set Method . . . . . . . . . . . . . . . . . . . 299.4.4 Velocity Carried by the Normal . . . . . . . . . . . 309.4.5 Extension of the Level Set Equations . . . . . . . . 31

9.5 Objective Function Dened on the Whole Image . . . . . . . 329.5.1 Tikhonov Regularization/Smoothing . . . . . . . . . 329.5.2 Objective Function of Mumford and Shah . . . . . . 329.5.3 Relaxation of the (N 1)-Hausdor Measure . . . . 339.5.4 Relaxation to BV-, Hs-, and SBV-Functions . . . . 339.5.5 Cracked Sets and Density Perimeter . . . . . . . . . 35

10 Shapes and Geometries: Background and Perspectives . . . . . . . . 3610.1 Parametrize Geometries by Functions or Functions by

Geometries? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.2 Shape Analysis in Mechanics and Mathematics . . . . . . . . 3910.3 Characteristic Functions: Surface Measure and Geometric

Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . 4110.4 Distance Functions: Smoothness, Normal, and Curvatures . . 4110.5 Shape Optimization: Compliance Analysis and Sensitivity

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.6 Shape Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 4410.7 Shape Calculus and Tangential Dierential Calculus . . . . . 4610.8 Shape Analysis in This Book . . . . . . . . . . . . . . . . . . 46

11 Shapes and Geometries: Second Edition . . . . . . . . . . . . . . . . 4711.1 Geometries Parametrized by Functions . . . . . . . . . . . . . 4811.2 Functions Parametrized by Geometries . . . . . . . . . . . . . 5011.3 Shape Continuity and Optimization . . . . . . . . . . . . . . 5211.4 Derivatives, Shape and Tangential Dierential Calculuses, and

Derivatives under State Constraints . . . . . . . . . . . . . . 53

2 Classical Descriptions of Geometries and Their Properties 551 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 Notation and Denitions . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 562.2 Abelian Group Structures on Subsets of a Fixed Holdall D . 56

2.2.1 First Abelian Group Structure on (P(D),) . . . . 572.2.2 Second Abelian Group Structure on (P(D),) . . . 58

2.3 Connected Space, Path-Connected Space, and GeodesicDistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.4 Bouligands Contingent Cone, Dual Cone, and Normal Cone 592.5 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.5.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . 60

Contents ix

2.5.2 The Space Wm,p0 () . . . . . . . . . . . . . . . . . . 612.5.3 Embedding of H10 () into H

10 (D) . . . . . . . . . . 62

2.5.4 Projection Operator . . . . . . . . . . . . . . . . . . 632.6 Spaces of Continuous and Dierentiable Functions . . . . . . 63

2.6.1 Continuous and Ck Functions . . . . . . . . . . . . 632.6.2 Holder (C0,) and Lipschitz (C0,1) Continuous

Functions . . . . . . . . . . . . . . . . . . . . . . . . 652.6.3 Embedding Theorem . . . . . . . . . . . . . . . . . 652.6.4 Identity Ck,1() = W k+1,(): From Convex to

Path-Connected Domains via the Geodesic Distance 663 Sets Locally Described by an Homeomorphism or a Dieomorphism 67

3.1 Sets of Classes Ck and Ck, . . . . . . . . . . . . . . . . . . . 673.2 Boundary Integral, Canonical Density, and Hausdor Measures 70

3.2.1 Boundary Integral for Sets of Class C1 . . . . . . . 703.2.2 Integral on Submanifolds . . . . . . . . . . . . . . . 713.2.3 Hausdor Measures . . . . . . . . . . . . . . . . . . 72

3.3 Fundamental Forms and Principal Curvatures . . . . . . . . . 734 Sets Globally Described by the Level Sets of a Function . . . . . . . 755 Sets Locally Described by the Epigraph of a Function . . . . . . . . 78

5.1 Local C0 Epigraphs, C0 Epigraphs, and Equi-C0 Epigraphsand the Space H of Dominating Functions . . . . . . . . . . . 79

5.2 Local Ck,-Epigraphs and Holderian/Lipschitzian Sets . . . . 875.3 Local Ck,-Epigraphs and Sets of Class Ck, . . . . . . . . . . 895.4 Locally Lipschitzian Sets: Some Examples and Properties . . 92

5.4.1 Examples and Continuous Linear Extensions . . . . 925.4.2 Convex Sets . . . . . . . . . . . . . . . . . . . . . . 935.4.3 Boundary Measure and Integral for Lipschitzian Sets 945.4.4 Geodesic Distance in a Domain and in Its Boundary 975.4.5 Nonhomogeneous Neumann and Dirichlet Problems 100

6 Sets Locally Described by a Geometric Property . . . . . . . . . . . 1016.1 Denitions and Main Results . . . . . . . . . . . . . . . . . . 1026.2 Equivalence of Geometric Segment and C0 Epigraph

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.3 Equivalence of the Uniform Fat Segment and the Equi-C0

Epigraph Properties . . . . . . . . . . . . . . . . . . . . . . . 1096.4 Uniform Cone/Cusp Properties and Holderian/Lipschitzian

Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.4.1 Uniform Cone Property and Lipschitzian Sets . . . 1146.4.2 Uniform Cusp Property and Holderian Sets . . . . . 115

6.5 Hausdor Measure and Dimension of the Boundary . . . . . 116

3 Courant Metrics on Images of a Set 1231 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232 Generic Constructions of Micheletti . . . . . . . . . . . . . . . . . . . 124

2.1 Space F() of Transformations of RN . . . . . . . . . . . . . 1242.2 Dieomorphisms for B(RN,RN) and C0 (RN,RN) . . . . . . 136

x Contents

2.3 Closed Subgroups G . . . . . . . . . . . . . . . . . . . . . . . 1382.4 Courant Metric on the Quotient Group F()/G . . . . . . . 1402.5 Assumptions for Bk(RN,RN), Ck(RN,RN), and Ck0 (RN,RN) 143

2.5.1 Checking the Assumptions . . . . . . . . . . . . . . 1432.5.2 Perturbations of the Identity and Tangent Space . . 147

2.6 Assumptions for Ck,1(RN,RN) and Ck,10 (RN,RN) . . . . . . 149

2.6.1 Checking the Assumptions . . . . . . . . . . . . . . 1492.6.2 Perturbations of the Identity and Tangent Space . . 151

3 Generalization to All Homeomorphisms and Ck-Dieomorphisms . . 153

4 Transformations Generated by Velocities 1591 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1592 Metrics on Transformations Generated by Velocities . . . . . . . . . 161

2.1 Subgroup G of Transformations Generated by Velocities . . 1612.2 Complete Metrics on G and Geodesics . . . . . . . . . . . . 1662.3 Constructions of Azencott and Trouve . . . . . . . . . . . . . 169

3 Semiderivatives via Transformations Generated by Velocities . . . . 1703.1 Shape Function . . . . . . . . . . . . . . . . . . . . . . . . . . 1703.2 Gateaux and Hadamard Semiderivatives . . . . . . . . . . . . 1703.3 Examples of Families of Transformations of Domains . . . . . 173

3.3.1 C-Domains . . . . . . . . . . . . . . . . . . . . . . 1733.3.2 Ck-Domains . . . . . . . . . . . . . . . . . . . . . . 1753.3.3 Cartesian Graphs . . . . . . . . . . . . . . . . . . . 1763.3.4 Polar Coordinates and Star-Shaped Domains . . . . 1773.3.5 Level Sets . . . . . . . . . . . . . . . . . . . . . . . . 178

4 Unconstrained Families of Domains . . . . . . . . . . . . . . . . . . . 1804.1 Equivalence between Velocities and Transformations . . . . . 1804.2 Perturbations of the Identity . . . . . . . . . . . . . . . . . . 1834.3 Equivalence for Special Families of Velocities . . . . . . . . . 185

5 Constrained Families of Domains . . . . . . . . . . . . . . . . . . . . 1935.1 Equivalence between Velocities and Transformations . . . . . 1935.2 Transformation of Condition (V2D) into a Linear

Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2006 Continuity of Shape Functions along Velocity Flows . . . . . . . . . 202

5 Metrics via Characteristic Functions 2091 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2092 Abelian Group Structure on Measurable Characteristic Functions . . 210

2.1 Group Structure on X(RN) . . . . . . . . . . . . . . . . . . 2102.2 Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2112.3 Complete Metric for Characteristic Functions in

Lp-Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 2123 Lebesgue Measurable Characteristic Functions . . . . . . . . . . . . 214

3.1 Strong Topologies and C-Approximations . . . . . . . . . . 2143.2 Weak Topologies and Microstructures . . . . . . . . . . . . . 2153.3 Nice or Measure Theoretic Representative . . . . . . . . . . . 220

Contents xi

3.4 The Family of Convex Sets . . . . . . . . . . . . . . . . . . . 2233.5 Sobolev Spaces for Measurable Domains . . . . . . . . . . . . 224

4 Some Compliance Problems with Two Materials . . . . . . . . . . . 2284.1 Transmission Problem and Compliance . . . . . . . . . . . . 2284.2 The Original Problem of Cea and Malanowski . . . . . . . . . 2354.3 Relaxation and Homogenization . . . . . . . . . . . . . . . . 239

5 Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 2406 Caccioppoli or Finite Perimeter Sets . . . . . . . . . . . . . . . . . . 244

6.1 Finite Perimeter Sets . . . . . . . . . . . . . . . . . . . . . . . 2456.2 Decomposition of the Integral along Level Sets . . . . . . . . 2516.3 Domains of Class W ,p(D), 0 < 1/p, p 1, and a Cascade

of Complete Metric Spaces . . . . . . . . . . . . . . . . . . . 2526.4 Compactness and Uniform Cone Property . . . . . . . . . . . 254

7 Existence for the Bernoulli Free Boundary Problem . . . . . . . . . . 2587.1 An Example: Elementary Modeling of the Water Wave . . . 2587.2 Existence for a Class of Free Boundary Problems . . . . . . . 2607.3 Weak Solutions of Some Generic Free Boundary Problems . . 262

7.3.1 Problem without Constraint . . . . . . . . . . . . . 2627.3.2 Constraint on the Measure of the Domain . . . . 264

7.4 Weak Existence with Surface Tension . . . . . . . . . . . . . 265

6 Metrics via Distance Functions 2671 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2672 Uniform Metric Topologies . . . . . . . . . . . . . . . . . . . . . . . 268

2.1 Family of Distance Functions Cd(D) . . . . . . . . . . . . . . 2682.2 PompeiuHausdor Metric on Cd(D) . . . . . . . . . . . . . . 2692.3 Uniform Complementary Metric Topology and Ccd(D) . . . . 2752.4 Families Ccd(E;D) and C

cd,loc(E;D) . . . . . . . . . . . . . . . 278

3 Projection, Skeleton, Crack, and Dierentiability . . . . . . . . . . . 2794 W 1,p-Metric Topology and Characteristic Functions . . . . . . . . . 292

4.1 Motivations and Main Properties . . . . . . . . . . . . . . . . 2924.2 Weak W 1,p-Topology . . . . . . . . . . . . . . . . . . . . . . . 296

5 Sets of Bounded and Locally Bounded Curvature . . . . . . . . . . . 2995.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

6 Reach and Federers Sets of Positive Reach . . . . . . . . . . . . . . 3036.1 Denitions and Main Properties . . . . . . . . . . . . . . . . 3036.2 Ck-Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 3106.3 A Compact Family of Sets with Uniform Positive Reach . . . 315

7 Approximation by Dilated Sets/Tubular Neighborhoods and CriticalPoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

8 Characterization of Convex Sets . . . . . . . . . . . . . . . . . . . . 3188.1 Convex Sets and Properties of dA . . . . . . . . . . . . . . . . 3188.2 Semiconvexity and BV Character of dA . . . . . . . . . . . . 3208.3 Closed Convex Hull of A and Fenchel Transform of dA . . . . 3228.4 Families of Convex Sets Cd(D), Ccd(D), Ccd(E;D), and

Ccd,loc(E;D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

xii Contents

9 Compactness Theorems for Sets of Bounded Curvature . . . . . . . . 3249.1 Global Conditions in D . . . . . . . . . . . . . . . . . . . . . 3259.2 Local Conditions in Tubular Neighborhoods . . . . . . . . . . 327

7 Metrics via Oriented Distance Functions 3351 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3352 Uniform Metric Topology . . . . . . . . . . . . . . . . . . . . . . . . 337

2.1 The Family of Oriented Distance Functions Cb(D) . . . . . . 3372.2 Uniform Metric Topology . . . . . . . . . . . . . . . . . . . . 339

3 Projection, Skeleton, Crack, and Dierentiability . . . . . . . . . . . 3444 W 1,p(D)-Metric Topology and the Family C0b (D) . . . . . . . . . . . 349

4.1 Motivations and Main Properties . . . . . . . . . . . . . . . . 3494.2 Weak W 1,p-Topology . . . . . . . . . . . . . . . . . . . . . . . 352

5 Boundary of Bounded and Locally Bounded Curvature . . . . . . . . 3545.1 Examples and Limit of Tubular Norms as h Goes to Zero . . 355

6 Approximation by Dilated Sets/Tubular Neighborhoods . . . . . . . 3587 Federers Sets of Positive Reach . . . . . . . . . . . . . . . . . . . . . 361

7.1 Approximation by Dilated Sets/Tubular Neighborhoods . . . 3617.2 Boundaries with Positive Reach . . . . . . . . . . . . . . . . . 363

8 Boundary Smoothness and Smoothness of bA . . . . . . . . . . . . . 3659 Sobolev or Wm,p Domains . . . . . . . . . . . . . . . . . . . . . . . . 37310 Characterization of Convex and Semiconvex Sets . . . . . . . . . . . 375

10.1 Convex Sets and Convexity of bA . . . . . . . . . . . . . . . . 37510.2 Families of Convex Sets Cb(D), Cb(E;D), and

Cb,loc(E;D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37910.3 BV Character of bA and Semiconvex Sets . . . . . . . . . . . 380

11 Compactness and Sets of Bounded Curvature . . . . . . . . . . . . . 38111.1 Global Conditions on D . . . . . . . . . . . . . . . . . . . . . 38211.2 Local Conditions in Tubular Neighborhoods . . . . . . . . . . 382

12 Finite Density Perimeter and Compactness . . . . . . . . . . . . . . 38513 Compactness and Uniform Fat Segment Property . . . . . . . . . . . 387

13.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 38713.2 Equivalent Conditions on the Local Graph Functions . . . . . 391

14 Compactness under the Uniform Fat Segment Property and a Boundon a Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39314.1 De Giorgi Perimeter of Caccioppoli Sets . . . . . . . . . . . . 39314.2 Finite Density Perimeter . . . . . . . . . . . . . . . . . . . . . 394

15 The Families of Cracked Sets . . . . . . . . . . . . . . . . . . . . . . 39416 A Variation of the Image Segmentation Problem of Mumford

and Shah . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40016.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 40016.2 Cracked Sets without the Perimeter . . . . . . . . . . . . . . 401

16.2.1 Technical Lemmas . . . . . . . . . . . . . . . . . . . 40116.2.2 Another Compactness Theorem . . . . . . . . . . . 40216.2.3 Proof of Theorem 16.1 . . . . . . . . . . . . . . . . . 402

16.3 Existence of a Cracked Set with Minimum Density Perimeter 405

Contents xiii

16.4 Uniform Bound or Penalization Term in the ObjectiveFunction on the Density Perimeter . . . . . . . . . . . . . . . 407

8 Shape Continuity and Optimization 4091 Introduction and Generic Examples . . . . . . . . . . . . . . . . . . . 409

1.1 First Generic Example . . . . . . . . . . . . . . . . . . . . . . 4111.2 Second Generic Example . . . . . . . . . . . . . . . . . . . . . 4111.3 Third Generic Example . . . . . . . . . . . . . . . . . . . . . 4111.4 Fourth Generic Example . . . . . . . . . . . . . . . . . . . . . 412

2 Upper Semicontinuity and Maximization of the First Eigenvalue . . 4123 Continuity of the Transmission Problem . . . . . . . . . . . . . . . . 4174 Continuity of the Homogeneous Dirichlet Boundary Value Problem . 418

4.1 Classical, Relaxed, and Overrelaxed Problems . . . . . . . . . 4184.2 Classical Dirichlet Boundary Value Problem . . . . . . . . . . 4214.3 Overrelaxed Dirichlet Boundary Value Problem . . . . . . . . 423

4.3.1 Approximation by Transmission Problems . . . . . . 4234.3.2 Continuity with Respect to X(D) in the

Lp(D)-Topology . . . . . . . . . . . . . . . . . . . . 4244.4 Relaxed Dirichlet Boundary Value Problem . . . . . . . . . . 425

5 Continuity of the Homogeneous Neumann Boundary Value Problem 4266 Elements of Capacity Theory . . . . . . . . . . . . . . . . . . . . . . 429

6.1 Denition and Basic Properties . . . . . . . . . . . . . . . . . 4296.2 Quasi-continuous Representative and H1-Functions . . . . . . 4316.3 Transport of Sets of Zero Capacity . . . . . . . . . . . . . . . 432

7 Crack-Free Sets and Some Applications . . . . . . . . . . . . . . . . 4347.1 Denitions and Properties . . . . . . . . . . . . . . . . . . . . 4347.2 Continuity and Optimization over L(D, r,O, ) . . . . . . . . 437

7.2.1 Continuity of the Classical Homogeneous DirichletBoundary Condition . . . . . . . . . . . . . . . . . . 437

7.2.2 Minimization/Maximization of the FirstEigenvalue . . . . . . . . . . . . . . . . . . . . . . . 438

8 Continuity under Capacity Constraints . . . . . . . . . . . . . . . . . 4409 Compact Families Oc,r(D) and Lc,r(O, D) . . . . . . . . . . . . . . . 447

9.1 Compact Family Oc,r(D) . . . . . . . . . . . . . . . . . . . . 4479.2 Compact Family Lc,r(O, D) and Thick Set Property . . . . . 4509.3 Maximizing the Eigenvalue A() . . . . . . . . . . . . . . . 4529.4 State Constrained Minimization Problems . . . . . . . . . . . 4539.5 Examples with a Constraint on the Gradient . . . . . . . . . 454

9 Shape and Tangential Dierential Calculuses 4571 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4572 Review of Dierentiation in Topological Vector Spaces . . . . . . . . 458

2.1 Denitions of Semiderivatives and Derivatives . . . . . . . . . 4582.2 Derivatives in Normed Vector Spaces . . . . . . . . . . . . . . 4612.3 Locally Lipschitz Functions . . . . . . . . . . . . . . . . . . . 4652.4 Chain Rule for Semiderivatives . . . . . . . . . . . . . . . . . 465

xiv Contents

2.5 Semiderivatives of Convex Functions . . . . . . . . . . . . . . 4672.6 Hadamard Semiderivative and Velocity Method . . . . . . . . 469

3 First-Order Shape Semiderivatives and Derivatives . . . . . . . . . . 4713.1 Eulerian and Hadamard Semiderivatives . . . . . . . . . . . . 4713.2 Hadamard Semidierentiability and Courant Metric

Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4763.3 Perturbations of the Identity and Gateaux and Frechet

Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4763.4 Shape Gradient and Structure Theorem . . . . . . . . . . . . 479

4 Elements of Shape Calculus . . . . . . . . . . . . . . . . . . . . . . . 4824.1 Basic Formula for Domain Integrals . . . . . . . . . . . . . . 4824.2 Basic Formula for Boundary Integrals . . . . . . . . . . . . . 4844.3 Examples of Shape Derivatives . . . . . . . . . . . . . . . . . 486

4.3.1 Volume of and Surface Area of . . . . . . . . . 4864.3.2 H1()-Norm . . . . . . . . . . . . . . . . . . . . . . 4874.3.3 Normal Derivative . . . . . . . . . . . . . . . . . . . 488

5 Elements of Tangential Calculus . . . . . . . . . . . . . . . . . . . . 4915.1 Intrinsic Denition of the Tangential Gradient . . . . . . . . 4925.2 First-Order Derivatives . . . . . . . . . . . . . . . . . . . . . 4955.3 Second-Order Derivatives . . . . . . . . . . . . . . . . . . . . 4965.4 A Few Useful Formulae and the Chain Rule . . . . . . . . . . 4975.5 The Stokes and Green Formulae . . . . . . . . . . . . . . . . 4985.6 Relation between Tangential and Covariant Derivatives . . . 4985.7 Back to the Example of Section 4.3.3 . . . . . . . . . . . . . . 501

6 Second-Order Semiderivative and Shape Hessian . . . . . . . . . . . 5016.1 Second-Order Derivative of the Domain Integral . . . . . . . 5026.2 Basic Formula for Domain Integrals . . . . . . . . . . . . . . 5046.3 Nonautonomous Case . . . . . . . . . . . . . . . . . . . . . . 5056.4 Autonomous Case . . . . . . . . . . . . . . . . . . . . . . . . 5106.5 Decomposition of d2J(;V (0),W (0)) . . . . . . . . . . . . . 515

10 Shape Gradients under a State Equation Constraint 5191 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5192 Min Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

2.1 An Illustrative Example and a Shape Variational Principle . 5212.2 Function Space Parametrization . . . . . . . . . . . . . . . . 5222.3 Dierentiability of a Minimum with Respect to a

Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5232.4 Application of the Theorem . . . . . . . . . . . . . . . . . . . 5262.5 Domain and Boundary Integral Expressions of the Shape

Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5303 Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 5324 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

4.1 Transport of Hk0 () by Wk,-Transformations of RN . . . . 536

4.2 Laplacian and Bi-Laplacian . . . . . . . . . . . . . . . . . . . 5374.3 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 546

Contents xv

5 Saddle Point Formulation and Function Space Parametrization . . . 5515.1 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . 5515.2 Saddle Point Formulation . . . . . . . . . . . . . . . . . . . . 5525.3 Function Space Parametrization . . . . . . . . . . . . . . . . 5535.4 Dierentiability of a Saddle Point with Respect to a

Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5555.5 Application of the Theorem . . . . . . . . . . . . . . . . . . . 5595.6 Domain and Boundary Expressions for the Shape Gradient . 561

6 Multipliers and Function Space Embedding . . . . . . . . . . . . . . 5626.1 The Nonhomogeneous Dirichlet Problem . . . . . . . . . . . . 5626.2 A Saddle Point Formulation of the State Equation . . . . . . 5636.3 Saddle Point Expression of the Objective Function . . . . . . 5646.4 Verication of the Assumptions of Theorem 5.1 . . . . . . . . 566

Elements of Bibliography 571

Index of Notation 615

Index 619

List of Figures

1.1 Graph of J(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Column of height one and cross section area A under the load . . . 51.3 Triangulation and basis function associated with node Mi. . . . . . . 81.4 Fixed domain D and its partition into 1 and 2. . . . . . . . . . . 111.5 Heat spreading scheme for high-power solid-state devices. . . . . . . 141.6 (A) Volume and its boundary ; (B) Surface A generating ;

(C) Surface D generating . . . . . . . . . . . . . . . . . . . . . . . . 151.7 Volume and its cross section. . . . . . . . . . . . . . . . . . . . . . 191.8 Volume and its generating surface A. . . . . . . . . . . . . . . . . 201.9 Image I of objects and their segmentation in the frame D. . . . . . . 221.10 Image I containing black curves or cracks in the frame D. . . . . . . 221.11 Example of a two-dimensional strongly cracked set. . . . . . . . . . . 351.12 Example of a surface with facets associated with a ball. . . . . . . . 37

2.1 Dieomorphism gx from U(x) to B. . . . . . . . . . . . . . . . . . . 682.2 Local epigraph representation (N = 2). . . . . . . . . . . . . . . . . . 792.3 Domain 0 and its image T (0) spiraling around the origin. . . . . . 912.4 Domain 0 and its image T (0) zigzagging towards the origin. . . . 922.5 Examples of arbitrary and axially symmetrical O around the

direction d = Ax(0, eN ). . . . . . . . . . . . . . . . . . . . . . . . . . 1092.6 The cone x+AxC(, ) in the direction AxeN . . . . . . . . . . . . . 1142.7 Domain for N = 2, 0 < < 1, e2 = (0, 1), = 1/6, = (1/6),

h() = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182.8 f(x) = dC(x)1/2 constructed on the Cantor set C for 2k + 1 = 3. . . 119

4.1 Transport of by the velocity eld V . . . . . . . . . . . . . . . . . . 171

5.1 Smiling sun and expressionless sun . . . . . . . . . . . . . . . . . 2205.2 Disconnected domain = 0 1 2. . . . . . . . . . . . . . . . . 2275.3 Fixed domain D and its partition into 1 and 2. . . . . . . . . . . 2285.4 The function f(x, y) = 56 (1 |x| |y|)6. . . . . . . . . . . . . . . . 2345.5 Optimal distribution and isotherms with k1 = 2 (black) and k2 = 1

(white) for the problem of section 4.1. . . . . . . . . . . . . . . . . . 2355.6 Optimal distribution and isotherms with k1 = 2 (black) and k2 = 1

(white) for the problem of Cea and Malanowski. . . . . . . . . . . . 239

xvii

xviii List of Figures

5.7 The staircase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.1 Skeletons Sk (A), Sk (A), and Sk (A) = Sk (A) Sk (A). . . . . . 2806.2 Nonuniqueness of the exterior normal. . . . . . . . . . . . . . . . . . 2866.3 Vertical stripes of Example 4.1. . . . . . . . . . . . . . . . . . . . . . 2936.4 dA for Examples 5.1, 5.2, and 5.3. . . . . . . . . . . . . . . . . . . 3016.5 Set of critical points of A. . . . . . . . . . . . . . . . . . . . . . . . . 318

7.1 bA for Examples 5.1, 5.2, and 5.3. . . . . . . . . . . . . . . . . . . . 3567.2 W 1,p-convergence of a sequence of open subsets {An : n 1} of R2

with uniformly bounded density perimeter to a set with emptyinterior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

7.3 Example of a two-dimensional strongly cracked set. . . . . . . . . . . 3967.4 The two-dimensional strongly cracked set of Figure 7.3 in an open

frame D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4007.5 The two open components 1 and 2 of the open domain for

N = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

Preface

1 Objectives and Scope of the BookThe objective of this book is to give a comprehensive presentation of mathematicalconstructions and tools that can be used to study problems where the modeling,optimization, or control variable is no longer a set of parameters or functions butthe shape or the structure of a geometric object. In that context, a good analyticalframework and good modeling techniques must be able to handle the occurrence ofsingular behaviors whenever they are compatible with the mechanics or the physicsof the problems at hand. In some optimization problems, the natural intuitivenotion of a geometric domain undergoes mutations into relaxed entities such asmicrostructures. So the objects under consideration need not be smooth open do-mains, or even sets, as long as they still makes sense mathematically.

This book covers the basic mathematical ideas, constructions, and methodsthat come from dierent elds of mathematical activities and areas of applicationsthat have often evolved in parallel directions. The scope of research is frighteninglybroad because it touches on areas that include classical geometry, modern partial dif-ferential equations, geometric measure theory, topological groups, and constrainedoptimization, with applications to classical mechanics of continuous media such asuid mechanics, elasticity theory, fracture theory, modern theories of optimal de-sign, optimal location and shape of geometric objects, free and moving boundaryproblems, and image processing. Innovative modeling or new issues raised in someapplications force a new look at the fundamentals of well-established mathematicalareas such as geometry, to relax basic notions of volume, perimeter, and curvatureor boundary value problems, and to nd suitable relaxations of solutions. In thatspirit, Henri Lebesgue was probably a pioneer when he relaxed the intuitive notionof volume to the one of measure on an equivalence class of measurable sets in 1907.He was followed in that endeavor in the early 1950s by the celebrated work of E. DeGiorgi, who used the relaxed notion of perimeter dened on the class of Caccioppolisets to solve Plateaus problem of minimal surfaces.

The material that is pertinent to the study of geometric objects and the en-tities and functions that are dened on them would necessitate an encyclopedicinvestment to bring together the basic theories and their elds of applications. Thisobjective is obviously beyond the scope of a single book and two authors. The

xix

xx Preface

coverage of this book is more modest. Yet, it contains most of the important fun-damentals at this stage of evolution of this expanding eld.

Even if shape analysis and optimization have undergone considerable and im-portant developments on the theoretical and numerical fronts, there are still culturalbarriers between areas of applications and between theories. The whole eld is ex-tremely active, and the best is yet to come with fundamental structures and toolsbeginning to emerge. It is hoped that this book will help to build new bridges andstimulate cross-fertilization of ideas and methods.

2 Overview of the Second EditionThe second edition is almost a new book. All chapters from the rst edition havebeen updated and, in most cases, considerably enriched with new material. Manychapters or parts of chapters have been completely rewritten following the devel-opments in the eld over the past 10 years. The book went from 9 to 10 chapterswith a more elaborate sectioning of each chapter in order to produce a much moredetailed table of contents. This makes it easier to nd specic material.

A series of illustrative generic examples has been added right at the begin-ning of the introductory Chapter 1 to motivate the reader and illustrate the basicdilemma: parametrize geometries by functions or functions by geometries? This isfollowed by the big picture: a section on background and perspectives and a moredetailed presentation of the second edition.

The former Chapter 2 has been split into Chapter 2 on the classical descrip-tions and properties of domains and sets and a new Chapter 3, where the importantmaterial on Courant metrics and the generic constructions of A. M. Micheletti havebeen reorganized and expanded. Basic denitions and material have been addedand regrouped at the beginning of Chapter 2: Abelian group structure on subsetsof a set, connected and path-connected spaces, function spaces, tangent and dualcones, and geodesic distance. The coverage of domains that verify some segmentproperty and have a local epigraph representation has been considerably expanded,and Lipschitzian (graph) domains are now dealt with as a special case.

The new Chapter 3 on domains and submanifolds that are the image of axed set considerably expands the material of the rst edition by bringing up thegeneral assumptions behind the generic constructions of A. M. Micheletti that leadto the Courant metrics on the quotient space of families of transformations bysubgroups of isometries such as identities, rotations, translations, or ips. Thegeneral results apply to a broad range of groups of transformations of the Euclideanspace and to arbitrary closed subgroups. New complete metrics on the whole spacesof homeomorphisms and Ck-dieomorphisms are also introduced to extend classicalresults for transformations of compact manifolds to general unbounded closed setsand open sets that are crack-free. This material is central in classical mechanicsand physics and in modern applications such as imaging and detection.

The former Chapter 7 on transformations versus ows of velocities has beenmoved right after the Courant metrics as Chapter 4 and considerably expanded.It now specializes the results of Chapter 3 to spaces of transformations that are

2. Overview of the Second Edition xxi

generated by the ow of a velocity eld over a generic time interval. One importantmotivation is to introduce a notion of semiderivatives as well as a tractable criterionfor continuity with respect to Courant metrics. Another motivation for the velocitypoint of view is the general framework of R. Azencott and A. Trouve starting in1994 with applications in imaging. They construct complete metrics in relationwith geodesic paths in spaces of dieomorphisms generated by a velocity eld.

The former Chapter 3 on the relaxation to measurable sets and Chapters 4 and5 on distance and oriented distance functions have become Chapters 5, 6, and 7.Those chapters have been renamed Metrics Generated by . . . in order to emphasizeone of the main thrusts of the book: the construction of complete metrics on shapesand geometries.1 Those chapters emphasize the function analytic description of setsand domains: construction of metric topologies and characterization of compactfamilies of sets or submanifolds in the Euclidean space. In that context, we are nowdealing with equivalence classes of sets that may or may not have an invariant openor closed representative in the class. For instance, they include Lebesgue measurablesets and Federers sets of positive reach. Many of the classical properties of sets canbe recovered from the smoothness or function analytic properties of those functions.

The former Chapter 6 on optimization of shape functions has been completelyrewritten and expanded as Chapter 8 on shape continuity and optimization. Withmeaningful metric topologies, we can now speak of continuity of a geometric objec-tive functional such as the volume, the perimeter, the mean curvature, etc., compactfamilies of sets, and existence of optimal geometries. The chapter concentrates oncontinuity issues related to shape optimization problems under state equation con-straints. A special family of state constrained problems are the ones for which theobjective function is dened as an inmum over a family of functions over a xeddomain or set such as the eigenvalue problems. We rst characterize the continuityof the transmission problem and the upper semicontinuity of the rst eigenvalue ofthe generalized Laplacian with respect to the domain. We then study the conti-nuity of the solution of the homogeneous Dirichlet and Neumann boundary valueproblems with respect to their underlying domain of denition since they requiredierent constructions and topologies that are generic of the two types of boundaryconditions even for more complex nonlinear partial dierential equations. An intro-duction is also given to the concepts and results from capacity theory from whichvery general families of sets stable with respect to boundary conditions can be con-structed. Note that some material has been moved from one chapter to another.For instance, section 7 on the continuity of the Dirichlet boundary problem in theformer Chapter 3 has been merged with the content of the former Chapter 4 in thenew Chapter 8.

The former Chapters 8 and 9 have become Chapters 9 and 10. They aredevoted to a modern version of the shape calculus, an introduction to the tangentialdierential calculus, and the shape derivatives under a state equation constraint. InChapters 3, 5, 6, and 7, we have constructed complete metric spaces of geometries.Those spaces are nonlinear and nonconvex. However, several of them have a group

1This is in line with current trends in the literature such as in the work of the 2009 Abel Prizewinner M. Gromov [1] and its applications in imaging by G. Sapiro [1] and F. Memoli and G.Sapiro [1] to identify objects up to an isometry.

xxii Preface

structure and, in some cases, it is possible to construct C1-paths in the groupfrom velocity elds. This leads to the notion of Eulerian semiderivative that issomehow the analogue of a derivative on a smooth manifold. In fact, two typesof semiderivatives are of interest: the weaker Gateaux style semiderivative and thestronger Hadamard style semiderivative. In the latter case, the classical chain ruleis still available even for nondierentiable functions. In order to prepare the groundfor shape derivatives, an enriched self-contained review of the pertinent material onsemiderivatives and derivatives in topological vector spaces is provided.

The important Chapter 10 concentrates on two generic examples often encoun-tered in shape optimization. The rst one is associated with the so-called complianceproblems, where the shape functional is itself the minimum of a domain-dependentenergy functional. The special feature of such functionals is that the adjoint statecoincides with the state. This obviously leads to considerable simplications in theanalysis. In that case, it will be shown that theorems on the dierentiability ofthe minimum of a functional with respect to a real parameter readily give explicitexpressions of the Eulerian semiderivative even when the minimizer is not unique.The second one will deal with shape functionals that can be expressed as the saddlepoint of some appropriate Lagrangian. As in the rst example, theorems on thedierentiability of the saddle point of a functional with respect to a real parameterreadily give explicit expressions of the Eulerian semiderivative even when the so-lution of the saddle point equations is not unique. Avoiding the dierentiation ofthe state equation with respect to the domain is particularly advantageous in shapeproblems.

3 Intended AudienceThe targeted audience is applied mathematicians and advanced engineers and sci-entists, but the book is also suitable for a broader audience of mathematicians as arelatively well-structured initiation to shape analysis and calculus techniques. Someof the chapters are fairly self-contained and of independent interest. They can beused as lecture notes for a mini-course. The material at the beginning of eachchapter is accessible to a broad audience, while the latter sections may sometimesrequire more mathematical maturity. Thus the book can be used as a graduate textas well as a reference book. It complements existing books that emphasize specicmechanical or engineering applications or numerical methods. It can be considereda companion to the book of J. Sokolowski and J.-P. Zolesio [9], Introductionto Shape Optimization, published in 1992.

Earlier versions of parts of this book have been used as lecture notes in grad-uate courses at the Universite de Montreal in 19861987, 19931994, 19951996,and 19971998 and at international meetings, workshops, or schools: Seminaire deMathematiques Superieures on Shape Optimization and Free Boundaries (Montreal,Canada, June 25 to July 13, 1990), short course on Shape Sensitivity Analysis(Kenitra, Morocco, December 1993), course of the COMETT MATARI EuropeanProgram on Shape Optimization and Mutational Equations (Sophia-Antipolis,France, September 27 to October 1, 1993), CRM Summer School on Boundaries,

4. Acknowledgments xxiii

Interfaces and Transitions (Ban, Canada, August 618, 1995), and CIME courseon Optimal Design (Troia, Portugal, June 1998).

4 AcknowledgmentsThe rst author is pleased to acknowledge the support of the Canada Council, whichinitiated the work presented in this book through a Killam Fellowship; the constantsupport of the National Sciences and Engineering Research Council of Canada;and the FQRNT program of the Ministe`re de lEducation du Quebec. Manythanks also to Louise Letendre and Andre Montpetit of the Centre de RecherchesMathematiques, who provided their technical support, experience, and talent overthe extended period of gestation of this book.

Michel DelfourJean-Paul Zolesio

August 13, 2009

Chapter 1

Introduction: Examples,Background, andPerspectives

1 Orientation1.1 Geometry as a Variable

The central object of this book1 is the geometry as a variable. As in the theoryof functions of real variables, we need a dierential calculus, spaces of geometries,evolution equations, and other familiar concepts in analysis when the variable isno longer a scalar, a vector, or a function, but is a geometric domain. This ismotivated by many important problems in science and engineering that involve thegeometry as a modeling, design, or control variable. In general the geometric objectswe shall consider will not be parametrized or structured. Yet we are not startingfrom scratch, and several building blocks are already available from many elds:geometric measure theory, physics of continuous media, free boundary problems,the parametrization of geometries by functions, the set derivative as the inverse ofthe integral, the parametrization of functions by geometries, the PompeiuHausdormetric, and so on.

As is often the case in mathematics, spaces of geometries and notions of de-rivatives with respect to the geometry are built from well-established elements offunctional analysis and dierential calculus. There are many ways to structurefamilies of geometries. For instance, a domain can be made variable by considering

1The numbering of equations, theorems, lemmas, corollaries, denitions, examples, and remarksis by chapter. When a reference to another chapter is necessary it is always followed by the wordsin Chapter and the number of the chapter. For instance, equation (2.18) in Chapter 9. Thetext of theorems, lemmas, and corollaries is slanted; the text of denitions, examples, and remarksis normal shape and ended by a square . This makes it possible to aesthetically emphasizecertain words especially in denitions. The bibliography is by author in alphabetical order. Foreach author or group of coauthors, there is a numbering in square brackets starting with [1]. Areference to an item by a single author is of the form J. Dieudonne [3] and a reference to anitem with several coauthors S. Agmon, A. Douglis, and L. Nirenberg [2]. Boxed formulae orstatements are used in some chapters for two distinct purposes. First, they emphasize certainimportant denitions, results, or identities; second, in long proofs of some theorems, lemmas, orcorollaries, they isolate key intermediary results which will be necessary to more easily follow thesubsequent steps of the proof.

1

2 Chapter 1. Introduction: Examples, Background, and Perspectives

the images of a xed domain by a family of dieomorphisms that belong to somefunction space over a xed domain. This naturally occurs in physics and mechan-ics, where the deformations of a continuous body or medium are smooth, or in thenumerical analysis of optimal design problems when working on a xed grid. Thisconstruction naturally leads to a group structure induced by the composition ofthe dieomorphisms. The underlying spaces are no longer topological vector spacesbut groups that can be endowed with a nice complete metric space structure byintroducing the Courant metric. The practitioner might or might not want to usethe underlying mathematical structure associated with his or her constructions, butit is there and it contains information that might guide the theory and inuencethe choice of the numerical methods used in the solution of the problem at hand.

The parametrization of a xed domain by a xed family of dieomorphismsobviously limits the family of variable domains. The topology of the images is simi-lar to the topology of the xed domain. Singularities that were not already presentthere cannot be created in the images. Other constructions make it possible to con-siderably enlarge the family of variable geometries and possibly open the doors topathological geometries that are no longer open sets with a nice boundary. Insteadof parametrizing the domains by functions or dieomorphisms, certain families offunctions can be parametrized by sets. A single function completely species a setor at least an equivalence class of sets. This includes the distance functions and thecharacteristic function, but also the support function from convex analysis. Per-haps the best known example of that construction is the PompeiuHausdor metrictopology. This is a very weak topology that does not preserve the volume of a set.When the volume, the perimeter, or the curvatures are important, such functionsmust be able to yield relaxed denitions of volume, perimeter, or curvatures. Thecharacteristic function that preserves the volume has many applications. It playeda fundamental role in the integration theory of Henri Lebesgue at the beginning ofthe 20th century. It was also used in the 1950s by E. De Giorgi to dene a relaxednotion of perimeter in the theory of minimal surfaces.

Another technique that has been used successfully in free or moving boundaryproblems, such as motion by mean curvature, shock waves, or detonation theory,is the use of level sets of a function to describe a free or moving boundary. Suchfunctions are often the solution of a system of partial dierential equations. This isanother way to build new tools from functional analysis. The choice of familiesof function parametrized sets or of families of set parametrized functions, or otherappropriate constructions, is obviously problem dependent, much like the choiceof function spaces of solutions in the theory of partial dierential equations oroptimization problems. This is one aspect of the geometry as a variable. Anotheraspect is to build the equivalent of a dierential calculus and the computationaland analytical tools that are essential in the characterization and computation ofgeometries. Again, we are not starting from scratch and many building blocks arealready available, but many questions and issues remain open.

This book aims at covering a small but fundamental part of that program. Wehad to make dicult choices and refer the reader to appropriate books and referencesfor background material such as geometric measure theory and specialized topicssuch as homogenization theory and microstructures which are available in excellent

2. A Simple One-Dimensional Example 3

books in English. It was unfortunately not possible to include references to theconsiderable literature on numerical methods, free and moving boundary problems,and optimization.

1.2 Outline of the Introductory Chapter

We rst give a series of generic examples where the shape or the geometry is themodeling, control, or optimization variable. They will be used in the subsequentchapters to illustrate the many ways such problems can be formulated. The rstexample is the celebrated problem of the optimal shape of a column formulated byLagrange in 1770 to prevent buckling. The extremization of the eigenvalues hasalso received considerable attention in the engineering literature. The free inter-face between two regions with dierent physical or mechanical properties is anothergeneric problem that can lead in some cases to a mixing or a microstructure. Twotypical problems arising from applications to condition the thermal environment ofsatellites are described in sections 7 and 8. The rst one is the design of a thermaldiuser of minimal weight subject to an inequality constraint on the output thermalpower ux. The second one is the design of a thermal radiator to eectively radiatelarge amounts of thermal power to space. The geometry is a volume of revolutionaround an axis that is completely specied by its height and the function which spec-ies its lateral boundary. Finally, we give a glimpse at image segmentation, whichis an example of shape/geometric identication problems. Many chapters of thisbook are of direct interest to imaging sciences.

Section 10 presents some background and perspectives. A fundamental issue isto nd tractable and preferably analytical representations of a geometry as a variablethat are compatible with the problems at hand. The generic examples suggest twotypes of representations: the ones where the geometry is parametrized by functionsand the ones where a family of functions is parametrized by the geometry. As isalways the case, the choice is very much problem dependent. In the rst case, thetopology of the variable sets is xed; in the second case the families of sets are muchlarger and topological changes are included. The book presents the two points ofview. Finally, section 11 sketches the material in the second edition of the book.

2 A Simple One-Dimensional ExampleA general feature of minimization problems with respect to a shape or a geometrysubject to a state equation constraint is that they are generally not convex andthat, when they have a solution, it is generally not unique. This is illustrated inthe following simple example from J. Cea [2]: minimize the objective function

J(a) def= a0

|ya(x) 1|2 dx,

where a 0 and ya is the solution of the boundary value problem (state equation)d2yadx2

(x) = 2 in a def= (0, a), dyadx

(0) = 0, ya(a) = 0. (2.1)


Here the one-dimensional geometric domain a = ]0, a[ is the minimizing variable.We recognize the classical structure of a control problem, except that the minimizingvariable is no longer under the integral sign but in the limits of the integral sign.One consequence of this dierence is that even the simplest problems will usuallynot be convex or convexiables. They will require a special analysis.

In this example it is easy to check that the solution of the state equation is

ya(x) = a2 x2 and J(a) = 815a5 4

3a3 + a.

The graph of J , shown in Figure 1.1, is not the graph of a convex function. Itsglobal minimum in a0 = 0, local maximum in a1, and local minimum in a2,

a1def=

34

(1 1

3

), a2

def=

34

(1 +

13

),

are all dierent.

a1 a2

Figure 1.1. Graph of J(a).

To avoid a trivial solution, a strictly positive lower bound must be put on a.A unique minimizing solution is obtained for a a1 where the gradient of J is zero.For 0 < a < a2, the minimum will occur at the preset lower or upper bound on a.

3 Buckling of ColumnsThe next example illustrates the fact that even simple problems can be nondier-entiable with respect to the geometry. This is generic of all eigenvalue problemswhen the eigenvalue is not simple.

One of the early optimal design problems was formulated by J. L. Lagrange[1] in 1770 (cf. I. Todhunter and K. Pearson [1]) and later studied by theDanish mathematician and astronomer T. Clausen [1] in 1849. It consists innding the best prole of a vertical column of xed volume to prevent buckling.

3. Buckling of Columns 5

It turns out that this problem is in fact a hidden maximization of an eigenvalue.Many incorrect solutions had been published until 1992. This problem and otherproblems related to columns have been revisited in a series of papers by S. J.Cox [1], S. J. Cox and M. L. Overton [1], S. J. Cox [2], and S. J. Cox andC. M. McCarthy [1]. Since Lagrange many authors have proposed solutions, buta complete theoretical and numerical solution for the buckling of a column was givenonly in 1992 by S. J. Cox and M. L. Overton [1]. The diculty was that theeigenvalue is not simple and hence not dierentiable with respect to the geometry.

Consider a normalized column of unit height and unit volume (see Figure 1.2).Denote by the magnitude of the normalized axial load and by u the resultingtransverse displacement. Assume that the potential energy is the sum of the bendingand elongation energies 1

0EI |u|2 dx

10

|u|2 dx,

where I is the second moment of area of the columns cross section and E is itsYoungs modulus. For suciently small load the minimum of this potential energywith respect to all admissible u is zero. Eulers buckling load of the column is thelargest for which this minimum is zero. This is equivalent to nding the followingminimum:

def= inf

0=uV

10 EI |u|2 dx 1

0 |u|2 dx, (3.1)

where V = H20 (0, 1) corresponds to the clamped case, but other types of bound-ary conditions can be contemplated. This is an eigenvalue problem with a specialRayleigh quotient.

Assume that E is constant and that the second moment of area I(x) of thecolumns cross section at the height x, 0 x 1, is equal to a constant c times its

0

x

1

normalized load

cross section areaA(x)

Figure 1.2. Column of height one and cross section area A under the load .


cross-sectional area A(x),

I(x) = cA(x) and 10

A(x) dx = 1.

Normalizing by cE and taking into account the engineering constraints

0 < A0 < A1, x [0, 1], 0 < A0 A(x) A1,we nally get

supAA

(A), (A) def= inf0=uV

10 A |u|2 dx 10 |u|2 dx

, (3.2)

A def={A L2(0, 1) : A0 A A1 and

10

A(x) dx = 1}. (3.3)

4 Eigenvalue ProblemsLet D be a bounded open Lipschitzian domain in RN and A L(D;L(RN,RN))be a matrix function dened on D such that

A = A and I A I (4.1)for some coercivity and continuity constants 0 < and A is the transpose ofA. Consider the minimization or the maximization of the rst eigenvalue

supA(D)

A()

infA(D)

A()

A() def= inf0=H10 () A dx

||2 dx, (4.2)

where A(D) is a family of admissible open subsets of D (cf., for instance, sections 2,7, and 9 of Chapter 8).

In the vectorial case, consider the following linear elasticity problem: ndU H10 ()3 such that

W H10 ()3,C(U) (W ) dx =

F W dx (4.3)

for some distributed loading F L2()3 and a constitutive law C which is a bilinearsymmetric transformation of

Sym3def={ L(R3;R3) : = }, def=

1i,j3ij ij

(L(R3;R3) is the space of all linear transformations of R3 or 3 3-matrices) underthe following assumption.

Assumption 4.1.The constitutive law is a transformation C Sym3 for which there exists a constant > 0 such that C for all Sym3.

5. Optimal Triangular Meshing 7

For instance, for the Lame constants > 0 and 0, the special constitutivelaw C = 2 + tr I satises Assumption 4.1 with = 2.

The associated bilinear form is

a(U,W )def=C(U) (W ) dx,

where U is a vector function, D(U) is the Jacobian matrix of U , and

(U) def=12(D(U) + D(U))

is the strain tensor. The rst eigenvalue is the minimum of the Rayleigh quotient

() = inf{

a(U,U) |U |2 dx

: U H10 ()3, U = 0}

.

A typical problem is to nd the sensitivity of the rst eigenvalue with respect tothe shape of the domain . In 1907, J. Hadamard [1] used displacements alongthe normal to the boundary of a C-domain to compute the derivative of therst eigenvalue of the clamped plate. As in the case of the column, this problem isnot dierentiable with respect to the geometry when the eigenvalue is not simple.

5 Optimal Triangular MeshingThe shape calculus that will be developed in Chapters 9 and 10 for problems gov-erned by partial dierential equations (the continuous model) will be readily ap-plicable to their discrete model as in the nite element discretization of ellipticboundary value problems. However, some care has to be exerted in the choice ofthe formula for the gradient, since the solution of a nite element discretizationproblem is usually less smooth than the solution of its continuous counterpart.

Most shape objective functionals will have two basic formulas for their shapegradient: a boundary expression and a volume expression. The boundary expressionis always nicer and more compact but can be applied only when the solution of theunderlying partial dierential equation is smooth and in most cases smoother thanthe nite element solution. This leads to serious computational errors. The rightformula to use is the less attractive volume expression that requires only the samesmoothness as the nite element solution. Numerous computational experimentsconrm that fact (cf., for instance, E. J. Haug and J. S. Arora [1] or E. J. Haug,K. K. Choi, and V. Komkov [1]). With the volume expression, the gradient ofthe objective function with respect to internal and boundary nodes can be readilyobtained by plugging in the right velocity eld.

A large class of linear elliptic boundary value problems can be expressed asthe minimum of a quadratic function over some Hilbert space. For instance, let be a bounded open domain in RN with a smooth boundary . The solution u ofthe boundary value problem

u = f in , u = 0 on


is the minimizing element in the Sobolev space H10 () of the energy functional

E(v,) def=|v|2 2f v dx,

J() def= infvH10 ()

E(v,) = E(u,) = |u|2 dx.

The elements of this problem are a Hilbert space V , a continuous symmetrical co-ercive bilinear form on V , and a continuous linear form on V . With this notation

u V, E(u) = infvV

E(v), E(v) def= a(v, v) 2 (v)

and u is the unique solution of the variational equation

u U, v V, a(u, v) = (v).In the nite element approximation of the solution u, a nite-dimensional

subspace Vh of V is used for some small mesh parameter h. The solution of theapproximate problem is given by

uh Vh, E(uh) = infvhVh

E(vh), uh Uh, vh Vh, a(uh, vh) = (vh).

It is easy to show that the error can be expressed as follows:

a(u uh, u uh) = u uh2V = 2 [E(uh) E(u)] .Assume that is a polygonal domain in RN. In the nite element method, thedomain is partitioned into a set h of small triangles by introducing nodes in

Mdef= {Mi : 1 i p}

Mdef= {Mi : p+ 1 i p+ q}

and M def= M M

for some integers p N + 1 and q 1 (see Figure 1.3). Therefore the triangular-ization h = h(M), the solution space Vh = Vh(M), and the solution uh = uh(M)are functions of the positions of the nodes of the set M . Assuming that the total

Mi 10

0

0

0 0

0

Figure 1.3. Triangulation and basis function associated with node Mi.

5. Optimal Triangular Meshing 9

number of nodes is xed, consider the following optimal triangularization problem:

infM

j(M), j(M) def= E(uh(h(M)),) = infvhVh(M)

E(vh,),

u uh2V =|(u uh)|2 dx = 2

[E(uh,(h(M))) E(u,)

]= 2

[J((h(M))) J()

],

J((h(M)))def= inf

vhVhE(vh,(h(M))) = E(uh,(h(M))) =

|uh|2 dx.

The objective is to compute the partial derivative of j(M) with respect to theth component (Mi) of the node Mi:

j

(Mi)(M).

This partial derivative can be computed by using the velocity method for the specialvelocity eld (cf. M. C. Delfour, G. Payre, and J.-P. Zolesio [3])

Vi(x) = bMi(x)e,

where bMi Vh is the (piecewise P 1) basis function associated with the node Mi:bMi(Mj) = ij for all i, j. In that method each point X of the plane is movedaccording to the solution of the vector dierential equation

dx

dt(t) = V (x(t)), x(0) = X.

This yields a transformation X Tt(X) def= x(t;X) : R2 R2 of the plane, and itis natural to introduce the following notion of semiderivative:

dJ(;V ) def= limt0

J(Tt()) J()t

.

For t 0 small, the velocity eld must be chosen in such a way that trianglesare moved onto triangles and the point Mi is moved in the direction e:

Mi Mit = Mi + te.This is achieved by choosing the following velocity eld:

Vi(t, x) = bMit(x)e,

where bMit is the piecewise P1 basis function associated with node Mit: bMit(Mj) =

ij for all i, j. This yields the family of transformations

Tt(x) = x+ t bMi(x)e

which moves the node Mi to Mi + te and hence

j

(Mi)(M) = dJ(;Vi).


Going back to our original example, introduce the shape functional

J() def= infvH10 ()

E(, v) = |u|2 dx, E(, v) =

|v|2 2 f v dx.

In Chapter 9, we shall show that we have the following boundary and volume ex-pressions for the derivative of J():

dJ(;V ) =

un2 V nd,

dJ(;V ) =A(0)u u 2 [div V (0)f +f V (0)]u dx,

A(0) = div V (0) I DV (0) DV (0).

For a P 1-approximation

Vhdef={v C0() : v|K P 1(K), K h

}and the trace of the normal derivative on is not dened. Thus, it is necessary touse the volume expression. For the velocity eld Vi

DVi = e bMi , divDVi = e bMi ,A(0) = e bMi I e bMi bMi e.

Since

j

(Mi)(M) = dJ(;Vi),

we nally obtain the formula for the derivative of the function j(M) with respectto node Mi in the direction e:

j

(Mi)(M) =

[e bMi I e bMi bMi e];uh uh

2 [e bMi f +f e bMi ]uh dx.Since the support of bMi consists of the triangles having Mi as a vertex, the gradientwith respect to the nodes can be constructed piece by piece by visiting each node.

6 Modeling Free Boundary ProblemsThe rst step towards the solution of a shape optimization is the mathematicalmodeling of the problem. Physical phenomena are often modeled on relativelysmooth or nice geometries. Adding an objective functional to the model will usuallypush the system towards rougher geometries or even microstructures. For instance,in the optimal design of plates the optimization of the prole of a plate led to highlyoscillating proles that looked like a comb with abrupt variations ranging from zero

6. Modeling Free Boundary Problems 11

to maximum thickness. The phenomenon began to be understood in 1975 with thepaper of N. Olhoff [1] for circular plates with the introduction of the mechanicalnotion of stieners. The optimal plate was a virtual plate, a microstructure, thatis a homogenized geometry. Another example is the Plateau problem of minimalsurfaces that experimentally exhibits surfaces with singularities. In both cases, itis mathematically natural to replace the geometry by a characteristic function, afunction that is equal to 1 on the set and 0 outside the set. Instead of optimizingover a restricted family of geometries, the problem is relaxed to the optimizationover a set of measurable characteristic functions that contains a much larger familyof geometries, including the ones with boundary singularities and/or an arbitrarynumber of holes.

6.1 Free Interface between Two Materials

Consider the optimal design problem studied by J. Cea and K. Malanowski [1]in 1970, where the optimization variable is the distribution of two materials withdierent physical characteristics within a xed domain D. It cannot a priori beassumed that the two regions are separated by a smooth interface and that eachregion is connected. This problem will be covered in more details in section 4 ofChapter 5.

Let D RN be a bounded open domain with Lipschitzian boundary D.Assume for the moment that the domain D is partitioned into two subdomains 1and 2 separated by a smooth interface 1 2 as illustrated in Figure 1.4.Domain 1 (resp., 2) is made up of a material characterized by a constant k1 > 0(resp., k2 > 0). Let y be the solution of the transmission problem

k1y = f in 1 and k2y = f in 2,y = 0 on D and k1

y

n1+ k2

y

n2= 0 on 1 2,

(6.1)

where n1 (resp., n2) is the unit outward normal to 1 (resp., 2) and f is a givenfunction in L2(D). Assume that k1 > k2. The objective is to maximize the equiva-lent of the compliance

J(1) = D

fy dx (6.2)

21

Figure 1.4. Fixed domain D and its partition into 1 and 2.


over all domains 1 in D subject to the following constraint on the volume ofmaterial k1 which occupies the part 1 of D:

m(1) , 0 < < m(D) (6.3)

for some constant .If denotes the characteristic function of the domain 1,

(x) = 1 if x 1 and 0 if x / 1,the compliance J() = J(1) can be expressed as the inmum over the Sobolevspace H10 (D) of an energy functional dened on the xed set D:

J() = minH10 (D)

E(, ), (6.4)

E(, ) def=D

(k1 + k2 (1 )) ||2 2fdx. (6.5)

J() can be minimized or maximized over some appropriate family of characteristicfunctions or with respect to their relaxation to functions between 0 and 1 that wouldcorrespond to microstructures. As in the eigenvalue problem, the objective functionis an inmum, but here the inmum is over a space that does not depend on thefunction that species the geometric domain. This will be handled by the specialtechniques of Chapter 10 for the dierentiation of the minimum of a functional.

6.2 Minimal Surfaces

The celebrated Plateaus problem, named after the Belgian physicist and profes-sor J. A. F. Plateau [1] (18011883), who did experimental observations on thegeometry of soap lms around 1873, also provides a nice example where the geome-try is a variable. It consists in nding the surface of least area among those boundedby a given curve. One of the diculties in studying the minimal surface problem isthe description of such surfaces in the usual language of dierential geometry. Forinstance, the set of possible singularities is not known.

Measure theoretic methods such as k-currents (k-dim surfaces) were used byE. R. Reifenberg [1, 2, 3, 4] around 1960, H. Federer and W. H. Fleming [1]in 1960 (normals and integral currents), F. J. Almgren, Jr. [1] in 1965 (varifolds),and H. Federer [5] in 1969.

In the early 1950s, E. De Giorgi [1, 2, 3] and R. Caccioppoli [1] considereda hypersurface in the N -dimensional Euclidean space RN as the boundary of aset. In order to obtain a boundary measure, they restricted their attention to setswhose characteristic function is of bounded variation. Their key property is anassociated natural notion of perimeter that extends the classical surface measure ofthe boundary of a smooth set to the larger family of Caccioppoli sets named afterthe celebrated Neapolitan mathematician Renato Caccioppoli.2

2In 1992 his tormented personality was remembered in a lm directed by Mario Martone, TheDeath of a Neapolitan Mathematician (Morte di un matematico napoletano).

7. Design of a Thermal Diuser 13

Caccioppoli sets occur in many shape optimization problems (or free boundaryproblems), where a surface tension is present on the (free) boundary, such as in thefree interface water/soil in a dam (C. Baiocchi, V. Comincioli, E. Magenes,and G. A. Pozzi [1]) in 1973 and in the free boundary of a water wave (M. Souliand J.-P. Zolesio [1, 2, 3, 4, 5]) in 1988. More details will be given in Chapter 5.

7 Design of a Thermal DiuserShape optimization problems are everywhere in engineering, physics, and medicine.We choose two illustrative examples that were proposed by the Canadian SpaceProgram in the 1980s. The rst one is the design of a thermal diuser to condi-tion the thermal environment of electronic devices in communication satellites; thesecond one is the design of a thermal radiator that will be described in the next sec-tion. There are more and more design and control problems coming from medicine.For instance, the design of endoprotheses such as valves, stents, and coils in bloodvessels or left ventricular assistance devices (cardiac pumps) in interventional car-diology helps to improve the health of patients and minimize the consequences andcosts of therapeutical interventions by going to mini-invasive procedures.

7.1 Description of the Physical Problem

This problem arises in connection with the use of high-power solid-state devices(HPSSD) in communication satellites (cf. M. C. Delfour, G. Payre, and J.-P. Zolesio [1]). An HPSSD dissipates a large amount of thermal power (typ.> 50 W) over a relatively small mounting surface (typ. 1.25 cm2). Yet, its junctiontemperature is required to be kept moderately low (typ. 110C). The thermalresistance from the junction to the mounting surface is known for any particularHPSSD (typ. 1C/W), so that the mounting surface is required to be kept ata lower temperature than the junction (typ. 60C). In a space application thethermal power must ultimately be dissipated to the environment by the mechanismof radiation. However, to radiate large amounts of thermal power at moderately lowtemperatures, correspondingly large radiating areas are required. Thus we have therequirement to eciently spread the high thermal power ux (TPF) at the HPSSDsource (typ. 40 W/cm2) to a low TPF at the radiator (typ. 0.04 W/cm2) so thatthe source temperature is maintained at an acceptably low level (typ. < 60C)at the mounting surface. The ecient spreading task is best accomplished usingheatpipes, but the snag in the scheme is that heatpipes can accept only a limitedmaximum TPF from a source (typ. max 4 W/cm2).

Hence we are led to the requirement for a thermal diuser. This device isinserted between the HPSSD and the heatpipes and reduces the TPF at the source(typ. > 40 W/cm2) to a level acceptable to the heatpipes (typ. > 4 W/cm2). Theheatpipes then suciently spread the heat over large space radiators, reducing theTPF from a level at the diuser (typ. 4 W/cm2) to that at the radiator (typ. 0.04W/cm2). This scheme of heat spreading is depicted in Figure 1.5.

It is the design of the thermal diuser which is the problem at hand. Wemay assume that the HPSSD presents a uniform thermal power ux to the diuser


heatpipesaddle

heatpipes

radiator to spaceschematic drawingnot to scale

thermaldiuser high-power

solid-state diuser

Figure 1.5. Heat spreading scheme for high-power solid-state devices.

at the HPSSD/diuser interface. Heatpipes are essentially isothermalizing devices,and we may assume that the diuser/heatpipe saddle interface is indeed isothermal.Any other surfaces of the diuser may be treated as adiabatic.

7.2 Statement of the Problem

Assume that the thermal diuser is a volume symmetrical about the z-axis(cf. Figure 1.6 (A)) whose boundary surface is made up of three regular pieces:the mounting surface 1 (a disk perpendicular to the z-axis with center in (r, z) =(0, 0)), the lateral adiabatic surface 2, and the interface 3 between the dif-fuser and the heatpipe saddle (a disk perpendicular to the z-axis with center in(r, z) = (0, L)).

The temperature distribution over this volume is the solution of the station-ary heat equation kT = 0 (T , the Laplacian of T ) with the following boundaryconditions on the surface = 1 2 3 (the boundary of ):

kT

n= qin on 1, k

T

n= 0 on 2, T = T3 (constant) on 3, (7.1)

where n always denotes the outward unit normal to the boundary surface andT/n is the normal derivative to the boundary surface ,

T

n= T n (T = the gradient of T ). (7.2)

The parameters appearing in (7.1) are

k = thermal conductivity (typ. 1.8W/cmC),

qin = uniform inward thermal power ux at the source (positive constant).


The radius R0 of the mounting surface 1 is xed so that the boundary surface 1is already given in the design problem.

For practical considerations, we assume that the diuser is solid without in-terior hollows or cutouts. The class of shapes for the diuser is characterized bythe design parameter L > 0 and the positive function R(z), 0 < z L, withR(0) = R0 > 0. They are volumes of revolution about the z-axis generated bythe surface A between the z-axis and the function R(z) (cf. Figure 1.6 (B)), that is,

def={(x, y, z) : 0 < z < L, x2 + y2 < R(z)2

}. (7.3)

So the shape of is completely specied by the length L > 0 and the functionR(z) > 0 on the interval [0, L].

(A) (B) (C)

z z

L 3

2

1

L

0 0

1

0y y 0 0 0R0 1

AR(z) S4

S3

S2

S1

D

Figure 1.6. (A) Volume and its boundary ; (B) Surface A generating; (C) Surface D generating .

Assuming that the diuser is made up of a homogeneous material of uniformdensity (no hollow) the design objective is to minimize the volume

J() def=dx =

L0

R(z)2 dz (7.4)

subject to a uniform constraint on the outward thermal power ux at the interface3 between the diuser and the heatpipe saddle:

supp3

k Tz

(p) qout or k Tn

+ qout 0 on 3, (7.5)

where qout is a specied positive constant.It is readily seen that the minimization problem (7.4) subject to the constraint

(7.5) (where T is the solution of the heat equation with the boundary conditions(7.1)) is independent of the xed temperature T3 on the boundary 3. In otherwords the optimal shape , if it exists, is independent of T3. As a result, from nowon we set T3 equal to 0.


7.3 Reformulation of the Problem

In a shape optimization problem the formulation is important from both the the-oretical and the numerical viewpoints. In particular condition (7.5) is dicult tonumerically handle since it involves the pointwise evaluation of the normal derivativeon the piece of boundary 3. This problem can be reformulated as the minimizationof T on 3, where T is now the solution of a variational inequality. Consider thefollowing minimization problem over the subspace of functions that are positive orzero on 3:

V +() def={v H1() : v|3 0

}, (7.6)

infvV +()

12|v|2 dx

1

qin v d+3

qout v d. (7.7)

H1() is the usual Sobolev space on the domain , and the inequality on 3has to be interpreted quasi-everywhere in the capacity sense. Leaving aside thosetechnicalities, the minimizing solution of (7.7) is characterized by

kT = 0 in , k Tn

= qin on 1, kT

n= 0 on 2,

T 0,(kT

n+ qout

) 0, T

(kT

n+ qout

)= 0 on 3.

(7.8)

The former constraint (7.5) is veried and replaced by the new constraint

T = 0 on 3. (7.9)

If there exists a nonempty domain of the form (7.3) such that T = 0 on 3, theproblem is feasible.

In this formulation the pointwise constraint on the normal derivative of thetemperature on 3 has been replaced by a pointwise constraint on the less de-manding trace of the temperature on 3. Yet, we now have to solve a variationalinequality instead of a variational equation for the temperature T .

7.4 Scaling of the Problem

In the above formulations the shape parameter L and the shape function R are notindependent of each other since the function R is dened on the interval [0, L]. Thismotivates the following changes of variables and the introduction of the dimension-less temperature y:

x 1 = xR0

, y 2 = yR0

, z = zL, 0 1,

L =L

R0, R() =

R(L)R0

,

y(1, 2, ) =k

LqinT (R01, R02, L),

Ddef={(1, 2, ) : 0 < < 1, 21 +

22 < R(z)

2}.


The parameter L now appears as a coecient in the partial dierential equation

L2(2y

21+

2y

22

)+

2y

2= 0 in D

with the following boundary conditions on the boundary S = S1 S2 S3 of D:y

A= 1 on S1,

y

A= 0 on S2, y = 0 on S3, (7.10)

where denotes the outward normal to the boundary surface S and y/A is theconormal derivative to the boundary surface S,

y

A= L2

(1

y

1+ 2

y

2

)+ 3

y

.

Finally, the optimal design problem depends only on the ratio q = qout/qin throughthe constraint

y

A+

qoutqin

0 on S3.

The design variables are the parameter L > 0 and the function R > 0 now denedon the xed interval [0, 1].

7.5 Design Problem

The fact that this specic design problem can be reduced to nding a parameterand a function gives the false or unfounded impression that it can now be solvedby standard mathematical programming and numerical methods. Early work onsuch problems revealed a dierent reality, such as oscillating boundaries and con-vergence towards nonphysical designs. Clearly, the geometry refused to be handledby standard methods without a better understanding of the underlying physics andits inception in the modeling of the geometric variable.

At the theoretical level, the existence of solution requires a concept of conti-nuity with respect to the geometry of the solution of either the heat equation withan inequality constraint on the TPF or the variational inequality with an equalityconstraint on the temperature. The other element is the lower semicontinuity of theobjective functional that is not too problematic for the volume functional as long asthe chosen topology on the geometry preserves the continuity of the volume func-tional. For instance, the classical Hausdor metric topology does not preserve thevolume. In the context of uid mechanics (cf., for instance, O. Pironneau [1]), itmeans that a drag minimizing sequence of sets with constant volume may convergeto a set with twice the volume (cf. Example 4.1 in Chapter 6). A wine makingindustry exploiting the convergence in the Hausdor metric topology could yieldmiraculous prots.

Other serious issues are, for instance, the lack of dierentiability of the solutionof a variational inequality at the continuous level that will inadvertently aect thedierentiability or the evolution of a gradient method at the discrete level. We shall


see that there is not only one topology for shapes but a whole range that selectivelypreserve some but not all of the geometrical features. Again the right choice isproblem dependent, much like the choice of the right Sobolev space in the theoryof partial dierential equations.

8 Design of a Thermal RadiatorCurrent trends indicate that future communications satellites and spacecrafts willgrow ever larger, consume ever more electrical power, and dissipate larger amountsof thermal energy. Various techniques and devices can be deployed to conditionthe thermal environment for payload boxes within a spacecraft, but it is desirableto employ those which oer good performance for low cost, low weight, and highreliability. A thermal radiator (or radiating n) which accepts a given TPF from apayload box and radiates it directly to space can oer good performance and highreliability at low cost. However, without careful design, such a radiator can be un-necessarily bulky and heavy. It is the mass-optimized design of the thermal radiatorwhich is the problem at hand (cf. M. C. Delfour, G. Payre, and J.-P. Zole-sio [2]). We may assume that the payload box presents a uniform TPF (typ. 0.1 to1.0 W/cm2) into the radiator at the box/radiator interface. The radiating surfaceis a second surface mirror which consists of a sheet of glass whose inner surface hassilver coating. We may assume that the TPF out of the radiator/space interfaceis governed by the T 4 radiation law, although we must account also for a constantTPF (typ. 0.01 W/cm2) into this interface from the sun. Any other surfaces ofthe radiator may be treated as adiabatic. Two constraints restrict freedom in thedesign of the thermal radiator:

(i) the maximum temperature at the box/radiator interface is not to exceed someconstant (typ. 50); and

(ii) no part of the radiator is to be thinner than some constant (typ. 1 mm).

8.1 Statement of the Problem

Assume that the radiator is a volume symmetrical about the z-axis (cf. Figure 1.7)whose boundary surface is made up of three regular pieces: the contact surface 1(a disk perpendicular to the z-axis with center at the point (r, z) = (0, 0)), thelateral adiabatic surface 2, and the radiating surface 3 (a disk perpendicular tothe z-axis with center at (r, z) = (0, L)). More precisely

1 ={(x, y, z) : z = 0 and x2 + y2 R20

},

2 ={(x, y, z) : x2 + y2 = R(z)2, 0 z L} ,

3 ={(x, y, z) : z = L and x2 + y2 R(L)2} , (8.1)

where the radius R0 > 0 (typ. 10 cm), the length L > 0, and the function

R : [0, L] R, R(0) = R0, R(z) > 0, 0 z L, (8.2)are given (R, the eld of real numbers).

8. Design of a Thermal Radiator 19

z z

L

0

3

2

1R00

ry

Figure 1.7. Volume and its cross section.

The temperature distribution (in Kelvin degrees) over the volume is thesolution of the stationary heat equation

T = 0 (the Laplacian of T ) (8.3)

with the following boundary conditions on the surface = 1 2 3 (theboundary of ):

kT

n= qin on 1, k

T

n= 0 on 2, k

T

n= T |T |3 + qs on 3, (8.4)

where n denotes the outward normal to the boundary surface , T/n is thenormal derivative on the boundary surface , and

T

n= T n (T = the gradient of T ).

The parameters appearing in (8.1)(8.4) are

k = thermal conductivity (typ. 1.8W/cmC),qin = uniform inward thermal power ux at the source (typ. 0.1 to 1.0 W/cm2),

= Boltzmanns constant (5.67 108 W/m2K4), = surface

[Michael C. Delfour, Jean-Paul Zolésio] Shapes an(BookFi.org)

Documents

Transcript of [Michael C. Delfour, Jean-Paul Zolésio] Shapes an(BookFi.org)