(Advances in design and control) michael c delfour, jean paul zolésio shapes and geometries metrics, analysis, differential calculus, and optimization, second edition (advances in design

Topics of interest include shape optimization, multidisciplinary design, trajectory optimization, feedback, and optimal control.. and Zolésio, J.-P., Shapes and Geometries: Analysis, Dif

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

Ralph C Smith, North Carolina State University

Editorial Board

Athanasios C Antoulas, Rice University

Siva Banda, Air Force Research Laboratory

Belinda A Batten, Oregon State University

John 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, Davis

Kirsten Morris, University of Waterloo

Richard Murray, California Institute of Technology

Ekkehard Sachs, University of Trier

Speyer, Jason L and Chung, Walter H., Stochastic Processes, Estimation, and Control

Krstic, Miroslav and Smyshlyaev, Andrey, Boundary Control of PDEs: A Course on Backstepping Designs Ito, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and Applications Xue, Dingyü, Chen, YangQuan, and Atherton, Derek P., Linear Feedback Control: Analysis and Design with MATLAB

Hanson, 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 Approach

Ioannou, Petros and Fidan, Barıs, Adaptive Control Tutorial

Bhaya, Amit and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix Problems Robinett III, Rush D., Wilson, David G., Eisler, G Richard, and Hurtado, John E., Applied Dynamic Programming for Optimization of Dynamical Systems

Huang, J., Nonlinear Output Regulation: Theory and Applications

Haslinger, J and Mäkinen, R A E., Introduction to Shape Optimization: Theory, Approximation, and Computation Antoulas, Athanasios C., Approximation of Large-Scale Dynamical Systems

Gunzburger, Max D., Perspectives in Flow Control and Optimization

Delfour, M C and Zolésio, 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-Iulian, eds., Advances in Linear Matrix Inequality Methods in Control Helton, J William and James, Matthew R., Extending H 1 Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives

¸

Society for Industrial and Applied Mathematics

National Center for Scientific Research (CNRS) and

National Institute for Research in Computer Science and Control (INRIA) Sophia Antipolis

France

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 Ministère de l’Éducation du Québec

Library of Congress Cataloging-in-Publication Data

Delfour, Michel C.,

Shapes and geometries : metrics, analysis, differential calculus, and optimization / M C Delfour, J.-P Zolésio 2nd ed

p cm

Includes bibliographical references and index

ISBN 978-0-898719-36-9 (hardcover : alk paper)

1 Shape theory (Topology) I Zolésio, J.-P II Title

QA612.7.D45 2011

514’.24 dc22

2010028846

This book is dedicated to Alice, Jeanne, Jean, and Roger

1 Objectives and Scope of the Book xix

2 Overview of the Second Edition xx

3 Intended Audience xxii

4 Acknowledgments xxiii

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

1.1 Geometry as a Variable 1

1.2 Outline of the Introductory Chapter 3

2 A Simple One-Dimensional Example 3

3 Buckling of Columns 4

4 Eigenvalue Problems 6

5 Optimal Triangular Meshing 7

6 Modeling Free Boundary Problems 10

6.1 Free Interface between Two Materials 11

6.2 Minimal Surfaces 12

7 Design of a Thermal Diffuser 13

7.1 Description of the Physical Problem 13

7.2 Statement of the Problem 14

7.3 Reformulation of the Problem 16

7.4 Scaling of the Problem 16

7.5 Design Problem 17

8 Design of a Thermal Radiator 18

8.1 Statement of the Problem 18

8.2 Scaling of the Problem 20

9 A Glimpse into Segmentation of Images 21

9.1 Automatic Image Processing 21

9.2 Image Smoothing/Filtering by Convolution and Edge Detectors 22 9.2.1 Construction of the Convolution of I 23

9.2.2 Space-Frequency Uncertainty Relationship 23

9.2.3 Laplacian Detector 25

vii

9.3 Objective Functions Defined on the Whole Edge 26

9.3.1 Eulerian Shape Semiderivative 26

9.3.2 From Local to Global Conditions on the Edge 27

9.4 Snakes, Geodesic Active Contours, and Level Sets 28

9.4.1 Objective Functions Defined on the Contours 28

9.4.2 Snakes and Geodesic Active Contours 28

9.4.3 Level Set Method 29

9.4.4 Velocity Carried by the Normal 30

9.4.5 Extension of the Level Set Equations 31

9.5 Objective Function Defined on the Whole Image 32

9.5.1 Tikhonov Regularization/Smoothing 32

9.5.2 Objective Function of Mumford and Shah 32

9.5.3 Relaxation of the (N − 1)-Hausdorff Measure 33

9.5.4 Relaxation to BV-, H s-, and SBV-Functions 33

9.5.5 Cracked Sets and Density Perimeter 35

10 Shapes and Geometries: Background and Perspectives 36

10.1 Parametrize Geometries by Functions or Functions by Geometries? 36

10.2 Shape Analysis in Mechanics and Mathematics 39

10.3 Characteristic Functions: Surface Measure and Geometric Measure Theory 41

10.4 Distance Functions: Smoothness, Normal, and Curvatures 41

10.5 Shape Optimization: Compliance Analysis and Sensitivity Analysis 43

10.6 Shape Derivatives 44

10.7 Shape Calculus and Tangential Differential Calculus 46

10.8 Shape Analysis in This Book 46

11 Shapes and Geometries: Second Edition 47

11.1 Geometries Parametrized by Functions 48

11.2 Functions Parametrized by Geometries 50

11.3 Shape Continuity and Optimization 52

11.4 Derivatives, Shape and Tangential Differential Calculuses, and Derivatives under State Constraints 53

2 Classical Descriptions of Geometries and Their Properties 55 1 Introduction 55

2 Notation and Definitions 56

2.1 Basic Notation 56

2.2 Abelian Group Structures on Subsets of a Fixed Holdall D 56 2.2.1 First Abelian Group Structure on (P(D),
) 57

2.2.2 Second Abelian Group Structure on (P(D), ) 58

2.3 Connected Space, Path-Connected Space, and Geodesic Distance 58

2.4 Bouligand’s Contingent Cone, Dual Cone, and Normal Cone 59 2.5 Sobolev Spaces 60

2.5.1 Definitions 60

2.5.2 The Space W0m,p(Ω) 61

2.5.3 Embedding of H1(Ω) into H1(D) 62

2.5.4 Projection Operator 63

2.6 Spaces of Continuous and Differentiable Functions 63

2.6.1 Continuous and C k Functions 63

2.6.2 H¨older (C 0,
) and Lipschitz (C 0,1) Continuous Functions 65

2.6.3 Embedding Theorem 65

2.6.4 Identity C k,1 (Ω) = W k+1, ∞(Ω): From Convex to Path-Connected Domains via the Geodesic Distance 66 3 Sets Locally Described by an Homeomorphism or a Diffeomorphism 67 3.1 Sets of Classes C k and C k,
67

3.2 Boundary Integral, Canonical Density, and Hausdorff Measures 70 3.2.1 Boundary Integral for Sets of Class C1 70

3.2.2 Integral on Submanifolds 71

3.2.3 Hausdorff Measures 72

3.3 Fundamental Forms and Principal Curvatures 73

4 Sets Globally Described by the Level Sets of a Function 75

5 Sets Locally Described by the Epigraph of a Function 78

5.1 Local C0 Epigraphs, C0 Epigraphs, and Equi-C0 Epigraphs and the SpaceH of Dominating Functions 79

5.2 Local C k,
-Epigraphs and H¨olderian/Lipschitzian Sets 87

5.3 Local C k,
-Epigraphs and Sets of Class C k,
89

5.4 Locally Lipschitzian Sets: Some Examples and Properties 92

5.4.1 Examples and Continuous Linear Extensions 92

5.4.2 Convex Sets 93

5.4.3 Boundary Measure and Integral for Lipschitzian Sets 94 5.4.4 Geodesic Distance in a Domain and in Its Boundary 97 5.4.5 Nonhomogeneous Neumann and Dirichlet Problems 100 6 Sets Locally Described by a Geometric Property 101

6.1 Definitions and Main Results 102

6.2 Equivalence of Geometric Segment and C0 Epigraph Properties 104

6.3 Equivalence of the Uniform Fat Segment and the Equi-C0 Epigraph Properties 109

6.4 Uniform Cone/Cusp Properties and H¨olderian/Lipschitzian Sets 113

6.4.1 Uniform Cone Property and Lipschitzian Sets 114

6.4.2 Uniform Cusp Property and H¨olderian Sets 115

6.5 Hausdorff Measure and Dimension of the Boundary 116

3 Courant Metrics on Images of a Set 123 1 Introduction 123

2 Generic Constructions of Micheletti 124

2.1 SpaceF(Θ) of Transformations of RN 124

2.2 Diffeomorphisms forB(RN, RN) and C ∞(RN, RN) 136

2.3 Closed Subgroups G 138

2.4 Courant Metric on the Quotient Group F(Θ)/G 140

2.5 Assumptions forB k(RN, RN), C k(RN, RN), and C k(RN, RN) 143 2.5.1 Checking the Assumptions 143

2.5.2 Perturbations of the Identity and Tangent Space 147

2.6 Assumptions for C k,1(RN, RN) and C0k,1(RN, RN) 149

2.6.1 Checking the Assumptions 149

2.6.2 Perturbations of the Identity and Tangent Space 151

3 Generalization to All Homeomorphisms and C k-Diffeomorphisms 153

4 Transformations Generated by Velocities 159 1 Introduction 159

2 Metrics on Transformations Generated by Velocities 161

2.1 Subgroup GΘ of Transformations Generated by Velocities 161

2.2 Complete Metrics on GΘ and Geodesics 166

2.3 Constructions of Azencott and Trouv´e 169

3 Semiderivatives via Transformations Generated by Velocities 170

3.1 Shape Function 170

3.2 Gateaux and Hadamard Semiderivatives 170

3.3 Examples of Families of Transformations of Domains 173

3.3.1 C ∞-Domains 173

3.3.2 C k-Domains 175

3.3.3 Cartesian Graphs 176

3.3.4 Polar Coordinates and Star-Shaped Domains 177

3.3.5 Level Sets 178

4 Unconstrained Families of Domains 180

4.1 Equivalence between Velocities and Transformations 180

4.2 Perturbations of the Identity 183

4.3 Equivalence for Special Families of Velocities 185

5 Constrained Families of Domains 193

5.1 Equivalence between Velocities and Transformations 193

5.2 Transformation of Condition (V2D) into a Linear Constraint 200

6 Continuity of Shape Functions along Velocity Flows 202

5 Metrics via Characteristic Functions 209 1 Introduction 209

2 Abelian Group Structure on Measurable Characteristic Functions 210

2.1 Group Structure on Xµ(RN) 210

2.2 Measure Spaces 211

2.3 Complete Metric for Characteristic Functions in L p-Topologies 212

3 Lebesgue Measurable Characteristic Functions 214

3.1 Strong Topologies and C ∞-Approximations 214

3.2 Weak Topologies and Microstructures 215

3.3 Nice or Measure Theoretic Representative 220

3.4 The Family of Convex Sets 223

3.5 Sobolev Spaces for Measurable Domains 224

4 Some Compliance Problems with Two Materials 228

4.1 Transmission Problem and Compliance 228

4.2 The Original Problem of C´ea and Malanowski 235

4.3 Relaxation and Homogenization 239

5 Buckling of Columns 240

6 Caccioppoli or Finite Perimeter Sets 244

6.1 Finite Perimeter Sets 245

6.2 Decomposition of the Integral along Level Sets 251

6.3 Domains of Class W ε,p (D), 0 ≤ ε < 1/p, p ≥ 1, and a Cascade of Complete Metric Spaces 252

6.4 Compactness and Uniform Cone Property 254

7 Existence for the Bernoulli Free Boundary Problem 258

7.1 An Example: Elementary Modeling of the Water Wave 258

7.2 Existence for a Class of Free Boundary Problems 260

7.3 Weak Solutions of Some Generic Free Boundary Problems 262

7.3.1 Problem without Constraint 262

7.3.2 Constraint on the Measure of the Domain Ω 264

7.4 Weak Existence with Surface Tension 265

6 Metrics via Distance Functions 267 1 Introduction 267

2 Uniform Metric Topologies 268

2.1 Family of Distance Functions C d (D) 268

2.2 Pomp´eiu–Hausdorff Metric on C d (D) 269

2.3 Uniform Complementary Metric Topology and C c d (D) 275

2.4 Families C c d (E; D) and C c d,loc (E; D) 278

3 Projection, Skeleton, Crack, and Differentiability 279

4 W 1,p-Metric Topology and Characteristic Functions 292

4.1 Motivations and Main Properties 292

4.2 Weak W 1,p-Topology 296

5 Sets of Bounded and Locally Bounded Curvature 299

5.1 Examples 301

6 Reach and Federer’s Sets of Positive Reach 303

6.1 Definitions and Main Properties 303

6.2 C k-Submanifolds 310

6.3 A Compact Family of Sets with Uniform Positive Reach 315

7 Approximation by Dilated Sets/Tubular Neighborhoods and Critical Points 316

8 Characterization of Convex Sets 318

8.1 Convex Sets and Properties of d A 318

8.2 Semiconvexity and BV Character of d A 320

8.3 Closed Convex Hull of A and Fenchel Transform of d A 322

8.4 Families of Convex SetsC d (D), C c d (D), C c d (E; D), and C c (E; D) 323

9 Compactness Theorems for Sets of Bounded Curvature 324

9.1 Global Conditions in D 325

9.2 Local Conditions in Tubular Neighborhoods 327

7 Metrics via Oriented Distance Functions 335 1 Introduction 335

2 Uniform Metric Topology 337

2.1 The Family of Oriented Distance Functions C b (D) 337

2.2 Uniform Metric Topology 339

3 Projection, Skeleton, Crack, and Differentiability 344

4 W 1,p (D)-Metric Topology and the Family C b0(D) 349

4.1 Motivations and Main Properties 349

4.2 Weak W 1,p-Topology 352

5 Boundary of Bounded and Locally Bounded Curvature 354

5.1 Examples and Limit of Tubular Norms as h Goes to Zero 355

6 Approximation by Dilated Sets/Tubular Neighborhoods 358

7 Federer’s Sets of Positive Reach 361

7.1 Approximation by Dilated Sets/Tubular Neighborhoods 361

7.2 Boundaries with Positive Reach 363

8 Boundary Smoothness and Smoothness of b A 365

9 Sobolev or W m,p Domains 373

10 Characterization of Convex and Semiconvex Sets 375

10.1 Convex Sets and Convexity of b A 375

10.2 Families of Convex SetsC b (D), C b (E; D), and C b,loc (E; D) 379

10.3 BV Character of b Aand Semiconvex Sets 380

11 Compactness and Sets of Bounded Curvature 381

11.1 Global Conditions on D 382

11.2 Local Conditions in Tubular Neighborhoods 382

12 Finite Density Perimeter and Compactness 385

13 Compactness and Uniform Fat Segment Property 387

13.1 Main Theorem 387

13.2 Equivalent Conditions on the Local Graph Functions 391

14 Compactness under the Uniform Fat Segment Property and a Bound on a Perimeter 393

14.1 De Giorgi Perimeter of Caccioppoli Sets 393

14.2 Finite Density Perimeter 394

15 The Families of Cracked Sets 394

16 A Variation of the Image Segmentation Problem of Mumford and Shah 400

16.1 Problem Formulation 400

16.2 Cracked Sets without the Perimeter 401

16.2.1 Technical Lemmas 401

16.2.2 Another Compactness Theorem 402

16.2.3 Proof of Theorem 16.1 402 16.3 Existence of a Cracked Set with Minimum Density Perimeter 405

16.4 Uniform Bound or Penalization Term in the Objective

Function on the Density Perimeter 407

8 Shape Continuity and Optimization 409 1 Introduction and Generic Examples 409

1.1 First Generic Example 411

1.2 Second Generic Example 411

1.3 Third Generic Example 411

1.4 Fourth Generic Example 412

2 Upper Semicontinuity and Maximization of the First Eigenvalue 412

3 Continuity of the Transmission Problem 417

4 Continuity of the Homogeneous Dirichlet Boundary Value Problem 418 4.1 Classical, Relaxed, and Overrelaxed Problems 418

4.2 Classical Dirichlet Boundary Value Problem 421

4.3 Overrelaxed Dirichlet Boundary Value Problem 423

4.3.1 Approximation by Transmission Problems 423

4.3.2 Continuity with Respect to X(D) in the L p (D)-Topology 424

4.4 Relaxed Dirichlet Boundary Value Problem 425

5 Continuity of the Homogeneous Neumann Boundary Value Problem 426 6 Elements of Capacity Theory 429

6.1 Definition and Basic Properties 429

6.2 Quasi-continuous Representative and H1-Functions 431

6.3 Transport of Sets of Zero Capacity 432

7 Crack-Free Sets and Some Applications 434

7.1 Definitions and Properties 434

7.2 Continuity and Optimization over L(D, r, O, λ) 437

7.2.1 Continuity of the Classical Homogeneous Dirichlet Boundary Condition 437

7.2.2 Minimization/Maximization of the First Eigenvalue 438

8 Continuity under Capacity Constraints 440

9 Compact FamiliesO c,r (D) and L c,r(O, D) 447

9.1 Compact FamilyO c,r (D) 447

9.2 Compact Family L c,r(O, D) and Thick Set Property 450

9.3 Maximizing the Eigenvalue λ A(Ω) 452

9.4 State Constrained Minimization Problems 453

9.5 Examples with a Constraint on the Gradient 454

9 Shape and Tangential Differential Calculuses 457 1 Introduction 457

2 Review of Differentiation in Topological Vector Spaces 458

2.1 Definitions of Semiderivatives and Derivatives 458

2.2 Derivatives in Normed Vector Spaces 461

2.3 Locally Lipschitz Functions 465

2.4 Chain Rule for Semiderivatives 465

2.5 Semiderivatives of Convex Functions 467

2.6 Hadamard Semiderivative and Velocity Method 469

3 First-Order Shape Semiderivatives and Derivatives 471

3.1 Eulerian and Hadamard Semiderivatives 471

3.2 Hadamard Semidifferentiability and Courant Metric Continuity 476

3.3 Perturbations of the Identity and Gateaux and Fr´echet Derivatives 476

3.4 Shape Gradient and Structure Theorem 479

4 Elements of Shape Calculus 482

4.1 Basic Formula for Domain Integrals 482

4.2 Basic Formula for Boundary Integrals 484

4.3 Examples of Shape Derivatives 486

4.3.1 Volume of Ω and Surface Area of Γ 486

4.3.2 H1(Ω)-Norm 487

4.3.3 Normal Derivative 488

5 Elements of Tangential Calculus 491

5.1 Intrinsic Definition of the Tangential Gradient 492

5.2 First-Order Derivatives 495

5.3 Second-Order Derivatives 496

5.4 A Few Useful Formulae and the Chain Rule 497

5.5 The Stokes and Green Formulae 498

5.6 Relation between Tangential and Covariant Derivatives 498

5.7 Back to the Example of Section 4.3.3 501

6 Second-Order Semiderivative and Shape Hessian 501

6.1 Second-Order Derivative of the Domain Integral 502

6.2 Basic Formula for Domain Integrals 504

6.3 Nonautonomous Case 505

6.4 Autonomous Case 510

6.5 Decomposition of d2J (Ω; V (0), W (0)) 515

10 Shape Gradients under a State Equation Constraint 519 1 Introduction 519

2 Min Formulation 521

2.1 An Illustrative Example and a Shape Variational Principle 521 2.2 Function Space Parametrization 522

2.3 Differentiability of a Minimum with Respect to a Parameter 523

2.4 Application of the Theorem 526

2.5 Domain and Boundary Integral Expressions of the Shape Gradient 530

3 Buckling of Columns 532

4 Eigenvalue Problems 535

4.1 Transport of H k (Ω) by W k, ∞-Transformations of RN 536

4.2 Laplacian and Bi-Laplacian 537

4.3 Linear Elasticity 546

5 Saddle Point Formulation and Function Space Parametrization 551

5.1 An Illustrative Example 551

5.2 Saddle Point Formulation 552

5.3 Function Space Parametrization 553

5.4 Differentiability of a Saddle Point with Respect to a Parameter 555

5.5 Application of the Theorem 559

5.6 Domain and Boundary Expressions for the Shape Gradient 561 6 Multipliers and Function Space Embedding 562

6.1 The Nonhomogeneous Dirichlet Problem 562

6.2 A Saddle Point Formulation of the State Equation 563

6.3 Saddle Point Expression of the Objective Function 564

6.4 Verification of the Assumptions of Theorem 5.1 566

List of Figures

1.1 Graph of J (a) . 4

1.2 Column of height one and cross section area A under the load . 5

1.3 Triangulation and basis function associated with node M i 8

1.4 Fixed domain D and its partition into Ω1and Ω2 11

1.5 Heat spreading scheme for high-power solid-state devices 14

1.6 (A) Volume Ω and its boundary Σ; (B) Surface A generating Ω; (C) Surface D generating
Ω 15

1.7 Volume Ω and its cross section 19

1.8 Volume
Ω and its generating surface A. 20

1.9 Image I of objects and their segmentation in the frame D . 22

1.10 Image I containing black curves or cracks in the frame D . 22

1.11 Example of a two-dimensional strongly cracked set 35

1.12 Example of a surface with facets associated with a ball 37

2.1 Diffeomorphism g x from U (x) to B. 68

2.2 Local epigraph representation (N = 2) . 79

2.3 Domain Ω0and its image T (Ω0) spiraling around the origin 91

2.4 Domain Ω0and its image T (Ω0) zigzagging towards the origin 92

2.5 Examples of arbitrary and axially symmetricalO around the direction d = A x (0, e N) 109

2.6 The cone x + A x C(λ, ω) in the direction A x e N 114

2.7 Domain Ω for N = 2, 0 < α < 1, e2 = (0, 1), ρ = 1/6, λ = (1/6) α, h(θ) = θ α 118

2.8 f (x) = d C (x) 1/2 constructed on the Cantor set C for 2k + 1 = 3 119

4.1 Transport of Ω by the velocity field V 171

5.1 Smiling sun Ω and expressionless sun
Ω 220

5.2 Disconnected domain Ω = Ω0∪ Ω1∪ Ω2 227

5.3 Fixed domain D and its partition into Ω1and Ω2 228

5.4 The function f (x, y) = 56 (1 − |x| − |y|)6 234

5.5 Optimal distribution and isotherms with k1= 2 (black) and k2= 1 (white) for the problem of section 4.1 235

5.6 Optimal distribution and isotherms with k1= 2 (black) and k2= 1 (white) for the problem of C´ea and Malanowski 239

xvii

5.7 The staircase 248

6.1 Skeletons Sk (A), Sk ( A), and Sk (∂A) = Sk (A) ∪ Sk (A) 280

6.2 Nonuniqueness of the exterior normal 286

6.3 Vertical stripes of Example 4.1 293

6.4 ∇d Afor Examples 5.1, 5.2, and 5.3 301

6.5 Set of critical points of A 318

7.1 ∇b A for Examples 5.1, 5.2, and 5.3 356

7.2 W 1,p-convergence of a sequence of open subsets {A n : n ≥ 1} of R2 with uniformly bounded density perimeter to a set with empty interior 387

7.3 Example of a two-dimensional strongly cracked set 396

7.4 The two-dimensional strongly cracked set of Figure 7.3 in an open frame D 400

7.5 The two open components Ω1and Ω2 of the open domain Ω for N = 2 406

The 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 physics

of the problems at hand In some optimization problems, the natural intuitive

notion of a geometric domain undergoes mutations into relaxed entities such as

microstructures So the objects under consideration need not be smooth open mains, or even sets, as long as they still makes sense mathematically

do-This book covers the basic mathematical ideas, constructions, and methodsthat come from different fields 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 asfluid 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 curvature

or boundary value problems, and to find suitable relaxations of solutions In thatspirit, Henri Lebesgue was probably a pioneer when he relaxed the intuitive notion

of 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 De

Giorgi, who used the relaxed notion of perimeter defined on the class of Caccioppoli

sets to solve Plateau’s problem of minimal surfaces

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

en-xix

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

fun-Even if shape analysis and optimization have undergone considerable and portant developments on the theoretical and numerical fronts, there are still culturalbarriers between areas of applications and between theories The whole field 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

The second edition is almost a new book All chapters from the first 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 field 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 find specific material

A series of illustrative generic examples has been added right at the 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

begin-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 important

material on Courant metrics and the generic constructions of A M Micheletti have

been reorganized and expanded Basic definitions and material have been addedand regrouped at the beginning of Chapter 2: Abelian group structure on subsets

of 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 afixed set considerably expands the material of the first edition by bringing up thegeneral assumptions behind the generic constructions of A M Micheletti that lead

to the Courant metrics on the quotient space of families of transformations by

subgroups of isometries such as identities, rotations, translations, or flips 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 spaces

of homeomorphisms and C k-diffeomorphisms 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 flows of velocities has been

moved 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

generated by the flow of a velocity field over a generic time interval One important

motivation 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 Trouv´e starting in

1994 with applications in imaging They construct complete metrics in relation

with geodesic paths in spaces of diffeomorphisms generated by a velocity field The former Chapter 3 on the relaxation to measurable sets and Chapters 4 and

5 on distance and oriented distance functions have become Chapters 5, 6, and 7 Those chapters have been renamed Metrics Generated by in order to emphasize

one of the main thrusts of the book: the construction of complete metrics on shapesand geometries.1 Those chapters emphasize the function analytic description of sets

and 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 open

or closed representative in the class For instance, they include Lebesgue measurablesets and Federer’s sets of positive reach Many of the classical properties of sets can

be recovered from the smoothness or function analytic properties of those functions.The former Chapter 6 on optimization of shape functions has been completely

rewritten and expanded as Chapter 8 on shape continuity and optimization With

meaningful metric topologies, we can now speak of continuity of a geometric 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 defined as an infimum over a family of functions over a fixeddomain or set such as the eigenvalue problems We first characterize the continuity

objec-of the transmission problem and the upper semicontinuity objec-of the first eigenvalue objec-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 definition since they requiredifferent constructions and topologies that are generic of the two types of boundaryconditions even for more complex nonlinear partial differential equations An intro-

duction is also given to the concepts and results from capacity theory from which

very general families of sets stable with respect to boundary conditions can be structed Note that some material has been moved from one chapter to another

con-For instance, section 7 on the continuity of the Dirichlet boundary problem in the

former 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 are

devoted to a modern version of the shape calculus, an introduction to the tangential

differential calculus, and the shape derivatives under a state equation constraint In

Chapters 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

1 This is in line with current trends in the literature such as in the work of the 2009 Abel Prize winner M Gromov [1] and its applications in imaging by G Sapiro [1] and F M´emoli and G.

[1] to identify objects up to an isometry.

structure and, in some cases, it is possible to construct C1-paths in the group

from velocity fields This leads to the notion of Eulerian semiderivative that is

somehow the analogue of a derivative on a smooth manifold In fact, two types

of semiderivatives are of interest: the weaker Gateaux style semiderivative and the stronger Hadamard style semiderivative In the latter case, the classical chain rule

is still available even for nondifferentiable 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 first one is associated with the so-called compliance

problems, where the shape functional is itself the minimum of a domain-dependent

energy functional The special feature of such functionals is that the adjoint statecoincides with the state This obviously leads to considerable simplifications in theanalysis In that case, it will be shown that theorems on the differentiability 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 first example, theorems on thedifferentiability 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 differentiation ofthe state equation with respect to the domain is particularly advantageous in shapeproblems

The targeted audience is applied mathematicians and advanced engineers and entists, but the book is also suitable for a broader audience of mathematicians as arelatively well-structured initiation to shape analysis and calculus techniques Some

sci-of the chapters are fairly self-contained and sci-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 text

as well as a reference book It complements existing books that emphasize specificmechanical or engineering applications or numerical methods It can be considered

a companion to the book of J Soko
lowski and J.-P Zol´esio [9], Introduction

to Shape Optimization, published in 1992.

Earlier versions of parts of this book have been used as lecture notes in uate courses at the Universit´e de Montr´eal in 1986–1987, 1993–1994, 1995–1996,and 1997–1998 and at international meetings, workshops, or schools: S´eminaire deMath´ematiques Sup´erieures on Shape Optimization and Free Boundaries (Montr´eal,

grad-Canada, June 25 to July 13, 1990), short course on Shape Sensitivity Analysis

(K´enitra, Morocco, December 1993), course of the COMETT MATARI European

Program on Shape Optimization and Mutational Equations (Sophia-Antipolis, France, September 27 to October 1, 1993), CRM Summer School on Boundaries,

Interfaces and Transitions (Banff, Canada, August 6–18, 1995), and CIME course

on Optimal Design (Troia, Portugal, June 1998).

The first 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 Minist`ere de l’ ´Education du Qu´ebec Manythanks also to Louise Letendre and Andr´e Montpetit of the Centre de RecherchesMath´ematiques, who provided their technical support, experience, and talent overthe extended period of gestation of this book

Michel Delfour

Jean-Paul Zol´esio

August 13, 2009

The central object of this book1 is the geometry as a variable As in the theory

of functions of real variables, we need a differential calculus, spaces of geometries,evolution equations, and other familiar concepts in analysis when the variable is

no 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 objects

we shall consider will not be parametrized or structured Yet we are not startingfrom scratch, and several building blocks are already available from many fields: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 Pomp´eiu–Hausdorffmetric, and so on

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

de-1The numbering of equations, theorems, lemmas, corollaries, definitions, examples, and remarks

is by chapter When a reference to another chapter is necessary it is always followed by the words

in Chapter and the number of the chapter For instance, “equation (2.18) in Chapter 9.” The

text of theorems, lemmas, and corollaries is slanted; the text of definitions, examples, and remarks

is normal shape and ended by a square This makes it possible to aesthetically emphasize certain words especially in definitions The bibliography is by author in alphabetical order For each author or group of coauthors, there is a numbering in square brackets starting with [1] A

reference to an item by a single author is of the form J Dieudonn´ e [3] and a reference to an

item with several coauthors S Agmon, A Douglis, and L Nirenberg [2] Boxed formulae or

statements are used in some chapters for two distinct purposes First, they emphasize certain

important definitions, results, or identities; second, in long proofs of some theorems, lemmas, or corollaries, they isolate key intermediary results which will be necessary to more easily follow the subsequent steps of the proof.

1

the images of a fixed domain by a family of diffeomorphisms that belong to somefunction space over a fixed 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 fixed grid Thisconstruction naturally leads to a group structure induced by the composition ofthe diffeomorphisms The underlying spaces are no longer topological vector spacesbut groups that can be endowed with a nice complete metric space structure by

introducing the Courant metric The practitioner might or might not want to use

the underlying mathematical structure associated with his or her constructions, but

it is there and it contains information that might guide the theory and influencethe choice of the numerical methods used in the solution of the problem at hand.The parametrization of a fixed domain by a fixed family of diffeomorphismsobviously limits the family of variable domains The topology of the images is simi-lar to the topology of the fixed 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 Instead

of parametrizing the domains by functions or diffeomorphisms, certain families offunctions can be parametrized by sets A single function completely specifies a set

or 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 Pomp´eiu–Hausdorff 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 definitions of volume, perimeter, or curvatures Thecharacteristic function that preserves the volume has many applications It played

a 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 define a relaxed

notion of perimeter in the theory of minimal surfaces.

Another technique that has been used successfully in free or moving boundary

problems, 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 differential equations This isanother way to build new tools from functional analysis The choice of families

of function parametrized sets or of families of set parametrized functions, or otherappropriate constructions, is obviously problem dependent, much like the choice

of function spaces of solutions in the theory of partial differential equations oroptimization problems This is one aspect of the geometry as a variable Anotheraspect is to build the equivalent of a differential 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 difficult choices and refer the reader to appropriate books and references

for background material such as geometric measure theory and specialized topics

such as homogenization theory and microstructures which are available in excellent

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

We first 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 firstexample 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 different physical or mechanical properties is another

generic problem that can lead in some cases to a mixing or a microstructure Two

typical problems arising from applications to condition the thermal environment ofsatellites are described in sections 7 and 8 The first one is the design of a thermaldiffuser of minimal weight subject to an inequality constraint on the output thermalpower flux The second one is the design of a thermal radiator to effectively radiatelarge amounts of thermal power to space The geometry is a volume of revolutionaround an axis that is completely specified by its height and the function which spec-

ifies its lateral boundary Finally, we give a glimpse at image segmentation, which

is an example of shape/geometric identification problems Many chapters of this

book are of direct interest to imaging sciences.

Section 10 presents some background and perspectives A fundamental issue is

to find tractable and preferably analytical representations of a geometry as a variable

that are compatible with the problems at hand The generic examples suggest two

types of representations: the ones where the geometry is parametrized by functions and the ones where a family of functions is parametrized by the geometry As is

always the case, the choice is very much problem dependent In the first case, thetopology of the variable sets is fixed; 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

A 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 C´ea[2]: minimize the objective function

Here the one-dimensional geometric domain Ωa = ]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 difference is that even the simplest problems will usuallynot be convex or convexifiables They will require a special analysis

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

=

34

=

34

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 ≥ a1where the gradient of J is zero For 0 < a < a2, the minimum will occur at the preset lower or upper bound on a.

The next example illustrates the fact that even simple problems can be entiable with respect to the geometry This is generic of all eigenvalue problemswhen the eigenvalue is not simple

nondiffer-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 the Danish mathematician and astronomer T Clausen [1] in 1849 It consists in

finding the best profile of a vertical column of fixed volume to prevent buckling

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 and

C M McCarthy[1] Since Lagrange many authors have proposed solutions, but

a complete theoretical and numerical solution for the buckling of a column was given

only in 1992 by S J Cox and M L Overton [1] The difficulty was that the

eigenvalue is not simple and hence not differentiable 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 resulting

transverse displacement Assume that the potential energy is the sum of the bendingand elongation energies

 1 0

EI |u  |2dx − 

 1 0

|u  |2dx,

where I is the second moment of area of the column’s cross section and E is its Young’s modulus For sufficiently small load  the minimum of this potential energy with respect to all admissible u is zero Euler’s buckling load λ of the column is the largest  for which this minimum is zero This is equivalent to finding the following

where V = H2(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 the column’s cross section at the height x, 0 ≤ x ≤ 1, is equal to a constant c times its

cross-sectional area A(x),

I(x) = c A(x) and

 1 0

A(x) dx = 1



Let D be a bounded open Lipschitzian domain in RNand A ∈ L ∞ (D; L(RN, RN))

be a matrix function defined on D such that

∗ A = A and αI ≤ A ≤ βI (4.1)

for some coercivity and continuity constants 0 < α ≤ β and ∗ A is the transpose of

A Consider the minimization or the maximization of the first eigenvalue

Cε(U ) ·· ε(W ) dx =

Ω

(L(R3; R3) is the space of all linear transformations of R3or 3× 3-matrices) under

the following assumption

Assumption 4.1.

The constitutive law is a transformation C ∈ Sym3for which there exists a constant

α > 0 such that Cτ ·· τ ≥ α τ·· τ for all τ ∈ Sym

For instance, for the Lam´e constants µ > 0 and λ ≥ 0, the special constitutive law Cτ = 2µ τ + λ tr τ I satisfies Assumption 4.1 with α = 2µ.

The associated bilinear form is

aΩ(U, W )def=

Ω

A typical problem is to find the sensitivity of the first eigenvalue with respect to

the shape of the domain Ω In 1907, J Hadamard [1] used displacements along

the normal to the boundary Γ of a C ∞-domain to compute the derivative of the

first eigenvalue of the clamped plate As in the case of the column, this problem isnot differentiable with respect to the geometry when the eigenvalue is not simple

The shape calculus that will be developed in Chapters 9 and 10 for problems erned by partial differential equations (the continuous model ) will be readily ap- plicable to their discrete model as in the finite element discretization of elliptic

gov-boundary value problems However, some care has to be exerted in the choice ofthe formula for the gradient, since the solution of a finite element discretizationproblem is usually less smooth than the solution of its continuous counterpart.Most shape objective functionals will have two basic formulas for their shape

gradient: a boundary expression and a volume expression The boundary expression

is always nicer and more compact but can be applied only when the solution of the

underlying partial differential equation is smooth and in most cases smoother than

the finite 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 finite element solution Numerous computational experiments

confirm 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 of

the objective function with respect to internal and boundary nodes can be readilyobtained by plugging in the right velocity field

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 Ω

be a bounded open domain in RN with a smooth boundary Γ The solution u of

the boundary value problem

−∆u = f in Ω, u = 0 on Γ

is the minimizing element in the Sobolev space H1(Ω) of the energy functional

E(v, Ω)def=

Ω

|∇v|2− 2f v dx,

J (Ω)def= inf

v ∈H1(Ω)E(v, Ω) = E(u, Ω) = −

Ω

approximate problem is given by

∃u h ∈ V h , E(u h) = inf

v h ∈V h

E(v h ), ⇒ ∃u h ∈ U h , ∀v h ∈ V h , a(u h , v h ) = (v h ).

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

a(u − u h , u − u h) =u − u h 2

V = 2 [E(u h)− E(u)]

Assume that Ω is a polygonal domain in RN In the finite element method, the

domain is partitioned into a set τ h of small triangles by introducing nodes in ¯Ω

M def={M i ∈ Ω : 1 ≤ i ≤ p}

∂M def= {M i ∈ ∂Ω : p + 1 ≤ i ≤ p + q}and M

def

for some integers p ≥ N + 1 and q ≥ 1 (see Figure 1.3) Therefore the

triangular-ization τ h = τ h (M ), the solution space V h = V h (M ), and the solution u h = u h (M ) are functions of the positions of the nodes of the set M Assuming that the total

00

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

|∇(u − u h)|2dx = 2 E(u h , Ω(τ h (M ))) − E(u, Ω)

|∇u h |2dx.

The objective is to compute the partial derivative of j(M ) with respect to the

th component (M i)
of the node M i:

where b M i ∈ V h is the (piecewise P1) basis function associated with the node M i:

b M i (M j ) = δ ij for all i, j In that method each point X of the plane is moved

according to the solution of the vector differential equation

dx

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

This yields a transformation X → T t (X)def= x(t; X) : R2→ R2of the plane, and it

is natural to introduce the following notion of semiderivative:

dJ (Ω; V )def= lim

t 0

J (T t(Ω))− J(Ω)

t .

For t ≥ 0 small, the velocity field must be chosen in such a way that triangles

M i → M it = M i e

This is achieved by choosing the following velocity field:

V i
(t, x) = b M it e
,

where b M it is the piecewise P1basis function associated with node M it : b M it (M j) =

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

T t (x) = x + t b M i e

which moves the node M i to M i e
and hence

∂j

∂(M)
(M ) = dJ (Ω; V i
).

Going back to our original example, introduce the shape functional

J (Ω)def= inf

v ∈H1(Ω)E(Ω, v) = −

Ω

|∇u|2dx, E(Ω, v) =

Ω

|∇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 (Ω):

dJ (Ω; V ) = −

Γ

∂u

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 P1-approximation

V hdef= v ∈ C0( ¯Ω) : v | K ∈ P1(K), ∀K ∈ τ h

and the trace of the normal derivative on Γ is not defined Thus, it is necessary to

use the volume expression For the velocity field V i

The first 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 profile of a plate led to highlyoscillating profiles that looked like a comb with abrupt variations ranging from zero

to maximum thickness The phenomenon began to be understood in 1975 with the

paper of N Olhoff [1] for circular plates with the introduction of the mechanical

notion of stiffeners The optimal plate was a virtual plate, a microstructure, that

is a homogenized geometry Another example is the Plateau problem of minimalsurfaces that experimentally exhibits surfaces with singularities In both cases, it

is 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 family

of geometries, including the ones with boundary singularities and/or an arbitrarynumber of holes

Consider the optimal design problem studied by J C´eaand K Malanowski [1]

in 1970, where the optimization variable is the distribution of two materials with

different physical characteristics within a fixed domain D It cannot a priori be

assumed 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 Ω1and Ω2 separated by a smooth interface ∂Ω1∩ ∂Ω2 as illustrated in Figure 1.4.Domain Ω1(resp., Ω2) is made up of a material characterized by a constant k1> 0

(resp., k2> 0) Let y be the solution of the transmission problem

over all domains Ω1 in D subject to the following constraint on the volume of material k1which occupies the part Ω1 of D:

m(Ω1)≤ α, 0 < α < m(D) (6.3)

for some constant α.

If χ denotes the characteristic function of the domain Ω1,

χ(x) = 1 if x ∈ Ω1and 0 if x / ∈ Ω1,

the compliance J (χ) = J (Ω1) can be expressed as the infimum over the Sobolev

space H1(D) of an energy functional defined on the fixed set D:

J (χ) can be minimized or maximized over some appropriate family of characteristic

functions or with respect to their relaxation to functions between 0 and 1 that wouldcorrespond to microstructures As in the eigenvalue problem, the objective function

is an infimum, but here the infimum is over a space that does not depend on the

function χ that specifies the geometric domain This will be handled by the special

techniques of Chapter 10 for the differentiation of the minimum of a functional

The celebrated Plateau’s problem, named after the Belgian physicist and

profes-sor J A F Plateau [1] (1801–1883), who did experimental observations on the

geometry of soap films around 1873, also provides a nice example where the try is a variable It consists in finding the surface of least area among those bounded

geome-by a given curve One of the difficulties in studying the minimal surface problem is

the description of such surfaces in the usual language of differential geometry Forinstance, the set of possible singularities is not known

Measure theoretic methods such as k-currents (k-dim surfaces) were used by

E 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] considered

a hypersurface in the N -dimensional Euclidean space RN as the boundary of a

set In order to obtain a boundary measure, they restricted their attention to sets

whose characteristic function is of bounded variation Their key property is an

associated natural notion of perimeter that extends the classical surface measure of the boundary of a smooth set to the larger family of Caccioppoli sets named after

the celebrated Neapolitan mathematician Renato Caccioppoli.2

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

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 Souli

and J.-P Zol´esio[1, 2, 3, 4, 5]) in 1988 More details will be given in Chapter 5.

Shape 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 first one is the design of a thermal diffuser 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

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 Zol´esio [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 110◦C) The thermal

resistance from the junction to the mounting surface is known for any particularHPSSD (typ 1◦C/W), so that the mounting surface is required to be kept at

a lower temperature than the junction (typ 60◦C) In a space application the

thermal power must ultimately be dissipated to the environment by the mechanism

of radiation However, to radiate large amounts of thermal power at moderately lowtemperatures, correspondingly large radiating areas are required Thus we have therequirement to efficiently spread the high thermal power flux (TPF) at the HPSSDsource (typ 40 W/cm2) to a low TPF at the radiator (typ 0.04 W/cm2) so that

the source temperature is maintained at an acceptably low level (typ < 60 ◦C)

at the mounting surface The efficient 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 diffuser 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 sufficiently spread the heat over large space radiators, reducing theTPF from a level at the diffuser (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 diffuser which is the problem at hand Wemay assume that the HPSSD presents a uniform thermal power flux to the diffuser

saddle

heatpipes

radiator to spaceschematic drawing

not to scale

thermal

solid-state diffuser

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

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

Assume that the thermal diffuser is a volume Ω symmetrical about the z-axis

(cf Figure 1.6 (A)) whose boundary surface is made up of three regular pieces:the mounting surface Σ1 (a disk perpendicular to the z-axis with center in (r, z) = (0, 0)), the lateral adiabatic surface Σ2, and the interface Σ3 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 Ω is the solution of the

station-ary heat equation k∆T = 0 (∆T , the Laplacian of T ) with the following boundstation-ary

conditions on the surface Σ = Σ1∪ Σ2∪ Σ3(the boundary of Ω):

k ∂T

∂n = q inon Σ1, k

∂T

∂n = 0 on Σ2, T = T3 (constant) on Σ3, (7.1)

where n always denotes the outward unit normal to the boundary surface Σ and

∂T /∂n is the normal derivative to the boundary surface Σ,

∂T

∂n =∇T · n (∇T = the gradient of T ). (7.2)The parameters appearing in (7.1) are

k = thermal conductivity (typ 1.8W/cm × ◦C),

q = uniform inward thermal power flux at the source (positive constant)

The radius R0of the mounting surface Σ1is fixed so that the boundary surface Σ1

is already given in the design problem

For practical considerations, we assume that the diffuser is solid without terior hollows or cutouts The class of shapes for the diffuser is characterized by

in-the design parameter L > 0 and in-the positive function R(z), 0 < z ≤ L, with R(0) = R0 > 0 They are volumes of revolution Ω about the z-axis generated by

the surface A between the z-axis and the function R(z) (cf Figure 1.6 (B)), that is,

dx = π

 L

0

subject to a uniform constraint on the outward thermal power flux at the interface

Σ3 between the diffuser and the heatpipe saddle:

where q out is a specified 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 fixed temperature T3 on the boundary Σ3 In other

words the optimal shape Ω, if it exists, is independent of T3 As a result, from now

on we set T equal to 0

7.3 Reformulation of the Problem

In a shape optimization problem the formulation is important from both the oretical and the numerical viewpoints In particular condition (7.5) is difficult tonumerically handle since it involves the pointwise evaluation of the normal derivative

the-on the piece of boundary Σ3 This problem can be reformulated as the minimization

of T on Σ3, where T is now the solution of a variational inequality Consider the

following minimization problem over the subspace of functions that are positive orzero on Σ3:

V+(Ω)def= v ∈ H1(Ω) : v |Σ 3 ≥ 0, (7.6)inf

v ∈V+(Ω)

Ω

H1(Ω) is the usual Sobolev space on the domain Ω, and the inequality on Σ3

has to be interpreted quasi-everywhere in the capacity sense Leaving aside those

technicalities, the minimizing solution of (7.7) is characterized by

inequality instead of a variational equation for the temperature T

In the above formulations the shape parameter L and the shape function R are not independent of each other since the function R is defined on the interval [0, L] This

motivates the following changes of variables and the introduction of the

and G A Pozzi [1]) in 1973 and in the free