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

Format:

Book

Author/Creator:

Delfour, Michel C., 1943-

Contributor:

Zolésio, J.-P.

Series:

Advances in design and control

Language:

English

Subjects (All):

Shape theory (Topology).

Physical Description:

xvii, 482 pages : illustrations ; 27 cm.

Place of Publication:

Philadelphia : Society for Industrial and Applied Mathematics, 2001.

Contents:

1 Geometry as a Variable 1

2 Shape Analysis 3

3 Geometric Measure Theory 5

4 Distance Functions, Smoothness, Curvatures 5

5 Shape Optimization 6

6 Shape Derivatives 8

7 Shape Calculus and Tangential Differential Calculus 9

8 Shape Analysis in This Book 10

9 Overview of the Book 11

9.1 Analysis, Shape Spaces, and Optimization 11

9.2 Continuity, Derivatives, Shape Calculus, and Tangential Differential Calculus 14

2 Classical Descriptions and Properties of Domains 17

2 Notation and Definitions 18

2.2 Continuous and C[superscript k] Functions 18

2.3 Holder and Lipschitz Continuous Functions 19

3 Smoothness of Domains, Boundary Integral, Boundary Curvatures 21

3.1 C[superscript k] and Holderian Sets 21

3.2 Boundary Integral, Canonical Density, and Hausdorff Measures 24

3.3 Fundamental Forms and Principal Curvatures 27

4 Domains as Level Sets of a Function 28

5 Domains as Local Epigraphs 32

5.1 Sets That Are Locally Lipschitzian Epigraphs 32

5.2 Local Epigraphs and Sets of Class C[superscript k,e] 38

5.3 Boundary Integral from the Graph of a Function 41

5.4 Volume and Surface Measures of the Boundary 43

5.5 Convex Sets 44

5.6 Nonhomogeneous Neumann and Dirichlet Problems 47

6 Uniform Cone Property for Lispchitzian Domains 49

7 Segment Property and C[superscript 0]-Epigraphs 59

8 Courant Metric Topology on Images of a Domain 68

8.1 The Complete Metric Group of Diffeomorphisms F[superscript k subscript 0] 68

8.2 The Courant Metric on the Images 76

8.3 Perturbations of the Identity 78

9 The Generic Framework of Micheletti 79

9.1 B[superscript k] (R[superscript N], R[superscript N])-Mappings 80

9.2 Lipschitzian Mappings 81

9.2.1 The Murat-Simon Approach 81

9.2.2 The Micheletti Approach 82

9.3 B[superscript k] (D, R[superscript N])- and C[superscript k subscript 0] (D, R[superscript N])-Mappings 87

10 Domains in a Closed Submanifold of R[superscript N] 88

3 Relaxation to Measurable Domains 91

2 Characteristic Functions in L[superscript p]-Topologies 92

2.1 Strong Topologies and C[superscript [infinity]-Approximations 92

2.2 Weak Topologies and Microstructures 95

2.3 Nice or Measure Theoretic Representative 100

2.4 The Family of Convex Sets 103

2.5 Sobolev Spaces for Measurable Domains 105

3 Some Compliance Problems with Two Materials 108

3.1 Transmission Problem and Compliance 108

3.2 The Original Problem of Cea and Malanowski 116

3.3 Relaxation and Homogenization 120

4 Buckling of Columns 120

5 Caccioppoli or Finite Perimeter Sets 125

5.1 Finite Perimeter Sets 125

5.2 Decomposition of the Integral along Level Sets 131

5.3 Domains of Class W[superscript [varepsilon],p] (D), 0 [less than or equal] [varepsilon] < 1/p, p [greater than or equal] 1 132

5.4 Compactness and Uniform Cone Property 135

6 Existence for the Bernoulli Free Boundary Problem 138

6.1 An Example: Elementary Modeling of the Water Wave 138

6.2 Existence for a Class of Free Boundary Problems 141

6.3 Weak Solutions of Some Generic Free Boundary Problems 143

6.3.1 Problem without Constraint 143

6.3.2 Constraint on the Measure of the Domain [Omega] 144

6.4 Weak Existence with Surface Tension 146

7 Continuity of the Dirichlet Boundary Value Problem 147

7.1 Continuity of Transmission Problems 148

7.2 Approximation by Transmission Problems 149

7.3 Continuity of the Dirichlet Problem 150

4 Topologies Generated by Distance Functions 153

2 Hausdorff Metric Topologies 154

2.1 The Family of Distance Functions C[subscript d](D) 154

2.2 Hausdorff Metric Topology 155

2.3 Hausdorff Complementary Metric Topology and C[superscript c subscript d](D) 160

3 Projection, Skeleton, Crack, and Differentiability 163

4 W[superscript 1,p]-Topology and Characteristic Functions 172

5 Sets of Bounded and Locally Bounded Curvature 178

5.1 Definitions and Properties 178

6 Characterization of Convex Sets 185

7 Federer's Sets of Positive Reach 192

8 Compactness Theorems 194

8.1 Global Conditions on D 194

8.2 Local Conditions in Tubular Neighborhoods 196

5 Oriented Distance Function and Smoothness of Sets 205

2 Uniform Metric Topology 206

2.1 The Family of Oriented Distance Functions C[subscript b](D) 206

2.2 Uniform Metric Topology 209

3 Projection, Skeleton, and Differentiability of b[subscript A] 213

4 Boundary Smoothness and Smoothness of b[subscript A] 218

5 W[superscript 1,p](D)-Topology and the Family C[superscript 0 subscript b](D) 225

6 Sets of Bounded and Locally Bounded Curvature 228

6.1 Definitions and Main Properties 229

6.2 Examples and Limits of the Tubular Norms as h Goes to Zero 231

6.3 (m,p)-Sobolev Domains (or W[superscript m,p]-Domains) 234

7 Characterization of Convex and Semiconvex Sets 236

8 Federer's Sets of Positive Reach 243

9 Compactness Theorems for Sets of Bounded Curvature 248

9.1 Global Conditions on D 249

9.2 Local Conditions in Tubular Neighborhoods 250

10 Compactness and Uniform Cone Property 252

11 Compactness and Uniform Cusp Property 256

6 Optimization of Shape Functions 259

1 Introduction and Generic Examples 259

2 Embedding of H[superscript 1 subscript 0]([Omega]) into H[superscript 1 subscript 0](D) and First Eigenvalue 261

3 Hausdorff Complementary Topology 264

4 Continuity of u[subscript [Omega] with Respect to [Omega] for the Dirichlet Problem 264

5 Continuity of the Neumann Problem 267

6 Optimization over Families of Lipschitzian or Convex Domains 269

6.1 Minimization of an Objective Function under State Equation Constraint 270

6.2 Optimization of the First Eigenvalue 271

7 Elements of Capacity Theory 272

7.2 Quasi-Continuous Representative and H[superscript 1]-Functions 273

7.3 Transport of Sets of Zero Capacity 274

8 Continuity under Capacity Constraints 276

9 Flat Cone Condition and the Compact Family O[subscript c,r](D) 281

10 Examples with a Constraint on the Gradient 284

7 Transformations versus Flows of Velocities 287

2 Shape Functions and Choice of Shape Derivatives 288

3 Families of Transformations of Domains 292

3.1 C[infinity]-Domains 292

3.2 C[superscript k]-Domains 294

3.3 Cartesian Graphs 295

3.4 Polar Coordinates 296

3.5 Level Sets 297

4 Unconstrained Families of Domains 299

4.1 Equivalence between Velocities and Transformations 299

4.2 Perturbations of the Identity 303

4.3 Equivalence for Special Families of Velocities 304

5 Constrained Families of Domains 312

5.1 Equivalence between Velocities and Transformations 313

5.2 Transformation of Condition (V2[subscript D]) into a Linear Constraint 319

6 Continuity of Shape Functions 321

6.1 Courant Metrics and Flows of Velocities 321

6.2 Shape Continuity and Velocity Method 326

8 Shape Derivatives and Calculus, and Tangential Differential Calculus 329

2 Review of Differentiation in Banach Spaces 330

2.1 Definitions of Semiderivatives and Derivatives 330

2.2 Locally Lipschitz Functions 332

2.3 Chain Rule for Semiderivatives 333

2.4 Semiderivatives of Convex Functions 335

2.5 Conditions for Frechet Differentiability 336

2.6 Hadamard Semiderivative and Velocity Method 338

3 First-Order Semiderivatives and Shape Gradient 340

3.1 Definitions of Semiderivatives and Derivatives 341

3.2 Perturbations of the Identity and Frechet Derivative 345

3.3 Shape Gradient and Structure Theorem 348

4 Elements of Shape Calculus 350

4.1 Basic Formula for Domain Integrals 351

4.2 Basic Formula for Boundary Integrals 353

4.3 Examples of Shape Derivative 355

4.3.1 Volume of [Omega] and Area of [Gamma] 355

4.3.2 H[superscript 1]([Omega])-Norm 356

4.3.3 Normal Derivative 357

5 Elements of Tangential Calculus 360

5.1 Intrinsic Definition of the Tangential Gradient 361

5.2 First-Order Derivatives 364

5.3 Second-Order

Derivatives 365

5.4 A Few Useful Formulae and the Chain Rule 366

5.5 The Stokes and Green Formulae 367

5.6 Relation between Tangential and Covariant Derivatives 367

5.7 Back to the Example of Section 4.3.3 370

6 Second-Order Semiderivative and Shape Hessian 370

6.1 Second-Order Derivative of the Domain Integral 371

6.2 Basic Formula for Domain Integrals 373

6.3 Nonautonomous Case 374

6.4 Autonomous Case 379

6.5 Decomposition of d[superscript 2] J([Omega]; V(0), W(0)) 384

9 Shape Gradients under a State Equation Constraint 389

2 Min Formulation 391

2.1 An Illustrative Example and a Shape Variational Principle 391

2.2 Function Space Parametrization 392

2.3 Differentiability of a Minimum with Respect to a Parameter 393

2.4 Application of the Theorem 396

2.5 Domain and Boundary Integral Expressions of the Shape Gradient 400

3 Buckling of Columns 402

4 Eigenvalue Problems 405

4.1 Transport of H[superscript k subscript 0]([Omega]) by W[superscript k,[infinity]-Transformations of R[superscript N] 406

4.2 Laplacian and Bi-Laplacian 407

4.3 Linear Elasticity 416

5 Saddle Point Formulation and Function Space Parametrization 421

5.1 An Illustrative Example 421

5.2 Saddle Point Formulation 422

5.3 Function Space Parametrization 423

5.4 Differentiability of a Saddle Point with Respect to a Parameter 425

5.5 Application of the Theorem 429

5.6 Domain and Boundary Expressions for the Shape Gradient 431

6 Multipliers and Function Space Embedding 433

6.1 The Nonhomogeneous Dirichlet Problem 433

6.2 A Saddle Point Formulation of the State Equation 433

6.3 Saddle Point Expression of the Objective Function 434

6.4 Verification of the Assumptions of Theorem 5.1 437.

Notes:

Includes bibliographical references and index.

ISBN:

0898714893

OCLC:

46320886