This considerably enriched new edition provides a self-contained presentation of the mathematical foundations, constructions, and tools necessary for studying problems where the modeling, optimization, or control variable is the shape or the structure of a geometric object. Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition presents the latest ground-breaking theoretical foundation to shape optimization in a form that can be used by the engineering and scientific communities. It also clearly explains the state-of-the-art developments in a mathematical language that will attract mathematicians to open questions in this important field. A series of generic examples has been added to the introduction and special emphasis has been put on the construction of important metrics. Advanced engineers in various application areas use basic ideas of shape optimization, but often encounter difficulties due to the sophisticated mathematical foundations for the field. This new version of the book challenges these difficulties by showing how the mathematics community has made extraordinary progress in establishing a rational foundation for shape optimization. This area of research is very broad, rich, and fascinating from both theoretical and numerical standpoints. It is applicable in many different areas such as fluid mechanics, elasticity theory, modern theories of optimal design, free and moving boundary problems, shape and geometric identification, image processing, and design of endoprotheses in interventional cardiology. Audience: This book is intended for applied mathematicians and advanced engineers and scientists, but the book is also structured as an initiation to shape analysis and calculus techniques for a broader audience of mathematicians. Some chapters are self-contained and can be used as lecture notes for a minicourse. The material at the beginning of each chapter is accessible to a broad audience, while the subsequent sections may sometimes require more mathematical maturity. Contents: List of Figures; Preface; Chapter 1: Introduction: Examples, Background, and Perspectives; Chapter 2: Classical Descriptions of Geometries and Their Properties; Chapter 3: Courant Metrics on Images of a Set; Chapter 4: Transformations Generated by Velocities; Chapter 5: Metrics via Characteristic Functions; Chapter 6: Metrics via Distance Functions; Chapter 7: Metrics via Oriented Distance Functions; Chapter 8: Shape Continuity and Optimization; Chapter 9: Shape and Tangential Differential Calculuses; Chapter 10: Shape Gradients under a State Equation Constraint; Elements of Bibliography; Index of Notation; Index.
Author(s): Michael C. Delfour, Jean-Paul Zolésio
Series: Advances in Design and Control
Edition: Second Edition
Publisher: SIAM-Sociecty for Industrial and Applied Mathematics
Year: 2010
Language: English
Pages: 646
Tags: Математика;Методы оптимизации;
Contents......Page 8
List of Figures......Page 18
1 Objectives and Scope of the Book......Page 20
2 Overview of the Second Edition......Page 21
3 Intended Audience......Page 23
4 Acknowledgments......Page 24
1 Orientation......Page 25
2 A Simple One-Dimensional Example......Page 27
3 Buckling of Columns......Page 28
4 Eigenvalue Problems......Page 30
5 Optimal Triangular Meshing......Page 31
6 Modeling Free Boundary Problems......Page 34
7 Design of a Thermal Di.user......Page 37
8 Design of a Thermal Radiator......Page 42
9 A Glimpse into Segmentation of Images......Page 45
10 Shapes and Geometries: Background and Perspectives......Page 60
11 Shapes and Geometries: Second Edition......Page 71
1 Introduction......Page 79
2 Notation and Definitions......Page 80
3 Sets Locally Described by an Homeomorphism or a Di.eomorphism......Page 91
4 Sets Globally Described by the Level Sets of a Function......Page 99
5 Sets Locally Described by the Epigraph of a Function......Page 102
6 Sets Locally Described by a Geometric Property......Page 125
1 Introduction......Page 147
2 Generic Constructions of Micheletti......Page 148
3 Generalization to All Homeomorphisms and Ck-Diffeomorphisms......Page 177
1 Introduction......Page 183
2 Metrics on Transformations Generated by Velocities......Page 185
3 Semiderivatives via Transformations Generated by Velocities......Page 194
4 Unconstrained Families of Domains......Page 204
5 Constrained Families of Domains......Page 218
6 Continuity of Shape Functions along Velocity Flows......Page 227
1 Introduction......Page 233
2 Abelian Group Structure on Measurable Characteristic Functions......Page 234
3 Lebesgue Measurable Characteristic Functions......Page 238
4 Some Compliance Problems with Two Materials......Page 252
5 Buckling of Columns......Page 264
6 Caccioppoli or Finite Perimeter Sets......Page 268
7 Existence for the Bernoulli Free Boundary Problem......Page 282
1 Introduction......Page 291
2 Uniform Metric Topologies......Page 292
3 Projection, Skeleton, Crack, and Differentiability......Page 303
4 W1,p-Metric Topology and Characteristic
Functions......Page 316
5 Sets of Bounded and Locally Bounded Curvature......Page 323
6 Reach and Federer’s Sets of Positive Reach......Page 327
7 Approximation by Dilated Sets/Tubular Neighborhoods and Critical Points......Page 340
8 Characterization of Convex Sets......Page 342
9 Compactness Theorems for Sets of Bounded Curvature......Page 348
1 Introduction......Page 359
2 Uniform Metric Topology......Page 361
3 Projection, Skeleton, Crack, and Di.erentiability......Page 368
4 W1,p(D)-Metric Topology and the Family C0
b (D)......Page 373
5 Boundary of Bounded and Locally Bounded Curvature......Page 378
6 Approximation by Dilated Sets/Tubular Neighborhoods......Page 382
7 Federer’s Sets of Positive Reach......Page 385
8 Boundary Smoothness and Smoothness of......Page 389
9 Sobolev or Wm,p Domains......Page 397
10 Characterization of Convex and Semiconvex Sets......Page 399
11 Compactness and Sets of Bounded Curvature......Page 405
12 Finite Density Perimeter and Compactness......Page 409
13 Compactness and Uniform Fat Segment Property......Page 411
14 Compactness under the Uniform Fat Segment Property and a Bound on a Perimeter......Page 417
15 The Families of Cracked Sets......Page 418
16 A Variation of the Image Segmentation Problem of Mumford and Shah......Page 424
1 Introduction and Generic Examples......Page 433
2 Upper Semicontinuity and Maximization of the First Eigenvalue......Page 436
3 Continuity of the Transmission Problem......Page 441
4 Continuity of the Homogeneous Dirichlet Boundary Value Problem......Page 442
5 Continuity of the Homogeneous Neumann Boundary Value Problem......Page 450
6 Elements of Capacity Theory......Page 453
7 Crack-Free Sets and Some Applications......Page 458
8 Continuity under Capacity Constraints......Page 464
9 Compact Families Oc,r(D) and Lc,r(O,D)......Page 471
1 Introduction......Page 481
2 Review of Differentiation in Topological Vector Spaces......Page 482
3 First-Order Shape Semiderivatives and Derivatives......Page 495
4 Elements of Shape Calculus......Page 506
5 Elements of Tangential Calculus......Page 515
6 Second-Order Semiderivative and Shape Hessian......Page 525
1 Introduction......Page 543
2 Min Formulation......Page 545
3 Buckling of Columns......Page 556
4 Eigenvalue Problems......Page 559
5 Saddle Point Formulation and Function Space Parametrization......Page 575
6 Multipliers and Function Space Embedding......Page 586
Elements of Bibliography......Page 595
Index of Notation......Page 639
Index......Page 643
