Shapes and Geometries: Metrics, Analysis, Differential Calculus

647 Pages·2011·4.31 MB·English

Shapes and

Geometries
Advances in Design and Control
SIAM’s Advances in Design and Control series consists of texts and monographs dealing with all areas of
design and control and their applications. Topics of interest include shape optimization, multidisciplinary
design, trajectory optimization, feedback, and optimal control. The series focuses on the mathematical and
computational aspects of engineering design and control that are usable in a wide variety of scientific and
engineering disciplines.
Editor-in-Chief
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
Series Volumes
Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Metrics, Analysis, Differential Calculus, and
Optimization, Second Edition
Hovakimyan, Naira and Cao, Chengyu, L Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation
1
Speyer, Jason L. and Jacobson, David H., Primer on Optimal Control Theory
Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second
Edition
Shima, Tal and Rasmussen, Steven, eds., UAV Cooperative Decision and Control: Challenges and Practical
Approaches
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, Barı¸s, 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 Control to Nonlinear Systems: Control of Nonlinear
∞
Systems to Achieve Performance Objectives
Shapes and
Geometries
Metrics, Analysis, Differential
Calculus, and Optimization
SECOND EDITION
M. C. Delfour
Université de Montréal
Montréal, Québec
Canada
J.-P. Zolésio
National Center for Scientific Research (CNRS) and
National Institute for Research in Computer Science and Control (INRIA)
Sophia Antipolis
France
Society for Industrial and Applied Mathematics
Philadelphia
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 Ministère de
l’Éducation du Québec.
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. 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
is a registered trademark.
This book is dedicated to
Alice, Jeanne, Jean, and Roger
!

Contents
List of Figures xvii
Preface xix
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 ImageSmoothing/FilteringbyConvolutionandEdgeDetectors 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
viii Contents
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-, Hs-, 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,ShapeandTangentialDifferentialCalculuses,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 ( (D),△) . . . . 57
P
2.2.2 Second Abelian Group Structure on ( (D),▽) . . . 58
P
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
Contents ix
2.5.2 The Space Wm,p(Ω) . . . . . . . . . . . . . . . . . . 61
0
2.5.3 Embedding of H1(Ω) into H1(D) . . . . . . . . . . 62
0 0
2.5.4 Projection Operator . . . . . . . . . . . . . . . . . . 63
2.6 Spaces of Continuous and Differentiable Functions . . . . . . 63
2.6.1 Continuous and Ck Functions . . . . . . . . . . . . 63
2.6.2 H¨older (C0,ℓ) and Lipschitz (C0,1) Continuous
Functions . . . . . . . . . . . . . . . . . . . . . . . . 65
2.6.3 Embedding Theorem . . . . . . . . . . . . . . . . . 65
2.6.4 Identity Ck,1(Ω) = Wk+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 Ck and Ck,ℓ . . . . . . . . . . . . . . . . . . . 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 Space of Dominating Functions . . . . . . . . . . . 79
H
5.2 Local Ck,ℓ-Epigraphs and Ho¨lderian/Lipschitzian Sets . . . . 87
5.3 Local Ck,ℓ-Epigraphs and Sets of Class Ck,ℓ . . . . . . . . . . 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 Ho¨lderian/Lipschitzian
Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.4.1 Uniform Cone Property and Lipschitzian Sets . . . 114
6.4.2 Uniform Cusp Property and Ho¨lderian 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 Space (Θ) of Transformations of RN . . . . . . . . . . . . . 124
F
2.2 Diffeomorphisms for (RN,RN) and C (RN,RN) . . . . . . 136
B 0∞