Shapes and geometries : analysis, differential calculus, and optimization

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

資料種別:

図書

出版情報:

Philadelphia : Society for Industrial and Applied Mathematics, c2001

形態:

xvii, 482 p. ; 26 cm

シリーズ名:

Advances in design and control / editor-in-chief, John A. Burns <BA45082828>

著者名:

ISBN:

9780898714890 [0898714893]

書誌ID:

BA54194006

子書誌情報

フルテキスト

所蔵情報

他の版・巻

書誌詳細

注記:

Includes bibliographical references (p.441-473) and index

言語:

英語

目次情報:

List of Figures

Preface

Introduction / 1：

Geometry as a Variable

Shape Analysis / 2：

Geometric Measure Theory / 3：

Distance Functions, Smoothness, Curvatures / 4：

Shape Optimization / 5：

Shape Derivatives / 6：

Shape Calculus and Tangential Differential Calculus / 7：

Shape Analysis in This Book / 8：

Overview of the Book / 9：

Analysis, Shape Spaces, and Optimization / 9.1：

Continuity, Derivatives, Shape Calculus, and Tangential Differential Calculus / 9.2：

Classical Descriptions and Properties of Domains

Notation and Definitions

Basic Notation / 2.1：

Continuous and C[superscript k] Functions / 2.2：

Holder and Lipschitz Continuous Functions / 2.3：

Smoothness of Domains, Boundary Integral, Boundary Curvatures

C[superscript k] and Holderian Sets / 3.1：

Boundary Integral, Canonical Density, and Hausdorff Measures / 3.2：

Fundamental Forms and Principal Curvatures / 3.3：

Domains as Level Sets of a Function

Domains as Local Epigraphs

Sets That Are Locally Lipschitzian Epigraphs / 5.1：

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

Boundary Integral from the Graph of a Function / 5.3：

Volume and Surface Measures of the Boundary / 5.4：

Convex Sets / 5.5：

Nonhomogeneous Neumann and Dirichlet Problems / 5.6：

Uniform Cone Property for Lispchitzian Domains

Segment Property and C[superscript 0]-Epigraphs

Courant Metric Topology on Images of a Domain

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

The Courant Metric on the Images / 8.2：

Perturbations of the Identity / 8.3：

The Generic Framework of Micheletti

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

Lipschitzian Mappings

The Murat-Simon Approach / 9.2.1：

The Micheletti Approach / 9.2.2：

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

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

Relaxation to Measurable Domains

Characteristic Functions in L[superscript p]-Topologies

Strong Topologies and C[superscript [infinity]-Approximations

Weak Topologies and Microstructures

Nice or Measure Theoretic Representative

The Family of Convex Sets / 2.4：

Sobolev Spaces for Measurable Domains / 2.5：

Some Compliance Problems with Two Materials

Transmission Problem and Compliance

The Original Problem of Cea and Malanowski

Relaxation and Homogenization

Buckling of Columns

Caccioppoli or Finite Perimeter Sets

Finite Perimeter Sets

Decomposition of the Integral along Level Sets

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

Compactness and Uniform Cone Property

Existence for the Bernoulli Free Boundary Problem

An Example: Elementary Modeling of the Water Wave / 6.1：

Existence for a Class of Free Boundary Problems / 6.2：

Weak Solutions of Some Generic Free Boundary Problems / 6.3：

Problem without Constraint / 6.3.1：

Constraint on the Measure of the Domain [Omega] / 6.3.2：

Weak Existence with Surface Tension / 6.4：

Continuity of the Dirichlet Boundary Value Problem

Continuity of Transmission Problems / 7.1：

Approximation by Transmission Problems / 7.2：

Continuity of the Dirichlet Problem / 7.3：

Topologies Generated by Distance Functions

Hausdorff Metric Topologies

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

Hausdorff Metric Topology

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

Projection, Skeleton, Crack, and Differentiability

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

Sets of Bounded and Locally Bounded Curvature

Definitions and Properties

Examples

Characterization of Convex Sets

Federer's Sets of Positive Reach

Compactness Theorems

Global Conditions on D

Local Conditions in Tubular Neighborhoods

Oriented Distance Function and Smoothness of Sets

Uniform Metric Topology

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

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

Boundary Smoothness and Smoothness of b[subscript A]

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

Definitions and Main Properties

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

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

Characterization of Convex and Semiconvex Sets

Compactness Theorems for Sets of Bounded Curvature

Compactness and Uniform Cusp Property / 11：

Optimization of Shape Functions

Introduction and Generic Examples

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

Hausdorff Complementary Topology

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

Continuity of the Neumann Problem

Optimization over Families of Lipschitzian or Convex Domains

Minimization of an Objective Function under State Equation Constraint

Optimization of the First Eigenvalue

Elements of Capacity Theory

Definition and Basic Properties

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

Transport of Sets of Zero Capacity

Continuity under Capacity Constraints

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

Examples with a Constraint on the Gradient

Transformations versus Flows of Velocities

Shape Functions and Choice of Shape Derivatives

Families of Transformations of Domains

C[infinity]-Domains

C[superscript k]-Domains

Cartesian Graphs

Polar Coordinates / 3.4：

Level Sets / 3.5：

Unconstrained Families of Domains

Equivalence between Velocities and Transformations / 4.1：

Equivalence for Special Families of Velocities / 4.2：

Constrained Families of Domains

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

Continuity of Shape Functions

Courant Metrics and Flows of Velocities

Shape Continuity and Velocity Method

Shape Derivatives and Calculus, and Tangential Differential Calculus

Review of Differentiation in Banach Spaces

Definitions of Semiderivatives and Derivatives

Locally Lipschitz Functions

Chain Rule for Semiderivatives

Semiderivatives of Convex Functions

Conditions for Frechet Differentiability

Hadamard Semiderivative and Velocity Method / 2.6：

First-Order Semiderivatives and Shape Gradient

Perturbations of the Identity and Frechet Derivative

Shape Gradient and Structure Theorem

Elements of Shape Calculus

Basic Formula for Domain Integrals

Basic Formula for Boundary Integrals

Examples of Shape Derivative

Volume of [Omega] and Area of [Gamma] / 4.3.1：

H[superscript 1]([Omega])-Norm / 4.3.2：

Normal Derivative / 4.3.3：

Elements of Tangential Calculus

Intrinsic Definition of the Tangential Gradient

First-Order Derivatives

Second-Order Derivatives

A Few Useful Formulae and the Chain Rule

The Stokes and Green Formulae

Relation between Tangential and Covariant Derivatives

Back to the Example of Section 4.3.3 / 5.7：

Second-Order Semiderivative and Shape Hessian

Second-Order Derivative of the Domain Integral

Nonautonomous Case

Autonomous Case

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

Shape Gradients under a State Equation Constraint

Min Formulation

An Illustrative Example and a Shape Variational Principle

Function Space Parametrization

Differentiability of a Minimum with Respect to a Parameter

Application of the Theorem

Domain and Boundary Integral Expressions of the Shape Gradient

Eigenvalue Problems

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

Laplacian and Bi-Laplacian

Linear Elasticity

Saddle Point Formulation and Function Space Parametrization

An Illustrative Example

Saddle Point Formulation

Differentiability of a Saddle Point with Respect to a Parameter

Domain and Boundary Expressions for the Shape Gradient

Multipliers and Function Space Embedding

The Nonhomogeneous Dirichlet Problem

A Saddle Point Formulation of the State Equation

Saddle Point Expression of the Objective Function

Verification of the Assumptions of Theorem 5.1

Elements of Bibliography

Index of Notation

Index

List of Figures

Preface

Introduction / 1：

Geometry as a Variable

Shape Analysis / 2：

Geometric Measure Theory / 3：

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