Holdings: Shapes and geometries

Shapes and geometries: metrics, analysis, differential calculus, and optimization

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Bibliographic Details
Main Authors:	Delfour, Michel C. 1943- (Author), Zolésio, Jean P. 1947- (Author)
Format:	Book
Language:	English
Published:	Philadelphia Society for Industrial and Applied Mathematics 2011
Edition:	2. ed.
Series:	Advances in design and control 22
Subjects:	Shape theory (Topology) Form Strukturoptimierung Formgebung Gestaltoptimierung Geometrie
Online Access:	Inhaltsverzeichnis
Item Description:	Includes bibliographical references and index
Physical Description:	XXIII, 622 S. Ill.
ISBN:	9780898719369

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Record in the Search Index

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adam_text	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 Image Smoothing/Filtering by Convolution and Edge Detectors 22 9.2.1 Construction of the Convolution of ƒ ........ 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, Shape and Tangential Differential Calculuses, and Derivatives under State Constraints .............. 53 2 Classiceli 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 {V{D), Δ) .... 57 2.2.2 Second Abelian Group Structure on (V{D), V) . . . 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 Contents ¡x 2.5.2 The Space W™ * (п) .................. 61 2.5.3 Embedding of H ¿(íl) into #¿(D) .......... 62 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 Holder (C0-) and Lipschitz (C0·1) Continuous Functions ........................ 65 2.6.3 Embedding Theorem ................. 65 2.6.4 Identity Ск г(ЈГ) = Wk+1 °°{íl): 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 e ................... 67 3.2 Boundary Integral, Canonical Density, and Hausdorff Measures 70 3.2.1 Boundary Integral for Sets of Class С l ....... 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 C° Epigraphs, C° Epigraphs, and Equi-C0 Epigraphs and the Space Ή of Dominating Functions ........... 79 5.2 Local C^-Epigraphs and Hölderian/Lipschitzian Sets .... 87 5.3 Local C^-Epigraphs and Sets of Class СкЛ .......... 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 C° 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 Hölderian/Lipschitzian Sets ................................ 113 6.4.1 Uniform Cone Property and Lipschitzian Sets . . . 114 6.4.2 Uniform Cusp Property and Hölderian 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 T{Q) of Transformations of RN ............. 124 2.2 Diffeomorphismsforß(RN,RN) and C£°(RN,RN) ...... 136 Contents 2.3 Closed Subgroups Ç ....................... 138 2.4 Courant Metric on the Quotient Group T(ßi)IQ ....... 140 2.5 Assumptions for ßfe(RN,RN), C^R^R), and Cofc(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 Cfe 1 (RÑ Rn) and C^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^-Diffeomorphisms . . 153 Transformations Generated by Velocities 159 1 Introduction ................................ 159 2 Metrics on Transformations Generated by Velocities ......... 161 2.1 Subgroup Ge of Transformations Generated by Velocities . . 161 2.2 Complete Metrics on Gq and Geodesies ............ 166 2.3 Constructions of Azencott and Trouvé ............. 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^-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 (V2ß) into a Linear Constraint ............................. 200 6 Continuity of Shape Functions along Velocity Flows ......... 202 Metrics via Characteristic Functions 209 1 Introduction ................................ 209 2 Abelian Group Structure on Measurable Characteristic Functions . . 210 2.1 Group Structure on XM(RN) .................. 210 2.2 Measure Spaces .......................... 211 2.3 Complete Metric for Characteristic Functions in ¿^-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 Contents x¡ 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 Céa 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^P(D), 0 < ε < λ/ρ, ρ > 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 Cd(D) .............. 268 2.2 Pompéiu-Hausdorŕf Metric on Cd(D) .............. 269 2.3 Uniform Complementary Metric Topology and Cd(D) .... 275 2.4 Families Ccd(E; D) and Ccd]oc{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^-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^-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 гід ................ 318 8.2 Semiconvexity and BV Character of (¿a ............ 320 8.3 Closed Convex Hull of A and Fenchel Transform of dA . . . . 322 8.4 Families of Convex Sets Cd{D), Ccd{D), Ccd{E D), and ChocWD) ............................ 323 XII Contents 9 Compactness Theorems for Sets of Bounded Curvature ........ 324 9.1 Global Conditions in D ..................... 325 9.2 Local Conditions in Tubular Neighborhoods .......... 327 Metrics via Oriented Distance Functions 335 1 Introduction ................................ 335 2 Uniform Metric Topology ........................ 337 2.1 The Family of Oriented Distance Functions Cb(D) ...... 337 2.2 Uniform Metric Topology .................... 339 3 Projection, Skeleton, Crack, and Differentiability ........... 344 4 ^1 P(JD)-Metric Topology and the Family C^(D) ........... 349 4.1 Motivations and Main Properties ................ 349 4.2 Weak H^ 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 Ьа ............. 365 9 Sobolev or Wm>p Domains ........................ 373 10 Characterization of Convex and Semiconvex Sets ........... 375 10.1 Convex Sets and Convexity of bj ................ 375 10.2 Families of Convex Sets Cb(D), Cb(E; D), and Cb,lûc(E;D) ............................ 379 10.3 BV Character of Ьа and 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 Contents xiii 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 Lp(£>)-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 7/ ^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 Families OCtr{D) and Lc,r(O,D) ............... 447 9.1 Compact Family Oc,r(D) .................... 447 9.2 Compact Family LC,T(C D) and Thick Set Property ..... 450 9.3 Maximizing the Eigenvalue λ4(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 Semiderivat i ves and Derivatives ......... 458 2.2 Derivatives in Normed Vector Spaces .............. 461 2.3 Locally Lipschitz Functions ................... 465 2.4 Chain Rule for Semiderivatives................. 465 xiv Contents 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 Fréchet 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 tf^j-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{Q]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 Н$(п) by ^^-Transformations of RN .... 536 4.2 Laplacian and Bi-Laplacian ................... 537 4.3 Linear Elasticity ......................... 546 Contents xv 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 Elements of Bibliography 571 Index of Notation 615 Index 619
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id	DE-604.BV036930837
illustrated	Illustrated
indexdate	2024-07-09T22:50:59Z
institution	BVB
isbn	9780898719369
language	English
lccn	2010028846
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-020845893
oclc_num	706929721
open_access_boolean
owner	DE-703 DE-824 DE-706 DE-11 DE-19 DE-BY-UBM DE-20 DE-355 DE-BY-UBR
owner_facet	DE-703 DE-824 DE-706 DE-11 DE-19 DE-BY-UBM DE-20 DE-355 DE-BY-UBR
physical	XXIII, 622 S. Ill.
publishDate	2011
publishDateSearch	2011
publishDateSort	2011
publisher	Society for Industrial and Applied Mathematics
record_format	marc
series	Advances in design and control
series2	Advances in design and control
spelling	Delfour, Michel C. 1943- Verfasser (DE-588)17252931X aut Shapes and geometries metrics, analysis, differential calculus, and optimization M. C. Delfour ; J.-P. Zolésio 2. ed. Philadelphia Society for Industrial and Applied Mathematics 2011 XXIII, 622 S. Ill. txt rdacontent n rdamedia nc rdacarrier Advances in design and control 22 Includes bibliographical references and index Shape theory (Topology) Form (DE-588)4154987-9 gnd rswk-swf Strukturoptimierung (DE-588)4183811-7 gnd rswk-swf Formgebung (DE-588)4113598-2 gnd rswk-swf Gestaltoptimierung (DE-588)4329076-0 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Gestaltoptimierung (DE-588)4329076-0 s DE-604 Formgebung (DE-588)4113598-2 s Strukturoptimierung (DE-588)4183811-7 s 1\p DE-604 Form (DE-588)4154987-9 s Geometrie (DE-588)4020236-7 s 2\p DE-604 Zolésio, Jean P. 1947- Verfasser (DE-588)121134121 aut Advances in design and control 22 (DE-604)BV021715022 22 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020845893&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Delfour, Michel C. 1943- Zolésio, Jean P. 1947- Shapes and geometries metrics, analysis, differential calculus, and optimization Advances in design and control Shape theory (Topology) Form (DE-588)4154987-9 gnd Strukturoptimierung (DE-588)4183811-7 gnd Formgebung (DE-588)4113598-2 gnd Gestaltoptimierung (DE-588)4329076-0 gnd Geometrie (DE-588)4020236-7 gnd
subject_GND	(DE-588)4154987-9 (DE-588)4183811-7 (DE-588)4113598-2 (DE-588)4329076-0 (DE-588)4020236-7
title	Shapes and geometries metrics, analysis, differential calculus, and optimization
title_auth	Shapes and geometries metrics, analysis, differential calculus, and optimization
title_exact_search	Shapes and geometries metrics, analysis, differential calculus, and optimization
title_full	Shapes and geometries metrics, analysis, differential calculus, and optimization M. C. Delfour ; J.-P. Zolésio
title_fullStr	Shapes and geometries metrics, analysis, differential calculus, and optimization M. C. Delfour ; J.-P. Zolésio
title_full_unstemmed	Shapes and geometries metrics, analysis, differential calculus, and optimization M. C. Delfour ; J.-P. Zolésio
title_short	Shapes and geometries
title_sort	shapes and geometries metrics analysis differential calculus and optimization
title_sub	metrics, analysis, differential calculus, and optimization
topic	Shape theory (Topology) Form (DE-588)4154987-9 gnd Strukturoptimierung (DE-588)4183811-7 gnd Formgebung (DE-588)4113598-2 gnd Gestaltoptimierung (DE-588)4329076-0 gnd Geometrie (DE-588)4020236-7 gnd
topic_facet	Shape theory (Topology) Form Strukturoptimierung Formgebung Gestaltoptimierung Geometrie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020845893&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV021715022
work_keys_str_mv	AT delfourmichelc shapesandgeometriesmetricsanalysisdifferentialcalculusandoptimization AT zolesiojeanp shapesandgeometriesmetricsanalysisdifferentialcalculusandoptimization

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