Shape Variation and Optimization : A Geometrical Analysis

381 Pages·2018·2.537 MB·English

by  Antoine Henrot, Michel Pierre

Tracts in Mathematics 28 MA Tracts in Mathematics 28
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Antoine Henrot o Antoine Henrot
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Michel Pierre
Michel Pierre
Shape Variation and Optimization
A Geometrical Analysis
Optimizing the shape of an object to make it the most efficient, resistant, S Shape Variation
streamlined, lightest, noiseless, stealthy or the cheapest is clearly a very h
old task. But the recent explosion of modeling and scientific computing a
have given this topic new life. Many new and interesting questions have p
and Optimization
been asked. A mathematical topic was born – shape optimization (or e
optimum design). V
a
This book provides a self-contained introduction to modern mathematical r
approaches to shape optimization, relying only on undergraduate level ia A Geometrical Analysis
prerequisite but allowing to tackle open questions in this vibrant field. The t
analytical and geometrical tools and methods for the study of shapes are io
developed. In particular, the text presents a systematic treatment of n
shape variations and optimization associated with the Laplace operator
a
and the classical capacity. Emphasis is also put on differentiation with
n
respect to domains and a FAQ on the usual topologies of domains is
d
provided. The book ends with geometrical properties of optimal shapes,

including the case where they do not exist. O
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ISBN 978-3-03719-178-1
www.ems-ph.org
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Antoine Henrot
Michel Pierre
Shape Variation
and Optimization
A Geometrical Analysis
Authors:
Antoine Henrot Michel Pierre
Institut Elie Cartan, UMR CNRS 7502 Univ Rennes
Université de Lorraine École Normale Supérieure de Rennes
Boulevard des Aiguillettes, B.P. 70239 Institut de Recherche Mathématique de Rennes
54506 Vandœuvre-lès-Nancy Cedex Campus de Ker Lann
France 35170 Bruz
E-mail: [email protected] France
E-mail: [email protected]
2010 Mathematical Subject Classification: 49Q10, 49Q05, 49Q12, 49K20, 49K40, 53A10, 35R35,
35J20, 58E25, 31B15, 65K10, 93B27, 74P20, 74P15, 74G65, 76M30.
Key words: Shape optimization, optimum design, calculus of variations, variations of domains, Haus-
dorff convergence, continuity with respect to domains, G-convergence, shape derivative, geometry of
optimal shapes, Laplace-Dirichlet problem, Neumann problem, overdetermined problems, isoperimet-
ric inequality, capacity, potential theory, spectral theory, homogenization.
ISBN 978-3-03719-178-1
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and
the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
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casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use
permission of the copyright owner must be obtained.
© 2018 European Mathematical Society
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Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany
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9 8 7 6 5 4 3 2 1
Foreword
ThisbookisessentiallytheEnglishversionoftheonepublishedinFrenchbythesame
authors [188], except for some additions and updates. It originated from graduate
courses given in past years at the Universities of Nancy, Besançon, and Rennes in
France.
Thegoalofthesegraduatecourseswastoprovideanintroductiontomodernap-
proachestoshapeoptimization,relyingonlyonundergraduatelevelasaprerequisite,
butreachingactual,currentopenquestionsofthisveryactivedomain.
Thebookiswritteninthissameinitialspirit. Somespecificdirectionsaremore
particularly developed, but all necessary mathematical tools and proofs are then
providedinordertoofferaself-containedpresentation.
For instance, we provide all necessary knowledge on the classical capacity as-
sociated with the energy space H1. We also develop the particular case of shape
optimizationassociatedwiththeDirichletproblemfortheLaplaceoperator: thisisa
simplebuttypicalexamplethatissignificantamongallthemainquestionsthatarise
in shape optimization associated with more complex systems of partial differential
equations. As apparent from this model example, it is important to understand the
behaviorofthesesystemsundervariationsoftheirunderlyingdomains. Thisexplains
whythetwokeywords“variation”and“optimization”appearinthetitleofthisbook.
In the same spirit, we chose to devote one full and extended chapter to the
mainquestionofdifferentiationwithrespecttoshapes. Thisisaratherdifficultbut
unavoidabletopicthatcanrapidlybecometechnical. Weaimedatamathematically
rigorouspresentationwhilebeingatthesametimeconcernedwithprovidingefficient
tools for the actual computations of the shape derivatives (rigorous analysis and
efficientcalculusaresomehowantagonisticinthistopic).
WehavealsodescribedallthevarioustopologiesonopensubsetsofRN,whichare
mostlyusedinthevariationofshapesandincontinuitypropertiesfortheassociated
PDEs. Wetriedtoprovidesomekindof“FAQ”onthisquestion.
The last two chapters address different questions. One is about qualitative geo-
metricpropertiesofoptimalshapes: Wechosetopresentseveralexplicitexamplesin
ordertodescribeasmanydifferentmethodsaspossible. Theotheronecontainsan
introductiontoquitedifferentpointsofviewinshapeoptimizationwhicharerecent
andstillinprogress.
And we thought it was interesting to add a bibliographical footnote each time a
new(noncontemporary)mathematicianwasquoted. Amongothersources,weused
the excellent book [168] by B. Hauchecorne and D. Suratteau, as well as the rich
websitehttp://www-history.mcs.st-and.ac.uk/Indexes/Full_Alph.html.
Wewouldliketothankallthecolleagueswhohelpeduswiththeirremarksonthe
Frenchversionandonpreliminaryversionsofthisone,inparticularMarcDambrine,
vi Foreword
Evans Harrell, Antoine Laurain, Morgan Pierre, Yannick Privat, Takéo Takahashi,
MichielvandenBergandhisgroupinBristol.
BruzandNancy,April,2017.
AntoineHenrot MichelPierre
UniversitédeLorraine UnivRennes
InstitutElieCartan ÉcoleNormaleSupérieuredeRennes
InstitutdeRechercheMathématiquedeRennes
Contents
Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 Introductionandexamples. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Someacademicexamples . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Isoperimetricproblems . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Minimalsurfacesandcapillarysurfaces . . . . . . . . . . . 5
1.2.3 Eigenvalueproblems . . . . . . . . . . . . . . . . . . . . . 6
1.3 Someotherexampleswithapplications. . . . . . . . . . . . . . . . 13
1.3.1 Electromagneticshapingofliquidmetal . . . . . . . . . . . 13
1.3.2 Optimizationofamagnet. . . . . . . . . . . . . . . . . . . 15
1.3.3 Imagesegmentation . . . . . . . . . . . . . . . . . . . . . 16
1.3.4 Identificationofcracksordefects . . . . . . . . . . . . . . 17
1.3.5 Reinforcementorinsulationproblems . . . . . . . . . . . . 18
1.3.6 Compositematerialsandstructuraloptimization . . . . . . 21
1.3.7 Examplesinaeronauticsorfluidmechanics . . . . . . . . . 22
2 TopologiesondomainsofRN . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Whydoweneedatopology? . . . . . . . . . . . . . . . . . . . . . 25
2.2 Differenttopologiesondomains . . . . . . . . . . . . . . . . . . . 26
2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2 Convergenceofcharacteristicfunctions . . . . . . . . . . . 27
2.2.3 Hausdorffconvergenceofopensets . . . . . . . . . . . . . 30
2.2.4 Compactconvergence . . . . . . . . . . . . . . . . . . . . 38
2.2.5 Linksbetweenthedifferentnotionsofconvergence . . . . . 39
2.2.6 Compactnessresults . . . . . . . . . . . . . . . . . . . . . 42
2.3 Sequenceofsetswithboundedperimeter . . . . . . . . . . . . . . 47
2.3.1 Definitionoftheperimeter,properties . . . . . . . . . . . . 47
2.3.2 Continuityandcompactness . . . . . . . . . . . . . . . . . 50
2.4 Sequencesofuniformlyregularopensets . . . . . . . . . . . . . . 54
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Continuitywithrespecttodomains. . . . . . . . . . . . . . . . . . . . . 67
3.1 TheDirichletproblem. . . . . . . . . . . . . . . . . . . . . . . . . 68
3.1.1 ThespaceH1 anditsdualspaceH−1 . . . . . . . . . . . . 68
0
3.1.2 Lip◦H1 ⊂ H1 . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.1.3 ThePoincaréinequality . . . . . . . . . . . . . . . . . . . 76
3.1.4 TheDirichletproblemfortheLaplacian . . . . . . . . . . . 79
viii Contents
3.2 ContinuityfortheDirichletproblem . . . . . . . . . . . . . . . . . 81
3.2.1 Problemstatement . . . . . . . . . . . . . . . . . . . . . . 81
3.2.2 Someeasyfacts . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2.3 Thesituationdoesnotdependon f . . . . . . . . . . . . . 84
3.2.4 Nondecreasingsequences . . . . . . . . . . . . . . . . . . 85
3.2.5 Thedimension-1case . . . . . . . . . . . . . . . . . . . . 86
3.2.6 Counterexamplestocontinuityindimension2. . . . . . . . 87
3.2.7 SequenceofuniformlyLipschitzopensets . . . . . . . . . 89
3.3 CapacityassociatedwiththeH1-norm . . . . . . . . . . . . . . . . 92
3.3.1 Definitionsandfirstproperties . . . . . . . . . . . . . . . . 92
3.3.2 Relativecapacityandcapacitarypotential . . . . . . . . . . 96
3.3.3 Howtocomputecapacitiesofsets: Someexamples . . . . . 101
3.3.4 Quasi-continuityandquasi-opensets . . . . . . . . . . . . 105
3.3.5 AnewdefinitionofH1(Ω) . . . . . . . . . . . . . . . . . . 111
0
3.4 BacktotheDirichletproblem. . . . . . . . . . . . . . . . . . . . . 113
3.4.1 Localperturbation . . . . . . . . . . . . . . . . . . . . . . 114
3.4.2 Compactconvergenceandstableopensets . . . . . . . . . 115
3.4.3 Capacitary-typeconstraints . . . . . . . . . . . . . . . . . . 117
3.5 Theγ-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.5.2 LinkwithMoscoconvergence . . . . . . . . . . . . . . . . 121
3.5.3 MoreoperatorsassociatedwiththeH1γ-convergence . . . . 123
0
3.5.4 Remarksaboutnonlinearoperators . . . . . . . . . . . . . 125
3.6 Quantitativeestimates . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.7 ContinuityfortheNeumannproblem . . . . . . . . . . . . . . . . . 128
3.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.7.2 Amainconvergenceresult . . . . . . . . . . . . . . . . . . 129
3.7.3 Moreconvergenceresults. . . . . . . . . . . . . . . . . . . 131
3.7.4 γ-convergenceandtheNeumannproblem . . . . . . . . . . 133
3.8 Thebi-Laplacianoperator. . . . . . . . . . . . . . . . . . . . . . . 136
3.8.1 H2-capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.8.2 Continuitywithrespecttodomains . . . . . . . . . . . . . 138
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4 Existenceofoptimalshapes. . . . . . . . . . . . . . . . . . . . . . . . . 145
4.1 Somegeometricproblems . . . . . . . . . . . . . . . . . . . . . . 145
4.1.1 Isoperimetricproblems . . . . . . . . . . . . . . . . . . . . 145
4.1.2 Ageneralization . . . . . . . . . . . . . . . . . . . . . . . 146
4.1.3 Capillarysurfaces . . . . . . . . . . . . . . . . . . . . . . 147
4.2 Examplesofnonexistence. . . . . . . . . . . . . . . . . . . . . . . 149
4.3 Uniformregularityofadmissibleshapes . . . . . . . . . . . . . . . 156
Contents ix
4.4 Constraintsofcapacitarytype . . . . . . . . . . . . . . . . . . . . 158
4.5 MinimizationoftheDirichletenergy . . . . . . . . . . . . . . . . . 159
4.6 Effectsofperimeterconstraints . . . . . . . . . . . . . . . . . . . . 166
4.7 Monotonicityofthefunctional . . . . . . . . . . . . . . . . . . . . 169
4.8 Unboundedclassofdomains . . . . . . . . . . . . . . . . . . . . . 178
4.8.1 Aconcentration–compactnessargument . . . . . . . . . . . 179
4.8.2 Notionofshapesubsolution . . . . . . . . . . . . . . . . . 183
4.8.3 Asurgeryargument . . . . . . . . . . . . . . . . . . . . . . 184
4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5 Differentiatingwithrespecttodomains. . . . . . . . . . . . . . . . . . . 189
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
5.2 Integralsonvariabledomains . . . . . . . . . . . . . . . . . . . . . 191
5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.2.2 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.2.3 Thedifferentiationformula . . . . . . . . . . . . . . . . . . 194
5.2.4 Theproofs . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.2.5 Differentiatingonintervalsandfirstapplications . . . . . . 199
5.3 AmodelPDEproblem . . . . . . . . . . . . . . . . . . . . . . . . 201
5.3.1 Presentationoftheproblem . . . . . . . . . . . . . . . . . 201
5.3.2 Aformalcomputation . . . . . . . . . . . . . . . . . . . . 201
5.3.3 Thetwomainresults . . . . . . . . . . . . . . . . . . . . . 203
5.3.4 Theproofs . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.3.5 Differentiabilityofhigherorder . . . . . . . . . . . . . . . 208
5.3.6 Differentiabilityinregularfunctionalspaces . . . . . . . . . 209
5.4 Integralsonmovingboundaries. . . . . . . . . . . . . . . . . . . . 211
5.4.1 Boundaryintegrals: Definitionsandproperties . . . . . . . 212
5.4.2 Afirststatement . . . . . . . . . . . . . . . . . . . . . . . 214
5.4.3 Somedifferentialgeometry . . . . . . . . . . . . . . . . . . 216
5.4.4 Extensionoftheunitnormalvectortoavariabledomain . . 222
5.4.5 Ageneralformulaforboundarydifferentiation . . . . . . . 223
5.5 DifferentiatingtheNeumannproblem . . . . . . . . . . . . . . . . 226
5.6 Howtodifferentiateboundaryvalueproblems . . . . . . . . . . . . 230
5.7 Differentiationofasimpleeigenvalue . . . . . . . . . . . . . . . . 232
5.8 Useoftheadjointstate . . . . . . . . . . . . . . . . . . . . . . . . 237
5.9 Structureofshapederivatives . . . . . . . . . . . . . . . . . . . . . 241
5.9.1 Introductionandnotation . . . . . . . . . . . . . . . . . . . 241
5.9.2 Afirststructureresult. . . . . . . . . . . . . . . . . . . . . 242
5.9.3 Aselectedlistoffirstshapederivatives . . . . . . . . . . . 243
5.9.4 Thestructuretheoremanditscorollaries. . . . . . . . . . . 245
5.9.5 Theproofs . . . . . . . . . . . . . . . . . . . . . . . . . . 247