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101Shapederivativesinsmoothdomains
13
Thelevelsetmethodisusedforshapeoptimizationofellipticequations.The
topologicalderivativesareemployedtoproducetheholesintheintegrationdo-
mains.Shapesensitivityanalysisisperformedfortheenergyfunctionalofthe
secondorderellipticequation.Thetopologicalderivativesoftheenergyfunc-
tionalarederivedforsemilinearellipticequations.Thelevelsetmethodisapplied
fornumericalsolutionoftheshapeoptimizationproblemsforlinearandnonlinear
ellipticequations.
101
Shapederivativesinsmoothdomains
10101Introduction
Thedescriptionofthespeedmethodinshapeoptimizationcanbefoundin[34].
Inthelevelsetmethodtheshapegradientoffunctionaltobeminimizedisused
asthecoefficientfortheHamilton-Jacobiequation.Therefore,theadditionalre-
quirementforthelevelsetmethodistheassumptionthattheshapegradientinthe
formofafunction,thisisinfactaregularityresultwhichcanbesatisfiedfortheel-
lipticboundaryvalueproblems.Itissufficienttoassumethatthedataareregular.
Inparticulartheassumptionisverifiedfortheenergytypefunctionalsdescribedto
someextendinthefirstchapter,wereferthereaderto[34]fortheproofsandthe
relatedresultsforlinearpartialdifferentialequationsandvariationalinequalities.
10102Thespeedmethod
Thefirstordershapesensitivityanalysisresultsintheshapegradientsandleads
tothegradienttypenumericalmethodsforsolutionoftherelatedshapeopti-
mizationproblems.Wepresentheretheclassicalshapesensitivityanalysiswhich
isdescribedin[35],wheretheso-calledspeedmethodisproposedandtheshape
derivativesforbroadclassesofshapefunctionalsareobtained.Wereferthereader
tothemonographfortheproofsoftheresultsinthesmoothcase.
LetΩ⊂R2beaboundeddomainwithsmoothboundaryF=aΩ(C2regular-
ity)andfbeasmoothfunction,e.g.f∈C2(R;R).Weconsiderthefollowing
problem:
{−∆u=f,inΩ,
u=0,onF.
(1.1)
Thevariationalformulation,orweakformulation,oftheproblem(1.1)isgivenby:
∫
Ω
⟨∇u,∇w⟩
R2dy=∫
Ω
fwdy,
∀w∈H1
0(Ω),
(1.2)
