Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
16
TopologicalderivativesandLevelsetmethodinshapeoptimization
For(1.3)theshapederivativeu′(vν)satisfies:
(
4
l
−∆u′=0
u′=−vν
au
aν
inΩ,
onF,
wherevν=⟨V,ν⟩.
102Topologicalderivativesforsemilinearproblems
Theformoftopologicalderivativesforintegralshapefunctionalisderivedfora
classofsemilinearellipticequations.Theproofsofallresultsinsection1.2can
befoundin[9].
10201Introduction
Topologicalderivativesareintroducedforlinearproblemsin[37],andforvari-
ationalinequalitiesin[35].Themathematicaltheoryofasymptoticanalysisis
appliedin[23],[27]forderivationoftopologicalderivativesinshapeoptimiza-
tionofellipticboundaryvaluesproblems.Thenumericalsolutionsoftheshape
optimizationproblemsforvariationalinequalitiesobtainedbythelevelsetmethod
combinedwiththetopologicalderivativesarepresentedin[6].
Inthischapter,wepresentthetopologicalderivativesforsemilinearelliptic
boundaryvalueproblems.Inthefirstpart,theasymptoticanalysisisperformed
ofaclassofboundaryvalueproblemsforasecondordersemilineardifferential
equation.Inthesecondparttheconvergenceofthefiniteelementapproximation
forthetopologicalderivativesisproved,andresultofnumericalexperimentsare
presentedaswell.
Thetopologicalsensitivityanalysisaimstoprovideanasymptoticexpansion
ofashapefunctionalwithrespecttothesizeofasmallholecreatedinsidethe
domain.Foracriterionj(Ω)=J(uΩ;Ω)whereΩ⊂RN(N=2or3)anduΩis
asolutionofasetofpartialdifferentialequationsdefinedoverΩ,thisexpansion
canbegenerallywrittenintheform:
j(Ω\(O+ωE))−j(Ω)=f(E)TΩ(O,ω)+o(f(E)).
(1.7)
HereEandOdenoterespectivelythediameterandthecenterofthehole,ωisa
fixeddomaincontainingtheoriginOandf(E)isapositivefunctiontendingto
zerowithE.ThecoefficientTΩiscommonlycalledtopologicalderivative.
