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A NEW ALGORITHM FOR TOPOLOGY OPTIMIZATION USING A LEVEL SET METHOD

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Niveau: Supérieur, Doctorat, Bac+8
A NEW ALGORITHM FOR TOPOLOGY OPTIMIZATION USING A LEVEL-SET METHOD SAMUEL AMSTUTZ AND HEIKO ANDRA Abstract. The level-set method has been recently introduced in the field of shape optimiza- tion, enabling a smooth representation of the boundaries on a fixed mesh and therefore leading to fast numerical algorithms. However, most of these algorithms use a Hamilton-Jacobi equa- tion to connect the evolution of the level-set function with the deformation of the contours, and consequently they can hardly create new holes in the domain (at least in 2D). In this work, we propose an evolution equation for the level-set function based on a generalization of the con- cept of topological gradient. This results in a new algorithm allowing for all kinds of topology changes. 1. Introduction Many methods have been worked out for the automatic optimization of elastic structures. The oldest and most popular one, the so-called classical shape optimization method [21, 28], is based on the computation of the sensitivity of the criterion of interest with respect to a smooth variation of the boundary. Its main drawback is that it does not allow any topology changes. To overcome this limitation, relaxed formulations using e.g. the homogenization theory have been introduced [1, 2, 6, 11, 12, 18]. However, these methods are mainly restricted to linear elasticity and particular objective functions.

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  • has been

  • design variable

  • optimization using

  • domain

  • shape optimization

  • local minima

  • gradient


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ANEWALGORITHMFORTOPOLOGYOPTIMIZATIONUSINGALEVEL-SETMETHODSAMUELAMSTUTZANDHEIKOANDRA¨Abstract.Thelevel-setmethodhasbeenrecentlyintroducedinthefieldofshapeoptimiza-tion,enablingasmoothrepresentationoftheboundariesonafixedmeshandthereforeleadingtofastnumericalalgorithms.However,mostofthesealgorithmsuseaHamilton-Jacobiequa-tiontoconnecttheevolutionofthelevel-setfunctionwiththedeformationofthecontours,andconsequentlytheycanhardlycreatenewholesinthedomain(atleastin2D).Inthiswork,weproposeanevolutionequationforthelevel-setfunctionbasedonageneralizationofthecon-ceptoftopologicalgradient.Thisresultsinanewalgorithmallowingforallkindsoftopologychanges.1.IntroductionManymethodshavebeenworkedoutfortheautomaticoptimizationofelasticstructures.Theoldestandmostpopularone,theso-calledclassicalshapeoptimizationmethod[21,28],isbasedonthecomputationofthesensitivityofthecriterionofinterestwithrespecttoasmoothvariationoftheboundary.Itsmaindrawbackisthatitdoesnotallowanytopologychanges.Toovercomethislimitation,relaxedformulationsusinge.g.thehomogenizationtheoryhavebeenintroduced[1,2,6,11,12,18].However,thesemethodsaremainlyrestrictedtolinearelasticityandparticularobjectivefunctions.Despitetheirhighcomputationalcost,stochasticalgorithms(likegeneticalgorithms,seee.g.[19])canbeusedtodealwithmoregeneralsituations,orwhenpracticalreasonsmakedifficultasensitivitycomputation(forinstancetheadjointstatemaynotbeeasilycomputable).Thelevel-setmethod,whichhasseveraladvantages,wasinstigatedbyOsherandSethian[23]fornumericallytrackingfrontsandfreeboundaries,andrecentlyintroducedinthefieldofshapeoptimization[4,5,13,22,25,30].First,itsmainfeatureistoenableanaccuratedescriptionoftheboundariesonafixedmesh.Thereforeitleadstofastnumericalalgorithms.Second,itsrangeofapplicationisverywide,sincethefrontvelocitycanbederivedfromtheclassicalshapesensitivity.Finally,itcanhandlesometopologychanges.Indeed,withintheusualframeworkofthecontrolofthelevel-setfunctionbyaHamilton-Jacobiequation,themergingandcancellationofholesprovetooccurinanaturalway.Conversely,thenucleationofnewholesseemstoberatherunlikelyinpracticalsituations.In3D,holescanstillappearbypinchingtwoboundaries,butthisprocessisimpossiblein2D.Itfollowsthattheobtaineddesignisstronglydependentontheinitialguessinthiscase.Besides,thenotionoftopologicalgradient[16,20,24,27]hasbeendevisedtomeasurethesensitivityofacriterionwithrespecttothesizeofasmallholecreatedaroundagivenpointofthedomain.Thisconceptgaverisetoanotherclassofoptimaldesignalgorithms.Instructuraloptimization,oneusuallyusesafixedpointmethodofthetypeΩk+1:={xΩk,gk(x)>ck},k=0,1,2,...,(1)wheregkdenotesthetopologicalgradientcomputedinthedomainΩkandck<0isathresholdplayingtheroleofastepsize(see[16,15]).Themaindrawbackofthisprocedureisitsinabilitytoaddmatterinsomeplaceswhereithasbeenremoved“bymistake”atpreviousiterations.Inthispaper,weproposeamodificationof(1)basedonageneralizationoftheconceptoftopologicalgradientandtherepresentationofthedomainbyalevel-setfunctionpossibly1