A high-level programming-language implementation o

arXiv:physics/0410086v2[physics.flu-dyn]1Feb2006Ahigh-levelprogramming-languageimplementationoftopologyoptimizationappliedtosteady-stateNavier–StokesflowLauritsHøjgaardOlesen,FridolinOkkels,andHenrikBruusMIC–DepartmentofMicroandNanotechnology,TechnicalUniversityofDenmark,DK-2800KongensLyngby,Denmark(Dated:26June2005)Wepresentaversatilehigh-levelprogramming-languageimplementationofnonlineartopologyoptimization.OurimplementationisbasedonthecommercialsoftwarepackageFemlab,anditallowsawiderangeofoptimizationobjectivestobedealtwitheasily.Weexemplifyourmethodbystudiesofsteady-stateNavier–Stokesflowproblems,thusextendingtheworkbyBorrvallandPeterssonontopologyoptimizationoffluidsinStokesflow[Int.J.Num.Meth.Fluids2003;41:77–107].Weanalyzethephysicalaspectsofthesolutionsandhowtheyareaffectedbydifferentparametersoftheoptimizationalgorithm.AcompleteexampleofourimplementationisincludedasFemlabcodeinanappendix.Keywords:topologyoptimization,Navier–Stokesflow,inertialeffects,FemlabI.INTRODUCTIONThematerialdistributionmethodintopologyoptimizationwasoriginallydevelopedforstiffnessdesignofmechanicalstructures[2]buthasnowbeenextendedtoamultitudeofdesignproblemsinstructuralmechanicsaswellastoopticsandacoustics[3,4,5,6].RecentlyBorrvallandPeterssonintroducedthemethodforfluidsinStokesflow[1].However,itisdesirabletoextendthemethodtofluidsdescribedinafullNavier–Stokesflow;adirectionpioneeredbytheworkofSigmundandGersborg-Hansen[7,8,9].Inthepresentworkwepresentsuchanextensionbyintroducingaversatilehigh-levelprogramming-languageimplementationofnonlineartopologyoptimization,basedonthecommercialsoftwarepackageFemlab.IthasawiderrangeofapplicabilitythantheNavier–Stokesproblemsstudiedhere,andmoreoveritallowsawiderangeofoptimizationobjectivestobedealtwitheasily.Extendingthetopologyoptimizationmethodtonewphysicaldomainsgenerallyinvolvessomerethinkingofthedesignproblemandsome”trialanderror”todeterminesuitabledesignobjectives.Italsorequiresthenumericalanalysisandimplementationoftheproblem,e.g.,usingthefiniteelementmethod(FEM).Thisprocessisacceleratedalotbyusingahigh-levelFEMlibraryorpackagethatallowsdifferentphysicalmodelstobejoinedandeasesthetasksofgeometrysetup,meshgeneration,andpostprocessing.Thedisadvantageisthathigh-levelpackagestendtohaverathercomplexdatastructure,noteasilyaccessibletotheuser.Thiscancomplicatetheactualimplementationoftheproblembecausethesensitivityanalysisistraditionallyformulatedinalow-levelmanner.Inthisworkwehaveusedthecommercialfinite-elementpackageFemlabbothforthesolutionoftheflowproblemandforthesensitivityanalysisrequiredbytheoptimizationalgorithm.Weshowhowthissensitivityanalysiscanbeperformedinasimplewaythatisalmostindependentoftheparticularphysicalproblemstudied.Thisapproachprovesevenmoreusefulformulti-fieldextensions,wheretheflowproblemiscoupledto,e.g.,heatconduction,convection-diffusionofsolutes,anddeformationofelasticchannelwallsinvalvesandflowrectifiers[10].Thepaperisorganizedasfollows:InSec.IIweintroducethetopologyoptimizationmethodforfluidsinNavier–Stokesflow,anddiscusstheobjectiveofdesigningfluidicdevicesorchannelnetworksforwhichthepowerdissipationisminimized.InSec.IIIweexpresstheNavier–StokesequationsinagenericdivergenceformthatallowsthemtobesolvedwithFemlab.Thisformencompassesawiderangeofphysicalproblems.Wealsoworkoutthesensitivityanalysisforaclassofintegral-typeoptimizationobjectivesinsuchawaythatthebuilt-insymbolicdifferentiationtoolsofFemlabcanbeexploited.InSec.IVwepresentourtwonumericalexamplesthatillustratesdifferentaspectsandproblemstoconsider:Thefirstexampledealswithdesigningastructurethatcanguidetheflowinthereversedirectionofanappliedpressuredrop.ThegeneraloutcomeoftheoptimizationisanS-shapedchannel,buttheexampleillustrateshowthedetailedstructuredependsonthechoiceoftheparametersofthealgorithm.ThesecondexampledealswithafourterminaldevicewherethefluidicchanneldesignthatminimizesthepowerdissipationshowsaReynoldsnumberdependence.AstheReynoldsnumberisincreasedatransitionoccursbetweentwotopologicallydifferentsolutions,andwediscusshowthepositionofthetransitiondependsonthechoiceofinitialconditions.FinallyintheappendixweincludeatranscriptofourFemlabcoderequiredforsolvingthesecondnumericalexample.Thecodeamountsto111lines–excludingtheoptimizationalgorithmthatcanbeobtainedbycontactingK.Svanberg[11,12,13].2II.TOPOLOGYOPTIMIZATIONFORNAVIER–STOKESFLOWINSTEADYSTATEAlthoughourhigh-levelprogramming-languageimplementationisgenerallyapplicablewehavechosentostartontheconcretelevelbytreatingthebasicequationsforourmainexample:thefullsteady-stateNavier–Stokesflowproblemforincompressiblefluids.WeconsideragivencomputationaldomainΩwithappropriateboundaryconditionsfortheflowgivenonthedomainboundary∂Ω.ThegoaloftheoptimizationistodistributeacertainamountofsolidmaterialinsideΩsuchthatthemateriallayoutdefinesafluidicdeviceorchannelnetworkthatisoptimalwithrespecttosomeobjective,formulatedasafunctionofthevariables,e.g.,minimizationofthepowerdissipatedinsidethedomain.Thebasicprincipleinthematerialdistributionmethodfortopologyoptimizationistoreplacetheoriginaldiscrete