On the stability of the finite element immersed bo

OnthestabilityoftheFiniteElementImmersedBoundaryMethod1LucaHeltai2DipartimentodiMatematica“F.Casorati”,ViaFerrata1,I-27100Pavia,ItalyAbstractTheimmersedboundary(IB)methodisamathematicalformulationforfluid-structurein-teractionproblems,whereimmersedincompressiblevisco-elasticbodiesorboundariesin-teractwithanincompressiblefluid.TheoriginalnumericalschemeassociatedtotheIBmethodrequiresasmoothedapprox-imationoftheDiracdeltadistributiontolinkthemovingLagrangiandomainwiththefixedEulerianone.Wepresentastabilityanalysisofthefiniteelementimmersedboundarymethod,wheretheDiracdeltadistributionistreatedvariationally,inageneralizedvisco-elasticframeworkandfortwodifferenttimesteppingschemes.Keywords:immersedboundarymethod,finiteelementmethod,numericalstability,CFLcondition.1IntroductionOneofthemaindifficultiesthatariseswhendealingwithvisco-elasticityandfluidstructureinteractionproblemsisthefactthatfluidsandelasticmaterialshaveadifferent“natural”framework.TheusualwayofcharacterizingafluidmotionistheEulerianframework,wherethesystemisdescribedusingthevelocityandpressurefields.Ontheotherhand,whendealingwithelasticity,itiscustomarytoexpressthestressasafunctionEmailaddress:[e-mail](LucaHeltai).1ThisworkwaspartiallysupportedbyIMATI-CNR,Pavia.2TheauthorwaspartiallysupportedbyaFulbrightgrant.PreprintsubmittedtoElsevierScience5September2007ofthedisplacementsofthematerialparticlesfromtheirreference,orLagrangian,position,whichisnotdirectlyavailableintheEulerianformulation.TheImmersedBoundarymethodgivesonewaytolinkthetwoframeworksto-getheranddeploythestrengthsofbothformulationsatthesametime.TheoriginalIBformulation(see[25]foranintroductiononthesubject)wasintendedtosimplifythestudyoftheinteractionbetweenthinmembranesundergoinglargedeformationsandfluidsdescribedbytheNavier-Stokesequations,bymeansofanapproximationoftheDiracdeltadistribution,whichwasusedasaninterpolationkernelbetweenthetwoframeworks.Afiniteelementformulationoftheproblemwasfirstintroducedin[21]andlaterdevelopedin[32,34],wherethediscretizationofthefluidisdoneviathefiniteelementmethodandthepassagefromtheEuleriantotheLagrangiandomainisdoneviatheReproducingKernelParticleMethod,toprovideanapproximationoftheDiracdeltadistributionsuitableforthefiniteelementmethod.Avariationalapproachtotheproblemwasintroducedin[2,3],wheretheDiracdeltadistributionisnolongerneededasitistreatedvariationallythroughitsac-tiononthetestfunctions.ThevariationalapproachtranslatesnaturallyinafiniteelementformulationoftheIBmethodwhichwasfurtherdevelopedin[4].Theoriginaldiscretizationonoftheinteractionequations,asproposedin[25],pre-servesmass,momentum,angularmomentum,torqueandpower,ensuringthatintheconversionbetweenthetwoframeworksnospuriouscreationordestructionofmass,momentumorenergyisinducedbythenumericalapproximation.Theintro-ductionofthetimediscretizationhoweverdisruptstheseconservationproperties.In[31]theauthorspresentacomparisonbetweenthreedifferenttimeapproxima-tionschemes,highlightingthedifficultiesrelatedtothenonlinearityofthecoupledproblem.Afirstattempttoanalyzethestabilitypropertiesoftheseapproximationschemeswasintroducedin[30]wheretheatuthorspresentastabilityanalysisbasedonthestudyofthemodesofoscillationofasinglestraightenedone-dimensionalfiberimmersedinatwodimensionalfluid.Theeffectofthesemodesofoscillationonthetimesteppingschemeswasfurtheranalyzedbythesameauthorsin[29].In[5,7,8,6]theauthorspresentastabilityanalysisthattakesadvantageofthefiniteelementformulationoftheproblemandofthenaturalenergyestimatesinheritedbythevariationalanalysisofthecoupledsystem.Theideaisbasedontherequirementthattheenergyofthesystemdecreasesateachtimestep,providinganeffectiveCFLconditionforthecoupledproblemtoremainstablewithrespecttothefluid-structureinteractioncharacteristics.Theextensiontomoregeneralfluidstructureinteractionproblemsusingthefor-mulationderivedin[25]waslimitedtoanisotropicelasticityduetothelackofatermintheformulationthattakescareofthecontinuityofthestressbetweenthe2solidbodyandthefluid.In[10]theauthorsrecognizedthisproblemandproposedaderivationoftheIBmethodbasedonclassicalhyper-elasticitytheory(see,forexample,[15])wherethemissingtermwasfoundtobeofthesamecharacteroftheoriginalsingulartermintroducedforthestudyofthinmembranes.InthispaperwereviewtheFiniteElementImmersedBoundary(FEIB)methodasintroducedin[2–4,10]andwegeneralizetheresultspresentedin[8]totakeintoaccountgeneralhyper-elasticmaterials,bothinthethincaseofco-dimensiononestructuresinteractingwithtwo-orthree-dimensionalfluidsaswellasinthemoregeneralframeworkoftwo-orthree-dimensionalstructuresinteractingwithtwo-orthree-dimensionalfluids.Sections2and3presentbrieflytheIBmethodandthehyper-elasticmodelsthatwillbeused.Section4presentthevariationalandfiniteelementformulationoftheIBmethod,whileinsections5and6wepresentthetimediscretizationandthegeneralizedstabilityanalysisforthefullydiscreteproblem.Section7and8presentsomenumericalexperimentsandconclusions.2TheImmersedBoundaryMethodTheformulationoftheImmersedBoundaryMethodthatweuseinthefollowingstabilityanalysisistheonethatwasder