NUMERICAL STABILITY OF THE FINITE ELEMENT IMMERSED

ConsiglioNazionaledelleRicercheIstitutodiMatematicaApplicataeTecnologieInformatichePUBBLICAZIONID.Boffi,L.Gastaldi,L.HeltaiNUMERICALSTABILITYOFTHEFINITEELEMENTIMMERSEDBOUNDARYMETHODN.38-PV2006IstitutodiMatematicaApplicataeTecnologieInformatichedelC.N.R.SedediPaviaViaFerrata,1-27100PAVIA(Italy)Tel.[phone]Fax[phone]SezionediGenovaViaDeMarini,6(TorrediFrancia)-16149GENOVA(Italy)Tel.[phone]Fax[phone]SezionediMilanoViaE.Bassini,15-20133MILANO(Italy)Tel.[phone]Fax[phone]URL::50WSPC/INSTRUCTIONFILEibm˙stabNUMERICALSTABILITYOFTHEFINITEELEMENTIMMERSEDBOUNDARYMETHOD∗DANIELEBOFFIDipartimentodiMatematica”F.Casorati”,ViaFerrata1,I-27100Pavia,Italydaniele.boffi@unipv.itLUCIAGASTALDIDipartimentodiMatematica,ViaValotti9,I-25133Brescia,[e-mail]diMatematica”F.Casorati”,ViaFerrata1,I-27100Pavia,[e-mail]hodisbothamathematicalformulationandanumericalmethod.Initscontinuousversionitisafullynon-linearlycoupledformulationforthestudyoffluidstructureinteractions.Asitiscommoninthesecases,manynumericalmethodshavebeenintroducedtoreducethedifficultiesrelatedtothenon-linearcouplingbetweenthestructureandthefluidevolution,howevernumericalinstabilitiesarisewhenexplicitorsemi-implicitmethodsareconsidered.Inthisworkwepresentastabilityanalysisbasedonenergyestimatesofthevariationalformulationoftheimmersedboundarymethod.Atwodimensionalincompressiblefluidandaboundaryintheformofasimpleclosedcurveareconsidered.WeusealinearizationoftheNavier-Stokesequationsandalinearelasticitymodeltoprovetheunconditionalstabilityofthefullyimplicitdiscretization,achievedwiththeuseofabackwardEulermethodforboththefluidandthestructureevolution,andaCFLconditionforthesemi-implicitmethodwherethefluidtermsaretreatedimplicitlywhilethestructureistreatedexplicitly.Wepresentsomenumericalteststhatshowgoodaccordancebetweentheobservedstabilitybehaviorandtheonepredictedbyourresults.Keywords:immersedboundarymethod;finiteelementmethod;numericalstability;CFLcondition.AMSSubjectClassification:65N30,65N12,74F10∗ThisworkwaspartiallysupportedbyIMATI-CNR,Pavia.1October10,200610:50WSPC/INSTRUCTIONFILEibm˙stab2DanieleBoffi,LuciaGastaldi,LucaHeltai1.IntroductionTheseverestiffnesswhicharisesinimmersedboundaryproblemsforthepresenceofsingularforcesandthecorrespondinglystricttimesteplimitincomputationshasbeenwell-documentedintheliterature.16,20Thiswouldsuggesttheuseofafullyimplicitmethodfortheresolutionofthestronglynonlinearcoupledsystemofequations.IndeedonesuchamethodwasimplementedbyTuandPeskinforStokesflow,20butwhileitappearedtobeunconditionallystable,itwashoweverfartooexpensiveforpracticalcomputations.Ontheotherhandnumericalinstabilitiesarisewhencomputationsarecarriedonusingsemi-implicitorexplicittime-steppingtechniqueswhichrequireacarefulchoiceofthediscretizationparameters.Thenonstandardcouplingbetweentheimmersedmaterialandthefluidtogetherwiththestrongnon-linearityoftheproblemcontributetomakingthetheoreticalanalysisofthismethodquitedifficult.Uptonowtherearesomeexistenceresultsforsimplifiedone-dimensionalmodels,2,3butverylittleanalysishasbeencarriedoutonhigherdimensionalmodels.Astabilityanalysisbasedonthemodesofoscillationofaone-dimensionalfiberimmersedinatwo-dimensionalfluidwasintroducedbyStockieandWettoninRef.19anditsinfluenceonthechoiceofthetimestepsizeforvariousdiscretizationschemeswasfurtheranalyzedbythesameauthorsinRef.18.Themainaimofthisworkistoaddressthestabilityaspectofimmersedbound-arycomputationstakingadvantageofthenaturalenergyestimatesthatarisefromtheuseofavariationalapproachtotheimmersedboundarymethod,asintroducedinRefs.3,4,5,6.Thestrongnon-linearityofthecoupledproblemandthedifficultiesconnectedtotheNavier-Stokesequationsthemselvesmadeusconcentratethetheoreticalanalysismainlyonasimpletwo-dimensionalmodel,whereweneglectedthenon-lineartermoftheNavier-Stokesequationsandweconsideredasimpleelasticitymodelfortheimmersedstructure.Wederiveouranalysisfromtheenergyconservationpropertiesofthecontinuousproblemandinparticularweaskournumericalmethodtoadheretothesameprinciples,obtainingsomeanswersonhowtochoosethediscretizationparametersinordertocontroltheinstabilitiesthatmayariseinthenumericalcomputations.InSection2webrieflyintroducetheideabehindtheimmersedboundarymethod,asitwasintroducedbyPeskin.13Section3isdedicatedtothederiva-tionoftheeasiestpossibleelasticitymodelwhileSections4and5reviewthefiniteelementapproachasitwaspresentedinRefs.3,4,5,6.InSection6weconcentrateonthetimesteppingscheme.Whileitispossibletochoosebetweenavarietyofmethodstoadvanceintime,werestrictouranalysistotwoextremelysimplecases.TheFE/BEandBE/BEschemes,thatisForwardEuler/BackwardEulerandBackwardEuler/BackwardEuler,wherethedifferenceisthetreatmentofthestructureevolutioninanexplicitversusimplicitway.InOctober10,200610:50WSPC/INSTRUCTIONFILEibm˙stabNumericalStabilityoftheFiniteElementImmersedBoundaryMethod3Section7wepresentthestabilityanalysisforthecontinuousproblemandweshowhowthefiniteelementapproachwe