Computer Methods in Applied Mechanics and Engineering

一些ME专业提升的论文。

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一些ME专业提升的论文。

Author's personal copy

Available online at

Comput.MethodsAppl.Mech.Engrg.197(2008)

2131–2146

/locate/cma

Afastimmersedboundarymethodusinganullspaceapproach

andmulti-domainfar- eldboundaryconditions

TimColonius*,KunihikoTaira

DivisionofEngineeringandAppliedScience,CaliforniaInstituteofTechnology,CA91125,USAReceived21March2007;receivedinrevisedform3August2007;accepted6August2007

Availableonline12September2007

Abstract

Wereportonthecontinueddevelopmentofaprojectionapproachforimplementingtheimmersedboundarymethodforincompress-ible owsintwoandthreedimensions.BoundaryforcesandpressureareregardedasLagrangemultipliersthatenabletheno-slipanddivergence-freeconstraintstobeimplicitlydeterminedtoarbitraryprecisionwithnoassociatedtime-steprestrictions.Inordertoaccel-eratethemethod,wefurtherimplementanullspace(discretestreamfunction)methodthatallowsthedivergence-freeconstrainttobeautomaticallysatis edtomachineroundo .Byemployingafastsinetransformtechnique,thelinearsystemtodeterminetheforcescanbesolvede cientlywithdirectoriterativetechniques.Amulti-domaintechniqueisdevelopedinordertoimprovefar- eldboundaryconditionsthatarecompatiblewiththefastsinetransformandaccountfortheextensivepotential owinducedbythebodyaswellasvorticitythatadvects/di usestolargedistancefromthebody.Themulti-domainandfasttechniquesarevalidatedbycomparingtotheexactsolutionsforthepotential owinducedbystationaryandpropagatingOseenvorticesandbyanimpulsively-startedcircularcyl-inder.Speed-upsofmorethananorder-of-magnitudeareachievedwiththenewmethod.Ó2007ElsevierB.V.Allrightsreserved.

1991MSC:76D05;76M12PACS:47.11.+j

Keywords:Immersedboundarymethod;Fractionalstepmethod;Projectionmethod;Nullspacemethod;Vorticity/streamfunctionformulation;Far- eldboundaryconditions;Multi-domainmethod;FastPoissonsolver;Finitevolumemethod;Incompressibleviscous ow

1.Introduction

Intheimmersedboundarymethod(IBmethod),immersedsurfacesaregeneratedbyforcesatasetofLagrangianpoints[29,20,19].The owissolvedonanEuleriangridthatdoesnotconformtothebodygeometry–typicallyauniformCartesiangridisused.Theboundaryforcesthatexistassingularfunctionsalongthesurfaceinthecontinuousequationsaredescribedbydiscretedeltafunctionsthatsmear(regularize)theforcinge ectovertheneighboringEuleriancells.

Correspondingauthor.

E-mailaddresses:colonius@caltech.edu(T.Colonius),kunihiko@cal-tech.edu(K.Taira).

0045-7825/$-seefrontmatterÓ2007ElsevierB.V.Allrightsreserved.doi:10.1016/j.cma.2007.08.014

*

IntheoriginalIBmethod,surfaceswereviewedas ex-ibleelasticmembraneswithaconstitutiverelation(e.g.Hooke’slaw)relatingtheforcestothemotionoftheLagrangianpoints[28].Thistechniquewaslaterextendedtosurfaceswithprescribedmotion(andinparticularrigidbodies)bytakingthespringconstanttobelarge[2,16].Goldsteinetal.[9]appliedtheconceptoffeedbackcontroltocomputetheforceontherigidimmersedsurface.Thedi erencebetweenthevelocitysolutionandtheboundaryvelocityisusedinaproportional-integralcontroller.Con-stitutivelawsareeliminatedinthedirectforcingmethod[21,7];forcinginthemomentumequationisdeterminedbypenalizingtheslipatthe(interpolated)surface.Fortheaforementionedtechniquesthatutilizeconstitutiverela-tions,thechoiceofgain(sti ness)rgegain

一些ME专业提升的论文。

2132T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–2146

resultsinrestrictionsonthetimestep,whilesmallgainresultsinsliperror.1Directforcingmethodssimilarlyresultinasliperroratthesurface.Whilethesliperrorisreportedtobesmall[7],themagnitudecannotbeestimatedinadeductivemanner.FurtherinformationregardingtheIBmethodandhigher-orderextensionsaregiveninarecentreview[20].

AnalternativeistoregardtheboundaryforcesasLagrangemultiplierswhosevaluesarechosentosatisfytheno-slipconstraint[8,36].Byintroducingappropriateregularizationandinterpolationoperatorsandgroupingthepressureandforceunknownstogether,thediscretizedincompressibleNavier–Stokesequationscanbeformulatedwithastructurealgebraicallyidenticaltothetraditionalfractionalstepmethod[36].Thepressureandforceunknownsarefoundbysolvinga(modi ed)Poissonequa-tion.Inwhatfollows,werefertothismethodastheimmersedboundaryprojectionmethod(IBPM).

TheprincipleadvantagesoftheIBPMtechniquearethatthecontinuityandno-slipconstraintscanbesatis ed(toarbitraryaccuracy)implicitlyatthenexttimelevel,andthattheCourantnumberisonlylimitedbythechoiceoftimemarchingschemesfortheviscousandadvectiontermsinthemomentumequation.Further,itispossibletoarrangealloperationssothatthemethodisuniformlysec-ond-orderaccurateintime,andsothatthematrixarisingfromimplicittreatmentoftheviscoustermsinthemomen-tumequationaswellasthemodi edPoissonmatrixarebothsymmetricandpositivede nite.Consequentlytheconjugate-gradientmethodcanbeusedtosolvethelinearsystems.However,iterativesolutionofthelinearsystemsresultsinaconvergenceerror.Thispresentsnodi cultyinthemomentumequationwherethesolutionneedonlybeconvergedtotheextentthatitissmallerthanotherdis-cretizationerrors.Butinthemodi edPoissonequation,convergenceerrorsdirectlyimpacttheaccuracytowithwhichthedivergence-freeandno-slipconstraintsaresatis- ed.Whiletheerrorscanbemadearbitrarilysmall,largenumbersofiterationsmayberequired.

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