Computer Methods in Applied Mechanics and Engineering(3)

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.

Inthepresentpaper,werevisitthismethodandproposesomeimprovementstoacceleratetheIBPM.InSection2,wereviewtheoriginalformulationandpresentsomenewresultsfromarecentextensionofthemethodtothree-dimensional ows.InSection3,weimplementanullspace(discretestreamfunction)method[11,4]thatallowsthedivergence-freeconstrainttobeautomaticallysatis edtomachineroundo .Weshowthatifthegridiskeptuniformthroughoutspace(withequalspacinginalldirections),thePoisson-likeequationfortheforcescanbee cientlysolvedeitherdirectlyforstationarybodiesoriterativelyformov-ingbodiesthroughtheuseofafastsinetransform.Whileuniformgridspacingisinfactrequiredinthevicinityof

1

Sti nessissuesarealsoobservedwithelasticsurfaces.Recently,stablesemi-andfully-implicittemporaldiscretizationstocouplethevelocity eldandtheboundaryforceforelasticboundarieshavebeenproposedby[24,22].

thebodybythediscretedeltafunctionthatisusedtoreg-ularizethesurfaceforce,itisrelativelyine cientforexter-nal owswherethedomainneedstoextendtolargedistancefromthebody.IntheoriginalIBPM,thisdi -cultyisovercomebystretchingthemeshawayfromthebody,butthisisincompatiblewiththenullspace/fastsinetransformformulationintroducedhere.Toovercomethisrestriction,wederiveinSection4improvedfar- eldboundaryconditionsthatarecompatiblewiththefastmethodandallowthedomaintobemoresnugaroundthebody.Thenewboundaryconditionsaccountfortheextensivepotential owinducedbythebodyaswellasvor-ticitythatadvects/di usestolargedistancefromthebody.Theboundaryconditionsrelyonamulti-domainapproachwherebythePoissonequationissolved(withthefastsinetransform)onaseriesofincreasinglylarger,butcoarser,computationaldomains.ValidationexamplespresentedinSections5and6demonstratethee cacyandimprovede ciency,respectively,oftherevisedformulation.2.Immersedboundaryprojectionmethod2.1.Projectionapproach

WeconsidertheincompressibleNavier–Stokesequa-tionswithasingularboundaryforcefaddedtothemomentumequationasacontinuousanalogoftheimmersedboundaryformulation:

ouotþuÁru¼Àrpþ12

Z

Reruþfðnðs;tÞÞdðnÀxÞds;ð1ÞsrÁu¼0;uðnðs;tÞÞ¼

Z

ð2Þ

uðxÞdðxÀnÞdx¼uBðnðs;tÞÞ;ð3Þx

whereuandparethevelocityandpressurevariables,

respectively.Notethatweexpresstheno-slipconditionusingadeltafunctionconvolutionalongtheimmersedsur-face.Here,non-dimensionalizationisperformedtoyieldasingleparameterofReynoldsnumber,Re.Spatialvariablexrepresentspositioninthe ow eld,D,andndenotescoordinatesalongtheimmersedboundary,oBhavingavelocityofuB.ThegeometryoftheimmersedobjectBisconsideredtobeofarbitraryshape.Inthepresentdevelop-ment,therearenoforcesinteriortothebodyandanymo-tionordeformation2ofthebodyisprescribed.Furthergeneralizationsofthemethodarepossiblebutawaitfuturework.

Theabovesystemisdiscretizedwithastandardstag-geredCartesiangrid nitevolumemethod.ThemeshandvariablelocationsaredepictedinFig.1.Thecomputa-tionaldomain,D,isrepresentedbyaCartesiangrid,(xi,yi),andtheimmersedboundary,oBisdescribedbyasetofLagrangianpoints,(nk,gk),whichcanbeafunctionof

2

Forexamplefullycoupled uid–structureinteractionviaanimmersedcontinuummethod[38].