Computer Methods in Applied Mechanics and Engineering(2)

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].

一些ME专业提升的论文。

T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462133

time.ThebodyBisassumedtohaveaprescribedsurfacemotion.Followingthematrix–vectornotationof[4],wecanwriteEqs.(1)–(3)semi-discretelyasM

dq

dt

þGpÀHf¼NðqÞþLqþbc1;ð4ÞDq¼0þbc2;

ð5ÞEq¼unþ1

B;

ð6Þ

whereq,p,andfarethediscretevelocity uxvector,pres-sure,andboundaryforce.Thediscretevelocity,u,canberelatedtoqbymultiplyingthecellfaceareanormaltothevector,i.e.,q=(qu,i,qv,i)=(uiDyi,viDxi).Theabove rst,second,andthirdequationsrepresentthediscretizedmomentumequation,continuityequation,andno-slipcon-ditionalongoB.Discretizednon-linearconvectivetermÀuÆ$uisdenotedbyNðqÞandoperatorsMandLarethe(diagonal)massmatrixanddiscreteLaplacian,respectively.

Wenotethatallofthematricesintheabove(andallthatfollow)aresparseandaremoste cientlycodedaspoint-operators-subroutinesreturnthematrix–vectormul-tiplysuchthatthematricesareneverexplicitlyformed.Forconvenience,point-operatorrepresentations(forthecaseofauniformgrid)aregiveninAppendixA.

OperatorsGandDarethediscretegradientanddiver-genceoperatorsTandcanbeformulatedsuchthatG=ÀD[27,4].TheremainingoperatorsofEandHaretheinterpolationandregularizationoperatorsresultingfromtheregularizationoftheDiracdeltafunctionsinEqs.(1)and(3).Theno-slipconstraintisenforcedbyequatingtheboundaryvelocity,uB,tothevelocityvaluealongoBinterpolatedbyEfromtheneighboringcells.Ontheotherhand,theregularizationoperatorsmearsthee ectofthesingularboundaryforcealongoBtotheCartesiangrid.Topreservesymmetryinthe nalalgo-

rithm,weconstructtheseoperatorstosatisfyE=ÀHT;see[36]forfurtherdiscussion.WementionthatmatricesG,D,E,andHarenotsquare.Consequently,Eqs.(4)–(6)2canbewritten30asasystemofalgebraicequations:nþ110n16AGÀHr0bc11

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