Computer Methods in Applied Mechanics and Engineering(5)

"CTACCTE

T#

snþ1 CTrn!1EC0~f¼unBþ1:ð17ÞTheleft-handsidematrixissymmetricbutingeneralindef-inite,makingadirectsolutionlesse cient.Theprojection

(fractionalstep)approachmimicsEqs.(9)–(11),andweobtain

CTACsüCTrn1;

ð18ÞECðCTACÞÀ1ðECÞT~f¼ECsÃÀunþ1B

;ð19Þsnþ1¼sÃÀðCTACÞÀ1ðECÞT~f

;ð20Þ

wherewehaveasnotyetinsertedanapproximationfortheinverseofCTAC.Directsolutionofthissysteminthegen-eralcaserequiresanestediterationtosolvethemodi edPoissonequation.Thismaybefeasibleingeneral(aroughoperationcountindicatethattheworkissimilartoEqs.(9)–(11)).Inthecasewherethebodyisnotmoving,itismoreoverpossibletoperformaCholeskydecompositionofEC(CTAC)À1(EC)Tonceandforall,sincethedimensionofthesystemscaleswiththenumberofforcesfortheim-mersedboundary.Inthiscaseasystemofequationsof

theformCTACx=bneedbesolvedonceforeachLagrangianforceatthebeginningofthecomputation.3.3.Fastmethodforuniformgridandsimpleboundaryconditions

Inthissectionwereverttothesemi-discretemomentumequation,

M

dqþGpþETdt

~f¼NðqÞþLqþbc1;ð21Þ

wheresymbolsareasde nedpreviously.Thedivergence

freeandno-slipconstraintsareunchanged.

Wenowshowthatwithsimpli cation,asimilarsystemtoEqs.(9)–(11)maybesolvedusingfastsinetransforms,resultinginasigni cantreductionincomputationalwork.Whenthegridisuniform(withequalgridspacinginallcoordinatedirections),themassmatrixMistheidentitymatrix.Weassumeforthemomentthatthevaluesofthevelocityareknownintheregionoutsidethecomputationaldomain.WeapplysimpleDirichletboundaryconditionstothevelocitynormaltothesides/edgesofthecomputationaldomain,ckingfurtherinformation,onecouldspecify,forexample,ano-penetrationBCforthenormalcomponentofvelocityandazerovorticity(orno-stress)conditionfortheremainingtangentcomponents.Thesearenaturalboundaryconditionsforanexternal owaroundthebody,providedthedomainislarge.Inthenextsectionwewillshowhowimprovedestimatesforthevelo-citiesoutsidethecomputationaldomaincanbeobtainedviaamulti-domainapproach.

Withthesesimpli cationsweoperateonEq.(21)withCT

(whicheliminatesthepressure)andweobtaindcþCTdt

ET~f¼ÀbCTCcþCTNðqÞþbcc:ð22Þ

InderivingthisequationwehaveusedthatLq=ÀbCCTq=ÀbCcprovidedthatDq=0.Herebisacon-stantequalto1/(ReD2),whereDistheuniformgridspac-ing.2Thisidentitymimicsthecontinuousidentity$u=$($Æu)À$·$·u=À$·$·u.

Withuniformgridandtheaforementionedboundaryconditions,thematrixÀbCTCisthestandarddiscreteLaplacianoperatorona5-or7-pointstencilintwoandthreespatialdimensions,respectively.Theboundarycondi-tionsdiscussedaboveresultinzeroDirichletboundaryconditionsforc.ThisdiscreteLaplacianisdiagonalizedbyasinetransformthatcanbecomputedinOðNlogc)[30].Wedenote2NÞoperations(whereNisthedimensionofherethesinetransformpair:^c¼Sc$c¼S^c;

ð23Þ

wherethecircum exdenotestheFouriercoe cients.Inwritingthetransformpair,wehaveusedthefactthatthesinetransformcanbenormalizedsothatitisidenticaltoitsinverse.Further,wemaywritesymbolicallyK=

一些ME专业提升的论文。

T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462137

SCTCS,whereKisadiagonalmatrixwiththeeigenvaluesofCTC.Thesearepositiveandknownanalytically(e.g.[30]),andwenotethatthereisnozeroeigenvalue(sincetheboundaryconditionsareDirichlet).

Applyingthesametime-marchingschemesusedprevi-ouslyS weobtainthetransformedsystem:IþbDtK Scü IÀbDt

CTC

cn22

þDtÀ2

3CT

NðqnÞÀCTNðqnÀ1ÞÁ þDtbcc;

ð24ÞECSKÀ1 IþbDt À1

!2

KSðECÞT~f¼ECSKÀ1ScÃÀunþ1B;ð25Þcnþ1

¼cÃÀS IþbDt2

K À1

SðECÞT~

f:ð26Þ

Thevelocity,neededforthenexttimestep,maybefoundbyintroducingthediscretestreamfunction:qn¼Csnþbcq;

sn¼SKÀ1Scnþbcs:

ð27Þ

Eachofthevectorsbcc,s,qinvolvestheassumedknownval-uesofvelocityattheedgeofthecomputationaldomain.Theirvaluesarediscussedindetailinthenextsection.Inthenewsystemofequations,onlyonelinearsystemneedbesolved,Eq.(25),withapositivede niteleft-handsideoperator.Thatthematrixispositivede nitecanbeseenbyinspection.ThedimensionsofthematrixarenowNf·Nf,andthusmanyfeweriterationsarerequiredthantheoriginalmodi edPoissonequation,Eq.(10).Tobemoreprecise,eachiterationonEq.(25)requiresOðNð2log2NþNbwþ4dÞÞoperations,whereNisthenum-berofvorticityunknownsandNbwisthebandwidth5ofthebody-forceregularization/interpolationoperators,anddisthedimensionalityofthe ow(2or3for2Dor3D,respectively).ForthediscreteDeltafunctionwithasup-portof3D,wehaveNbw=3d.FortheoriginalPoissonequation,Eq.(10),thecostperiterationisOðNÂðNbwþð2dþ1ÞjÞþ4dÞ,wherejistheorderoftheapproximateTaylor-seriesinverseofAandthefactor2d+1isthestencilofthediscreteLaplacian.Furthermore,usingstandardestimatesforthenumberofiterationsrequiredforconvergenceoftheconjugate-gradientTmethod[35]alongwiththeknowneigenvaluesofCC,wecanesti-matethattheoperationcountpertimestepforthePoissonsolutionhasbeenreducedfrom6

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