"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