Computer Methods in Applied Mechanics and Engineering(7)

一些ME专业提升的论文。

2136T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–2146

Fig.4.Locationofvariablesonstaggered3Dmesh.Velocitycomponentsarede nedatthecenterofeachedge.Streamfunctionandcirculationarede nedsimilarlyfortheVoronoicell–inthiscaseacellthatiso setbyhalfacelllengthineachdirection.

3.2.NullspaceapproachwithanimmersedboundaryInordertosatisfyboththeincompressibilityandtheno-slipconditionswiththenullspacetechnique,itwouldbenecessarytoderiveabasisforthenullspaceofQT.Although,asingularvaluedecompositionofQTcanbeperformedtonumericallydeterminethenullspace,theresultisnotingeneralasparserepresentationwhichisdesirableforcomputationalfeasibility.Ananalyticalderi-vationofthenullspaceoperatordoesnotseemtobeaneasytaskeither.Moreover,inthegeneralcasewherethebodyismoving,thenullspacerepresentationwouldneedtoberecomputedatleastoncepertimestep.

Tocircumventthisdi culty,weonceagainrelyonaprojectionapproach.TConsiderthesystemn+1thatisobtainedbyincorporatingCandqn+1=CstoEq.(8).TheincompressibilityconstraintandthepressurevariableareeliminatedandwearriveatanotherKTTsystem:

"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=