Computer Methods in Applied Mechanics and Engineering(8)

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

OðN1=2Nð7dþð2dþ1ÞjÞÞoperationcountforEq:ð10Þto

5

Wehaveusedthefactthat6HerethefactorsN1/2orN1=N2

f(Ninarrivingattheestimate.

faretheestimatednumberofiterationsoftheconjugategradientsolver,the2Nlog2Nfactorcomesfromtwo(fast)sinetransforms,the(2d+1)jfactorfromtheLaplacian,and7dfromtheinterpolation,regularization,CT,andCoperationstogether.

OðN1=2

fNð2log2Nþ7dÞÞ

operationcountforEq:ð25Þ:

Forexample,inathree-dimensionalcasewithN=1283,Nf=103,d=3,andj=3theestimatedspeedupisabout30.Foratwo-dimensionalcasewithN=1282,Nf=200,d=2andj=3,thespeedupisabout10.ThisisforthePoissonsolvealone.AdditionalspeedupoccursbecauseitisnolongernecessarytosolveasystemAx=bforthemomentumequation.NumericalexperimentsinSection6forthetwo-dimensionalcasecon rmatleasttheorder-of-magnitudeofthespeedup(theactualspeedupisfasterthanpredicted).Finally,werecallthatthenewsystemofequationsresultsinnoiterativeerrorinsatisfyingthedivergence-freeconstraint(itisautomaticallyzerotoround-o ).

Ifthebodyisstationary,thenthePoisson-likeequationfortheforcescanbee cientlysolvedusingatriangularCholeskydecomposition.Thisresultsinavastlylowerworkpertime-step,sincetheoperationcountforthePois-sonsolveissimplyOðN2fÞ.InthiscasethecomputationalspeedislimitedonlybythesolutionofEq.(24).

Tosummarize,ifthegridisuniformandsimplebound-aryconditionsareused,itisvastlypreferabletosolveEqs.(24)–(26).Werefertothisinwhatfollowsasthefastmethod.Unfortunately,forexternal ows,thesimpli edboundaryconditionsarenote ectiveunlessthecomputa-tionaldomainisquitelarge.Sincethegridisalsorequiredtobeuniform,evenfarawayfromthebody,thelargerdomainwouldquicklynegatethebene toffastmethod.However,inthenextsectionwediscussanalternativestrat-egyforimplementingboundaryconditionsinthefastmethodthathasamoremodestcostpenalty.

4.Far- eldboundaryconditions:amulti-domainapproachThefastmethodreliesonsimpli edfar- eldboundaryconditions,namelyknownvelocitynormaltotheboundaryandknownvorticity.Thesecanbesettozeroifthecompu-tationaldomainissu cientlylarge.Forsmallerdomains,thiswillleadtosigni canterrorsand,inparticular,theforcescomputedonthebodywillsu erasigni cantblock-ageerror.Theerrorarisesfromtwosources.The rstistheextensive,algebraicallydecayingpotential owinducedbythebody(orequivalently,thesystemofforces).Thesecondisthatvorticitymayadvectordi usethroughthebound-ary.InouroriginalmethoddiscussedinSection2,theseerrorsareminimizedbyusingalargedomainwithahighlystretchedCartesianmeshnearthefar- eldboundaries(butretaininguniformgridspacingnearthebody),aswellasbyusinganapproximateconvectiveout owboundarycondi-tion.Unfortunately,stretchedmeshesareincompatible7withdirectFouriermethodsforsolutionofthePoissonequation.Inthissection,weshowhowtoposeanaccurate

7

IncertainspecialcircumstancesstretchedmeshescanbecombinedwithFourier-transformmethodsforellipticequations,e.g.[3].