Computer Methods in Applied Mechanics and Engineering(9)

一些ME专业提升的论文。

2138T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–2146

far- eldboundaryconditionthatisalsocompatiblewiththefastmethoddescribedinthelastsection.

Westartbybrie yreviewingrelevantboundarycondi-tionsdesignedtoreduceoneorbothoftheaforementionederrors.Forerrorsassociatedwiththeslowlydecayingpotential ow,afewtechniqueshavebeenposedinthepasttopatchinthepotential owextendingfromthetruncatedcomputationalboundarytoin nity.RennichandLele[31]proposeatechniquefortwounboundeddirectionsandoneperiodicdirection.Theirmethodisbasedonmatchingthenumericalsolutiontoanalyticalrepresentationofthesolu-tiontoLaplaceequationoutsideacylindricalvolume.Theyreporta50%increasepertimestepforatypicallarge-scalecomputation,butthiscostismorethano setbytheabilitytousemuchmorecompactdomains.Wang[39]presentsasimilarapproachfortwo-dimensional owintheformofacorrectiontoatrialsolutionthatsatis esanincorrectDirichletboundarycondition.Vortexparticlemethodsinprincipleautomaticallyaccountfortheextensivepotential owgeneratedbythevorticity.However,inpracticeitisoftennecessarytoremoveparticlesthatadvecttolargedis-tancefromtheregionofinterest.Aninterestingtechniquetoreduceerrorsassociatedwithremovalofparticlesiscalledmerging,wherebythecirculationsofseveralvortexparticlesarecombinedintoasingleelementwhentheyaresu cientlyfarfromthebody[34,32].

Thesecondtypeoferrorassociatedwithvorticityadvectingordi usingthroughtheboundaryistypicallyhandledbyposingout owboundaryconditions.Forincompressible owtheseareusuallycalledconvectiveboundaryconditions,whereasincompressible owthetermnon-re ectingboundaryconditionisoftenused.Anothertechniqueistoselectivelyapplydampinginaregionnearthecomputationalboundary.Methodsthatemploythistechniquevaryfromadhocspeci cationoflayerwidth,dampingstrength,etc.,totechniquesthattheoreticallyspecifythedampingparametersaccordingtoamodel.Anexampleistheperfectlymatchedlayer[1]forlinearwaveequations(includinglinearizedcompressibleEulerequations[12])thatusesanalyticalsolutionstothegovern-ingequationstoderivedampingtermsthatpreventre ec-tionofwavesfromtheinterface.Anothertechniquecalledsuper-grid[6]isbasedonananalogywithturbulencemod-eling–thatthee ectoftheturbulencemodelistomodelscalestoo netoberesolvedinthecomputationalmesh,whereasthee ectoftheboundaryconditionistomodelscalestoolargetoberesolvedinthecomputationaldomain.Afulldiscussionofthesetechniquesisbeyondthescopeofthispaper;wereferthereadertosomerecentreferencesforfurtherdetails[33,15,25,5].Thesetechniquesaredesignedtoremovevorticityfromthedomainassmoothlyaspossibletherebypreventingundesirablere ec-tionsoraliasing.Mostdonotaccountforthevelocityinducedbyvorticitythathasalreadyexitedthedomain(anon-locale ect).

Wepresenthereanalternativetechniquethatsharessomefeatureswiththesepreviousmethods,especiallythoseof[31,34,6].Itisbasedonamulti-domainapproachthatalsosharessomeoperationswiththemultigridmethodforsolvingellipticequations.We rstdescribethemethodinwords.Thebasicideaistoconsiderthedomainasembeddedinalargerdomainbutwithacoarsermesh.Thecirculationontheinner(smaller, ner)meshistheninterpolatedorcoarsi edontotheouter(larger,coarser)mesh.ThePoissonequationissolved(withzeroboundaryconditions)ontheouterdomain.ThissolutionistheninterpolatedalongtheboundaryoftheinnermeshandthePoissonequationissolved,withthe‘‘corrected’’boundaryvaluespeci ed,ontheinnermesh.

Similartothevortexmergingmethoddiscussedabove,anyexistingcirculationintheouterportionofthelargerdomainisretainedfromtheprevioustimelevel.Inthisway,weapproximatelyaccountforcirculationthathasadvectedordi usedoutoftheinnerdomain.Clearly,thesolutiononthecoarsermeshcontainsalargertruncationerrorfortheevolutionofthisvorticity.However,inversionoftheLaplacianisasmoothingoperation.Highfrequencycomponentsofthesolutioninducedbycirculationintheoutermeshdecaymorerapidlythanlow-frequencycompo-nents.Beinginterestedinthe owinthevicinityofthebody(anditswake),wediscardthesolutionsintheouterregiononlyretainingthevelocityitinducesontheinnerdomain.

Weapplythistechniquerecursivelyanumberoftimes,enlarging(andcoarsening)thedomainineachgridlevel.Wechoosetokeepthetotalnumberofgridpointsineachdirection xedoneachmesh;wemagnifythedomainandcoarsenthegridbyafactorof2ateachgridlevel.Thepro-cedureisshownschematicallyinFig.5.Thevorticityisrepeatedlycoarsi edoneachprogressivegrid.ThePoissonequationisthensolvedonthelargestdomain,inturnpro-vidingaboundaryconditionforthenextsmallerdomain.Theprocessisthenrepeateduntilwereturntotheoriginaldomain.

Thevelocity elddecaysalgebraicallyinthefar- eldandwethusexpecterrorsassociatedwiththeboundaryconditiononthelargestdomaintodecreasegeometricallyasthesizeofthelargestdomainisincreased.Intheworstcaseofatwo-dimensional owwithnon-zerototalcircula-tion,thevelocitydecayswiththeinverseofthedistancetothevorticalregion.AnalyticalestimatesgiveninAppendixBshowthatweobtainafactorof4reductioninthebound-aryerrorwitheachprogressivelylargergrid.This,ofcourse,iswhatwouldbeobtainedbysimplyextendingtheoriginalgridtoadistanceequaltotheextentofthelargestgrid,butduetothecoarseningoperation,thecostincreaseslinearlywithincreasingextent,ratherthanqua-dratically(intwodimensions)orcubically(inthreedimensions).

Themethodcanthusbewrittenasfollows.Wede nethedomainofeachgridasDðkÞ,k=1,2,...,Ng,wherek=1referstotheoriginal(smallest)gridandk=Ngreferstothelargestone.Wethende nethemulti-domaininverseLaplacian