Computer Methods in Applied Mechanics and Engineering(6)

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].

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).

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