Computer Methods in Applied Mechanics and Engineering(7)

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

一些ME专业提升的论文。

T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462139

~s¼SKÀ1Sc~

;ð28Þ

where~cisanarbitraryinputvector(withlengthequaltothenumberofdiscretecirculationvaluesonthegrid),~sisthesolution(withlengthequaltothenumberofdiscretestreamfunctionvalues),andtheoperatorSKSimpliesthefollowingoperations:

~cð1Þ¼~c8

;

ð29Þ>~cðkÞwherex2DðkÞnDðkÀ1Þ;~cðkÞ¼<

:

PðkÀ1Þ!ðkÞð~cðkÀ1ÞÞwherex2DðkÀ1Þ>;

ð30Þk¼2;3;...;Ng;~sðNgþ1Þ¼0;

ð31Þ~sðkÞ¼SKÀ1Sc~

ðkÞþbcs½Pðkþ1Þ!ðkÞð~sðkþ1ÞÞ ;k¼Ng;NgÀ1;...;1;ð32Þ~s¼SKSc~

¼~sð1Þ:ð33Þ

HereP(kÀ1)!(k)(k)!(kÀ1)isa ne-to-coarseinterpolationoperator

andPisitscoarse-to- necounterpartrestrictedtooDðkÀ1Þbybcs.

InconstructingP,itwouldbedesirabletopreserve(tomachineroundo )certainmomentsofthecirculationdis-tributionsothatthevelocitydecayratefarfromthebodyiscorrect.Inthepresentimplementation,weattempttopreserveonlythetotalcirculation.Switchingfrommatrix/vectortopoint-operatornotation,wewrite,forthetwo-dimensionalcase,

PðkÀ1Þ!ðkÞðc~

ðkÀ1ÞÞðkÀ1Þ

2i;2j¼~ci;jþ1ðkÀ1Þ1ðkÀ1Þ

2~ciÀ1;jþ2~ciþ1;j

þ12~cðkÀ1Þi;jÀ1þ12~cðkÀ1Þi;jþ1þ14~c

ðkÀ1ÞiÀ1;jÀ1þ14~cðkÀ1Þ1ðkÀ1Þiþ1;jÀ1þ4~ciÀ1;jþ1þ1ðkÀ1Þ4~c

iþ1;jþ1

:ð34ÞThe9-pointstencilleadstoaconservationofthetotalcir-culationandissecond-orderaccuratebasedonaTaylor-seriesexpansion.Wenotethatthecoe cientsinEq.(34)sumto4sincethecirculationinthe(dual)cellisthevortic-itymultipliedbythearea,andcoarsifyingthegridbyafac-torof2resultsinafactorof4increaseincellarea.Thethree-dimensionalversionofEq.(34)consistsofaveragingEq.(34)overtwoadjacent(i,j)planesofdatanormaltothevorticitycomponent,foreachofthethreecomponents.Forthecoarse-to- neinterpolationattheboundaryofthenext- nermesh,weusethevaluefromthecoarsermeshforthosegridpointsthatcoincide,andamid-pointlinearinterpolation(againsecond-orderaccurate)forthosepointsinbetween.

百度搜索“77cn”或“免费范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，免费范文网，提供经典小说实用文档Computer Methods in Applied Mechanics and Engineering(7)在线全文阅读。