Computer Methods in Applied Mechanics and Engineering(14)

一些ME专业提升的论文。

T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462143

validatedinbothtwoandthreedimensions).Speed-upsforthree-dimensionalproblemsarelikelytobemoredramaticasdiscussedinSection3.3.

Next,wecomparethespeed-upfromforatranslatingcircularcylindersimulatedbymovingtheLagrangianboundarypoints.NowEq.(36)issolvediterativelywiththeconjugate-gradientmethod.Acylinderoriginallyattheoriginatt=0isimpulsivelytranslatedtotheleftwithunitvelocitywithRe=200.TheinnermostdomainisselectedasDð1Þ¼½À5;1 ½À1;1 withD=0.02DandweuseNg=4multi-domains.Insidethishighlycon nedDð1Þ,thetranslatingcylindergeneratestwocounterrotat-ingvorticesinthewakeasshowninFig.13fort=3.5.Thevorticitypro leisinaccordwithpreviousresultsreportedin[36].Comparedtoacomputationperformedwiththeoriginalapproach,thepresentcomputationisfoundtobe43.4timesfaster.Recallthataspeed-upof53.0isobservedforastationarycylinder(Table2),whichsuggeststhattheoverallalgorithmisstillsolvede cientlyevenwithamovingimmersedboundary.7.Summary

Wehavereportedonimprovementstotheimmersedboundaryprojectionmethodfor owovertwo-andthree-dimensionalbodieswithprescribedmotion.Inprevi-ouswork[36],weshowedthattheIBmethodcanbeformulatedinanalgebraicallyidenticalwaytotheincom-pressibleNavier–Stokesequationswithoutanimmersedboundary.ThisformulationenablestheclassicalfractionalstepmethodtobeappliedtotheIBequations,eliminatingtheneedforanyconstitutiverelationforthemotionofthebody(andhenceassociatedsti ness),andensuringthattheno-slipanddivergence-freeconstraintsaresatis edtoarbi-trarilyhighprecision.Inthispaperweshowedthatthesolutioncanbesubstantiallyacceleratedbyemployinganullspace(discretestreamfunction)methodtosatisfyingthedivergencefree-constraint,andbyrestrictingthecom-putationtoequally-spacedmeshes.

Inthisfastmethod,theviscousterms,divergence-free,andno-slipconstraintsarestilltreatedimplicitly,butthelinearsystemsassociatedwiththePoissonequationandimplicitviscoustermscanbesolvedirectlywithfastsinetransforms.Inthesolution,thedivergence-freeconstraintisautomaticallysatis edtomachineprecision.Forstation-

arybodies,theno-slipconstraintcanalsobeenforcedtomachineprecisionbydirectsolutionoftheequationforthebodyforcesbyusingaCholeskydecomposition.Formovingbodies,iterativesolutionofthelinearsystemforthebodyforcesisstillrequired,butthesizeofthesystemisproportionaltothenumberofLagrangiansurfacepoints;thematrixispositivede niteandtheconjugate-gradienttechniqueise cientforitssolution.

NeartheIB,therestrictiontouniformmeshisastan-dardrequirementofthediscretedeltafunction;however,farfromthebody,thiswouldingeneralbeoverlyrestric-tiveasitisusefultostretchthemeshsothatthedomaincanbemadelargetoapproachthesolutiononanunboundeddomain.Wepursuedanalternativestrategyofimprovingthefar- eldboundaryconditionstothepointwherethedomaincanenclosethebody(andtheportionofthewakeonewishestoresolve)snugly.Wederivedamulti-domaintechniquethatsolvesthePoissonequationonprogressivelylarger,butcoarser,meshes.Vorticityisallowedtoadvectanddi usefrom nertocoarsermesh.Theresulting owontheoriginaldomainthenaccountsforboth(i)theslowlydecayingpotential owinducedbythebodymotion,and(ii)theslowlydecayinginducedvelocityassociatedwithvorticitythathasadvectedtolargedistancefromthebody.Whilethereiscostpenaltyassoci-atedwiththemulti-domainsolution,theoverallschemeemployingthefastnullspacemethodandmulti-domainboundaryconditionsisstillmorethananorder-of-magni-tudefasterthanouroriginalmethodintwodimensions.Thespeed-upresultsfrombene tsassociatedwiththefastnullspacemethodaswellasbeingabletousemorecom-pactdomains.

Thefastnullspacemethodandmulti-domainboundaryconditionsareequallyvalidforbothtwo-andthree-dimensional ows.Two-dimensionaltestcasesincludingstationaryandadvectingOseenvorticesand owoverimpulsively-startedcylindersdemonstratetheaccuracyofthemulti-domainboundaryconditions.WenotethatthetechniquesaregenerallyapplicabletotheincompressibleNavier–Stokesequationsonunboundeddomainswithorwithoutimmersedboundaries.Themulti-domaintech-nique,inparticular,shouldproveusefulinsimulating owsthatinvolveweakinteractionsof nite-circulationvorticesthathaveplaguedmethodsemployingperiodicorothersimpli edboundaryconditionsinthepast(e.g.[10,14]).

Acknowledgements

TheauthorsthankProf.BlairPerotfortheenlighteningdiscussionsonthefractionalstepmethod.Theexperimen-taldatapresentedinSection2.2weregenerouslysharedbyDr.WilliamDickson.ThisworkwaspartiallysupportedbytheUnitedStatesAirForceO ceofScienti cResearch(AFOSR/MURIFA9550-05-1-0369)andtheNationalScienceFoundation

(DMS-0514414).