2023年11月29日发(作者：)

Comput.MethodsAppl.Mech.Engrg.193(2004)2139–2154

Astabilizededge-basedimplicitincompressible

flowformulation

€

hner,J.Cebral,F.CamelliO.Soto,R.Lo

*

SCS/LaboratoryforComputationalFluidDynamics,GeorgeMasonUniversity,MS4C7,4400UniversityDrive,Fairfax,

VA22030-4444,USA

Received21March2003;receivedinrevisedform9October2003;accepted2January2004

Abstract

Anedge-basedimplementationofanimplicit,monolithic,finiteelement(FE)schemeforthesolutionofthe

incompressibleNavier–Stokes(NS)equationsispresented.Theoriginalelementformulationisbasedonthepressure

stabilitypropertiesofanimplicitsecond-orderintimefractionalstep(FS)method,whichisconditionallystable.The

finalmonolithicschemepreservesthesecond-orderaccuracyoftheFSmethod,andisunconditionallystable.Fur-

thermore,itcanbedemonstratedthatthefinalpressurestabilizingtermispracticallythesamefourth-orderpressure

termaddedbysomeauthors(butfollowingdifferentarguments)toobtainhighorderaccurateresults,andthatthefinal

discretizedconvectivetermsareformallyasecond-orderdiscretizationoftherespectivecontinuousone.

Thedevelopmentoftheedgeimplementationissupportedbytwocriteria:thepropertiesoftheelementbasedone,

whichhasalreadybeenextensivelytestedandforwhichconvergenceandstabilityanalysishasalreadybeenpresented,

andontheenforcementofglobalconservationandsymmetryatthediscretelevel.Amonotonicitypreservingterm

whichdecreasesthediscretizationorderinsharpgradientregionstoavoidlocalizedoscillations(overshootsand

undershoots),isformulatedandtested.Somenumericalexamplesandexperimentalcomparisonsarepresented.

Ó2004ElsevierB.V.Allrightsreserved.

Keywords:Incompressibleflows;Stabilizedmethods;Edge-basedschemes

1.Introduction

AmongtheschemesdevelopedoverthelastdecadeforthesolutionoftheincompressibleNSequations

(monolithicschemes[13,16],projectionorfractionalstep(FS)schemes[3,8,18,20–22,25],artificialcom-

pressibility(AC)[7,15,17,23,24,28],pre-conditioningofthecompressibleNSequations[6,34,35],etc.)the

FSschemesyieldhighlyaccurate,pressure-stableresultsbyintegratinginanexplicitmannertheadvective

termsoftheNSequations.However,thetimestepimposedbythesmallestelementsmaybeordersof

*

Correspondingauthor.

E-mailaddress:sorlando@(O.Soto).

URL:/~rlohner/.

0045-7825/$-seefrontmatterÓ2004ElsevierB.V.Allrightsreserved.

doi:10.1016/.2004.01.018

2140O.Sotoetal./Comput.MethodsAppl.Mech.Engrg.193(2004)2139–2154

magnitudesmallerthanthetimesteprequiredtoobtaintime-accurateresults.Formanyclassesofprob-

lems,e.g.biologicalflows(bloodandairflow)andenvironmentalflows(contaminantrelease),thisimplies

tensofthousandsoftimestepspersimulation,renderingtheschemesimpractical.Mostoftheartificial

compressibilityandpre-conditionedschemessufferfromthesameshortcoming.

Ontheotherhand,themonolithicschemestreat,ingeneral,theadvectiveterminanimplicitmanner,

whichavoidsthesedisadvantages.Nevertheless,thesemethodsareveryexpensivefromacomputational

pointofview:thevelocityandpressurediscreteequationsarecoupled.Forthisreason,animplicit

monolithicbutuncoupledschemewasdeveloped,whichisunconditionallystable,andwhichpreservesthe

accuracyandstabilityofasecond-orderFSmethod[11,32].

Thismethodwasimplementedusinganelement-baseddiscretization,whichinvolvesaloopoverthe

elements,thecomputationsoftheelementcontributions(tothesystemmatrixandtotheforcevector),and

theassemblyofthesetotheglobalarrays.Suchprocedureimpliestherecalculationofthemass,Laplacian,

andgradientmatricesineachiterationofeachtimestep,whichishighlytimeconsuming.Intheimple-

mentationthatispresentedhere,themass,gradients,andLaplacianmatricesarecomputedandstoredonly

onceatthebeginningoftherun(oreachtimearemeshingisdone).Alltheleft-hand-side(LHS)andright-

hand-side(RHS)termsinvolvedinthefinalalgebraicsystemarecomputedbyloopingoverthepointsof

themesh,andthenoverthepointsconnectedtoagivenpoint.Thestorageofthefinalsystemofequations,

andofthedifferentedge-basedarrays(mass,Laplacianandgradients)isdonebyusingastandardcom-

pressedsparserow(CSR)format[29].Inthisway,thecomputationofthedifferenttermscanbeparallelized

inastraightforwardmanner,sincethemeshedgeijistouchedonlywhentheloopoverthepointsgoes

throughthepointi(inthisworktheedgejiisdifferentfromtheedgeij).

Therestofthepaperisorganizedasfollows:InSection2thestandardelement-basedstabilizedfor-

mulationisbrieflysummarized.InSection3theedge-basedimplementationispresented,andsomeaspects

ofitsdevelopmentarediscussed.Section4isdedicatedtoshowsome2Dand3Dnumericalexamples,and

somecomparisonswiththeelement-basedscheme.Finally,inSection5someconclusionsaredrawn.

2.Element-basedscheme

ThecontinuousincompressibleNSequationstobesolvedcanbewrittenas:

ou

þðuÁrÞuÀmDuþrp¼finXÂð0;t

ot

rÁu¼0inXÂð0;tÞð2Þ

f

f

Þ;ð1Þ

whereXistheflowdomain,tisthetimevariable,ð0;t

f

Þthetimeintervalforthesimulation,uthevelocity

field,rthegradientoperator,mthekinematicviscosity,DtheLaplacianoperator,pthepressureandfthe

externalbodyforces(i.e.thegravityandtheBoussinesqforces).

LetrbetheviscousstresstensorandutheunitoutwardnormaltotheboundaryoX.Denotingbyan

overbarprescribedvalues,theboundaryconditionsfor(2)tobeconsideredhereare:

u¼uonC