外文翻译英文版--反应注射成型过程中熔体流动前沿的PETROV-GALERKIN有限元分析.pdf

PergamonComputers&FluidsVol.24,No.1,pp.55-62,1995Copyright01995ElsevierScienceLtd0045-7930(94)00020-4PrintedinGreatBritain.Allrightsreserved0045-7930/95$9.50+0.00PETROV-GALERKINFINITEELEMENTANALYSISFORADVANCINGFLOWFRONTINREACTIONINJECTIONMOLDINGNITINR.ANTURKARFordResearchLaboratory,FordMotorCompany,P.O.Box2053,MD3198,Dearborn,MI48121-2053,U.S.A.(Received4August1993；inrevisedform4May1994)Abstract-Anumericalschemeforcomputingtheadvancementofaflowfrontandrelatedvelocity,pressure,confersionandtemperaturedistributionsduringmoldfillinginreactioninjectionmolding(RIM)isdescribedinthiswork.IntheRIMprocess,theconvectivetermintheenergyequationisdominant.Therefore,thenumericalschemehasincorporatedaPetrov-Galerkinfiniteelementmethodtosuppressspuriousoscillationsandtoimproveaccuracyofthecalculations.Theotherfeatureofthenumericalschemeisthattheflowfrontlocationsarecomputedsimultaneouslywithprimaryvariablesbyusingasurfaceparameterizationtechnique.Thenumericalresultscomparewellwiththereportedexperimentaldata.ImprovedaccuracyobtainedbythisnumericalschemeintheflowfrontregionisexpectedtoassistinthepredictionsofthefiberorientationsandthebubblegrowthinRIM,whicharedeterminedprimarilybytheflowfrontregion.I.INTRODUCTIONReactioninjectionmolding(RIM)isawidelyusedprocesstomanufactureexteriorfasciasintheautomobileindustry.Inthisprocess,aprepolymerizedisocyanateandapolyol/aminemixturearemixedtogether,andinjectedintoamold,wherepolymerizationoccurs.Afountainfloweffectintheadvancingflowfrontregionduringthemold-fillingstageplaysanimportantroleindeterminingtheresidencetimeofthefluidelementsandincontrollingthefiberorientationsinthefinalproduct11.Anaccurates:imulationofthisflowfront,however,posesachallengingproblem.Evolvingflowdomainwithadvancingflowfrontrequiresupdatingofthenumericalgridsandpredictionofthemovingboundaryateverytimestep.Lowthermalconductivityofthematerial,highflowratesintheRIMprocess,andhighlyexothermicrapidreactionsresultinconvection-dominatedenergytransportequation,whichneedsaspecialnumericaltreatment.Besides,movingcontactlinesnearthewallsneedsuitableboundaryconditionsthatdonotintroducenumericalinstability.AnumericalschemethatincorporatesallthesecomplexfeaturesoftheRIMprocessisrequiredforaccuratepredictionsneartheflowfrontregion.Previousstudieseitherhavemadesimplifyingassumptionsregardingtheflowfrontregion2&l,orhavenotcomparedtheirresultswiththeexperiments5,6.Inthispaper,wedescribeanumericalschemeindetail,whichwilladdresstheabove-mentionedcomplexities,and.presentthereleventresultsthathighlightthenumericalscheme(refertoourearlierwork7forthedetaileddiscussionofthegoverningequationsandadditionalresults).Noaprioriassumptionsaremadeinthenumericalschemeregardingtheshapeofthenewfrontorthevelocitydistributionintheflowdomain.Afree-surfaceparameterizationtechniqueisused,inwhichtheshapeoftheflowfrontiscalculatedsimultaneouslywithotherfieldvariables,suchaspressure,velocitiesandconversion,byincorporatingkinematicboundaryconditionatthesurfaceoftheflowfrontasoneofthegoverningequations.AconventionalGalerkinfinite-elementtechniqueisnotoriousforitsnumericalinstabilityinconvection-dominatedtransportproblems8.Theresultingspuriousoscillationscanbeusuallyeliminatedbymeshrefinement.However,fortransientproblemdescribedhere,meshrefinementisanimpracticalandexpensivealternative.Theotheralternativesincludevariousupwindingschemes9-121,amethodofcharacteristics6,13,141,andaGalerkin/least-squarestechnique151.Althoughthe“conservative”methods,suchasmethodsofcharacteristicsandGalerkin/least-squarestechniquesaremoreaccurate,asimplePetrov-Galerkinupwindingmethodiseasierto5556NITINR.ANTURKARimplementandcosteffective,particularlyforatransientprobleminvestigatedinthiswork.Therefore,suchaschemeisimplementedherefollowingAdornatoandBrown9tosuppressnumericalinstabilitywithoutresolvingtoextremelyrefinedmeshes.ThegoverningequationsarepresentedbrieflyinSection2,andthenumericalmethodisdescribedindetailinSection3.ThetypicalresultsofthemoldfillingstageoftheRIMprocessinatwo-dimensionalrectangularplaquearepresentedinSection4.Theresultsarealsocomparedwiththereportedexperimentaldata2,andwiththenumericalresultsobtainedbyusingconventionalGalerkinfiniteelementmethod.2.GOVERNINGEQUATIONSThelumpedkineticrateexpressionforpolymerizationreactionsinRIMis16,171:ri=-A,exp(-E,/RT)Cr,(1)where,Ciistheisocyanateconcentration,Tthetemperature,Rthegas-lawconstant,mtheorderofthereaction,E,theactivationenergyofthereaction,andA,therateconstant.Theviscositydependsontheconversionandtemperature,andisexpressedintheformofCastroMacoskoviscosityfunction2,(X,T)=rl(X)-II(T)=A,exp()(iBXi,(2)whereXistheisocyanateconversion,X,thegelconversion,andA,E,AandBaretheconstants.Forconstantthermalpropertiesanddensityofthereactivemixture,andfornegligiblemoleculardiffusion,thedimensionlessgoverningequationsare,continuityequation:v.v=o；(3)conservationofmomentumequation:Re$+v.Vv=-pV.I+v:(rcj)；Gz7,$+v-VX=Dak.(l-X)“；molebalanceequation:(4)(5)conservationofenergyequation:Gzg+vVT=V*T+Brrc(j:Vv)+Darc,(l-X)m；L.1(6)where,visthevelocityvector,qtherate-of-straintensor,tthetime,pthepressure,andk,isthedimensionlessrateconstant,definedasexp(-E,/R)(l/T-l/T,).TheequationsaremadedimensionlessusingtheaveragevelocityV,halfofthethicknessofthemoldH,andthetemperatureT,andtheviscosityqO(=r(X=0,T=T,)attheinletofthemold.AllthedimensionlessgroupsandtheirdefinitionsarelistedinTable1.Theboundaryconditionsintermsofdimensionlessvariablesare1.atthewalls:v,=0(no-slip),T=T,；2.atthemid-plane:aTjay=0,&Jay=0,V,=0；3.attheinlet:v=fullydevelopedflow,T=1,X=X,；4.atthecontactline:n*(-PI+2)=0(full-slip):5.attheflowfront:n.(-PI+2)=0(forcebalance),n.(v-ah/at)=0(kinematiccondition)；Table1.Dimensionlessgroupsingoverningequations,whereAH,istheheatofreaction,AT,theadiabatictemperaturerise,andC,theinitialconcentrationofisocyanateGZGraetznumberVHpC,lkReReynoldsnumberHVlrloKviscosityratio41%BrBrinkmannnumbertoV=lkT,DaDamkohlernumber(AH,H*C$/kT,)A,exp(-E,/RT)TadbadiabatictemperatureriseAT,IT,Flowfrontadvancementinreactioninjectionmolding57wherea.,andvYarethecomponentsofthevelocityvectorv,IItheunitnormalvector,rtheextrastresstensor,hthelocationvectoroftheflowfrontandTwal,thedimensionlesstemperatureatthemoldwall.Thedetailsofincorporatingtheboundaryconditionsinthenumericalanalysisareexplainedinthenextsection.3.NUMERICALANALYSISInthefiniteelementformulationtheunknownvelocities,temperatureandconversionareexpandedintermsofthebiquadraticbasisfunctions4,thepressureintermsofthebilinearbasisfunctionsll/iandtheflowfrontshapehintermsofthequadraticbasisfunctions:(7)wherelandqarethecoordinatesinisoparametrictransformation,definedasi=1i=lintheisoparametricdomain(-1<4<+1,-1<q<+1).Here,n,npandn,arethenumberofvelocity,pressureandfreesurfacenodes,respectively.Theunknownnodalcoefficientsofthevariablesaswellasthex-coordinateofeachnodedependontime.Notethatbiquadratic-bilinear(v,p)elementdoesnotsatisfythecelebratedBrezzi-Babuskastabilitycondition18,191,andtherefore,leadstooverallmassbalance,butcannotensurelocal,elementlevelmassbalance.Althoughseveralconvergentcombinationsofvelocityandpressureelementshavebeendeveloped20,21thatsatisfyBrezzi-Babuskacondition,theirinterpolationpatternsareinconvenienttoimplementinFE:Mcodes.Theabovebiquadratic-bilinear(v,p)elementexhibitsnospuriouspressuremodes22,23andhasbeenwidelyusedwithlimitednumericalstabilityproblems.Besides,thePetrov-Galerkinformulationisexpectedtoenhancethenumericalstability24.They-coordinatesofallthenodesarefixed,whereasthex-coordinatesareadjustedinproportiontothefreesurfacelocations.Since,theflowisalongthex-direction,theflowfrontadvances,andtheflowdomainexpands.Whenthelengthoftheelementalongx-directionexceedsapredeterminedvalue,meshisregeneratedbydividingeachelementintotwoelementsofequalsizealongtheflowdirection.Interpolationofthevariablestothenewnodesissimple,andcanberapidlycomputedforbiquadraticelements.InconventionalGalerkinfinite-elementformulation,thebasisfunctionsthemselvesareusedasweightingfunctionsincomputingtheresidualsofthegoverningequationsintheflowdomain.However,themethodisnotrobust.Itiswellknown8thatwhenGalerkinformulationisusedforsolvingconvectiondominatedequations,anumericalinstabilityandspuriousoscillationsareintroducedinthesolution.Theenergyequationintroducedearlierisdominatedbyconvectiontermsinreactioninjectionmolding.Onealternativeistorefinethemeshtosuppresstheoscillations.However,thisapproachisimpracticalfortransientproblemswithalargenumberofunknowns,suchastheonesolvedhere.Theotheralternativeapproachistomodifytheweightingfunctionsbyintroducinganartificialdiffusivity.Theformulationcanbeadaptedconvenientlyforhigher-orderelements25,9.Theweightingfunctionis,W$(LV,v,D)=9(5,rl)+LZj(L?,v,D),(9)where,theartificialdiffusivityisintroducedthroughthefunctionc,whichdependsonthelocalvelocityfieldandtheappropriatediffusivityDassociatedwitheachgoverningequation.Thefunctionalformofcisbasedonone-dimensionalconvection-diffusionproblem25,andisexpressedas,i(S,v,D)=-YPe>L(5),where,PeisthelocalelementPecletnumber(=VA/D),Atheelementsizeandc,(s)isthecubicpolynomial(=(5/8)5(<-l)(t+1).Theindexi=1correspondstothevertexnodes,andi=258NITINR.ANTIJRKARcorrespondstothecentroidnodesintheelement.Thestandardone-dimensionalconvec-tiondiffusionproblemhasexactsolutionatthenodesif25,9c(Pe)=2tanh(Pe/2)l+(3/Pe)coth(Pe/4)-(X/Pe)-coth(Pe/4),(1la)c2=(16/Pe)-4coth(Pe/4).(1lb)Inatwo-dimensionalproblem,thetensorialproductofequations(10)and(11)providesthefunctioncintheweightingfunctionsdescribedinequation(9).ThelocalPecletnumberiscomputedforeachthree-nodegroupbasedontheaveragevelocitiesattherelevantboundariesinthetwo-dimensionalelement9.Therearesixsuchgroups(threeinthex-direction,andthreeinthey-direction)andthus,thereare12upwindingparametersE.ThecalculationsofthePecletnumberinvolvelineardistances,whichessentiallyneglectthecurvilinearsidesoftheelements.However,itisagoodapproximationsinceflowfrontisnotseverelydeformedinourproblem.Thediffusivitiesarel/GzfortheenergyequationandisK/Rforthemomentumequation.ThePetrov-Galerkinweightedresidualequationsare,-R:=(V.v)$dl=O,s-RL=IvReg+v.VvWfdV(12)+y-PI+(K+)VWidV-ssn.-pI+(lcf)WdS=O,(13)s-Brrc(j:Vv)-Dak,(l-X)”WdV1+sVT.VWdV-s(n.VT)WdS=O,(15)VS-RI=sn.(v-ah/&)4(+=1)dS=0.(16)swhere,VistheflowdomainandStheflowboundary.Theboundarytermsappearintheenergyandmomentumequationsbecausedivergencetheoremisappliedtothehigher-orderterms.TheresidualsR,R,R,R,andR,correspondtothevariablesp,v,X,Tandh,respectively.ThePetrov-Galerkinweightingfunctionsareusedonlyformomentumandenergyequationsduetothepresenceofconvectiontermsintheseequations.Beforeintegratingtheaboveequationsusinganine-pointGaussianquadrature,theequationsaremappedintheisoparametricdomain(referto26fordetails)andtheboundaryconditionsareapplied.TheessentialboundaryconditionsforvandTatthewalls；forv,TandXattheinletofthemold；andforvYatthemid-plane(axisofsymmetry)areappliedbysubstitutingtheboundaryconditionsfortheequations.Thenaturalboundaryconditions,namelythesymmetryconditionsatthemid-plane,thefull-slip(zerofriction)conditionatthecontactpoint,andthezeroforceatthefreesurfaceareimplementedbysubstitutingtheboundarytermsintheresidualequations.Thekinematicboundaryconditionattheflowfrontisincorporatedasthegoverningequationforpredictingtheflowfrontlocations.Theweakformofenergyequationisextendedtotheflowfrontboundarybyevaluatingtheboundaryterms,insteadofbyimposinganyunknownessentialornaturalboundaryconditions27.Such“freeboundarycondition”,asdenotedbyPapanastasiouetal.27,minimizestheenergyfunctionalamongallpossiblechoices,atleastforvarioustypesofcreepingflows,andhasbeensuccessfullyusedinseveralapplications,includingthosewithhighReynoldsnumbers.Thespatialdiscretizationreducesthetime-dependentequations(12)<16)toasystemofordinarydifferentialequations,M2+R(q)=0,(17)