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

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PERGAMONCOMPUTERSFLUIDSVOL24,NO1,PP5562,1995COPYRIGHT01995ELSEVIERSCIENCELTD0045793094000204PRINTEDINGREATBRITAINALLRIGHTSRESERVED00457930/95950000PETROVGALERKINFINITEELEMENTANALYSISFORADVANCINGFLOWFRONTINREACTIONINJECTIONMOLDINGNITINRANTURKARFORDRESEARCHLABORATORY,FORDMOTORCOMPANY,POBOX2053,MD3198,DEARBORN,MI481212053,USARECEIVED4AUGUST1993;INREVISEDFORM4MAY1994ABSTRACTANUMERICALSCHEMEFORCOMPUTINGTHEADVANCEMENTOFAFLOWFRONTANDRELATEDVELOCITY,PRESSURE,CONFERSIONANDTEMPERATUREDISTRIBUTIONSDURINGMOLDFILLINGINREACTIONINJECTIONMOLDINGRIMISDESCRIBEDINTHISWORKINTHERIMPROCESS,THECONVECTIVETERMINTHEENERGYEQUATIONISDOMINANTTHEREFORE,THENUMERICALSCHEMEHASINCORPORATEDAPETROVGALERKINFINITEELEMENTMETHODTOSUPPRESSSPURIOUSOSCILLATIONSANDTOIMPROVEACCURACYOFTHECALCULATIONSTHEOTHERFEATUREOFTHENUMERICALSCHEMEISTHATTHEFLOWFRONTLOCATIONSARECOMPUTEDSIMULTANEOUSLYWITHPRIMARYVARIABLESBYUSINGASURFACEPARAMETERIZATIONTECHNIQUETHENUMERICALRESULTSCOMPAREWELLWITHTHEREPORTEDEXPERIMENTALDATAIMPROVEDACCURACYOBTAINEDBYTHISNUMERICALSCHEMEINTHEFLOWFRONTREGIONISEXPECTEDTOASSISTINTHEPREDICTIONSOFTHEFIBERORIENTATIONSANDTHEBUBBLEGROWTHINRIM,WHICHAREDETERMINEDPRIMARILYBYTHEFLOWFRONTREGIONIINTRODUCTIONREACTIONINJECTIONMOLDINGRIMISAWIDELYUSEDPROCESSTOMANUFACTUREEXTERIORFASCIASINTHEAUTOMOBILEINDUSTRYINTHISPROCESS,APREPOLYMERIZEDISOCYANATEANDAPOLYOL/AMINEMIXTUREAREMIXEDTOGETHER,ANDINJECTEDINTOAMOLD,WHEREPOLYMERIZATIONOCCURSAFOUNTAINFLOWEFFECTINTHEADVANCINGFLOWFRONTREGIONDURINGTHEMOLDFILLINGSTAGEPLAYSANIMPORTANTROLEINDETERMININGTHERESIDENCETIMEOFTHEFLUIDELEMENTSANDINCONTROLLINGTHEFIBERORIENTATIONSINTHEFINALPRODUCT11ANACCURATESIMULATIONOFTHISFLOWFRONT,HOWEVER,POSESACHALLENGINGPROBLEMEVOLVINGFLOWDOMAINWITHADVANCINGFLOWFRONTREQUIRESUPDATINGOFTHENUMERICALGRIDSANDPREDICTIONOFTHEMOVINGBOUNDARYATEVERYTIMESTEPLOWTHERMALCONDUCTIVITYOFTHEMATERIAL,HIGHFLOWRATESINTHERIMPROCESS,ANDHIGHLYEXOTHERMICRAPIDREACTIONSRESULTINCONVECTIONDOMINATEDENERGYTRANSPORTEQUATION,WHICHNEEDSASPECIALNUMERICALTREATMENTBESIDES,MOVINGCONTACTLINESNEARTHEWALLSNEEDSUITABLEBOUNDARYCONDITIONSTHATDONOTINTRODUCENUMERICALINSTABILITYANUMERICALSCHEMETHATINCORPORATESALLTHESECOMPLEXFEATURESOFTHERIMPROCESSISREQUIREDFORACCURATEPREDICTIONSNEARTHEFLOWFRONTREGIONPREVIOUSSTUDIESEITHERHAVEMADESIMPLIFYINGASSUMPTIONSREGARDINGTHEFLOWFRONTREGION2L,ORHAVENOTCOMPAREDTHEIRRESULTSWITHTHEEXPERIMENTS5,6INTHISPAPER,WEDESCRIBEANUMERICALSCHEMEINDETAIL,WHICHWILLADDRESSTHEABOVEMENTIONEDCOMPLEXITIES,ANDPRESENTTHERELEVENTRESULTSTHATHIGHLIGHTTHENUMERICALSCHEMEREFERTOOUREARLIERWORK7FORTHEDETAILEDDISCUSSIONOFTHEGOVERNINGEQUATIONSANDADDITIONALRESULTSNOAPRIORIASSUMPTIONSAREMADEINTHENUMERICALSCHEMEREGARDINGTHESHAPEOFTHENEWFRONTORTHEVELOCITYDISTRIBUTIONINTHEFLOWDOMAINAFREESURFACEPARAMETERIZATIONTECHNIQUEISUSED,INWHICHTHESHAPEOFTHEFLOWFRONTISCALCULATEDSIMULTANEOUSLYWITHOTHERFIELDVARIABLES,SUCHASPRESSURE,VELOCITIESANDCONVERSION,BYINCORPORATINGKINEMATICBOUNDARYCONDITIONATTHESURFACEOFTHEFLOWFRONTASONEOFTHEGOVERNINGEQUATIONSACONVENTIONALGALERKINFINITEELEMENTTECHNIQUEISNOTORIOUSFORITSNUMERICALINSTABILITYINCONVECTIONDOMINATEDTRANSPORTPROBLEMS8THERESULTINGSPURIOUSOSCILLATIONSCANBEUSUALLYELIMINATEDBYMESHREFINEMENTHOWEVER,FORTRANSIENTPROBLEMDESCRIBEDHERE,MESHREFINEMENTISANIMPRACTICALANDEXPENSIVEALTERNATIVETHEOTHERALTERNATIVESINCLUDEVARIOUSUPWINDINGSCHEMES9121,AMETHODOFCHARACTERISTICS6,13,141,ANDAGALERKIN/LEASTSQUARESTECHNIQUE151ALTHOUGHTHE“CONSERVATIVE”METHODS,SUCHASMETHODSOFCHARACTERISTICSANDGALERKIN/LEASTSQUARESTECHNIQUESAREMOREACCURATE,ASIMPLEPETROVGALERKINUPWINDINGMETHODISEASIERTO5556NITINRANTURKARIMPLEMENTANDCOSTEFFECTIVE,PARTICULARLYFORATRANSIENTPROBLEMINVESTIGATEDINTHISWORKTHEREFORE,SUCHASCHEMEISIMPLEMENTEDHEREFOLLOWINGADORNATOANDBROWN9TOSUPPRESSNUMERICALINSTABILITYWITHOUTRESOLVINGTOEXTREMELYREFINEDMESHESTHEGOVERNINGEQUATIONSAREPRESENTEDBRIEFLYINSECTION2,ANDTHENUMERICALMETHODISDESCRIBEDINDETAILINSECTION3THETYPICALRESULTSOFTHEMOLDFILLINGSTAGEOFTHERIMPROCESSINATWODIMENSIONALRECTANGULARPLAQUEAREPRESENTEDINSECTION4THERESULTSAREALSOCOMPAREDWITHTHEREPORTEDEXPERIMENTALDATA2,ANDWITHTHENUMERICALRESULTSOBTAINEDBYUSINGCONVENTIONALGALERKINFINITEELEMENTMETHOD2GOVERNINGEQUATIONSTHELUMPEDKINETICRATEEXPRESSIONFORPOLYMERIZATIONREACTIONSINRIMIS16,171RIA,EXPE,/RTCR,1WHERE,CIISTHEISOCYANATECONCENTRATION,TTHETEMPERATURE,RTHEGASLAWCONSTANT,MTHEORDEROFTHEREACTION,E,THEACTIVATIONENERGYOFTHEREACTION,ANDA,THERATECONSTANTTHEVISCOSITYDEPENDSONTHECONVERSIONANDTEMPERATURE,ANDISEXPRESSEDINTHEFORMOFCASTROMACOSKOVISCOSITYFUNCTION2,X,TRLXIITA,EXPIBXI,2WHEREXISTHEISOCYANATECONVERSION,X,THEGELCONVERSION,ANDA,,,E,,AANDBARETHECONSTANTSFORCONSTANTTHERMALPROPERTIESANDDENSITYOFTHEREACTIVEMIXTURE,ANDFORNEGLIGIBLEMOLECULARDIFFUSION,THEDIMENSIONLESSGOVERNINGEQUATIONSARE,CONTINUITYEQUATIONVVO;3CONSERVATIONOFMOMENTUMEQUATIONREVVVPVIVRCJ;GZ7,,VVXDAKLX“‘;MOLEBALANCEEQUATION45CONSERVATIONOFENERGYEQUATIONGZGVVTVTBRRCJVVDARC,LXM;L16WHERE,VISTHEVELOCITYVECTOR,QTHERATEOFSTRAINTENSOR,TTHETIME,PTHEPRESSURE,ANDK,ISTHEDIMENSIONLESSRATECONSTANT,DEFINEDASEXPE,/RL/TL/T,THEEQUATIONSAREMADEDIMENSIONLESSUSINGTHEAVERAGEVELOCITYV,HALFOFTHETHICKNESSOFTHEMOLDH,ANDTHETEMPERATURET,ANDTHEVISCOSITYQORX0,TT,ATTHEINLETOFTHEMOLDALLTHEDIMENSIONLESSGROUPSANDTHEIRDEFINITIONSARELISTEDINTABLE1THEBOUNDARYCONDITIONSINTERMSOFDIMENSIONLESSVARIABLESARE1ATTHEWALLSV,,,,0NOSLIP,TT,,,,;2ATTHEMIDPLANEATJAY0,JAY0,V,0;3ATTHEINLETVFULLYDEVELOPEDFLOW,T1,XX,;4ATTHECONTACTLINENPI20FULLSLIP5ATTHEFLOWFRONTNPI20FORCEBALANCE,NVAH/AT0KINEMATICCONDITION;TABLE1DIMENSIONLESSGROUPSINGOVERNINGEQUATIONS,WHEREAH,ISTHEHEATOFREACTION,AT,,,THEADIABATICTEMPERATURERISE,ANDC,,THEINITIALCONCENTRATIONOFISOCYANATEGZGRAETZNUMBERVHPC,LKREREYNOLDSNUMBERHVLRLOKVISCOSITYRATIO41BRBRINKMANNNUMBER‘TOVLKT,DADAMKOHLERNUMBERAH,HC/KT,A,EXPE,/RTTADBADIABATICTEMPERATURERISEAT,,,IT,FLOWFRONTADVANCEMENTINREACTIONINJECTIONMOLDING57WHEREA,ANDVYARETHECOMPONENTSOFTHEVELOCITYVECTORV,IITHEUNITNORMALVECTOR,RTHEEXTRASTRESSTENSOR,HTHELOCATIONVECTOROFTHEFLOWFRONTANDTWAL,THEDIMENSIONLESSTEMPERATUREATTHEMOLDWALLTHEDETAILSOFINCORPORATINGTHEBOUNDARYCONDITIONSINTHENUMERICALANALYSISAREEXPLAINEDINTHENEXTSECTION3NUMERICALANALYSISINTHEFINITEELEMENTFORMULATIONTHEUNKNOWNVELOCITIES,TEMPERATUREANDCONVERSIONAREEXPANDEDINTERMSOFTHEBIQUADRATICBASISFUNCTIONS4’,THEPRESSUREINTERMSOFTHEBILINEARBASISFUNCTIONSLL/IANDTHEFLOWFRONTSHAPEHINTERMSOFTHEQUADRATICBASISFUNCTIONS7WHERELANDQARETHECOORDINATESINISOPARAMETRICTRANSFORMATION,DEFINEDASI1ILINTHEISOPARAMETRICDOMAIN141,1Q1HERE,N,,NPANDN,ARETHENUMBEROFVELOCITY,PRESSUREANDFREESURFACENODES,RESPECTIVELYTHEUNKNOWNNODALCOEFFICIENTSOFTHEVARIABLESASWELLASTHEXCOORDINATEOFEACHNODEDEPENDONTIMENOTETHATBIQUADRATICBILINEARV,PELEMENTDOESNOTSATISFYTHECELEBRATEDBREZZIBABUSKASTABILITYCONDITION18,191,ANDTHEREFORE,LEADST’OOVERALLMASSBALANCE,BUTCANNOTENSURELOCAL,ELEMENTLEVELMASSBALANCEALTHOUGHSEVERALCONVERGENTCOMBINATIONSOFVELOCITYANDPRESSUREELEMENTSHAVEBEENDEVELOPED20,21THATSATISFYBREZZIBABUSKACONDITION,THEIRINTERPOLATIONPATTERNSAREINCONVENIENTTOIMPLEMENTINFEMCODESTHEABOVEBIQUADRATICBILINEARV,PELEMENTEXHIBITSNOSPURIOUSPRESSUREMODES22,23ANDHASBEENWIDELYUSEDWITHLIMITEDNUMERICALSTABILITYPROBLEMSBESIDES,THEPETROVGALERKINFORMULATIONISEXPECTEDTOENHANCETHENUMERICALSTABILITY24THEYCOORDINATESOFALLTHENODESAREFIXED,WHEREASTHEXCOORDINATESAREADJUSTEDINPROPORTIONTOTHEFREESURFACELOCATIONSSINCE,THEFLOWISALONGTHEXDIRECTION,THEFLOWFRONTADVANCES,ANDTHEFLOWDOMAINEXPANDSWHENTHELENGTHOFTHEELEMENTALONGXDIRECTIONEXCEEDSAPREDETERMINEDVALUE,MESHISREGENERATEDBYDIVIDINGEACHELEMENTINTOTWOELEMENTSOFEQUALSIZEALONGTHEFLOWDIRECTIONINTERPOLATIONOFTHEVARIABLESTOTHENEWNODESISSIMPLE,ANDCANBERAPIDLYCOMPUTEDFORBIQUADRATICELEMENTSINCONVENTIONALGALERKINFINITEELEMENTFORMULATION,THEBASISFUNCTIONSTHEMSELVESAREUSEDASWEIGHTINGFUNCTIONSINCOMPUTINGTHERESIDUALSOFTHEGOVERNINGEQUATIONSINTHEFLOWDOMAINHOWEVER,THEMETHODISNOTROBUSTITISWELLKNOWN8THATWHENGALERKINFORMULATIONISUSEDFORSOLVINGCONVECTIONDOMINATEDEQUATIONS,ANUMERICALINSTABILITYANDSPURIOUSOSCILLATIONSAREINTRODUCEDINTHESOLUTIONTHEENERGYEQUATIONINTRODUCEDEARLIERISDOMINATEDBYCONVECTIONTERMSINREACTIONINJECTIONMOLDINGONEALTERNATIVEISTOREFINETHEMESHTOSUPPRESSTHEOSCILLATIONSHOWEVER,THISAPPROACHISIMPRACTICALFORTRANSIENTPROBLEMSWITHALARGENUMBEROFUNKNOWNS,SUCHASTHEONESOLVEDHERETHEOTHERALTERNATIVEAPPROACHISTOMODIFYTHEWEIGHTINGFUNCTIONSBYINTRODUCINGANARTIFICIALDIFFUSIVITYTHEFORMULATIONCANBEADAPTEDCONVENIENTLYFORHIGHERORDERELEMENTS25,9THEWEIGHTINGFUNCTIONIS,WLV,V,D9’5,RLLZJL,V,D,9WHERE,THEARTIFICIALDIFFUSIVITYISINTRODUCEDTHROUGHTHEFUNCTIONC,WHICHDEPENDSONTHELOCALVELOCITYFIELDANDTHEAPPROPRIATEDIFFUSIVITYDASSOCIATEDWITHEACHGOVERNINGEQUATIONTHEFUNCTIONALFORMOFCISBASEDONONEDIMENSIONALCONVECTIONDIFFUSIONPROBLEM25,ANDISEXPRESSEDAS,I’S,V,DYPEL5,WHERE,PEISTHELOCALELEMENTPECLETNUMBERVA/D,ATHEELEMENTSIZEANDC,SISTHECUBICPOLYNOMIAL5/85LT1THEINDEXI1CORRESPONDSTOTHEVERTEXNODES,ANDI258NITINRANTIJRKARCORRESPONDSTOTHECENTROIDNODESINTHEELEMENTTHESTANDARDONEDIMENSIONALCONVECTIONDIFFUSIONPROBLEMHASEXACTSOLUTIONATTHENODESIF25,9C‘PE2TANHPE/2L3/PECOTHPE/4X/PECOTHPE/4,1LAC216/PE4COTHPE/41LBINATWODIMENSIONALPROBLEM,THETENSORIALPRODUCTOFEQUATIONS10AND11PROVIDESTHEFUNCTIONCINTHEWEIGHTINGFUNCTIONSDESCRIBEDINEQUATION9THELOCALPECLETNUMBERISCOMPUTEDFOREACHTHREENODEGROUPBASEDONTHEAVERAGEVELOCITIESATTHERELEVANTBOUNDARIESINTHETWODIMENSIONALELEMENT9THEREARESIXSUCHGROUPSTHREEINTHEXDIRECTION,ANDTHREEINTHEYDIRECTIONANDTHUS,THEREARE12UPWINDINGPARAMETERSETHECALCULATIONSOFTHEPECLETNUMBERINVOLVELINEARDISTANCES,WHICHESSENTIALLYNEGLECTTHECURVILINEARSIDESOFTHEELEMENTSHOWEVER,ITISAGOODAPPROXIMATIONSINCEFLOWFRONTISNOTSEVERELYDEFORMEDINOURPROBLEMTHEDIFFUSIVITIESAREL/GZFORTHEENERGYEQUATIONANDISK/RFORTHEMOMENTUMEQUATIONTHEPETROVGALERKINWEIGHTEDRESIDUALEQUATIONSARE,RVV‘DL’O,S’RLIVREGVVVWFDV12YPIKVWIDVSSNPILCFW’DSO,13SBRRCJVVDAK,LX”W’DV1SVTVW’DVSNVTW’DSO,15VSRISNVAH/4’1DS016SWHERE,VISTHEFLOWDOMAINANDSTHEFLOWBOUNDARYTHEBOUNDARYTERMSAPPEARINTHEENERGYANDMOMENTUMEQUATIONSBECAUSEDIVERGENCETHEOREMISAPPLIEDTOTHEHIGHERORDERTERMSTHERESIDUALSR,,R,,R,,R,ANDR,CORRESPONDTOTHEVARIABLESP,V,X,TANDH,RESPECTIVELYTHEPETROVGALERKINWEIGHTINGFUNCTIONSAR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