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陈敬虎：POM水龙头壳体注塑模设计与制造52附录NumericalFillingSimulationofInjectionMoldingUsingThree—DimensionalModelAbstract：Mostinjectionmoldedpartsarethree-dimensional,withcomplexgeometricalconfigurationsandthick／thinwallsections．A3Dsimulationmodelwillpredictmoreaccuratelythefillingprocessthana2.5Dmode1.Thispapergivesamathematicalmodelandnumericmethodbasedon3Dmodel，inwhichanequal-ordervelocity-pressureinterpolationmethodisemployedsuccessfully．Therelationbetweenvelocityandpressureisobtainedfromthediscretizedmomentumequationsinordertoderivethepressureequation．A3Dcontrolvolumeschemeisemployedtotracktheflowfront．Thevalidityofthemodelhasbeentestedthroughtheananalysisoftheflowincavity．Keywords：threedimension；equal-orderinterpolation；simulation；injectionmolding1IntroductionDuringinjectionmolding，thetheologicalresponseofpolymermeltsisgenerallynon-Newtonianandnoisothermalwiththepositionofthemovingflowfront．Becauseoftheseinherentfactors，itisdifficulttoanalyzethefillingprocess．Therefore，simplificationsusuallyareused．Forexample，inmiddle-planetechniqueanddualdomaintechnique[1],becausethemostinjectionmoldedpartshavethecharacteristicofbeingthinbutgenerallyofcomplexshape，theHele-Shawapproximation[2]isusedwhileananalyzingtheflow,i.e.．Thevariationsofvelocityandpressureinthegapwise(thickness)dimensionareneglected．Sothesetwotechniquesareboth2.5Dmoldfillingmodels，inwhichthefillingofamoldcavitybecomesa2Dprobleminflowdirectionanda1Dprobleminthicknessdirection．However,becauseoftheuseoftheHele-Shawapproximation，theinformationthat2.5Dmodelscangenerateislimitedandincomplete．Thevariationinthegapwise(thickness)dimensionofthephysicalquantitieswiththeexceptionofthetemperature，whichissolvedbyfinitedifferencemethod，isneglected．Withthedevelopmentofmoldingtechniques，molded华东交通大学理工学院毕业设计（论文）53partswillhavemoreandmorecomplexgeometryandthedifferenceinthethicknesswillbemoreandmorenotable，sothechangeinthegapwise(thickness)dimensionofthephysicalquantitiescannotbeneglected．Inaddition，theflowsimulatedlooksunrealisticinasmuchasthemeltpolymerflowsonlyonsurfacesofcavity,whichappearsmoreobviouswhentheflowsimulationisdisplayedinamouldcavity．3Dsimulationmodelhasbeenaresearchdirectionandhotspotinthescopeofsimulationforplasticinjectionmolding．In3Dsimulationmodel，velocityinthegapwise(thickness)dimensionisnotneglectedandthepressurevariesinthedirectionofpartthickness，and3Dfiniteelementsareusedtodiscretizethepartgeometry．Aftercalculating，completedataareobtained(notonlysurfacedatabutalsointernaldataareobtained)．Therefore,a3Dsimulationmodelshouldbeabletogeneratecomplementaryandmoredetailedinformationrelatedtotheflowcharacteristicsandstressdistributionsinthinmoldedpartsthantheoneobtainedwhenusinga2.5Dmodel(basedontheHele-Shawapproximation)．Ontheotherhand，a3Dmodelwillpredictmoreaccuratelythecharacteristicsofmoldedpartshavingthickwalledsectionssuchasencounteredingasassistedinjectionmolding．Severalflowbehaviorsattheflowfront．suchas“fountainflow”．which2.5Dmodelcannotpredict,canbepredictedby3Dmode1.Meanwhile,theflowsimulationlooksmorerealisticinasmuchastheoverallananalysisresultisdirectlydisplayedin3Dpartgeometryortransparentmouldcavity．ThisPaperpresentsa3Dfiniteelementmodeltodealwiththethree—dimensionalflow,whichemploysanequa1-ordervelocity-pressureformulationmethod[3,4]．Therelationbetweenvelocityandpressureisobtainedfromthediscretizedmomentumequations,thensubstitutedintothecontinuityequationtoderivepressureequation．A3Dcontrolvolumeschemeisemployedtotracktheflowfront．Thevalidityofthemodelhasbeentestedthroughtheanalysisoftheflowincavity．2GoverningEquationsThepressureofmeltisnotverybigduringfillingthecavity,inaddition，reasonablemoldstructurecanavoidoverbigpressure，sothemeltisconsideredincompressible．Inertiaandgravitationareneglectedascomparedtotheviscousforce．Withtheaboveapproximation，thegoverningequations，expressedincartesiancoordinates，areasfollowing：陈敬虎：POM水龙头壳体注塑模设计与制造54MomentumequationsContinuityequationEnergyequationwhere,x，y，zarethreedimensionalcoordinatesandu,v，warethevelocitycomponentinthex,y,zdirections．P，T，ρandηdenotepressure，temperature,densityandviscostyrespectively．Crossviscositymodelhasbeenusedforthesimulations：where，n,γ，rarenon-Newtonianexponent，shearrateandmaterialconstantrespectively．Becausethereisnonotablechangeinthescopeoftemperatureofthemeltpolymerduringfilling，Anheniusmodel[5]forη0isemployedasfollowing：whereB，Tb，βarematerialconstants.3NumericalSimulationMethod3.1Velocity-PressureRelationIna3Dmodel，sincethechangeofthephysicalquantitiesarenotneglectedinthegapwise(thickness)dimension，themomentumequationsaremuchmorecomplexthanthoseina2.5Dmode1．Itisimpossibletoobtainthevelocity—pressurerelationbyintegratingthemomentum华东交通大学理工学院毕业设计（论文）55equationsinthegapwisedimension，whichisdoneina2.5Dmodel.Themomentumequationsmustbefirstdiscretized，andthentherelationbetweenvelocityandpressureisderivedfromit.Inthispaper,themomentumequationsarediscretizedusingGalerkin’smethodwithbilinearvelocity-pressureformulation．Theelementequationsareassembledintheconventionalmannertoformthediscretizedglobalmomentumequationsandthevelocitymaybeexpressedasfollowingwherethenodalpressurecoefficientsaredefinedaswhererepresentglobalvelocitycoefficientmatricesinthedirectionofx,y,zcoordinaterespectively.denotethenodalpressurecoefficientsthedirectionofx,y,zcoordinaterespectively.Thenodalvaluesforareobtainedbyassemblingtheelement-by-elementcontributionsintheconventionalmanner.N，iselementinterpolationandimeansglobalnodenumberandj,isforanode,theamountofthenodesaroundit.3.2PressureEquationSubstitutionofthevelocityexpressions(2)intodiscretizedcontinuityequation,whichisdiscretizedusingGalerkinmethod，yieldselementequationforpressure:

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