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GJournalofMechanicalScienceandTechnology212007789798JournalofMechanicalScienceandTechnologyMicroGeneticAlgorithmBasedOptimalGatePositioninginInjectionMoldingDesignJongsooLee,JonghunKimSchoolofMechanicalEngineeringYonseiUniversity,Seoul120749KoreaManuscriptReceivedDecember12,2006RevisedMarch26,2007AcceptedMarch26,2007AbstractThepaperdealswiththeoptimizationofrunnersystemininjectionmoldingdesign.Thedesignobjectiveistolocategatepositionsbyminimizingbothmaximuminjectionpressureattheinjectionportandmaximumpressuredifferenceamongallthegatesonaproductwithconstraintsonshearstressand/orweldline.Theanalysisoffillingprocessisconductedbyafiniteelementbasedprogramforpolymerflow.MicrogeneticalgorithmmGAisusedasaglobaloptimizationtoolduetothenatureofinherentnonlinearlityinflowanalysis.Fourdifferentdesignapplicationsininjectionmoldsareexploredtoexaminetheproposeddesignstrategies.ThepapershowstheeffectivenessofmGAinthecontextofoptimizationofrunnersystemininjectionmoldingdesign.GKeywordsMicrogeneticalgorithmDesignoptimizationFillinginjectionmold1.IntroductionInjectionmoldingprocesshasbeenrecognizedasoneofthemostefficientmanufacturingtechnologiessincehighperformancepolymermaterialscanbeutilizedtoaccuratelymanufactureaproductwithcomplicatedshapeChiang,etal.,1991ChangandYang,2001Himasekhar,etal.,1992KwonandPark,2004.Also,thedemandoninjectionmoldedproductssuchasfromconventionalplasticgoodstomicroopticaldevicesisbeingdramaticallyincreasedovertherecentyearsPiotter,etal.,2001Kang,etal.,2000.Ingeneral,theinjectionmoldprocessisinitiatedbythefillingstagewherethepolymermaterialsfillintoacavityundertheinjectiontemperature.Afterthecavityiscompletelyfilled,thepostfillingstage,thatis,thepackingstageisconductedtobeadditionallyfilledwiththehighpressurepolymer,therebyresultingintheavoidanceofmaterialshrinkage.Subsequently,thecoolingstageisrequiredforamoldedproducttobeejectedwithoutanydeformation.ItisimportanttoaccommodatethemoldingconditionsinthefillingstagesinceitisthefirststageintheoverallinjectionmoldingdesignZhouandD.Li,2001.Afterthat,onecansuccessfullyexpectmoreimprovedmoldingconditionsduringpostfillingstagessuchaspacking,coolingstages.Thepaperdealswithoptimalconditionsofthefillinginjectionmoldingdesigninwhichtheflowpatternandpressureforthepolymermaterialstobefilledthroughgatesofarunnerareofsignificant.Thatis,oneofdesignrequirementsaresuchthatwhenthepolymercomesintoacavitythroughanumberofgateslocatedatdifferentpositions,pressurelevelsonthesurfaceofaproductshouldbeasuniformaspossible.SuchdesigncanbeperformedthroughtheintelligentgatepositioningtogeneratethemoreCorrespondingauthor.Tel.82221234474Fax.8223622736Emailaddressjleejyonsei.ac.kr790JongsooLeeandJonghunKim/JournalofMechanicalScienceandTechnology212007740749uniformdistributionofinjectionpressureovertheproductsurface.TherehavebeenanumberofstudiesofoptimalgatelocationinthecontextofCAEfillinginjectionmoldingdesignproblemswherevariouskindsofoptimizerhavebeenemployedtoconductdesignoptimizationKimetal.,1996Young,1994PandelidisandZou,2004Lin,2001LiandShen,1995.ThepaperexploresthedesignofinjectionmoldsystemusingmicrogeneticalgorithmmGA.GeneticalgorithmconventionalGAisbasedontheDarwinstheoryofthesurvivalofthefittest,andadoptstheconceptofnaturalevolutionthecompetitivedesignswithmorefitaresurvivedbyselection,andthenewdesignsarecreatedbycrossoverandmutationLee,1996LeeandHajela,1996.AconventionalGAworkswithamultiplenumberofdesignsinapopulation.Handlingwithsuchdesignsresultsinincreasingahigherprobabilityoflocatingaglobaloptimumaswellasmultiplelocaloptima.GAisalsoadvantageouswhenthedesignproblemisrepresentedbyamixtureofinteger/discreteandcontinuousdesignvariables.Nevertheless,itrequiresexpensivecomputationalcostsespeciallywhencombiningwithfiniteelementbasedCAEanalysistools.AconventionalGAdeterminesthepopulationsizedependinguponthestringlengthofachromosomethatisacodedvalueofasetofdesignvariables.ThemaindifferencebetweenaconventionalGAandmGAresidesonthepopulationsize.ThepopulationsizeinmGAisbasedonGoldbergsconceptsuchthatEvolutionprocessispossiblewithsmallpopulationstoreducethecostoffitnessfunctionevaluationGoldberg,1988.ThisimpliesthatmGAemploysafewnumberofpopulationsforGAevolutionregardlessofthenumberofdesignvariablesandthecomplexityofdesignparametersKrishnakumar,1989DennisandDulikravich,2001.Thepaperdiscussesthedesignrequirementsoffillinginjectionmoldoptimizationtoconstructtheproperobjectivefunctionsanddesignconstraints.Fourdifferentdesignapplicationsininjectionmoldsareexploredtoexaminetheproposeddesignstrategies.ThepapershowstheeffectivenessofmGAinthecontextofoptimizationofrunnersystemininjectionmoldingdesign.2.MoldflowanalysisTheflowofapolymerininjectionmoldingprocessobeysthefollowinggoverningequations220ppSSxxyywwwwwwww1222pxyTTTTCktxyzUQQKJwwwwwwww2where,220hzSdzK³.Intheaboveequations,pisaflowpressure,Tisatemperatureofpolymer,andtisdenotedastime.ParametersK,J,andkareviscosity,shearrateandthermalconductivity,respectivelyLee,2003.Itisassumedthatpolymerisanoncompactionsubstanceinthefillinganalysis.TheflowanalysisinthepresentstudyisconductedbyComputerAidedPlasticsApplicationCAPAKoo,2003,afiniteelementbasedcommercialcodeforpolymerflowofinjectionmolding.Therunnersystemininjectionmoldcoversthepassageofmoltenpolymerfrominjectionporttogates.Thepresentstudydevelopstwodifferentrunnersystemswhereacoldsystemrequiresthechangeinpolymertemperature,andahotsystemkeepitunchangedwhiletheflowpassesthroughtherunner.ForthehotrunnersystemhasageometricallyconsistentthicknessduetotheconstanttemperatureasshowninFig.1a.However,theCAEresultofacoldrunnersystemdependsonthethicknessandshapeTable1.Tenbartrussdesignresults.microGAconventionalGACase1Case2Case3Case1Case2Case3Reference20X17.868.157.858.157.307.817.90X20.410.180.190.100.830.450.10X38.387.998.158.208.778.378.10X45.053.833.893.973.274.163.90X50.120.960.151.100.750.550.10X60.410.250.250.100.820.300.10X76.415.675.875.846.746.305.80X85.236.295.525.685.065.265.51X93.833.855.055.072.893.863.68OptimalareaX100.500.250.250.401.160.420.14Optimalweight1599158715881593159015851499offunctionevaluations575405423025335788946949773533JongsooLeeandJonghunKim/JournalofMechanicalScienceandTechnology212007789798791aHotrunnersystembColdrunnersystemFig.1.Modelingofrunnersystem.shapeofarunner.ThetypicalillustrationofthegeometricmodelinacoldrunnersystemisshowninFig.1bwheretherunnerthicknessischangedaccordingtothetemperaturegradient.3.Moldingdesignrequirements3.1ObjectivefunctionsOneofthemostsignificantfactorsconsideredintheinjectionmoldingdesignisaflowpattern,whichimpliesthatabalancedflowshouldbemaintainedwhileapolymerarrivesateachpartofadesignproduct.Oncetheimprovementonflowbalanceisobtained,theflowofmoltenpolymersmoothesandthemaximuminjectionpressureisdecreasedwiththesameoratleastevenlydistributedinjectionpressurelevelateachgate.Inacasewherethecertainpartofaproductwithinthemoldisfilledupearlierthanotherparts,eachpartwouldfallintooverpackingandunderpackingsituationsduringthefillingprocessofapolymerintomold.Suchproblemfurtherevokesamalformationliketwistingandbending,resultingfromthedifferenceincontractionrateduringthecourseofcoolingoff.Thedifferenceinpressuretriggerstheflowofpolymerduringthefillingprocess,inwhichthemaximuminjectionpressureisdetectedattheinjectionportofpolymer.Thepolymeralwaysflowsfromhighpressureregiontolowpressureone.Whenaflowpatternimproves,theflowofpolymergetssmootherwiththemaximuminjectionpressuredecreased.However,theflowinstabilitysometimeshappens,therebyrequiringahigherpressuretofillup.Thatis,themaximuminjectionpressureneedstobereducedinordertoimprovetheflowinstability.Thepressuregapi.e.,thehighestandlowestpressurevaluesamongallofgatesisalsotakenasanotherobjectivefunctiontodeterminewhetherthewholemoldisbeingfilledatonce.Mostcommonlyaccepteddesignstrategytoimprovetheflowpatternistheadjustmentofgatelocation.Thepresentstudycontrolstheflowpatternbydevelopingtheoptimalgatepositioningproblemswithproperobjectivefunctionsanddesignconstraints.ObjectivefunctionsforinjectionmoldingdesignareconsideredasbothmaximuminjectionpressureMIPandmaximumpressuredifferenceMPD.Itshouldbenotedthatthemaximuminjectionpressureiscalculatedattheinjectionportandthemaximumpressuredifferenceisanumericaldifferencebetweenthehighestandlowestvaluesofpressureamongallofgates.Theaforementionedstatementscouldbeinterpretedasamultiobjectivedesignproblem,hencethepresentstudysimplyemploysaweightingmethodasfollowsMIPxMPDxFxMIPMPDDE3where,DandEareweightingfactorsasDE1,andxisasetofdesignvariableswhichareCartesiancoordinatesofgatesonaproduct.Eachcomponentintheaboveequationisnormalizedbyoptimalsingleobjectivefunctionvalue,i.e.,MIP,MPD.Itismentionedthatthenumberofgatesisconsideredasaproblemparameterinthestudy.3.1ConstraintsWeldlinesareeasilydetectedwhenmorethantwoflowfrontshavingdifferenttemperaturevaluesmeetduringthefillingprocess.Theweldlineisoneoftheweakestpointsinmoldedproductitisvery792JongsooLeeandJonghunKim/JournalofMechanicalScienceandTechnology212007740749vulnerabletoashockandsubsequentlycausesexternaldefectsofaveryglossypolymer.Theweldlineshouldbemovedintoalessweakregionbyadjustingthewidthofaproduct,thesizeand/orshapeofgatesandrunners,andthepositionofgates,etc.Thepresentstudyconsidersthepositionofaweldlineasaconstraintinoptimalgatepositioningofmolddesign.Onceadesignerspecifiesareaswhereweldlinesshouldnotbegenerated,allofthefiniteelementnodesinsuchareasareconstrainednottoformtheweldlines.Shearstressisdefinedasashearforceimposedonthewallofamoldbytheshearflowofapolymer.Themagnitudeofshearstressisproportionaltothepressuregradientofeachposition.Ingeneral,theshearstressiszeroatthecenterofamoldedproduct,andreachesamaximumvalueonthewall.Highshearstresstriggersthemoleculecultivationonthesurfaceofamoldedproduct.Flowinstabilitysuchasmeltfracturehasacloserelationshipwiththeshearstress.Theclearsurfaceofamoldedproductcanbeobtainedbyreducingthemagnitudeofshearstress.Thatis,shearstressshouldbeminimizedduringthemoldfillingprocessinordertoimprovethequalityofamoldedproduct,particularlyonitssurface.Maximumallowableshearstressdependsonthekindsofpolymer,andisgenerallytakenas1oftensilestrengthofapolymer.Shearstressaffectingthequalityofendproductisconsideredasanotherconstraint.3.3FormulationofoptimizationproblemThestatementofamolddesignoptimizationproblemcanbewrittenasfollowsFind12,,{,,,,,,...,,,}Nxijkxijkxijkxijk4tominimizeMIPxMPDxFxMIPMPDDE5subjecttoshearstressi,j,kshearstressallowable6weldlinei,j,kdesignatedareasonly7where,lowerupperxxxddAsetofdesignvariables,xareCartesiancoordinatesi,j,kofgatesonthesurfaceofamoldedproduct,whereNisthenumberofgates.AtraditionalweightedsummethodinthecontextofmultiobjectiveoptimizationisemployedbyusingtwoweiFig.2.MicroGAprocess.ghtingfactorsofDandE,whereDE1.MultiobjectivefunctionsconsideredinthepresentstudyaremaximuminjectionpressureMIPmeasuredattheinjectionportandmaximumpressuredifferencePDamongallofgates.Theconstants,MIPandMPDareoptimalobjectivefunctionvaluesobtainedviasingleobjectiveoptimization.Thepermissionofweldlinestodesignatedareasonlyandtheupperlimitsonshearstressareimposedasdesignconstraints.TheflowpatternanalysisisperformedbyCAPAasmentionedintheearliersection,andtheoptimizationisconductedthroughmGA.ItshouldbenotedthatCartesiancoordinatesi,j,kisrecognizedasnodalpointswhenamoldedproductisdiscretizedbyfiniteelementsinCAPA.4.MicroGATheoverallprocessofmGAinthepresentstudyisdepictedinFig.2,andastepwiseprocedurecanbeexplainedasfollowsStep1Generateaninitialpopulationatrandom.Therecommendedpopulationsizeis3,5,or7.Step2PerformaconventionalGAevolutionuntilthenominalconvergenceissatisfied.Inthepresentstudy,thepopulationsizeisselectedas5,andatournamentselectionoperatorisused.ThecrossoverprobabilityinmGAis1.0duetothesmallsizeinpopulation,whileaconventionalGAispreferredtouseitlessthan1.0.Thenominalconvergencemeansthatthedifferenceof1sand/or0samongstringpositionsiswithin5outofthestringlength,therebyresultingintheconvergencetoalocalsolution.Step3Duringtheuserspecifiednumberofgenerations,anewpopulationisupdatedoneindividualisselectedbyelitismtheremainingindividualsina
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