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GJournalofMechanicalScienceandTechnology21(2007)789798JournalofMechanicalScienceandTechnologyMicroGeneticAlgorithmBasedOptimalGatePositioninginInjectionMoldingDesignJongsooLee*,JonghunKimSchoolofMechanicalEngineeringYonseiUniversity,Seoul120-749Korea(ManuscriptReceivedDecember12,2006；RevisedMarch26,2007；AcceptedMarch26,2007)-AbstractThepaperdealswiththeoptimizationofrunnersystemininjectionmoldingdesign.Thedesignobjectiveistolocategatepositionsbyminimizingbothmaximuminjectionpressureattheinjectionportandmaximumpressuredifferenceamongallthegatesonaproductwithconstraintsonshearstressand/orweld-line.Theanalysisoffillingprocessisconductedbyafiniteelementbasedprogramforpolymerflow.Microgeneticalgorithm(mGA)isusedasaglobaloptimizationtoolduetothenatureofinherentnonlinearlityinflowanalysis.Fourdifferentdesignapplicationsininjectionmoldsareexploredtoexaminetheproposeddesignstrategies.ThepapershowstheeffectivenessofmGAinthecontextofoptimizationofrunnersystemininjectionmoldingdesign.GKeywords:Microgeneticalgorithm；Designoptimization；Fillinginjectionmold-1.IntroductionInjectionmoldingprocesshasbeenrecognizedasoneofthemostefficientmanufacturingtechnologiessincehighperformancepolymermaterialscanbeutilizedtoaccuratelymanufactureaproductwithcomplicatedshape(Chiang,etal.,1991；ChangandYang,2001；Himasekhar,etal.,1992；KwonandPark,2004).Also,thedemandoninjectionmoldedproductssuchasfromconventionalplasticgoodstomicroopticaldevicesisbeingdramaticallyincreasedovertherecentyears(Piotter,etal.,2001；Kang,etal.,2000).Ingeneral,theinjectionmoldprocessisinitiatedbythefillingstagewherethepolymermaterialsfillintoacavityundertheinjectiontemperature.Afterthecavityiscompletelyfilled,thepost-fillingstage,thatis,thepackingstageisconductedtobeadditionallyfilledwiththehighpressurepolymer,therebyresultingintheavoidanceofmaterialshrinkage.Subsequently,thecoolingstageisrequiredforamoldedproducttobeejectedwithoutanydeformation.Itisimportanttoaccommodatethemoldingconditionsinthefillingstagesinceitisthefirststageintheoverallinjectionmoldingdesign(ZhouandD.Li,2001).Afterthat,onecansuccess-fullyexpectmoreimprovedmoldingconditionsduringpost-fillingstagessuchaspacking,coolingstages.Thepaperdealswithoptimalconditionsofthefillinginjectionmoldingdesigninwhichtheflowpatternandpressureforthepolymermaterialstobefilledthroughgatesofarunnerareofsignificant.Thatis,oneofdesignrequirementsaresuchthatwhenthepolymercomesintoacavitythroughanumberofgateslocatedatdifferentpositions,pressurelevelsonthesurfaceofaproductshouldbeasuniformaspossible.Suchdesigncanbeperformedthroughtheintelligentgatepositioningtogeneratethemore*Correspondingauthor.Tel.:+82221234474；Fax.:+8223622736E-mailaddress:jleejyonsei.ac.kr790JongsooLeeandJonghunKim/JournalofMechanicalScienceandTechnology21(2007)740749uniformdistributionofinjectionpressureovertheproductsurface.TherehavebeenanumberofstudiesofoptimalgatelocationinthecontextofCAEfillinginjectionmoldingdesignproblemswherevariouskindsofoptimizerhavebeenemployedtoconductdesignoptimization(Kimetal.,1996；Young,1994；Pan-delidisandZou,2004；Lin,2001；LiandShen,1995).Thepaperexploresthedesignofinjectionmoldsystemusingmicrogeneticalgorithm(mGA).Ge-neticalgorithm(conventionalGA)isbasedontheDarwinstheoryofthesurvivalofthefittest,andadoptstheconceptofnaturalevolution；thecompetitivedesignswithmorefitaresurvivedbyselection,andthenewdesignsarecreatedbycrossoverandmutation(Lee,1996；LeeandHajela,1996).AconventionalGAworkswithamultiplenumberofdesignsinapopulation.Handlingwithsuchdesignsresultsinincreasingahigherprobabilityoflocatingaglobaloptimumaswellasmultiplelocaloptima.GAisalsoadvantageouswhenthedesignproblemisrepresentedbyamixtureofinteger/dis-creteandcontinuousdesignvariables.Nevertheless,itrequiresexpensivecomputationalcostsespeciallywhencombiningwithfiniteelementbasedCAEanalysistools.AconventionalGAdeterminesthepopulationsizedependinguponthestringlengthofachromosomethatisacodedvalueofasetofdesignvariables.Themaindifferencebetweenaconven-tionalGAandmGAresidesonthepopulationsize.ThepopulationsizeinmGAisbasedonGoldbergsconceptsuchthatEvolutionprocessispossiblewithsmallpopulationstoreducethecostoffitnessfunctionevaluation(Goldberg,1988).ThisimpliesthatmGAemploysafewnumberofpopulationsforGAevolutionregardlessofthenumberofdesignvariablesandthecomplexityofdesignparameters(Krishnakumar,1989；DennisandDulikravich,2001).Thepaperdiscussesthedesignrequirementsoffillinginjectionmoldoptimizationtoconstructtheproperobjectivefunctionsanddesignconstraints.Fourdifferentdesignapplicationsininjectionmoldsareexploredtoexaminetheproposeddesignstrategies.ThepapershowstheeffectivenessofmGAinthecontextofoptimizationofrunnersystemininjectionmoldingdesign.2.MoldflowanalysisTheflowofapolymerininjectionmoldingprocessobeysthefollowinggoverningequations:22()()0ppSSxxyywwwwwwww(1)222()pxyTTTTCktxyzUQQKJwwwwwwww(2)where,220hzSdzK³.Intheaboveequations,pisaflowpressure,Tisatemperatureofpolymer,andtisdenotedastime.ParametersK,J,andkareviscosity,shearrateandthermalconductivity,respectively(Lee,2003).Itisassumedthatpolymerisanon-compactionsubstanceinthefillinganalysis.TheflowanalysisinthepresentstudyisconductedbyComputerAidedPlasticsApplication(CAPA)(Koo,2003),afiniteelementbasedcommercialcodeforpolymerflowofinjectionmolding.Therunnersystemininjectionmoldcoversthepassageofmoltenpolymerfrominjectionporttogates.Thepresentstudydevelopstwodifferentrunnersystemswhereacoldsystemrequiresthechangeinpolymertemperature,andahotsystemkeepitunchangedwhiletheflowpassesthroughtherunner.ForthehotrunnersystemhasageometricallyconsistentthicknessduetotheconstanttemperatureasshowninFig.1a.However,theCAEresultofacoldrunnersystemdependsonthethicknessandshapeTable1.Ten-bartrussdesignresults.microGAconventionalGACase1Case2Case3Case1Case2Case3Reference20X17.868.157.858.157.307.817.90X20.410.180.190.100.830.450.10X38.387.998.158.208.778.378.10X45.053.833.893.973.274.163.90X50.120.960.151.100.750.550.10X60.410.250.250.100.820.300.10X76.415.675.875.846.746.305.80X85.236.295.525.685.065.265.51X93.833.855.055.072.893.863.68OptimalareaX100.500.250.250.401.160.420.14Optimalweight1599158715881593159015851499#offunctionevaluations575405423025335788946949773533JongsooLeeandJonghunKim/JournalofMechanicalScienceandTechnology21(2007)789798791(a)Hotrunnersystem(b)ColdrunnersystemFig.1.Modelingofrunnersystem.shapeofarunner.ThetypicalillustrationofthegeometricmodelinacoldrunnersystemisshowninFig.1bwheretherunnerthicknessischangedaccordingtothetemperaturegradient.3.Moldingdesignrequirements3.1ObjectivefunctionsOneofthemostsignificantfactorsconsideredintheinjectionmoldingdesignisaflowpattern,whichimpliesthatabalancedflowshouldbemaintainedwhileapolymerarrivesateachpartofadesignproduct.Oncetheimprovementonflowbalanceisobtained,theflowofmoltenpolymersmoothesandthemaximuminjectionpressureisdecreasedwiththesameoratleastevenlydistributedinjectionpressurelevelateachgate.Inacasewherethecertainpartofaproductwithinthemoldisfilledupearlierthanotherparts,eachpartwouldfallintoover-packingandunder-packingsituationsduringthefillingprocessofapolymerintomold.Suchproblemfurtherevokesamalformationliketwistingandbending,resultingfromthedifferenceincontractionrateduringthecourseofcooling-off.Thedifferenceinpressuretriggerstheflowofpolymerduringthefillingprocess,inwhichthemaximuminjectionpressureisdetectedattheinjectionportofpolymer.Thepolymeralwaysflowsfromhigh-pressureregiontolow-pressureone.Whenaflowpatternimproves,theflowofpolymergetssmootherwiththemaximuminjectionpressuredecreased.However,theflowinstabilitysometimeshappens,therebyrequiringahigherpressuretofillup.Thatis,themaximuminjectionpressureneedstobereducedinordertoimprovetheflowinstability.Thepressuregap(i.e.,thehighestandlowestpressurevalues)amongallofgatesisalsotakenasanotherobjectivefunctiontodeterminewhetherthewholemoldisbeingfilledatonce.Mostcommonlyaccepteddesignstrategytoimprovetheflowpatternistheadjustmentofgatelocation.Thepresentstudycontrolstheflowpatternbydevelopingtheoptimalgatepositioningproblemswithproperobjectivefunction(s)anddesigncons-traints.Objectivefunctionsforinjectionmoldingdesignareconsideredasbothmaximuminjectionpressure(MIP)andmaximumpressuredifference(MPD).Itshouldbenotedthatthemaximuminjectionpressureiscalculatedattheinjectionportandthemaximumpressuredifferenceisanumericaldifferencebetweenthehighestandlowestvaluesofpressureamongallofgates.Theaforementionedstatementscouldbeinterpretedasamultiobjectivedesignproblem,hencethepresentstudysimplyemploysaweightingmethodasfollows:*()()()MIPxMPDxFxMIPMPDDE(3)where,DandEareweightingfactorsasD+E=1,andxisasetofdesignvariableswhichareCartesiancoordinatesofgatesonaproduct.Eachcomponentintheaboveequationisnormalizedbyoptimalsingle-objectivefunctionvalue,(i.e.,MIP*,MPD*).Itismentionedthatthenumberofgatesisconsideredasaproblemparameterinthestudy.3.1ConstraintsWeld-linesareeasilydetectedwhenmorethantwoflowfrontshavingdifferenttemperaturevaluesmeetduringthefillingprocess.Theweld-lineisoneoftheweakestpointsinmoldedproduct；itisvery792JongsooLeeandJonghunKim/JournalofMechanicalScienceandTechnology21(2007)740749vulnerabletoashockandsubsequentlycausesexternaldefectsofaveryglossypolymer.Theweld-lineshouldbemovedintoalessweakregionbyadjustingthewidthofaproduct,thesizeand/orshapeofgatesandrunners,andthepositionofgates,etc.Thepresentstudyconsidersthepositionofaweld-lineasaconstraintinoptimalgatepositioningofmolddesign.Onceadesignerspecifiesareaswhereweld-linesshouldnotbegenerated,allofthefiniteelementnodesinsuchareasareconstrainednottoformtheweld-lines.Shearstressisdefinedasashearforceimposedonthewallofamoldbytheshearflowofapolymer.Themagnitudeofshearstressisproportionaltothepressuregradientofeachposition.Ingeneral,theshearstressiszeroatthecenterofamoldedproduct,andreachesamaximumvalueonthewall.Highshearstresstriggersthemoleculecultivationonthesurfaceofamoldedproduct.Flowinstabilitysuchasmeltfracturehasacloserelationshipwiththeshearstress.Theclearsurfaceofamoldedproductcanbeobtainedbyreducingthemagnitudeofshearstress.Thatis,shearstressshouldbeminimizedduringthemoldfillingprocessinordertoimprovethequalityofamoldedproduct,particularlyonitssurface.Maximumallowableshearstressdependsonthekindsofpolymer,andisgenerallytakenas1%oftensilestrengthofapolymer.Shearstressaffectingthequalityofendproductisconsideredasanotherconstraint.3.3FormulationofoptimizationproblemThestatementofamolddesignoptimizationproblemcanbewrittenasfollows:Find12(,)(,),(,),.,(,)Nxijkxijkxijkxijk(4)tominimize*()()()MIPxMPDxFxMIPMPDDE(5)subjecttoshearstress(i,j,k)<shearstressallowable(6)weld-line(i,j,k)=designatedarea(s)only(7)where,lowerupperxxxddAsetofdesignvariables,xareCartesiancoordi-nates(i,j,k)ofgatesonthesurfaceofamoldedproduct,whereNisthenumberofgates.Atraditionalweighted-summethodinthecontextofmultiob-jectiveoptimizationisemployedbyusingtwowei-Fig.2.MicroGAprocess.ghtingfactorsofDandE,whereD+E=1.Multi-objectivefunctionsconsideredinthepresentstudyaremaximuminjectionpressure(MIP)measuredattheinjectionportandmaximumpressuredifference(PD)amongallofgates.Theconstants,MIP*andMPD*areoptimalobjectivefunctionvaluesobtainedviasingle-objectiveoptimization.Thepermissionofweld-linestodesignatedareasonlyandtheupperlimitsonshearstressareimposedasdesigncons-traints.TheflowpatternanalysisisperformedbyCAPAasmentionedintheearliersection,andtheoptimizationisconductedthroughmGA.ItshouldbenotedthatCartesiancoordinates(i,j,k)isrecognizedasnodalpointswhenamoldedproductisdiscretizedbyfiniteelementsinCAPA.4.MicroGATheoverallprocessofmGAinthepresentstudyisdepictedinFig.2,andastepwiseprocedurecanbeexplainedasfollows:Step-1)Generateaninitialpopulationatrandom.Therecommendedpopulationsizeis3,5,or7.Step-2)PerformaconventionalGAevolutionuntilthenominalconvergenceissatisfied.Inthepresentstudy,thepopulationsizeisselectedas5,andatournamentselectionoperatorisused.ThecrossoverprobabilityinmGAis1.0duetothesmallsizeinpopulation,whileaconventionalGAispreferredtouseitlessthan1.0.Thenominalconvergencemeansthatthedifferenceof1sand/or0samongstringpositionsiswithin5%outofthestringlength,therebyresultingintheconvergencetoalocalsolution.Step-3)Duringtheuser-specifiednumberofge-nerations,anewpopulationisupdated；oneindividualisselectedbyelitism；theremainingindividualsina