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Lietal./JZhejiangUnivSciA20078(7):1077-10831077Singlegateoptimizationforplasticinjectionmold*LIJi-quan,LIDe-qun,GUOZhi-ying,LVHai-yuan(DepartmentofPlasticityTechnology,ShanghaiJiaoTongUniversity,Shanghai200030,China)E-mail:hutli163.comReceivedNov.22,2006；revisionacceptedMar.19,2007Abstract:Thispaperdealswithamethodologyforsinglegatelocationoptimizationforplasticinjectionmold.Theobjectiveofthegateoptimizationistominimizethewarpageofinjectionmoldedparts,becausewarpageisacrucialqualityissueformostinjectionmoldedpartswhileitisinfluencedgreatlybythegatelocation.Featurewarpageisdefinedastheratioofmaximumdisplacementonthefeaturesurfacetotheprojectedlengthofthefeaturesurfacetodescribepartwarpage.Theoptimizationiscombinedwiththenumericalsimulationtechnologytofindtheoptimalgatelocation,inwhichthesimulatedannealingalgorithmisusedtosearchfortheoptimum.Finally,anexampleisdiscussedinthepaperanditcanbeconcludedthattheproposedmethodiseffective.Keywords:Injectionmold,Gatelocation,Optimization,Featurewarpagedoi:10.1631/jzus.2007.A1077Documentcode:ACLCnumber:TQ320.66INTRODUCTIONPlasticinjectionmoldingisawidelyused,com-plexbuthighlyefficienttechniqueforproducingalargevarietyofplasticproducts,particularlythosewithhighproductionrequirement,tighttolerance,andcomplexshapes.Thequalityofinjectionmoldedpartsisafunctionofplasticmaterial,partgeometry,moldstructureandprocessconditions.Themostimportantpartofaninjectionmoldbasicallyisthefollowingthreesetsofcomponents:cavities,gatesandrunners,andcoolingsystem.LamandSeow(2000)andJinandLam(2002)achievedcavitybalancingbyvaryingthewallthick-nessofthepart.Abalancefillingprocesswithinthecavitygivesanevenlydistributedpressureandtem-peraturewhichcandrasticallyreducethewarpageofthepart.Butthecavitybalancingisonlyoneoftheimportantinfluencingfactorsofpartqualities.Espe-cially,theparthasitsfunctionalrequirements,anditsthicknessesshouldnotbevariedusually.Fromthepointviewoftheinjectionmolddesign,agateischaracterizedbyitssizeandlocation,andtherunnersystembythesizeandlayout.Thegatesizeandrunnerlayoutareusuallydeterminedasconstants.Relatively,gatelocationsandrunnersizesaremoreflexible,whichcanbevariedtoinfluencethequalityofthepart.Asaresult,theyareoftenthedesignpa-rametersforoptimization.LeeandKim(1996a)optimizedthesizesofrunnersandgatestobalancerunnersystemformul-tipleinjectioncavities.Therunnerbalancingwasdescribedasthedifferencesofentrancepressuresforamulti-cavitymoldwithidenticalcavities,andasdifferencesofpressuresattheendofthemeltflowpathineachcavityforafamilymoldwithdifferentcavityvolumesandgeometries.Themethodologyhasshownuniformpressuredistributionsamongthecavitiesduringtheentiremoldingcycleofmultiplecavitiesmold.Zhaietal.(2005a)presentedthetwogateloca-tionoptimizationofonemoldingcavitybyaneffi-cientsearchmethodbasedonpressuregradient(PGSS),andsubsequentlypositionedweldlinestothedesiredlocationsbyvaryingrunnersizesforJournalofZhejiangUniversitySCIENCEAISSN1673-565X(Print)；ISSN1862-1775(Online)www.zju.edu.cn/jzus；www.springerlink.comE-mail:jzuszju.edu.cn*Project(No.50675080)supportedbytheNationalNaturalScienceFoundationofChinaLietal./JZhejiangUnivSciA20078(7):1077-10831078multi-gateparts(Zhaietal.,2006).Aslarge-volumepart,multiplegatesareneededtoshortenthemaxi-mumflowpath,withacorrespondingdecreaseininjectionpressure.Themethodispromisingforde-signofgatesandrunnersforasinglecavitywithmultiplegates.Manyofinjectionmoldedpartsareproducedwithonegate,whetherinsinglecavitymoldorinmultiplecavitiesmold.Therefore,thegatelocationofasinglegateisthemostcommondesignparameterforoptimization.Ashapeanalysisapproachwaspre-sentedbyCourbebaisseandGarcia(2002),bywhichthebestgatelocationofinjectionmoldingwasesti-mated.Subsequently,theydevelopedthismethodol-ogyfurtherandappliedittosinglegatelocationop-timizationofanLshapeexample(Courbebaisse,2005).Itiseasytouseandnottime-consuming,whileitonlyservestheturningofsimpleflatpartswithuniformthickness.PandelidisandZou(1990)presentedtheopti-mizationofgatelocation,byindirectqualitymeasuresrelevanttowarpageandmaterialdegradation,whichisrepresentedasweightedsumofatemperaturedif-ferentialterm,anover-packterm,andafrictionaloverheatingterm.Warpageisinfluencedbytheabovefactors,buttherelationshipbetweenthemisnotclear.Therefore,theoptimizationeffectisrestrictedbythedeterminationoftheweightingfactors.LeeandKim(1996b)developedanautomatedselectionmethodofgatelocation,inwhichasetofinitialgatelocationswereproposedbyadesignerandthentheoptimalgatewaslocatedbytheadjacentnodeevaluationmethod.Theconclusiontoagreatextentdependsmuchonthehumandesignersintuition,becausethefirststepofthemethodisbasedonthedesignersproposition.Sotheresultistoalargeex-tentlimitedtothedesignersexperience.LamandJin(2001)developedagatelocationoptimizationmethodbasedontheminimizationoftheStandardDeviationofFlowPathLength(SDL)andStandardDeviationofFillingTime(SDT)duringthemoldingfillingprocess.Subsequently,Shenetal.(2004a；2004b)optimizedthegatelocationdesignbyminimizingtheweightedsumoffillingpressure,fillingtimedifferencebetweendifferentflowpaths,temperaturedifference,andover-packpercentage.Zhaietal.(2005b)investigatedoptimalgatelocationwithevaluationcriteriaofinjectionpressureattheendoffilling.Theseresearcherspresentedtheobjec-tivefunctionsasperformancesofinjectionmoldingfillingoperation,whicharecorrelatedwithproductqualities.Butthecorrelationbetweentheperform-ancesandqualitiesisverycomplicatedandnoclearrelationshiphasbeenobservedbetweenthemyet.Itisalsodifficulttoselectappropriateweightingfactorsforeachterm.Anewobjectivefunctionispresentedheretoevaluatethewarpageofinjectionmoldedpartstooptimizegatelocation.Tomeasurepartqualitydi-rectly,thisinvestigationdefinesfeaturewarpagetoevaluatepartwarpage,whichisevaluatedfromthe“flowpluswarpage”simulationoutputsofMoldflowPlasticsInsight(MPI)software.Theobjectivefunc-tionisminimizedtoachieveminimumdeformationingatelocationoptimization.Simulatedannealingal-gorithmisemployedtosearchfortheoptimalgatelocation.Anexampleisgiventoillustratetheeffec-tivityoftheproposedoptimizationprocedure.QUALITYMEASURES:FEATUREWARPGEDefinitionoffeaturewarpageToapplyoptimizationtheorytothegatedesign,qualitymeasuresofthepartmustbespecifiedinthefirstinstance.Theterm“quality”maybereferredtomanyproductproperties,suchasmechanical,thermal,electrical,optical,ergonomicalorgeometricalprop-erties.Therearetwotypesofpartqualitymeasures:directandindirect.Amodelthatpredictstheproper-tiesfromnumericalsimulationresultswouldbecharacterizedasadirectqualitymeasure.Incontrast,anindirectmeasureofpartqualityiscorrelatedwithtargetquality,butitcannotprovideadirectestimateofthatquality.Forwarpage,theindirectqualitymeasuresinrelatedworksareoneofperformancesofinjectionmoldingflowingbehaviororweightedsumofthose.Theperformancesarepresentedasfillingtimedif-ferentialalongdifferentflowpaths,temperaturedif-ferential,over-packpercentage,andsoon.Itisob-viousthatwarpageisinfluencedbytheseperform-ances,buttherelationshipbetweenwarpageandtheseperformancesisnotclearandthedeterminationoftheseweightingfactorsisratherdifficult.Therefore,theoptimizationwiththeaboveobjectivefunctionLietal./JZhejiangUnivSciA20078(7):1077-10831079probablywillnotminimizepartwarpageevenwithperfectoptimizationtechnique.Sometimes,improperweightingfactorswillresultinabsolutelywrongre-sults.Somestatisticalquantitiescalculatedfromthenodaldisplacementswerecharacterizedasdirectqualitymeasurestoachieveminimumdeformationinrelatedoptimizationstudies.Thestatisticalquantitiesareusuallyamaximumnodaldisplacement,anav-erageoftop10percentilenodaldisplacements,andanoverallaveragenodaldisplacement(LeeandKim,1995；1996b).Thesenodaldisplacementsareeasytoobtainfromthesimulationresults,thestatisticalval-ues,tosomeextents,representingthedeformation.Butthestatisticaldisplacementcannoteffectivelydescribethedeformationoftheinjectionmoldedparts.Inindustry,designersandmanufacturersusuallypaymoreattentiontothedegreeofpartwarpageonsomespecificfeaturesthanthewholedeformationoftheinjectionmoldedparts.Inthisstudy,featurewarpageisdefinedtodescribethedeformationoftheinjectionparts.Thefeaturewarpageistheratioofthemaximumdisplacementofthefeaturesurfacetotheprojectedlengthofthefeaturesurface(Fig.1):100%,hL=×(1)whereisthefeaturewarpage,histhemaximumdisplacementonthefeaturesurfacedeviatingfromthereferenceplatform,andListheprojectedlengthofthefeaturesurfaceonareferencedirectionparallelingthereferenceplatform.Forcomplicatedfeatures(onlyplanefeaturediscussedhere),thefeaturewarpageisusuallysepa-ratedintotwoconstituentsonthereferenceplane,whicharerepresentedona2Dcoordinatesystem:100%,100%,xyhhLL=×=×(2)wherex,yaretheconstituentfeaturewarpagesintheX,Ydirection,andLx,LyaretheprojectedlengthsofthefeaturesurfaceonX,Ycomponent.EvaluationoffeaturewarpageAfterthedeterminationoftargetfeaturecom-binedwithcorrespondingreferenceplaneandpro-jectiondirection,thevalueofLcanbecalculatedimmediatelyfromthepartwiththecalculatingmethodofanalyticgeometry(Fig.2).Lisaconstantforanypartonthespecifiedfeaturesurfaceandpro-jecteddirection.Buttheevaluationofhismorecom-plicatedthanthatofL.Simulationofinjectionmoldingprocessisacommontechniquetoforecastthequalityofpartde-sign,molddesignandprocesssettings.Theresultsofwarpagesimulationareexpressedasthenodalde-flectionsonX,Y,Zcomponent(Wx,Wy,Wz),andthenodaldisplacementW.WisthevectorlengthofvectorsumofWx·i,Wy·j,andWz·k,wherei,j,karetheunitvectorsonX,Y,Zcomponent.Thehisthemaximumdisplacementofthenodesonthefeaturesurface,whichiscorrelatedwiththenormalorientationofthereferenceplane,andcanbederivedfromtheresultsofwarpagesimulation.Tocalculateh,thedeflectionofithnodeisevaluatedfirstlyasfollows:coscoscos(),iixiyiziAAiBBWWWWWW=+(3)whereWiisthedeflectioninthenormaldirectionofthereferenceplaneofithnode；Wix,Wiy,WizarethedeflectionsonX,Y,Zcomponentofithnode；,aretheanglesofnormalvectorofthereference；AandBaretheterminalnodesofthefeaturetoprojectingdirection(Fig.2)；WAandWBarethedeflectionsofnodesAandB:Fig.1ThedefinitionoffeaturewarpagehReferenceplaneSurfaceLFig.2TheprojectedlengthevaluationFeatureYXLxBALyLietal./JZhejiangUnivSciA20078(7):1077-10831080coscoscos,coscoscos,AAxAyAzBBxByBzWWWWWWWW=+=+(4)whereWAx,WAy,WAzarethedeflectionsonX,Y,ZcomponentofnodeA；WBx,WByandWBzarethede-flectionsonX,Y,ZcomponentofnodeB；iAandiBaretheweightingfactorsoftheterminalnodedeflec-tionscalculatedasfollows:1/,1,iAiAiBiALL=(5)whereLiAistheprojectordistancebetweenithnodeandnodeA.Ultimately,histhemaximumoftheabsolutevalueofWi:12max|,|,|.khWWW=(6)Inindustry,theinspectionofthewarpageiscarriedoutwiththehelpofafeelergauge,whilethemeasuredpartshouldbeplacedonareferenceplat-form.Thevalueofhisthemaximumnumericalreadingofthespacebetweenthemeasuredpartsur-faceandthereferenceplatform.GATELOCATIONOPTIMIZATIONPROBLEMFORMATIONThequalityterm“warpage”meanstheperma-nentdeformationofthepart,whichisnotcausedbyanappliedload.Itiscausedbydifferentialshrinkagethroughoutthepart,duetotheimbalanceofpolymerflow,packing,cooling,andcrystallization.Theplacementofagateinaninjectionmoldisoneofthemostimportantvariablesofthetotalmolddesign.Thequalityofthemoldedpartisgreatlyaf-fectedbythegatelocation,becauseitinfluencesthemannerthattheplasticflowsintothemoldcavity.Therefore,differentgatelocationsintroduceinho-mogeneityinorientation,density,pressure,andtemperaturedistribution,accordinglyintroducingdifferentvalueanddistributionofwarpage.Therefore,gatelocationisavaluabledesignvariabletominimizetheinjectionmoldedpartwarpage.Becausethecor-relationbetweengatelocationandwarpagedistribu-tionistoalargeextentindependentofthemeltandmoldtemperature,itisassumedthatthemoldingconditionsarekeptconstantinthisinvestigation.Theinjectionmoldedpartwarpageisquantifiedbythefeaturewarpagewhichwasdiscussedintheprevioussection.Thesinglegatelocationoptimizationcanthusbeformulatedasfollows:Minimize:minf(X)=；Subjectto:0()/10,gpp=X,1,2,.,iXiN=Xwhereisthefeaturewarpage；pistheinjectionpressureatthegateposition；p0istheallowablein-jectionpressureofinjectionmoldingmachineortheallowableinjectionpressurespecifiedbythedesignerormanufacturer；Xisthecoordinatevectorofthecandidategatelocations；Xiisthenodeonthefiniteelementmeshmodelofthepartforinjectionmoldingprocesssimulation；Nisthetotalnumberofnodes.Inthefiniteelementmeshmodelofthepart,everynodeisapossiblecandidateforagate.There-fore,thetotalnumberofthepossiblegatelocationNpisafunctionofthetotalnumberofnodesNandthetotalnumberofgatelocationstobeoptimizedn:p(1)(1).!NNNnNn=Inthisstudy,onlythesingle-gatelocationproblemisinvestigated.SIMULATEDANNEALINGALGORITHMThesimulatedannealingalgorithmisoneofthemostpowerfulandpopularmeta-heuristicstosolveoptimizationproblemsbecauseoftheprovisionofgoodglobalsolutionstoreal-worldproblems.ThealgorithmisbaseduponthatofMetropolisetal.(1953),whichwasoriginallyproposedasameanstofindanequilibriumconfigurationofacollectionofatomsatagiventemperature.Theconnectionbe-tweenthisalgorithmandmathematicalminimizationwasfirstnotedbyPincus(1970),butitwasKirkpatricketal.(1983)whoproposedthatitformedthebasisofanoptimizationtechniqueforcombina-tional(andother)problems.Toapplythesimulatedannealingmethodtoop-