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维普资讯 LfefalJZhejiangUnivSciA 20078(7)：10771083 1077 JournalofZhejiangUniversitySCIENCEA ISSN1673-565X(Print)；ISSN1862-1775(Online) wwwzjueducnjzus；wwwspringerlinkcom E-mail：jzuszjueduca Singlegateoptimizationforplasticinjectionmold LIJi-quan十LIDequn，GUO Zhiying，LV Haiyuan ， (DepartmentofPlasticityTechnology,ShanghaiJiaoTongUniversity，Shanghai200030,China) Email：hutli163corn ReceivedNov22，2006；revisionacceptedMar19，2007 Abstract：ThispaperdealswithamethodologyforsinglegatelocationoptimizationofrplasticinjectionmoldTheobjectiveof thegateoptimizationistominimizethewarpageofinjectionmoldedparts，becausewarpageisacrucialqualityissueofrmost injectionmoldedpartswhileitisinfluencedgreatlybythegatelocationFeaturewarpageisdefinedastheratioofmaximum displacementonthefeaturesurfacetotheprojectedlengthofthefeaturesurfacetodescribepartwarpageTheoptimizationis combinedwiththenumericalsimulationtechnologytofindtheoptimalgatelocation，inwhichthesimulatedannealingalgorithm isusedtosearchofrtheoptimumFinally，anexampleisdiscussedinthepaperanditcanbeconcludedthattheproposedmethod iseffective Keywords：Injectionmold，Gatelocation，Optimization，Featurewarpage doi：101631jzus2007A1077 Documentcode：A CLCnumber：TQ32066 INTRODUCTION Fromthepointviewoftheinjectionmolddesign， agateischaracterizedbyitssizeandlocationandthe Plasticinjectionmoldingisawidelyusedcom urnnersystem bythesizeand layoutThegatesize plexbuthighly efficienttechniqueforproducinga andurIlllerlayoutareusuallydeterminedasconstants largevariety ofplasticproducts，particularly those Relatively，gatelocationsandrunnersizesaremore withhighproductionrequirement，tighttolerance，and flexible whichcanbevariedtoinfluencethequality complexshapesThequalityofinjectionmoldedparts ofthepartAsaresult，theyareoftenthedesignpa isafunctionofplasticmaterial，partgeometry，mold rametersforoptimization structureandprocessconditionsThemostimportant LeeandKim rl996a)optimizedthesizesof partofaninjectionmoldbasicallyistheofllowing urnnersandgatestobalanceurnnersystem ofrmul threesetsofcomponents：cavities，gatesandurnners， tipleinjectioncavitiesTheurnnerbalancingwas andcoolingsystem describedasthedifferencesofentrancepressuresofr Lam andSeow (2000)andJinandLani (2002) amulticavitymoldwith identicalcavitiesandas achievedcavitybalancingbyvaryingthewallthick differencesofpressuresattheendofthemeltflow nessofthepartA balancefillingprocesswithinthe pathineachcavityofrafamilymoldwim different cavitygivesanevenlydistributedpressureandtem cavityvolumesandgeometriesThemethodologyhas perautrewhichcandrasticallyreducethewarpageof shown uniofmr pressure distributions among the thepartButthecavitybalancingisonlyoneofthe cavitiesduringtheentiremoldingcycleofmultiple importantinfluencing factorsofpartqualitiesEspe cavitiesmold cially。theparthasitsfunctionalrequirementsandits Zhaieta1(2005a1presentedthetwogateloca thicknessesshouldnotbevariedusually tion optimization ofonemolding cavitybyanef前一 cient search method based on pressure gradient Project(No50675080)supportedbytheNationalNaturalScience (PGSS)，andsubsequentlypositionedweldlinestothe FoundationofChina desired locations by varying urnner sizes ofr 维普资讯 1078 L，ea1dZhejiangUnivSciA 20078(7)：1077-1083 multigateparts(Zhaieta1，2006)Aslargevolume endoffillingTheseresearcherspresentedtheobiec part，multiplegatesareneededtoshortenthemaxi tivefunctionsasperfomr ancesofiniectionmolding mum flow path，with a corresponding decrease in filling operation，which arecorrelatedwith product injectionpressureThemethodispromisingforde qualitiesButthe correlation between theperofrii1 sign ofgatesandrunnersofra single cavity wich ancesandqualitiesisverycomplicatedandnoclear multiplegates relationshiphasbeenobservedbetweenthem yetItis Manyofinjectionmoldedpartsareproduced alsodifficulttoselectappropriateweightingfactors with onegatewhetherinsinglecavitymoldorin foreachtemr multiplecavitiesmoldThereofrethegatelocationof A new objectivefunctionispresentedhereto asinglegateisthemostcommondesignparameterofr evaluatethewarpageofinjectionmoldedpartsto optimizationA shape analysis approach waspre optimizegatelocationTomeasurepartqualitydi sentedbyCOHrbebaisseandGarcia(2002)，bywhich rectly，thisinvestigation definesfeaturewarpageto thebestgatelocationofiniectionmoldingwasesti evaluatepartwaprage，which isevaluated from the matedSubsequently，theydevelopedthismethodo1 “flow pluswarpagesimulationoutputsofMoldflow ogyfurtherandappliedittosinglegatelocationop PlasticsInsight(MPI)softwareTheobjectiveufnc timizationofanL shapeexample (Courbebaisse， tionisminimizedtoachieveminimum defomr ationin 2005)Itiseasytouseandnottimeconsumingwhile gatelocation optimizationSimulatedannealingal itonly servestheturning ofsimple flatpartswich gorithm isemployedto search ofrtheoptimalgate uniofrii1thickness locationAnexampleisgiventoillustratetheeffec PandelidisandZour1990)presentedtheopti tivityoftheproposedoptimizationprocedure mizationofgatelocationbyindirectqualitymeasures relevanttowarpageandmaterialdegradationwhich isrepresentedasweightedsum ofatemperautredif- QUALITYMEASURES：FEATUREWARPGE ferentialterman overpack temr anda frictional overheatingtemr W aprageisinfluencedbytheabove Definitionoffeaturewarpage factorsbuttherelationshipbetweenthem isnotclear Toapplyoptimizationtheorytothegatedesign， Thereofre，theoptimizationeffectisrestrictedbythe qualitymeasuresofthepartmustbespecifiedinthe detemr inationoftheweightingfactors firstinstanceThetemr “quality”maybereferredto LeeandKim rl996b)developedanautomated manyproductproperties，suchasmechanical，themr al， selectionmethodofgatelocation，inwhichasetof electrical，optical，ergonomicalorgeometricalprop initialgatelocationswereproposedbyadesignerand ertiesTherearetwotypesofpartqualitymeasures： thentheoptimalgatewaslocatedbytheaacentnode directandindirectA modelthatpredictstheproper evaluationmethodTheconclusiontoagreatextent ties rfom numericalsimulation resultswould be dependsmuch on thehumna designers inutition characterizedasadirectqualitymeasureIncontrast becausethefirststepofthemethodisbasedonthe anindirectmeasureofpartqualityiscorrelatedwith designerspropositionSotheresultistoalargeex targetquality，butitcannotprovideadirectestimate tentlimitedtothedesignersexperience Ofchatquality Lam andJin(200l、developedagatelocation Forwarpage，theindirectqualiyt measuresin optimizationmethodbasedontheminimizationofthe relatedworksareoneofperfomr ancesofiniection StandardDeviationofFlowPathLength(SDL)and moldingflowingbehaviororweightedsum ofthose StandardDeviationofFillingTime(SDT)during Theperofmr ancesarepresentedasfillingtimedif- themolding fillingprocessSubsequently，Shen et ferentialalongdifferentflow pathstemperaturedif- 口，004a；2004b)optimizedthegatelocationdesign ferential，overpack percentage，andsoonItisob byminimizingtheweightedsum offillingpressure， viousthatwaprageisinfluencedby theseperforii1 fillingtimedifferencebetweendifferentflow paths， ances，buttherelationshipbewt eenwaprageandthese temperautre difference，and overpack percentage perfomr ancesisnotclearand the detemr inationof Zhaieta1(2005b)investigatedoptimalgatelocation theseweightingfactorsisratherdifficultTherefore with evaluation criteriaofiniection pressure atthe theoptimization with theaboveObiectiveufnction 维普资讯 LfefalJZhejiangUnivSciA 20078(：1077-1083 1079 probablywillnotminimizepartwarpageevenwith where ，yyaretheconstiutentfeautrewarpagesinthe perfectoptimizationtechniqueSometimes，improper X YdirectionandLx，Laretheprojectedlengthsof weightingfactorswillresultinabsolutelywrongre thefeautresurfaceon Ycomponent sults Somestatisticalquantitiescalculated from the Evaluationoffeaturewarpage nodaldisplacementswere characterized as direct Atierthe detemr ination oftargetfeautrecom qualitymeasurestoachieveminimum deformationin binedwith correspondingreferenceplaneandpro relatedoptimizationstudiesThestatisticalquantities iection direction，thevalueofL canbecalculated areusually amaximum nodaldisplacement，anav immediately from the part with the calculating erageoftop10percentilenodaldisplacements，andan methodofanalyticgeometry(Fig2)Lisaconstant overallaveragenodaldisplacement(LeeandKim， foranypartonthespecifiedfeautresurfaceandpro l995；I996b)ThesenodaIdisplacementsareeasyto iecteddirectionButtheevaluationofhismorecom obtainfrom thesimulationresults，thestatisticalval plicatedthanthatofL ues，to someextents，representing thedefomr ation Butthe statisticaldisplacementcannoteffectively y describe the defomr ation of the in _ iectionmolded Feature parts Inindustrydesignersandmanufacutrersusually paymoreattentiontothedegreeofpartwarpageon somespecificfeaturesthanthewholedefomr ationof the in ， _ iectionmoldedparts Inthisstudy feature warpageisdefinedtodescribethedefomr ationofthe Fig2Theprojectedlengthevaluation injectionpartsThefeautrewarpageistheratioofthe maximum displacementofthefeautresurfacetothe Simulationofiniectionmoldingprocessisa projectedlengthofthefeaturesurface(Fig1)： commontechniquetoforecastthequalityofpart de sign，molddesignandprocesssettingsTheresultsof h waprage simulation areexpressedasthenodalde 100， ：_ flectionson Zcomponent( ， ， )，andthe L nodaldisplacement W isthevectorlengthofvector sum of ， ，andWzk，where ，karetheunit whereyisthefeautrewarpage，h isthemaximum displacementon thefeautre surface deviating from vectorson ZcomponentThehisthemaximum displacementofthenodeson the feautre surface， thereferenceplatfomr，andListheprojectedlengthof thefeaturesurfaceonareferencedirectionparalleling whichiscorrelatedwiththenomr alorientationofthe thereferenceplatfomr referenceplane，andcanbederivedfrom theresultsof warpagesimulation Surface To calculatehthe deflection of fth node is evaluatedfirstlyasfollows： ceplane = COSO+ cosfl+W,zcosy一( + )， Fig1 Thedefinitionoffeaturewarpage (3) Forcomplicatedfeautres (onlyplanefeautre where isthedeflectioninthenomr aldirectionof discussedhere)，thefeaturewarpageisusuallysepa thereferenceplaneofithnode；W ，WI1、，Wizarethe rated into two constiutentson thereferenceplane， deflectionsonX，Y，Zcomponentofithnode；伐，B， whicharerepresentedona2D coordinatesystem： aretheanglesofnormalvectorofthereference；A and Barethetemrinalnodesofthefeaturetoproiecting h l0 direction (Fig2)； and arethedeflectionsof = = l00oo， nodesAand ： 维普资讯 1080 LfefaldZhefiangUnivSciA 20078(1077-1083 f = COSa+ cosfl+ ：cosy， conditionsarekeptconstantinthisinvestigationThe injectionmoldedpartwarpageisquantifiedbythe l COSa+ cosfl+Gzcosy， featurewarpagewhichwasdiscussedintheprevious section whereWA，WAy， ：arethedeflectionson Z Thesinglegatelocationoptimizationcanthusbe componentofnode ；Wsx， andWezarethede ofmr ulatedasof11ows： flectionsonX，Y，Zcomponentofnode COiAandlB aretheweightingfactorsoftheterminalnodedeflec Minimize：mini(X)=)，； tionscalculatedasfollows： Subjectto：g(X)：PPo一10， X X ，i=1，2，N ， 2()iA=1一LiAL，co =1一J()iA， where)，ishtefeaturewarpage；P ishteinjection whereLiAistheprojectordistancebetweenfthnode pressureatthegateposition；P0istheallowablein andnodeAUltimately，h isthemaximum ofthe iectionpressureofinjectionmoldingmachineorthe absolutevalueof ： allowableinjectionpressurespecifiedbythedesigner ormanufacutrer；X isthe coordinatevectorofhte h=maxl I,I ，1 candidategatelocations； isthenodeonthefinite elementmeshmodelofthepartofrinjectionmolding In industry，the inspection ofthewarpage is processsimulation；N isthetotalnumberofnodes carriedoutwiththehelpofafeelergauge，whilehte In htefiniteelementmeshmodelofthepart， measuredpartshouldbeplacedonareferenceplat- everynodeisapossiblecandidateofragateThere fomr Thevalueofh isthemaximum numerical fore，thetotalnumberofthepossiblegatelocationn readingofthespacebetween themeasuredpartsur isafunctionofthetotalnumberofnodesN andthe faceandhtereferenceplatform totalnumberofgatelocationstobeoptimized ： N(N一1)(一 一1) GATE L0CAT10N 0PTIM IZAT10N PR0BLEM ! FORM ATION In this sutdy，only the singlegate location Thequality temr “warpage”meansthepemr a problem isinvestigated nentdefomr ationofthepart，whichisnotcausedby anappliedloadItiscausedbydifferentialshrinkage throughoutthepart，duetotheimbalanceofpolymer SIMULATED ANNEALING ALG0RITHM flow，packing，cooling，andcry stallization Theplacementofagateinaniniectionmoldis Thesimulatedannealingalgorithm isoneofthe oneofthemostimportantvariablesofthetotalmold mostpowerfulandpopularmeta-heuristicsto solve designThequalityofthemoldedpart isgreatlyaf- optimization problemsbecauseoftheprovision of fectedbythegatelocationbecauseitinfluencesthe good globalsolutionstorealworldproblemsThe mannerthattheplastic f1owsinto themold cavity algorithm isbasedupon thatofMetropoliseta1 Therefore，differentgate locationsintroduce inho (1953)，whichwasoriginallyproposedasameansto mogeneity in orientation， density，pressure，and findan equilibrium configurationofacollectionof temperature distribution， accordingly introducing atomsata given temperatureTheconnection be differentvalueanddistributionofwarpageTherefore tweenthisalgorithm andmathematicalminimization gatelocationisavaluabledesignvariabletominimize wasfirstnoted by Pincusr1970)butitwas theinjectionmoldedpartwarpageBecausethecor Kirkpatricketa1(1983)whoproposedthatitformed relationbewt eengatelocationandwapragedistribu thebasisofanoptimization techniqueofrcombina tionistoalargeextentindependentofthemeltand tional(andother)problems mold temperature itisassumed thatthemolding Toapplythesimulatedannealingmethodtoop 维普资讯 LfefaldZhejiangUnivSciA 20078(：1077-1083 1081 timizationproblemstheobiectivefunctionfisused f4)Thisprocessisrepeatedforalargeenough asan energY functionEInsteadoffindingalow numberofiterations enerate)ofr Thesequenceof energYconfiguration，theproblem becomestoseekan trialgatelocationsgeneratedinthiswayiskn ownas approximateglobaloptimalsolutionTheconfigura M arkovchain tionsOfmevaluesofdesignvariablesaresubstituted (5)A new Markov chain isthen generated f-ortheenergy cOnfiguratiOnsOfthebody and the (startingrfom thelastacceptedgatelocationinthe controlparameterfortheprocessissubstitutedofr previousMarkovchain)ofrareduced“temperature temperatureA random numbergeneratorisusedasa Tk+l=CTkand the sameprocesscontinuesofrde wayofgeneratingnewvaluesofrthedesignvariables creasingvaluesof “temperautreuntilthealgorithm Itisobviousthatthisalgorithm iusttakesthemini stops mizationproblemsintoaccountHencewhileper formingamaximizationproblem theobjectivefunc Systeminitialization【 tionismultipliedbyfl1toobtainacapableofrm f =O，朋 =O， Themajoradvantageofsimulatedannealing J generateX，ld J algorithm overomermethodsistheabilitytoavoid being trappedatlocalminimaThisalgorithm em 1 t=0 1 ploysa random search，which notonly accepts 一 i一 changesthatdecreaseoNectivefunction butalso ， acceptssomechangesthatincreaseitTheIatterare GenerateXnew acceptedwithaprobabilityP f一 involved In thecaseofgatelocation optimization，the implementationofthisalgorithm isillustratedinFig3， andthisalgorithm jsdetailedasofllows： (1)SAalgorithm startsfromaninitialgateloca tionXoldwithanassignedvalue ofthe “tempera Fig3 Theflow chartofthesimulatedannealingalgorithm ture”parameterT fthe “temperature”counterk is initiallysettozero)Propercontrolparameter (0c 1)inannealingprocessandMarkovchain enerae aregiven APPLICATION AND DISCUSSION (2)SAalgorithmgeneratesanewgatelocation XnewintheneighborhoodofXouandthevalueofthe Theapplicationtoacomplexindustrialpartis objectiveufnctionflX)iscalculated presented in this section to illustratetheproposed (3)Thenewgatelocationwillbeacceptedwith qualitymeasureandoptimizationmethodologyThe probabilitydetemr inedbvmeacceptancefunction partisprovidedbyamanufacturer，asshowninFig4 Inmisparttheflatnessofbasalsurfaceisthemost 唧：min1，exp一k(f(Xn。)一f(Xo) importantprofileprecision requirementThereofre， thefeaturewarpageisdiscussedonbasalsurface，in A uniform random variablePunifisgeneratedin whichreferenceplatfomr isspecifiedasahorizontal 0，1IfPunif尸accept，Xnewisaccepted；otherwiseitis planeattachedtothebasalsurfaceandthelongiut rejected dinaldirection is specified aspr0iected reference 维普资讯 1082 LjefaldZhejiangUnivSciA 20078(：10771083 directionTheparameterh isthemaximum basal theunevenorientationdistributionoftheglassfiber ， surfacedeflectiononthenormaldirection，namelythe asshowninFig6aFig6ashowsthattheglassfiber verticaldirection，andtheparameterListheprojected orientation changesfrom negativedirectiontoposi lengthofthebasalsurfacetothelongitudinaldirec tivedirectionbecauseofthelocationofthegatepar tion ticularlythegreatestchangeofthefiberorientation appearsnearthegateThegreatdiversification of fiberorientation causedbygatelocation introduces seriousdifferentialshrinkageAccordingly,thefea turewaprageisnotableandthegatelocationmustbe optimizedtoreducepartwarpage Fig4 Industrialpartprovidedbythemanufacturer Thematerialofthepart isNylon Zytel101L (30 EGF，DuPontEngineering Polymer)The molding conditionsin the simulation are listed in Fig5 Finiteelementmeshmodelofthepart Table1Fig5showsthefiniteelementmeshmodelof thepartemployedinthenumericalsimulationIthas Averagefiberorientation 。 1469nodesand2492elementsTheobjectivefunc 08613 tion，namelyfeaturewarpage，isevaluatedbyEqs(1)， (3)(6)Thehisevaluatedfrom theresultsof“Flow 07387 +Warp”AnalysisSequenceinMPIbyEq(1)，andthe 06161 ismeasured on the industrialpartimmediately, 。 J L=2050mm 7 。 矗1 Table1Themoldingconditionsinthesimulation y 14 (a) Conditions Values Averagefiberorientation 0 。 Filltime(s) 25 Melttemperature(。C) 295 8574 Moldtemperature(。C) 70 7344 Packingtime(s) 10 6l13 Packingpressure(offillingpressure)() 80 J MPIisthemostextensivesoftwareforthein iectionmoldingsimulation，whichcanrecommend y 14 thebestgatelocationbasedonbalancedflowGateloca tionanalysisisaneffectivetoolofrgatelocationde Fig6 Theorientationdistributionoftheglassfiberwith signbesidesempiricalmethodForthispart，thegate variedgatelocation locationanalysisofMPIrecommendsthatthebest (a)GatesetonN7459；(b)TheoptimalgatelocationN7379 gatelocationisnearnodeN7459asshowninFig5 Thepartwaprageissimulatedbasedonthisrecom Tooptimizethegatelocationthesimulatedan mendedgateandthusthefeautrewarpageisevaluated： nealingsearchingdiscussedinthesection “Simulated 7=515whichisagreatvalueIntrialmanufacutr annealing algorithm” isapplied to thispartThe ing，partwarpageisvisibleonthesampleworkpiece maximum numberofiterationsischosen as30 tO Thisisunacceptableofrthemanufacutrer ensure theprecision oftheoptimization，and the Thegreatwaprageonbasalsurfaceiscausedby maximum numberofrandom trialsallowedofreach 维普资讯 LfefalJZhejiangUnivSciA 20078(7)：1077-1083 1083 iterationischosenas10todecreasetheprobabilityof tionbysimulatedannealingScience，220(4598)：671680 nulliteration withoutan iterative solutionNode doi：lO1126science2204598671 LamYCSeow,LW 2000Cavity balanceforplasticin N7379(Fig5、isfoundtobetheoptimumgateloca jectionmoldingPolymerEngineering andScience， tionThefeaturewarpageisevaluatedfrom thewar 40(6)：12731280doi：1O1O02pen11255 pagesimulationresults )，097，which isless Lam，YC，Jin，S，2001Optimization ofgatelocation for thanthatoftherecommendedgatebyMPIAndthe plasticinjectionmoldingJournalofInjectionMolding partwarpagemeetsthemanufacturersrequirements Technology，5f3)：180一l92 Lee BH，Kim ，BH，1995Optimizationofpartwal1thick intrialmanufacturingFig6bshowsthefiberorien nessestoreducewarpageofinjectionmoldedpartsbased tationinthesimulationItisseenthattheoptimalgate on the modified complex methodPolymer-Plastics locationresultsintheevenglassfiberorientationand Technoloyg andEngineering34(5)：7938l1 thusintroducesgreatreduction ofshrinkagediffer Lee，BHKimBH 1996aAutomateddesignofrtherunner enceontheverticaldirectionalongthelongitudinal system ofinjectionmoldsbasedonpackingsimulation directionAccordingly,the featurewarpage isre Polymer-PlasticsTechnoloyg andEngineering，35(1)： 147168 duced Lee，BH，Kim ，BH，1996bAutomated selectionofgate 1ocationbasedondesiredqualityofinjectionmoldedpart Polymer-PlasticsTechnoloyg andEngineering，35(2)： CONCLUSION 253269 M etropolis，N，Rosenbluth，AW ，Rosenbluth，M N，Teller, AH TellerE，1953Equationsofstatecalculationsby Featurewaprageisdefinedtodescribethewar fastcomp