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IntJAdvManufTechnol(2001)17:2973042001Springer-VerlagLondonLimitedOptimumGateDesignofFreeFormInjectionMouldusingtheAbductiveNetworkJ.C.LinDepartmentofMechanicalDesignEngineering,NationalHu-WeiInstituteofTechnology,Yunlin,TaiwanThisstudyusestheinjectionpositionandsizeofthegateasthemajorcontrolparametersforasimulatedinjection-mould.Oncetheinjectionparameters(gatesizeandgateposition)aregiven,theproductperformance(deformation)canbeaccuratelypredictedbytheabductivenetworkdeveloped.Toavoidthenumerousinfluencingfactors,firstthepart-lineoftheparameterequationiscreatedbyanabductivenetworktolimittherangeofthegate.Theoptimalinjectionparameterscanbesearchedforbyasimulationannealing(SA)optimisa-tionalgorithm,withaperformanceindex,toobtainaperfectpart.Themajorpurposeissearchingfortheoptimalgatelocationonthepart-surface,andminimisingtheair-trapanddeformationafterpartformation.Thisstudyalsousesaprac-ticalexamplewhichhasbeenandprovedbyexperimenttoachieveasatisfactoryresult.Keywords:Abductivenetwork；Injectionmould；Simulationannealing(SA)1.IntroductionOwingtotherapiddevelopmentofindustryandcommerceinrecentyears,thereisaneedforrapidandhighvolumeproductionofgoods.Theproductsaremanufacturedusingmouldsinordertosavethetimeandcost.Plasticproductsarethemajority.Owingtotheseproductsnotrequiringcomplicatedprocessesitispossibletocopewithmarketdemandspeedilyandconveniently.Intraditionalplasticproduction,thedesignsoftheportionsofthemouldaredeterminedbyhumans.However,becauseoftheincreasedperformancerequirements,thecomplexityofplasticproductshasincreased.First,thegeometricshapesoftheplasticproductsaredifficulttodraw,andtheinternalshapeisoftencomplexwhichalsoaffectstheproductionoftheproduct.Injectionprocessingcanbedividedintothreestages:Correspondenceandoffprintrequeststo:Dr.J.C.Lin,DepartmentofMechanicalDesignEngineering,NationalHu-WeiInstituteofTechnology,Yunlin632,Taiwan.E-mail:linrcK.tw1.Heattheplasticmaterialtoamoltenstate.Then,byhighpressure,forcethematerialtotherunner,andthenintothemouldcavity.2.Whenthefillingofthemouldcavityiscompleted,moremoltenplasticshouldbedeliveredintothecavityathighpressuretocompensatefortheshrinkageoftheplastic.Thisensurescompletefillingofthemouldcavity.3.Takeouttheproductaftercooling.Thoughthefillingprocessisonlyasmallproportionofthecompleteformationcycle,itisveryimportant.Iffillinginincomplete,thereisnopressureholdingandcoolingisrequired.Thus,theflowoftheplasticfluidshouldbecontrolledthoroughlytoensurethequalityoftheproduct.TheisothermalfillingmodelofaNewtonianfluidisthesimplestinjectionmould-flowfillingmodel.Richardson1producedacompleteanddetailedconcept.Themajorconceptisbasedontheapplicationoflubricationtheory,andhesimplifiedthecomplex3Dflowtheoryto2DHele-Shawflow.TheHele-Shawflowwasusedtosimulatethepotentialflowandwasfurthermoreusedintheplasticityflowoftheplastic.Heassumedtheplasticityflowonanextremelythinplateandfullydevelopedtheflowbyignoringthespeedchangethroughthethickness.Kamaletal.usedsimilarmethodstoobtainthefillingconditionforarectangularmouldcavity,andtheanalyti-calresultobtainedwasalmostidenticaltotheexperimentalresult.PlasticmaterialfollowstheNewtonianfluidmodelforflowinamouldcavity,andBirdetal.24derivedmould-flowtheorybasedonthis.Whentheshapeofamouldiscomplicatedandthereisvariationinthickness,thentheequilibriumequa-tionschangestononlinearandhasnoanalyticalsolution.Thus,itcanbesolvedonlybyfinite-differenceornumericalsolutions2,5Ofcourse,asthepolymerisavisco-elasticfluid,itisbesttosolvetheflowproblembyusingvisco-elasticityequations.In1998,Goyaletal.usedtheWhiteMetznervisco-elasticitymodeltosimulatethediskmould-flowmodelforcentralpouring.Metzner,usingafinite-differencemethodtosolvethegoverningequation,fouldthevisco-elasticityeffectwouldnotchangethedistributionofspeedandtemperature.However,itaffectsthestressfieldverymuch.Ifitisapurevisco-elastic298J.C.Linflowmodel,thepopularGNFmodelisgenerallyusedtoperformnumericalsimulation.Currently,finite-elementmethodsaremostlyusedforthesolutionofmould-flowproblems.Othermethodsarepurevisco-elasticmodels,suchasC-FOLWandMOLD-FLOWsoftware.Weusedthismethodaswell.Somesoftwareemploysthevisco-elasticWhiteMetznermodel,butitislimitedto2Dmould-flowanalysis.SimplemouldflowanalysisislimitedbyCPUtime.Forthecomplicatedmouldshapes,Papthanasionetal.usedUCMfluidforfillinganalysis,usingafinite-differencemethodandBFCCcoordinationsystemapplicationforthesolutionofthemorecomplicatedmould-shapeandfillingproblem,butitwasnotcommercialised6.Manyfactorsaffectplasticmaterialinjection.Thefillingspeed,injectionpressureandmoltentemperature,holdingpress-ure712,coolingtube13,14andinjectiongateaffecttheaccuracyoftheplasticproduct,because,whentheinjectionprocessingiscompleted,theflowofmaterialinthemouldcavityresultsinuneventemperatureandpressure,andinducesresidualstressanddeformationoftheworkpieceaftercooling.Itisdifficulttodecideonthemouldpart-surfaceandgatepositions.Generally,themouldpart-surfaceislocatedatthewidestplaneofthemould.Searchingfortheoptimalgatepositiondependsonexperience.Minimalmodificationtothemouldisrequiredifyouarelucky.However,thetimeandcostrequiredforthemodificationofmostinjection-mouldsexceedstheoriginalcost,ifthechoiceofthepart-lineispoor.Forthemouldpartsurface,manyworkersusedvariousmethodstosearchfortheoptimalmouldpart-line,suchasgeometricshapeandfeature-baseddesign1517.Someworkersusedfinite-elementmethodsandabductivenetworkstolookfortheoptimalgatedesignforadie-castingmould18.Thisstudyusedanabductivenetworktoestablishthepara-meterrelationshipofthemouldpart-line,andusedthisformulaforsearchingfor22pointsontheinjectionmouldpart-linetoserveasthelocationforaninjectiongate.Abductivenetworksareusedtomatchinjectionpressureandpressureholdingtimetoperforminjectionformationanalysis,andtoestablisharelationshipbetweentheseparameters,andtheoutputresultoftheinjectionprocess.Ithasbeenshownthatpredictionaccuracyinabductivenetworksismuchhigherthanthatinothernetworks19.Abductivenetworksbasedontheabductivemodellingtech-niqueareabletorepresentcomplexanduncertainrelationshipsbetweenmould-flowanalysisresultsandinjectionparameters.Ithasbeeenshownthattheinjection-strainandinjection-stressinaproductcanbepredicted,withreasonableaccuracy,basedonthedevelopednetworks.Theabductivenetworkhasbeenconstructedoncetherelationshipsofgatelocationthatareinputandsimulatedhavebeendetermined；anappropriateoptimisationalgorithmwithaperformanceindexisthenusedtosearchfortheoptimallocationparameters.Inthispaper,anoptimisationmethodforsimulatedannealing20ispresented.Thesimulatedannealingalgorithmisasimulationoftheannealingprocessforminimisingtheperform-anceindex.Ithasbeensuccessfullyappliedtofilteringinimageprocessing21,VLSIlayoutgeneration22,discretetolerancedesign23,wireelectricaldischargemachining24,deep-drawclearance25,andcasting-dierunnerdesign26,etc.Itprovidesanexperimentalfoundationbasedontheoryforthedevelopmentandapplicationofthetechnologies.2.Mould-FlowTheoryThemouldflowanalysisincludefourmajorparts:1.Fillingstage.2.Pressureholdingstage.3.Coolingandsolidificationstage.4.Shrinkageandwarp,i.e.stressresiduestage.Thus,themajormouldflowequationsaredividedintofourgroups.Inthefillingstage,themouldcavityisfilledwithmoltenplasticfluidathighpresssure.Thus,thegoverningequationsinclude:1.Continuityequation.Theplasticdeformationorshapechangeaccompanytheflowduringthefillingprocess(massconservation):rt+=(=V)=0(1)r=plasticdensity；V=vectorvelocity2.Momentumequation.Newtonssecondlawisusedtoderivethemomentum(accelerationcondition)orforcebalancegeneratedbyplasticflow:rFVt+(V=)VG=-VP+=t+rf(2)P=flowpressure；f=bodyforce；t=stresstensor.3.Energyequation.Theenergyconservationofsystemandlawsofconservationofflowmaterial,ifthefluidisincom-pressible:rCPFTt+(V=)TG=-=q+t:=V(3)T=temperature；CP=specificheatofconstantpressure；q=heatflux4.Rheologyequationt=fn(g,T,P,%)(4)g=V+(=V)T(5)=V=deformtensor；(=V)T=transportvector.Holdingpressureanalysis.Theholdingpressureprocessistomaintainthepressureafterthemouldcavityisfilledinordertoinjectmoreplastic,tocompensatefortheshrinkageincooling.rV1t=-Px1+Ft11x1+t21x2+t31x3G(6)rV2t=-Px2+Ft12x1+t22x2+t32x3G(7)rV3t=-Px1+Ft13x1+t23x2+t33x3G(8)OptimumGateDesignofFreeFormInjectionMould299Coolinganalysis.Theanalysisofthecoolingprocesscon-siderstherelationshipoftheplasticflowdistributionandheattransmission.Thehomogenousmouldtemperatureandthesequenceoffillingwillbeaffectedbytheshrinkageoftheproductformed.Ifthetemperatureisdistributednon-uniformly,ittendstoproducewarp.Thisismainlyduetoheat-transferandcrystallisationheatoftheplastic.rCPTt=kF2Tx21+2Tx22+2Tx33G+rCPrDH(9)r=crystallisationrate；DH=crystallisationheat3.AbductiveNetworkSynthesisandEvaluationMiller22observedthathumanbehaviourlimitstheamountofinformationconsideredatatime.Theinputdataaresummar-isedandthenthesummarisedinformationispassedtoahigherreasoninglevel.Inanabductivenetwork,acomplexsystemcanbedecom-posedintosmaller,simplersubsystemsgroupedintoseverallayersusingpolynomialfunctionnodes.Thesenodesevaluatethelimitednumberofinputsbyapolynomialfunctionandgenerateanoutputtoserveasaninputtosubsequentnodesofthenextlayer.Thesepolynomialfunctionalnodesarespecifiedasfollows:1.NormaliserAnormalisertransformstheoriginalinputvariablesintoarelativelycommonregion.a1=q0+q1x1(10)Wherea1isthenormalisedinput,q0,q1arethecoefficientsofthenormaliser,andx1istheoriginalinput.2.WhitenodeAwhitenodeconsistsoflinearweightedsumsofalltheoutputsofthepreviouslayer.b1=r0+r1y1+r2y2+r3y3+%+rnyn(11)Wherey1,y2,y3,ynaretheinputofthepreviouslayer,b1istheoutputofthenode,andther0,r1,r2,r3,%,rnarethecoefficientsofthetriplenode.3.Single,double,andtriplenodesThesenamesarebasedonthenumberofinputvariables.Thealgebraicformofeachofthesenodesisshowninthefollowing:single:c1=s0+s1z1+s2z21+s3z31(12)double:d1=t0+(t1n1+t2n21+t3n31)+(t4n2+t5n22+t6n32)+(t7n1n2)(13)triple:e1=u0+(u1o1+u2o21+u3o31)+(u4o2+u5o22+u6o32)+(u7o3+u8o23+u9o33)+u10o1o2+u11o2o3+u12o1o3+u13o1o2o3(14)wherez1,z2,z3,%,zn,n1,n2,n3,%,nn,o1,o2,o3,%,onaretheinputofthepreviouslayer,c1,d1,ande1aretheoutputofthenode,andthes0,s1,s2,s3,%,sn,t0,s1,t2,t3,%,tn,u0,u1,u2,u3,%,unarethecoefficientsofthesingle,double,andtriplenodes.Thesenodesarethird-degreepolynomialEq.anddoublesandtripleshavecross-terms,allowinginteractionamongthenodeinputvariables.4.UnitiserOntheotherhand,aunitiserconvertstheoutputtoarealoutput.f1=v0+v1i1(15)Wherei1istheoutputofthenetwork,f1istherealoutput,andv0andv1arethecoefficientsoftheunitiser.4.Part-SurfaceModelThisstudyusesanactualindustrialproductasasample,Fig.1.Themouldpartsurfaceislocatedatthemaximumprojectionarea.AsshowninFig.1,thebottomisthewidestplaneandischosenasthemouldpartsurface.However,mostimportantisthesearchingofgatepositiononthepartsurface.Thisstudyestablishestheparameterequationbyusinganabductiveneuronnetwork,inordertoestablishthesimulatedannealingmethod(SA)tofindtheoptimalgatepathposition.Theparameterequationofapart-surfaceisexpressedbyF(Y)=X.First,useaCMMsystemtomeasuretheXYZcoordinatevalues(inthisstudyz=0)of22pointsonthemouldpart-lineonthemouldpart-surfaceasillustratedinTable1,andthegatepositioniscompletelyonthecurveinthisspace.Priortodevelopingaspace-curvemodel,adatabasehastobetrained,andagoodrelationshipmsutexistbetweenthecontrolpointandabductivenetworksystem.AcorrectandFig.1.Injection-mouldproduct.300J.C.LinTable1.X,Ycoordinate.SetnumberX-coordinateY-coordinate10.02-4.62-1.63-4.333-3.28-3.54-5.29-2.045-7.31-0.566-9.340.97-11.332.358-12.983.949-13.855.5710-14.127.3411-13.699.6712-12.9611.913-10.0021.0314-9.3323.1615-8.6425.2816-7.9827.3917-7.8728.3118-7.8029.2919-7.8330.3420-7.6031.3021-7.0732.1522-6.1132.49precisecurveEq.ishelpfulforfindingtheoptimalgatelocation.Tobuildacompleteabductivenetwork,thefirstrequirementistotrainthedatabase.Theinformationgivenbytheinputandoutputparametersmustbesufficient.Apredictedsquareerror(PSE)criterionisthenusedtodetermineautomaticallyanoptimalstructure23.ThePSEcriterionisusedtoselecttheleastcomplexbutstillaccuratenetwork.ThePSEiscomposedoftwoterms:PSE=FSE+KP(16)WhereFSEistheaveragesquareerrorofthenetworkforfittingthetrainingdataandKPisthecomplexpenaltyofthenetwork,shownbythefollowingequation:KP=CPM2s2pKN(17)WhereCPMisthecomplexpenaltymultiplier,KPisacoef-ficientofthenetwork；Nisthenumberoftrainingdatatobeusedands2pisapriorestimateofthemodelerrorvariance.Basedonthedevelopmentofthedatabaseandthepredictionoftheaccuracyofthepart-surface,athree-layerabductivenetwork,whichcompriseddesignfactors(input:variousYcoordinate)andoutputfactors(Xcoordinate)issynthesisedautomatically.Itiscapableofpredictingaccuratelythespacecurveatanypointundervariouscontrolparameters.Allpoly-nomialequationsusedinthisnetworkarelistedinAppendixA.(PSE=5.810-3).Table2comparestheerrorpredictedbytheabductivemodelandCMMmeasurementdata.Themeasurementdaaisexcludedfromthe22setsofCMMmeasurementdataforestablishingthemodel.Thissetofdataisusedtotesttheappropriatenessofthemodelestablishedabove.WecanseefromTable2thattheerrorisapproximately2.13%,whichshowsthatthemodelissuitableforthisspacecurve.Table2.CMMS-