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IntJAdvManufTechnol200117297–3042001SpringerVerlagLondonLimitedOptimumGateDesignofFreeFormInjectionMouldusingtheAbductiveNetworkJ.C.LinDepartmentofMechanicalDesignEngineering,NationalHuWeiInstituteofTechnology,Yunlin,TaiwanThisstudyusestheinjectionpositionandsizeofthegateasthemajorcontrolparametersforasimulatedinjectionmould.Oncetheinjectionparametersgatesizeandgatepositionaregiven,theproductperformancedeformationcanbeaccuratelypredictedbytheabductivenetworkdeveloped.Toavoidthenumerousinfluencingfactors,firstthepartlineoftheparameterequationiscreatedbyanabductivenetworktolimittherangeofthegate.TheoptimalinjectionparameterscanbesearchedforbyasimulationannealingSAoptimisationalgorithm,withaperformanceindex,toobtainaperfectpart.Themajorpurposeissearchingfortheoptimalgatelocationonthepartsurface,andminimisingtheairtrapanddeformationafterpartformation.Thisstudyalsousesapracticalexamplewhichhasbeenandprovedbyexperimenttoachieveasatisfactoryresult.KeywordsAbductivenetworkInjectionmouldSimulationannealingSA1.IntroductionOwingtotherapiddevelopmentofindustryandcommerceinrecentyears,thereisaneedforrapidandhighvolumeproductionofgoods.Theproductsaremanufacturedusingmouldsinordertosavethetimeandcost.Plasticproductsarethemajority.Owingtotheseproductsnotrequiringcomplicatedprocessesitispossibletocopewithmarketdemandspeedilyandconveniently.Intraditionalplasticproduction,thedesignsoftheportionsofthemouldaredeterminedbyhumans.However,becauseoftheincreasedperformancerequirements,thecomplexityofplasticproductshasincreased.First,thegeometricshapesoftheplasticproductsaredifficulttodraw,andtheinternalshapeisoftencomplexwhichalsoaffectstheproductionoftheproduct.InjectionprocessingcanbedividedintothreestagesCorrespondenceandoffprintrequeststoDr.J.C.Lin,DepartmentofMechanicalDesignEngineering,NationalHuWeiInstituteofTechnology,Yunlin632,Taiwan.EmaillinrcKsunws.nhit.edu.tw1.Heattheplasticmaterialtoamoltenstate.Then,byhighpressure,forcethematerialtotherunner,andthenintothemouldcavity.2.Whenthefillingofthemouldcavityiscompleted,moremoltenplasticshouldbedeliveredintothecavityathighpressuretocompensatefortheshrinkageoftheplastic.Thisensurescompletefillingofthemouldcavity.3.Takeouttheproductaftercooling.Thoughthefillingprocessisonlyasmallproportionofthecompleteformationcycle,itisveryimportant.Iffillinginincomplete,thereisnopressureholdingandcoolingisrequired.Thus,theflowoftheplasticfluidshouldbecontrolledthoroughlytoensurethequalityoftheproduct.TheisothermalfillingmodelofaNewtonianfluidisthesimplestinjectionmouldflowfillingmodel.Richardson1producedacompleteanddetailedconcept.Themajorconceptisbasedontheapplicationoflubricationtheory,andhesimplifiedthecomplex3Dflowtheoryto2DHeleShawflow.TheHeleShawflowwasusedtosimulatethepotentialflowandwasfurthermoreusedintheplasticityflowoftheplastic.Heassumedtheplasticityflowonanextremelythinplateandfullydevelopedtheflowbyignoringthespeedchangethroughthethickness.Kamaletal.usedsimilarmethodstoobtainthefillingconditionforarectangularmouldcavity,andtheanalyticalresultobtainedwasalmostidenticaltotheexperimentalresult.PlasticmaterialfollowstheNewtonianfluidmodelforflowinamouldcavity,andBirdetal.2–4derivedmouldflowtheorybasedonthis.Whentheshapeofamouldiscomplicatedandthereisvariationinthickness,thentheequilibriumequationschangestononlinearandhasnoanalyticalsolution.Thus,itcanbesolvedonlybyfinitedifferenceornumericalsolutions2,5Ofcourse,asthepolymerisaviscoelasticfluid,itisbesttosolvetheflowproblembyusingviscoelasticityequations.In1998,Goyaletal.usedtheWhiteMetznerviscoelasticitymodeltosimulatethediskmouldflowmodelforcentralpouring.Metzner,usingafinitedifferencemethodtosolvethegoverningequation,fouldtheviscoelasticityeffectwouldnotchangethedistributionofspeedandtemperature.However,itaffectsthestressfieldverymuch.Ifitisapureviscoelastic298J.C.Linflowmodel,thepopularGNFmodelisgenerallyusedtoperformnumericalsimulation.Currently,finiteelementmethodsaremostlyusedforthesolutionofmouldflowproblems.Othermethodsarepureviscoelasticmodels,suchasCFOLWandMOLDFLOWsoftware.Weusedthismethodaswell.SomesoftwareemploystheviscoelasticWhite–Metznermodel,butitislimitedto2Dmouldflowanalysis.SimplemouldflowanalysisislimitedbyCPUtime.Forthecomplicatedmouldshapes,Papthanasionetal.usedUCMfluidforfillinganalysis,usingafinitedifferencemethodandBFCCcoordinationsystemapplicationforthesolutionofthemorecomplicatedmouldshapeandfillingproblem,butitwasnotcommercialised6.Manyfactorsaffectplasticmaterialinjection.Thefillingspeed,injectionpressureandmoltentemperature,holdingpressure7–12,coolingtube13,14andinjectiongateaffecttheaccuracyoftheplasticproduct,because,whentheinjectionprocessingiscompleted,theflowofmaterialinthemouldcavityresultsinuneventemperatureandpressure,andinducesresidualstressanddeformationoftheworkpieceaftercooling.Itisdifficulttodecideonthemouldpartsurfaceandgatepositions.Generally,themouldpartsurfaceislocatedatthewidestplaneofthemould.Searchingfortheoptimalgatepositiondependsonexperience.Minimalmodificationtothemouldisrequiredifyouarelucky.However,thetimeandcostrequiredforthemodificationofmostinjectionmouldsexceedstheoriginalcost,ifthechoiceofthepartlineispoor.Forthemouldpartsurface,manyworkersusedvariousmethodstosearchfortheoptimalmouldpartline,suchasgeometricshapeandfeaturebaseddesign15–17.Someworkersusedfiniteelementmethodsandabductivenetworkstolookfortheoptimalgatedesignforadiecastingmould18.Thisstudyusedanabductivenetworktoestablishtheparameterrelationshipofthemouldpartline,andusedthisformulaforsearchingfor22pointsontheinjectionmouldpartlinetoserveasthelocationforaninjectiongate.Abductivenetworksareusedtomatchinjectionpressureandpressureholdingtimetoperforminjectionformationanalysis,andtoestablisharelationshipbetweentheseparameters,andtheoutputresultoftheinjectionprocess.Ithasbeenshownthatpredictionaccuracyinabductivenetworksismuchhigherthanthatinothernetworks19.Abductivenetworksbasedontheabductivemodellingtechniqueareabletorepresentcomplexanduncertainrelationshipsbetweenmouldflowanalysisresultsandinjectionparameters.Ithasbeeenshownthattheinjectionstrainandinjectionstressinaproductcanbepredicted,withreasonableaccuracy,basedonthedevelopednetworks.Theabductivenetworkhasbeenconstructedoncetherelationshipsofgatelocationthatareinputandsimulatedhavebeendeterminedanappropriateoptimisationalgorithmwithaperformanceindexisthenusedtosearchfortheoptimallocationparameters.Inthispaper,anoptimisationmethodforsimulatedannealing20ispresented.Thesimulatedannealingalgorithmisasimulationoftheannealingprocessforminimisingtheperformanceindex.Ithasbeensuccessfullyappliedtofilteringinimageprocessing21,VLSIlayoutgeneration22,discretetolerancedesign23,wireelectricaldischargemachining24,deepdrawclearance25,andcastingdierunnerdesign26,etc.Itprovidesanexperimentalfoundationbasedontheoryforthedevelopmentandapplicationofthetechnologies.2.MouldFlowTheoryThemouldflowanalysisincludefourmajorparts1.Fillingstage.2.Pressureholdingstage.3.Coolingandsolidificationstage.4.Shrinkageandwarp,i.e.stressresiduestage.Thus,themajormouldflowequationsaredividedintofourgroups.Inthefillingstage,themouldcavityisfilledwithmoltenplasticfluidathighpresssure.Thus,thegoverningequationsinclude1.Continuityequation.Theplasticdeformationorshapechangeaccompanytheflowduringthefillingprocessmassconservation›r›tV01rplasticdensityVvectorvelocity2.Momentumequation.NewtonssecondlawisusedtoderivethemomentumaccelerationconditionorforcebalancegeneratedbyplasticflowrF›V›tVVGVPtrf2Pflowpressurefbodyforcetstresstensor.3.Energyequation.Theenergyconservationofsystemandlawsofconservationofflowmaterial,ifthefluidisincompressiblerCPF›T›tVTGqtV3TtemperatureCPspecificheatofconstantpressureqheatflux4.Rheologyequationtfng,T,P,4gVVT5VdeformtensorVTtransportvector.Holdingpressureanalysis.Theholdingpressureprocessistomaintainthepressureafterthemouldcavityisfilledinordertoinjectmoreplastic,tocompensatefortheshrinkageincooling.r›V1›t›P›x1F›t11›x1›t21›x2›t31›x3G6r›V2›t›P›x2F›t12›x1›t22›x2›t32›x3G7r›V3›t›P›x1F›t13›x1›t23›x2›t33›x3G8OptimumGateDesignofFreeFormInjectionMould299Coolinganalysis.Theanalysisofthecoolingprocessconsiderstherelationshipoftheplasticflowdistributionandheattransmission.Thehomogenousmouldtemperatureandthesequenceoffillingwillbeaffectedbytheshrinkageoftheproductformed.Ifthetemperatureisdistributednonuniformly,ittendstoproducewarp.Thisismainlyduetoheattransferandcrystallisationheatoftheplastic.rCP›T›tkF›2T›x21›2T›x22›2T›x33GrCPrDH9rcrystallisationrateDHcrystallisationheat3.AbductiveNetworkSynthesisandEvaluationMiller22observedthathumanbehaviourlimitstheamountofinformationconsideredatatime.Theinputdataaresummarisedandthenthesummarisedinformationispassedtoahigherreasoninglevel.Inanabductivenetwork,acomplexsystemcanbedecomposedintosmaller,simplersubsystemsgroupedintoseverallayersusingpolynomialfunctionnodes.Thesenodesevaluatethelimitednumberofinputsbyapolynomialfunctionandgenerateanoutputtoserveasaninputtosubsequentnodesofthenextlayer.Thesepolynomialfunctionalnodesarespecifiedasfollows1.NormaliserAnormalisertransformstheoriginalinputvariablesintoarelativelycommonregion.a1q0q1x110Wherea1isthenormalisedinput,q0,q1arethecoefficientsofthenormaliser,andx1istheoriginalinput.2.WhitenodeAwhitenodeconsistsoflinearweightedsumsofalltheoutputsofthepreviouslayer.b1r0r1y1r2y2r3y3rnyn11Wherey1,y2,y3,ynaretheinputofthepreviouslayer,b1istheoutputofthenode,andther0,r1,r2,r3,,rnarethecoefficientsofthetriplenode.3.Single,double,andtriplenodesThesenamesarebasedonthenumberofinputvariables.Thealgebraicformofeachofthesenodesisshowninthefollowingsinglec1s0s1z1s2z21s3z3112doubled1t0t1n1t2n21t3n31t4n2t5n22t6n32t7n1n213triplee1u0u1o1u2o21u3o31u4o2u5o22u6o32u7o3u8o23u9o33u10o1o2u11o2o3u12o1o3u13o1o2o314wherez1,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.ThesenodesarethirddegreepolynomialEq.anddoublesandtripleshavecrossterms,allowinginteractionamongthenodeinputvariables.4.UnitiserOntheotherhand,aunitiserconvertstheoutputtoarealoutput.f1v0v1i115Wherei1istheoutputofthenetwork,f1istherealoutput,andv0andv1arethecoefficientsoftheunitiser.4.PartSurfaceModelThisstudyusesanactualindustrialproductasasample,Fig.1.Themouldpartsurfaceislocatedatthemaximumprojectionarea.AsshowninFig.1,thebottomisthewidestplaneandischosenasthemouldpartsurface.However,mostimportantisthesearchingofgatepositiononthepartsurface.Thisstudyestablishestheparameterequationbyusinganabductiveneuronnetwork,inordertoestablishthesimulatedannealingmethodSAtofindtheoptimalgatepathposition.TheparameterequationofapartsurfaceisexpressedbyFYX.First,useaCMMsystemtomeasuretheXYZcoordinatevaluesinthisstudyz0of22pointsonthemouldpartlineonthemouldpartsurfaceasillustratedinTable1,andthegatepositioniscompletelyonthecurveinthisspace.Priortodevelopingaspacecurvemodel,adatabasehastobetrained,andagoodrelationshipmsutexistbetweenthecontrolpointandabductivenetworksystem.AcorrectandFig.1.Injectionmouldproduct.300J.C.LinTable1.X,Ycoordinate.SetnumberXcoordinateYcoordinate10.024.621.634.3333.283.545.292.0457.310.5669.340.9711.332.35812.983.94913.855.571014.127.341113.699.671212.9611.91310.0021.03149.3323.16158.6425.28167.9827.39177.8728.31187.8029.29197.8330.34207.6031.30217.0732.15226.1132.49precisecurveEq.ishelpfulforfindingtheoptimalgatelocation.Tobuildacompleteabductivenetwork,thefirstrequirementistotrainthedatabase.Theinformationgivenbytheinputandoutputparametersmustbesufficient.ApredictedsquareerrorPSEcriterionisthenusedtodetermineautomaticallyanoptimalstructure23.ThePSEcriterionisusedtoselecttheleastcomplexbutstillaccuratenetwork.ThePSEiscomposedoftwotermsPSEFSEKP16WhereFSEistheaveragesquareerrorofthenetworkforfittingthetrainingdataandKPisthecomplexpenaltyofthenetwork,shownbythefollowingequationKPCPM2s2pKN17WhereCPMisthecomplexpenaltymultiplier,KPisacoefficientofthenetworkNisthenumberoftrainingdatatobeusedands2pisapriorestimateofthemodelerrorvariance.Basedonthedevelopmentofthedatabaseandthepredictionoftheaccuracyofthepartsurface,athreelayerabductivenetwork,whichcompriseddesignfactorsinputvariousYcoordinateandoutputfactorsXcoordinateissynthesisedautomatically.Itiscapableofpredictingaccuratelythespacecurveatanypointundervariouscontrolparameters.AllpolynomialequationsusedinthisnetworkarelistedinAppendixA.PSE5.8103.Table2comparestheerrorpredictedbytheabductivemodelandCMMmeasurementdata.Themeasurementdaaisexcludedfromthe22setsofCMMmeasurementdataforestablishingthemodel.Thissetofdataisusedtotesttheappropriatenessofthemodelestablishedabove.WecanseefromTable2thattheerrorisapproximately2.13,whichshowsthatthemodelissuitableforthisspacecurve.Table2.CMMScoordinateandneuralnetworkpredictcompareditisnotincludedinanyoriginal22setsdatabase.ItemsCMMSneuralnetworkErrorvaluescoordinatepredictCMMSpredict/coordinateCMMSCoordinate11.25,16.011.01,16.02.135.CreatetheInjectionMouldModelSimilarly,therelationshipisestablishedbetweeninputparametersgatelocationandgatesizeandtheoutputparameterdeformationduringtheinjectionprocess.Tobuildacompleteabductivenetwork,thefirstrequirementistotrainthedatabase.Theinformationgivenbytheinputandtheoutputdatamustbesufficient.Thus,thetrainingfactorgatelocationforabductivenetworktrainingshouldbesatisfactoryformakingdefectfreeproducts.Figure2showsthesimulationofFEMmouldflow.Table3showsthepositionofthegateandthemaximumdeformationoftheproductobtainedfrommouldflowanalysis.Basedonthedevelopmentoftheinjectionmouldmodel,threelayerabductivenetworks,whicharecomprisedofinjectionmouldconditionsandtheinjectionresultsdeformation,aresynthesisedautomatically.Theyarecapableofpredictingaccuratelytheproductstraintheresultofinjectionmouldedproductundervariouscontrolparameters.AllpolynomialequationsusedinthisnetworkarelistedinAppendixBPSE2.3105.Table4comparestheerrorpredictedbytheabductivemodelandthesimulationcase.Thesimulationcaseisexcludedfromthe22setsofsimulationcasesforestablishingthemodel.Thissetofdataisusedtotesttheappropriatenessofthemodelestablishedabove.WecanseefromTable4thattheerrorisFig.2.ThedeformationofFEMmouldflow.
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