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ORIGINAL ARTICLEOptimization of injection molding process parametersusing integrated artificial neural network modeland expected improvement function methodHuizhuo Shi&Yuehua Gao&Xicheng WangReceived: 16 October 2008 /Accepted: 24 September 2009 /Published online: 20 November 2009#Springer-Verlag London Limited 2009Abstract In this study, an adaptive optimization methodbased on artificial neural network model is proposed tooptimize the injection molding process. The optimizationprocess aims at minimizing the warpage of the injectionmolding parts in which process parameters are designvariables. Moldflow Plastic Insight software is used toanalyze the warpage of the injection molding parts. Themold temperature, melt temperature, injection time, packingpressure, packing time, and cooling time are regarded asprocess parameters. A combination of artificial neuralnetwork and design of experiment (DOE) method is usedto build an approximate function relationship betweenwarpage and the process parameters, replacing the expensivesimulation analysis in the optimization iterations. Theadaptive process is implemented by expected improvementwhich is an infilling sampling criterion. Although the DOEsize is small, this criterion can balance local and global searchand tend to the global optimal solution. As examples, acellular phone cover and a scanner are investigated. Theresults show that the proposed adaptive optimization methodcan effectively reduce the warpage of the injection moldingparts.Keywords Injectionmolding.Warpage.Optimization.Design ofexperiment.Artificialneuralnetwork.Expectedimprovementfunction1 IntroductionInjection molding is the most widely used process forproducingplastic products.The entireinjectionmoldingcyclecan be divided into three stages: filling, post-filling, and moldopening 1. During production, warpage is one of the mostimportantqualityproblems,especiallyforthethin-shellplasticproducts.Severalresearcheshavebeendevotedtothewarpageoptimization of thin-shell plastic parts 29. Warpage can bereduced by modifying the geometry of parts, or changing thestructure of molds, or adjusting the process parameters.The part design and mold design are usually determined inthe initial stage of product development, which cannot beeasily changed. Therefore, optimizing process parameters isthe most feasible and reasonable method.It is an important issue in plastic injection molding topredict and optimize the warpage before manufacturingtakes place. Many literatures have been devoted to warpageoptimization. Lee and Kim 10 optimized the wallthickness and process conditions using the modifiedcomplex method to reduce warpage and obtained areduction in warpage of over 70%. Sahu et al. 11optimized process conditions to reduce warpage by acombined implementation of the modified complex methodand design of experiments. Their results showed that thesemethods can effectively reduce warpage.Although these methods can reduce warpage effectively,they are costly and time-consuming because they performlots of expensive function evaluations. Compared to thesemethods, the Taguchi method 1214 is easier to performandcananalyze the effective factors,but itcanonlyobtainthebetter combination of process parameters, not the optimalsolution in the design space.The warpage is a nonlinear and implicit function of theprocess parameters, which is typically estimated by theH. Shi:Y. Gao:X. Wang (*)State Key Laboratory of Structural Analysis for IndustrialEquipment, Dalian University of Technology,Dalian,116024 Liaoning, Chinae-mail: Int J Adv Manuf Technol (2010) 48:955962DOI 10.1007/s00170-009-2346-7solution of finite element equations. In general, a complicatedtask often requires huge computational cost. Hence, in orderto reduce the computational cost in warpage optimization,many researchers have introduced some surrogate models,such as Kriging surrogate model, artificial neutral network(ANN), response surface method, and support vector regres-sion. Gao et al. 1517 optimized process conditions toreduce the warpage by combining the kriging surrogatemodel with modified rectangular grid approach or expectedimprovement (EI) function method. Kurtaran et al. combinedthe genetic algorithms with a neural network or responsesurface method to optimize the process parameters forreducing the warpage of plastic parts 18, 19. Zhou et al.20 optimized injection molding process using supportvector regression model and genetic algorithm. Their resultshave shown that the methods based on the surrogate modelcan reduce the high computational cost in the warpageoptimization, and the genetic algorithm can be used toapproach to the global optimal design effectively.In this study, the mold temperature, melt temperature,injection time, packing pressure, packing time, and coolingtime are considered as process parameters. A small-sizedesign of experiment is obtained by Latin hypercube design(LHD), and the warpage values are evaluated by MoldFlowPlastic Insight software. An adaptive optimization based onartificial neural network model is proposed. The adaptiveprocess is performed by an EI function, which canadaptively select the additional sample points to improvethe surrogate model and find the optimum value 17. Thismethod has been viewed as effective global optimization21. The numerical results show that this method canreduce warpage efficiently.2 Artificial neural networkANN is a powerful tool for the simulation and prediction ofnonlinear problems. A neural network comprises manyhighly interconnected processing units called neurons. Eachneuron sums weighted inputs and then applies a linear ornonlinear function to the resulting sum to determine theoutput, and all of them are arranged in layers and combinedthrough excessive connectivity.The typical ANN is a back propagation network (BPN)2226 which has been widely used in many researchfields. A BPN has hierarchical feed-forward networkarchitecture, and the output of each layer is sent directlyto each neuron in the layer above. Although a BPN canhave many layers, all pattern recognition and classificationtasks can be accomplished with a three-layer BPN 27. pac ktct packP Warpage1mel tTmoldT 6 int 6Fig. 1 Configuration of the ANN model Add the modified design as a new sample in set of samplesStartGenerate a set of samples Run Moldflow to generate corresponding warpage values Perform ANN simulation Optimize EI function Is the convergence criterion satisfied?Obtain optimal design End YNFig. 2 Flowchart of combining ANN/EI optimizationFig. 3 Mid-plane model of a cellular phone coverTable 1 Ranges of the process parametersParameterTmold(C)Tmelt(C)tin(s)Ppack(%)tpack(s)tc(s)Lower limit502600.26015Upper limit903000.890515956Int J Adv Manuf Technol (2010) 48:955962A BPN is trained by repeatedly presenting a series ofinput/output pattern sets to the network. The neural networkgradually “learns” the input/output relationship of interestby adjusting the weights between its neurons to minimizethe error between the actual and predicted output patterns ofthe training set. After training, a separate set of data whichis not in the training set is used to monitor the networksperformance. When the mean squared error (MSE) reachesa minimum, network training is considered complete andthe weights are fixed.In this paper, a three-layer ANN model with one hiddenlayer was used. The mold temperature (Tmold), melttemperature (Tmelt), injection time (tin), packing pressure(Ppack), packing time (tpack), and cooling time (tc) areregarded as input variables, and warpage is regarded asoutput variable. So the neuron numbers of the input layerand output layer of ANN are determined. The neuronnumber of the middle layer was determined by trials. Thetransfer function between the input layer and the hiddenlayer is “Logsig,” and the transfer function between thehidden layer and the output layer is “Purelin.” The trainfunction is trainlm, performance function is MSE, learningcycle is 50,000, learning rate is 0.05, and momentum factoris 0.9. The configuration of ANN used in this paper isshown in Fig. 1.3 EI methodANNs can be used as an arbitrary function approximationmechanism which “learns” from observed data. ANN is hereused to build an approximate function relationship betweenthe warpage and the process parameters, replacing theexpensive analysis and reanalysis of simulation programs inthe optimizationprocess.Ingeneral,the approximate functionmay have many extremum points, making the optimizationalgorithms employing such functions converge to a localminimum. EI algorithm is here introduced to close to theglobal optimization solution.EI involves computing the possible improvement at agiven point. It is a heuristic algorithm for a sequential designstrategy for detecting the global minimum of a deterministicfunction 17, 21. It can balance local and global search.Beforesamplingatsomepointx, the value of Y(x) is uncertain.Y(x) at a candidate point x is normally distributed with b yx,and variance 2given using the ANN predictor. If the currentbest function value is Ymin, then an improvement I Ymin? yx by the ANN predictor can be achieved. Thelikelihood of this improvement is given by the normal density:1ffiffiffiffiffi2pps x exp ?Ymin? I ?b y x 22s2x #:1Then, the expected value of the improvement is found byintegrating over this density:EI ZI1I01ffiffiffiffiffi2pps x exp ?Ymin? I ?b y x 22s2x #()dI:2Fig. 5 Warpage of the cover after optimizationFig. 4 Warpage of the cover before optimizationTable 2 Optimization resultsParameterTmold(C)Tmelt(C)tin(s)Ppack(%)tpack(s)tc(s)Warpage(mm)Beforeoptimization75.57 288.31 0.57 63.96 1.22 5.700.1941Afteroptimization73.86 298.99 0.20 60.00 1.00 9.480.0833Fig. 6 Model of a scannerInt J Adv Manuf Technol (2010) 48:955962957Using integration by parts, Eq. 2 can be written as:EI sx u6u fu?3where and f are the normal cumulative distributionfunction and density function, respectively, andu Ymin?b y x s x :4The first term of Eq. 3 is the difference between thecurrent minimum response value Yminand the predictedvalueb yx at x, penalized by the probability of improvement.Hence, the first term is large when b yx is small. The secondterm is a product of predicted error (x) and normal densityfunction f(u). The normal density function value is largewhen the error (x) is large and b yx is close to Ymin. Thus,the expected improvement will tend to be large at a pointwith the predicted value smaller than Yminand/or with muchpredicted uncertainty.This infilling sampling method has some advantages: (1)It can intelligently add sample points to improve the ANN,so it allows “learns” from observed data with a small size;(2) it can avoid searching the areas with relative largefunction values and reduce the computational cost; (3) itcan also avoid adding some points close to each other in thedesign space and keep the stability of ANN prediction.4 Warpage optimization based on improved ANNmethod4.1 Warpage optimization problemA warpage minimum design problem can be described asfollows:Findx1;x2;?;xmmaxmizeE I x1;x2;?;xm?Subjecttox?j? xj? xjj 1;2;?;m5where the process parameters x1;x2;?;xmare the designvariables and x?jand xjare the lower and upper limits of thejth design variable. The objective function E I x1;x2;? ? ?;xm?is given by Eqs. 3 and 4 inwhich Yminand yx are the currentminimum value and the predicted value of warpage, respectively.4.2 Convergence criterionThe convergence criterion is here to satisfy:E Ix?Ymin? $r6where r is a given convergence tolerance and Yminis theminimum function value in samples. The left-hand side is aratio between the maximum expected improvement and theminimum function value. Thus, r can be given withoutconsideration of the magnitudes, and r=0.1%.4.3 Implementation of optimization procedureImplementation of integrated ANN model and EI functionmethod is given in Fig. 2.5 Warpage optimization for a cellular phone coverand a scanner5.1 The optimization problemIn this section, the results of two warpage optimizationexamples are presented. These are intended to show theTable 3 Ranges of the process parametersParameterTmold(C)Tmelt(C)tin(s)Ppack(%)tpack(s)tc(s)Lower limit802600.26015Upper limit1203000.890515Table 4 Optimization resultsParameterTmold(C)Tmelt(C)tin(s)Ppack(%)tpack(s)tc(s)Warpage (mm)Before optimization92.95298.380.2585.492.8310.300.4805After optimization119.32300.000.2090.004.9215.000.2896Fig. 7 Warpage of the scanner before optimization958Int J Adv Manuf Technol (2010) 48:955962efficiency and accuracy of the integrated ANN model andEI function method.The first example is a cellular phone cover. It isdiscretized by 3,780 triangle elements, as shown in Fig. 3.Its length, width, height, and thickness are 130, 55, 11, and1 mm, respectively. The material is polycarbonate (PC)/acrylonitrile-butadiene-styrene.The mold temperature (Tmold), melt temperature (Tmelt),injection time (tin), packing pressure (Ppack), packing time(tpack), and cooling time (tc) are considered as designvariables. The objective function warpage(x) is quantifiedby the out-of-plane displacement, which is the sum of bothmaximum upward and downward deformations withreference to the default plane in Moldflow Plastics Insightsoftware. The constraints consist of the lower and upperbounds on the design variables given in Table 1. ANNmodel is here used to approximate warpage(x), i.e., b y x inEq. 2.The ranges of mold temperature and melt temperatureare based on the recommended values in Moldflow PlasticsInsight, and the ranges of injection time, packing pressure,packing time, and cooling time are determined by theexperience of the manufacturer.First, ten samples are selected by LHD, then the warpagevalue corresponding to every sample design is obtained byrunning Moldflow Plastics Insight software, and finally, anapproximate function relationship between warpage and theprocess parameters is constructed by means of ANN modelsimulation, replacing the expensive simulation analysis inthe optimization iterations.The optimization problem based on EI function is solvedhere using the sequential quadratic programming 28. Theexpected improvement surface may be highly multimodalFig. 8 Warpage of the scanner after optimization00.010.020.030.040.050.060.070.080.095060708090Mold temperature (oC)Warpage (mm)00.0260270280290300Melt temperature (oC)00.0Injection time(s)Warpage (mm)00.020.040.060.0460708090Packing pressure (MPa)Warpage (mm)Warpage (mm)00.020.040.060.041 23 45Packing time (s)Warpage (mm)0.0760.0780.080.0820.0840.0860.0880.090.0920.094579111315Cooling time (s)Warpage (mm)Fig. 9 Each factors individualeffect on the warpage of acellular phone coverInt J Adv Manuf Technol (2010) 48:955962959and thus difficult to optimize reliably. Firstly, 1,000 randompoints are selected, and EI function values computation areperformed by means of the constructed approximatemathematical function. The point with maximum EIfunction value is then selected to be one initial design. Inaddition, the point with minimum warpage value in samplepoints is selected to be another initial design, i.e., twooptimization processes are executed at each iteration. Incomparison with simulation analysis, these processesconsume very short time and can be ignored.Only 20 iterations were needed to obtain the optimizationsolution; the results are given in Table 3. Figures 4 and 5show the warpage values before and after optimization,respectively (Table 2).The second example is a scanner. The cover is discretizedby 8,046 triangle elements, as shown in Fig. 6. It is made ofPC. The mold temperature (Tmold), melt temperature (Tmelt),injection time (tin), packing pressure (Ppack), packing time(tpack), and cooling time (tc) are considered as designvariables. The objective function warpage(x) is quantifiedby the out-of-plane displacement, which is the sum of bothmaximum and minimum deformations with reference tothe default plane in Moldflow Plastics Insight software. Theconstraints consist of the lower and upper bounds on thedesign variables given in Table 3.The ranges of mold temperature and melt temperatureare based on the recommended values in Moldflow PlasticsInsight, and the ranges of injection time, packing pressure,packing time, and cooling time are determined by theexperience of the manufacturer.Initial ten samples are selected by LHD; the optimalsolution was obtained after 25 iterations. The results aregiven in Table 4. Figures 7 and 8 show the warpage beforeand after optimization, respectively.6 DiscussionsTables 2 and 4 show that several process parameters arelying in the boundaries of the limits. Figures 9 and 10 showeach factors effect on the warpage when all other factorsare kept at their optimal level, respectively.Figures 9 and 10 show that high melt temperature andshort injection time are desirable. The warpage valuedecreases nonlinearly as melt temperature changesfrom260C to 300C. This is because lower melt temper-ature has bad liquidity and can lead to early formation offrozen skin layer, which can generate higher shear stressand block flow. If there is no enough time to release theshear stress, the warpage will increase. However, the00.050.48090100110120Mold temperature (oC)Warpage (mm)0260270280290300Melt temperature (oC)Warpage (mm)00.050.40.40.70.8Injection time (s)Warpage (mm)0160708090Packing pressure (MPa)Warpage (mm)01 23 4 5Packing time(s)Warpage (mm)0.270.280.290.300.310.320.335 79111315Cooling time (s)Warpage (mm)Fig. 10 Each factors individualeffect on the warpage of ascanner960Int J Adv Manuf Technol (2010) 48:955962warpage value increases nonlinearly with the injection time.For the thin-wall injection molded parts, long injection timecan increase the ratio of the frozen skin layer to the moltencore layer. This can block badly the flow and lead to highershear stress and more molecular orientation in the material.The warpage value changes only a period of packing timeand almost is constant when packing time is longer thansome values. Figures 9 and 10 also show that the variationof warpage values is irregular when changing other processparameters such as packing pressure, cooling time, andmold temperature. The warpage value depends on thecombined efforts of all process parameters, and all theseprocess parameters should be provided by means ofoptimization.7 ConclusionsIn this study, an integrated ANN model and EI functionmethod is proposed to minimize the warpage of theinjection molding parts. This method aims at optimizingsome approximate functions trained by the ANN model.The optimization process can be started from anapproximate function trained by a set of a few samplepoints, then adding the best sample point into thetraining set by means of EI function. Every iteration ofthe optimization consists of training the approximatefunction and optimizing the EI function. The EI functioncan take the relatively unexpected space into consider-ation to improve the accuracy of the ANN model andquickly approach to the global optimization solution. Asthe applications, a cellular phone cover and a scanner,are investigated, only a small number of MoldflowPlastics Insight analysis are needed in optimizationsbecause the first iterations for both examples need a setof a few sample points (only ten sample points) andfollow-up of every iteration adds one sample point intothe set only. Numerical results show that the proposedoptimization method is efficient for reducing warpage ofinjection molded parts and can converge to the optimi-zation solution quickly. Although the design variables ofthese relatively examples are limited to the moldtemperature, melt temperature, injection time, packingpressure, packing time, and cooling time, the presentmethod is also applicable to more process parameters.However, there still are two problems. The first one isthe development of an efficient optimization algorithm.Because the EI function is multimodal with sharp peaks, soit would be difficult to find the optimum solution. Thesecond one is developed for some optimization methods todetermine some network parameters, such as learning cycle,learning rate, momentum factor, and number of hiddenneuron in the learning framework of the BPN, making theconvergence speed of the network quick and steady. Furtherdevelopments are planned.AcknowledgmentsThe authors gratefully acknowledge financialsupport for this work from the Major program (10590354) of theNational Natural Science Foundation of China and wish to thankMoldflow Corporation for making their simulation software availablefor this study.References1. Shen CY, Wang LX, Li Q (2007) Optimization of injectionmolding process parameters using combination of artificial neuralnetwork and genetic algorithm method. J Mater Process Technol138(23):4124182. Kabanemi KK, Vaillancourt H, Wang H, Salloum G (1998)Residual stresses, shrinkage, and warpage of complex injectionmolded products: numerical simulation and experimental validation.Polym Eng Sci 38(1):21373. Hiroyuki K, Kiyohito K (1996) Warpage anisotropy, and partthickness. Polym Eng Sci 36(10):132613354. Fan B, Kazmer DO, Bushko WC, Theriault RP, Poslinski A(2003) Warpage prediction of optical media. J Polym Sci Part B:Polym Phys 41(9):8598725. Akay M, Ozden S, Tansey T (1996) Prediction of process-inducedwarpage in injection molded thermoplastics. Polym Eng Sci 36(13):183918466. Hiroyuki K, Kiyohito K (1996) The relation between thicknessand warpage in a disk injection molded from fiber reinforcedPA66. Polym Eng Sci 36(10):131713257. Fahy EJ (1998) Modeling warpage in reinforced polymer disks.Polym Eng Sci 38(7):107210848. Santhanam N, Chiang HH, Himasekhar K, Tuschak P, Wang KK(1991) Postmolding and load-induced deformation analysis ofplastic parts in the injection molding process. Adv Polym Technol11(2):77899. Yuanxian GU, Haimei LI, Changyo S (2001) Numerical simulationof thermally induced stress and warpage in injection-moldedthermoplastics. Adv Polym Technol 20(1):142110. Lee BH, Kim BH (1995) Optimization of part wall thicknesses toreduce warpage of injection-molded parts based on the modifiedcomplex method. Polym Plast Technol Eng 34(5):79381111. Sahu R, Yao DG, Kim B (1997) Optimal mold design method-ology to minimize warpage in injection molded parts. Technicalpapers of the 55th SPE ANTEC Annual Technical Conference,Toronto, Canada, April/May 1997, vol 3, pp 3308331212. Tang SH, Tan YJ, Sapuan SM, Sulaiman S, Ismail N, Samin R(2007) The use of Taguchi method in the design of plasticinjection mould for reducing warpage. J Mater Process Technol182(13):41842613. Huang MC, Tai CC (2001) The effective factors in the warpageproblem of an injection-molded part with a thin shell feature. JMater Process Technol 110(1):1914. Liao SJ, Chang DY, Chen HJ, Tsou LS, Ho JR, Yau HT, Hsieh WH,WangJT,SuYC(2004)Optimalprocessconditionsofshrinkageandwarpage of thin-wall parts. Polym Eng Sci 44(5):91792815. Gao YH, Wang XC (2008) An effective warpage optimizationmethod in injection molding based on the Kriging model. Int JAdv Manuf Technol 37(910):95396016.
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