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ORIGINAL ARTICLEAn effective warpage optimization methodin injection molding based on the Kriging modelYuehua Gao&Xicheng WangReceived: 15 November 2006 /Accepted: 5 April 2007 /Published online: 15 June 2007#Springer-Verlag London Limited 2007Abstract In this paper, an effective optimization methodusing the Kriging model is proposed to minimize thewarpage in injection molding. The warpage deformationsare nonlinear, implicit functions of the process conditions,which are typically evaluated by the solution of finiteelement (FE) equations, a complicated task which ofteninvolves huge computational effort. The Kriging model canbuild an approximate function relationship between warp-age and the process conditions, replacing the expensive FEreanalysis of warpage in the optimization. In addition, a“space-filing” sampling strategy for the Kriging model,named rectangular grid, is modified. Moldflow Corpora-tions Plastics Insight software is used to analyze thewarpage deformations of the injection-molded parts. As anexample, the warpage of a cellular phone cover isinvestigated, where the mold temperature, melt tempera-ture, injection time, and packing pressure are regarded asthe design variables. The result shows that the proposedoptimization method can effectively decrease the warpagedeformations of the cellular phone cover and that theinjection time has the most important influence on warpagein the chosen range.Keywords Injectionmolding.Warpage deformation.Krigingmodel.Rectangulargrid.Modifiedrectangulargrid.Taguchimethod1 IntroductionWarpage is an important factor affecting product quality.Especially, as communication and electronic products havebeen developing towards the design concept of being light,thin, short, and small, reducing warpage to improve thequality of a part with a thin shell is becoming increasinglymore important. The causes of warpage are attributed to theuneven shrinkage of parts. We can reduce the warpage bychanging the geometry of parts, or modifying the structureof molds, or adjusting the process conditions. In fact,optimizing process conditions is the most feasible andreasonable method.Different process conditions will result in different non-uniformity. Several studies on the effective factors ofwarpage have been reported 15. According to theirconclusions, the packing pressure, mold temperature, andinjection time (or injection speed) have an important effecton the warpage of injection-molded parts. It is an importantissue in plastic injection molding to predict and optimizethe warpage deformations before manufacturing takesplace. There have been some publications devoted towarpage optimization. An early literature about warpageoptimization was proposed by Lee and Kim 6. Theyoptimized the wall thickness and process conditions usingthe modified complex method to reduce warpage andobtained a reduction in warpage of over 70%. Subsequently7, they optimize the gate location by a two-step searchmethod in order to improve the product qualities, includingwarpage, weld line, and percussive intensity. Sahu et al. 8optimized process conditions using a modified complexmethod, the Taguchi method, and genetic algorithms, andtheir results showed that the complex method attained thebest results for reducing warpage.Int J Adv Manuf Technol (2008) 37:953960DOI 10.1007/s00170-007-1044-6Y. Gao:X. Wang (*)The State Key Laboratory of Structural Analysis for IndustrialEquipment, Department of Engineering Mechanics,Dalian University of Technology,Dalian, 116024 Liaoning, Peoples Republic of Chinae-mail: The complex method can reduce warpage effectively, butit is time-consuming and costly because it performs toomuch reanalysis, which needs a lot of expensive functionevaluations. Reducing warpage using the Taguchi method15 is easy to perform, and can analyze the effectivefactors, but the “optimal process condition” obtained is notthe best in the design space; it is only the best combinationof factor levels.Recently, the response surface method and neuralnetwork models have turned up in the task of warpageoptimization. Shen et al. 9 combined a neural networkmodel and genetic algorithms to optimize the processconditions for reducing the difference between the maxi-mum and minimum volume shrinkage. Ozcelik, Erzurumlu,and Kurtaran optimized dimensional parameters 10 andprocess conditions 1113 to reduce the warpage of thin-shell plastic parts by combining genetic algorithms with theresponse surface method or a neural network model. Fromtheir results, both the response surface methodology and theneural network model can be considered as good ways toreduce the high computational cost in the warpageoptimization and the genetic algorithm can be used to findthe global optimal design effectively.In this study, the packing pressure, melt temperature,mold temperature, and injection time will be considered aseffective factors on warpage. The Kriging model 14, 15combining modified rectangular grid approach is applied tobuild the approximate relationship of warpage and theprocess parameters, and the optimization iterations arebased on the approximate relationship for reducing thehigh computational cost. Besides the approximate relation-ship, the Kriging model can provide some information foranalyzing the important factors.2 A sampling strategyA modified rectangular grid (MRG) approach is presentedto provide sample points for building the Kriging model.We define the ranges of m design variables as lj? xj?uj; j 1;.; m; and the number of levels in the jthdimension as qj(i.e., the number of sample points isQmj1qj: Then, the approach is performed as follows:1.Contract the ranges of the variables:lj? xj? b uj; b uj uj?12uj? ljqj? 1j 1;.; m12.Perform RG sampling in the contracted space. Thedistribution of sample points is defined by all differentcombinations of data in the different dimensions:xi j lj ki jb uj? ljqj? 1ki j 0; 1;.; qj? 1; j 1;.; m23.Add a stochastic movement to each dimension of eachsample point; the stochastic movement is:anj2uj? ljqj? 1j 1; 2;.; m; n 1; 2;.;Ymj1qj3where nj0, 1 is from a uniform distribution.Comparing with RG 14, MRG can move some pointslying on the boundary into the internal design region, whichwill provide more useful information for the Kriging model,and it can ensure that the points have less replicatedcoordinate values. Moreover, it can avoid the case that thesample points are spaced close to each other, which mayoccur using LHS 16, as the distance between two arbitrarypoints must satisfy:d ? min1?j?muj? lj2 qj? 1?1 ?1qj? 1?#4Figure 1 shows that the MRG approach is better than RGand LHS.abcRG02468100246810LHS02468100246810MRG02468100246810Fig. 1ac Sample distributionsfor three different methods. aRG. b MRG. c LHS954Int J Adv Manuf Technol (2008) 37:9539603 The Kriging modelThe Kriging model is described as a way of “modeling thefunction as a realization of a stochastic process,” so it isnamed a “stochastic process model.” In fact, the Krigingmodel is an interpolate technology, and the Krigingpredictor is a predictor that minimizes the expected squaredprediction error subject to: (i) being unbiased and (ii) beinga linear function of the observed response values.3.1 ModelThe Kriging model can be written as:b y xi? ?Xhhfhxi? ? z xi? ? fTxi? ? z xi? ?5where xixi1; xi2;.; xim?is the ith sample point with mvariables, b y xi is an approximate function fitted to the nsample points, fhxi is a linear or nonlinear function of xi,his the regression coefficient to be estimated, and z(xi) is astochastic function with mean zero and variance 2. Thespatial correlation function between stochastic functions isgiven by:corr z xi? ?; z xj? ? R; xi; xj?aml1exp ?lxil? xjl?2hi6The parameters h, 2, and lcan be estimated bymaximizing the likelihood of samples. The likelihoodfunction is:12n=22n=2Rj j1=2exp ?y ? fT?TR?1y ? fT?22#7In practice, they can be obtained by maximizing thelogarithm of the likelihood function, ignoring constantterms:?n2log 2?12logRj j ?y ? fT?TR?1y ? fT?228Let the derivatives of this expression with respect to 2and be equal to zero; then, we can obtain:b 2y ? fTb?TR?1y ? fTb?n9b fTR?1yfTR?1f10Substituting Eqs. 9 and 10 into Eq. 8, we can obtain the so-called “concentrated log-likelihood” function:?n2log b s2?12logRj j11It depends on R only and, hence, on the correlationparameters ls. By maximizing the function we can obtain:b min ? Rj j1=m2no12Then, the estimatesbb and b s2can be obtained from Eqs. 9and 10.3.2 PredictorThe function value b y x? at a new point x* can beapproximately estimated as a linear combination of theresponse values of samples Y:b y x? cTY13The error is:b y x? ? y x? cTY ? y x?14Substituting Eq. 1 into Eq. 14 gives:b y x? ? y x? cTF Z ?f x?T z? cTZ ? z FTc ? f x?T15where Z z1; z2;.; zn? and F f1; f2;.; fn?: Tomake the predictor unbiased for x*, the mean error at thispoint should be zero, i.e.:E b y x? ? y x? 016Then, we have:FTc x? f x?17The mean squared error (MSE) of the predictor shown inEq. 15 is: x? Eb y x? ? y x?2hi EcTZ ? z?2hi 21 cTRc ? 2cTr?18where:r x? R q; x1; x?;.; R q; xn; x?19Minimizing (x*) with the constraint shown in Eq. 17, wecan obtain:c R?1r ? Fe?;e FTR?1F?1FTR?1r ? f?20then:b y x? f x?b r x?T21Int J Adv Manuf Technol (2008) 37:953960955where: R?1Y ? Fb?22Thus, we can predict the function value b y x? at every newpoint x* by using Eq. 21.Simpson et al. 17 suggested that the Kriging model isthe best choice for deterministic and highly nonlinear in amoderate number of variables (less than 50). It has beenapplied early by a number of researchers in designingcomplex engineering 1820. Recently, Huang et al. 21have used the Kriging model to minimize die wear formetal-forming process design improvement. In addition,Hawe and Sykulski 22 have showed an application of theKriging model to electromagnetic device optimization.4 Warpage optimization based on the Kriging model4.1 Optimization model and optimization processA warpage minimum design problem can be stated asfollows:findx1; x2;.; xmminimize warpage x1; x2;.; xmsubject to xj? xj? xjj 1; 2;.; m23where x1, x2,., xmare the variables, representing processconditions, warpage(x1, x2,.xm) is a quantified warpagevalue, which will be replaced by an approximate functionbased on the Kriging model in optimization iterations, andxjand xjare the lower and upper limits of the jth designvariable.The optimization algorithm based on the Kriging modelis described as follows:1.Get a set of samples with n points (each pointcorresponding to a group of process conditions) usingthe MRG approach and run the Moldflow program toobtain the warpage values for the sample points. Then,select a group of process conditions corresponding tothe minimum warpage value as the initial design.2.Model the approximate relationship between warpageand the process parameters using the Kriging modelbased on the trial samples obtained.3.Minimize the warpage value to obtain a modifieddesign by means of the Kriging approximate function.Then, compute the corresponding warpage value by theMoldflow program.4.Check convergence: if convergence criteria of the nextsection are satisfied, then stop; else, add the modifieddesign into the set of samples and go to step 2. Notethat the initial design will be renewed if the modifieddesign is better than the former initial design.4.2 Convergence criteriaThe convergence criteria are used to satisfy the accuraciesof both optimization and the Kriging approximationsimultaneously, i.e.:warpagek? warpagek?1? 124Fig. 2 Mid-plane model of acellular phone cover956Int J Adv Manuf Technol (2008) 37:953960warpagek?b yk? 225where k is the optimization iteration index and b ykis theapproximate warpage value from the Kriging model.5 Warpage optimization for a cellular phone coverAs an example, a cellular phone cover is investigated. Itslength, width, height, and thickness are 130 mm, 55 mm,11 mm, and 1 mm, respectively. The cover is discretized by3,780 triangle elements, as shown in Fig. 2. It is made ofPC/ABS and its material properties are given in Table 1.The design variables are the mold temperature (A), melttemperature (B), injection time (C), and packing pressure(D). The warpage is quantified by the out-of planedisplacement, which is the sum of the maximum upwarddeformation and the maximum downward deformation withreference to the default plane in Moldflow. The ranges ofthe four variables are given in Table 2. We hope to find theoptimal design in a large feasible molding window. Thus,the ranges may be larger than those in practical manufac-turing. Besides, the ranges can avoid melt short shot. Therange of the mold temperature is based on the recommen-ded values in Moldflows Plastics Insight, which considersthe property of the material. The range of the melttemperature is 10C higher than the minimum values thatshould be used in Moldflow, as a lower melt temperaturemay result in melt-short shot. Those of injection time andpacking pressure are determined based on the experience ofthe manufacturer.Fifty-four process combinations are selected by theMRG approach. After FE simulations, the trial samplesare obtained and then the Kriging model is constructedusing the DACE toolbox. Under the condition of constantregression term and 1=2=1.0e-3, only five modificationswere needed to obtain the optimal solution, and the result isgiven in Table 3. It takes 11 hours of CPU time (runningMoldflow and executing optimization) on an Intel P4processor PC and the net time consumed by the optimiza-tion process is only 2.3s. Figure 3 shows the iterationhistories for the cellular phone cover optimization. Thesimulation values from the Kriging model approaches theanalysis values in Moldflow gradually as the iterationsincrease. Figures 4 and 5 show the warpage before and afteroptimization, respectively.6 Results and discussion6.1 Analysis of optimization resultsIn order to analyze the results in detail, each factors effecton the warpage will also be investigated by means of FEsimulation under the condition that all other factors are keptat their optimum level. The results are shown in Fig. 6.In general, if the mold temperature is low, higherresidual stress will occur because the melt in the cavityhas a high cooling rate. Therefore, from the view of quality,the highest mold temperature is best in its range. But Fig. 6shows that the mold temperature has very little effect onwarpage when all other factors are kept at their optimumvalue. This phenomenon results in that the optimal moldtemperature isnt the highest value in its range.Table 3 Optimization resultsParameterA (C)B (C)C (s)D(%)Warpage(mm)Beforeoptimization752651.080.00.8111Afteroptimization83.002299.320.25984600.13400.00.250.312345IterationsWapage(mm)KrigingMoldFlowFig. 3 Iteration histories of optimizationTable 1 Material properties of PC/ABSPropertiesMelt density0.98258 g/cm3Solid density1.1161 g/cm3Eject temperature99CMaximum shear stress0.4 MPaMaximum shear rate40,000 (1/s)Thermal conductivity0.27 W/mCElastic module2,780 MPaPoisson ratio0.23Table 2 Ranges of the process parametersParameterA (C)B (C)C (s)D (%)Lower limit502600.260Upper limit923001.090Int J Adv Manuf Technol (2008) 37:953960957Figure 6 shows that the warpage value decreases non-linearly as the melt temperature changes from 260C to300C. A lower melt temperature has bad liquidity, whichcan generate higher shear stress. If there isnt enough time torelease the shear stress, the warpage will increase. The resultshows that high melt temperature is desirable for minimizingwarpage, which agrees with the optimization results.A short injection time can induce fast melt flow in thecavity, which has contributions to residual stress and molec-ular orientation. On the other hand, a long injection time willcause the ratio of the frozen skin layer to the molten core layerto rise. This will cause a higher shear stress and moremolecular orientation in the material. Figure 6 shows thatthe latter effect may be more important in the chosen range.Fig. 5 Warpage of the coverafter optimizationFig. 4 Warpage of the coverbefore optimization958Int J Adv Manuf Technol (2008) 37:953960The packing pressure affects warpage in two aspects. Alow packing pressure cannot compact the plastic material inthe cavity, which can form volume shrinkage and inducelarge warpage. On the other hand, a high packing pressurecan generate higher residual-stress-induced flow and highpressure when transferring more melt into the cavity.Figure 6 shows that the latter effect is more important inthe chosen range because the warpage almost increaseslinearly when the packing pressure gets higher and higher.6.2 Result analysis of the Kriging modelTwo correlation functions for a design variable are shownin Fig. 7, corresponding to =1 and =5. The curve for =5drops off more rapidly with the change in the designvariable. This illustrates that the larger is, the more activethe variable is. Therefore, the parameter can beinterpreted as measuring the importance of thecorresponding variable 15. For this example, the numberof parameter ls is the same as the process parameters, soeach element lreflects the effect of corresponding processparameters on the warpage. Table 4 shows that the lvaluecorresponding to injection time is bigger than the others inthe Kriging model after optimization, so the injection timehas the most effect on warpage and it is also consistent withFigure 6.7 ConclusionsIn this study, a modified rectangular grid (MRG) isproposed. Comparing with RG, MRG moves some pointslying on the boundary into the internal design region, whichwill provide more useful information for the Kriging model.Moreover, it can ensure that the points have less replicatedcoordinate values. With the heritance of RG, it can avoidthe case that the points are spaced close to each other.Based on the MRG, an effective optimization method forminimizing warpage in injection molding is presented. ThisFig. 7 Correlation functions for a design variableTable 4 Corresponding lvalues for every parameterParameterABCDl0.07870.39692.00.1575Fig. 6 Each factors individualeffect on the warpageInt J Adv Manuf Technol (2008) 37:953960959method performs optimization based on an approximatefunction from the Kriging model instead of expensivewarpage analysis by means of Moldflow. The optimizationmethod has been used to minimize the warpage of a cellularphone cover and the results show that it has good accuracyand effectiveness for warpage optimization.The Kriging model can not only help to reduce thecomputational cost in optimization, but it is also conduciveto analyzing the effect of process parameters on warpage,especially to reflect their nonlinear relationship. As far as acellular phone cover is concerned, the injection time is theimportant factor of warpage in the chosen range.AcknowledgmentsThe authors gratefully acknowledge the finan-cial support for this work from the Major program (10590354) of theNational Natural Science Foundation of China and wish to thank theMoldflow Corporation (Framingham, MA) for making their simula-tion software available for this study.References1. Wang TJ, Yoon CK (2000) Shrinkage and warpage analysis ofinjection-molded parts. In: Proceedings of the SPE ANTECAnnual Technical Conference, Orlando, Florida, May 2000, pp6876922. 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):193. Liao SJ, Chang DY, Chen HJ, Tsou LS, Ho JR, Yau HT, Hsieh WH,Wang JT, Su YC (2004) Optimal process conditions of shrinkageand warpage of thin-wall parts. Polym Eng Sci 44(5):9179284. Wang TH, Young WB, Wang J (2002) Process design for reducingthe warpage in thin-walled injection molding. Int Polym Process17(2):1461525. Dong B-B, Shen C-Y, Liu C-T (2005) The effect of injectionprocess parameters on the shrinkage and warpage of PC/ABSspart. Polym Mater Sci Eng 21(4):2322356. 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):7938117. Lee BH, Kim BH (1996) Automated selection of gate locationbased on desired quality of injection molded part. Polym PlastTechnol Eng 35(2):2532698. 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 330833129. Shen C-Y, Wang L-X, Zhang Q-X (2005) Process optimization ofinjection molding by the combining ANN/HGA method. PolymMater Sci Eng 21(5):232710. Ozcelik B, Erzurumlu T (2005) Determination of effectingdimensional parame
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