An effective warpage optimization method.pdf

ORIGINAL ARTICLE An effective warpage optimization method in injection molding based on the Kriging model Yuehua Gao it is only the best combination of factor levels. Recently, the response surface method and neural network models have turned up in the task of warpage optimization. Shen et al. 9 combined a neural network model and genetic algorithms to optimize the process conditions for reducing the difference between the maxi- mum and minimum volume shrinkage. Ozcelik, Erzurumlu, and Kurtaran optimized dimensional parameters 10 and process conditions 1113 to reduce the warpage of thin- shell plastic parts by combining genetic algorithms with the response surface method or a neural network model. From their results, both the response surface methodology and the neural network model can be considered as good ways to reduce the high computational cost in the warpage optimization and the genetic algorithm can be used to find the global optimal design effectively. In this study, the packing pressure, melt temperature, mold temperature, and injection time will be considered as effective factors on warpage. The Kriging model 14, 15 combining modified rectangular grid approach is applied to build the approximate relationship of warpage and the process parameters, and the optimization iterations are based on the approximate relationship for reducing the high computational cost. Besides the approximate relation- ship, the Kriging model can provide some information for analyzing the important factors. 2 A sampling strategy A modified rectangular grid (MRG) approach is presented to 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 jth dimension as qj(i.e., the number of sample points is Q m j1 qj: Then, the approach is performed as follows: 1.Contract the ranges of the variables: lj? xj? b uj; b uj uj? 1 2 uj? lj qj? 1 j 1;.; m1 2.Perform RG sampling in the contracted space. The distribution of sample points is defined by all different combinations of data in the different dimensions: x i j lj k i j b uj? lj qj? 1 k i j 0; 1;.; qj? 1; j 1;.; m 2 3.Add a stochastic movement to each dimension of each sample point; the stochastic movement is: anj 2 uj? lj qj? 1 j 1; 2;.; m; n 1; 2;.; Y m j1 qj 3 where nj0, 1 is from a uniform distribution. Comparing with RG 14, MRG can move some points lying on the boundary into the internal design region, which will provide more useful information for the Kriging model, and it can ensure that the points have less replicated coordinate values. Moreover, it can avoid the case that the sample points are spaced close to each other, which may occur using LHS 16, as the distance between two arbitrary points must satisfy: d ? min 1?j?m uj? lj 2 qj? 1 ?1 ? 1 qj? 1 ? # 4 Figure 1 shows that the MRG approach is better than RG and LHS. abc RG 0 2 4 6 8 10 0246810 LHS 0 2 4 6 8 10 0246810 MRG 0 2 4 6 8 10 0246810 Fig. 1ac Sample distributions for three different methods. a RG. b MRG. c LHS 954Int J Adv Manuf Technol (2008) 37:953960 3 The Kriging model The Kriging model is described as a way of “modeling the function as a realization of a stochastic process,” so it is named a “stochastic process model.” In fact, the Kriging model is an interpolate technology, and the Kriging predictor is a predictor that minimizes the expected squared prediction error subject to: (i) being unbiased and (ii) being a linear function of the observed response values. 3.1 Model The Kriging model can be written as: b y xi ? ? X h hfhxi ? ? z xi ? ? fTxi ? ? z xi? ? 5 where xixi1; xi2;.; xim ? is the ith sample point with m variables, b y xi is an approximate function fitted to the n sample points, fhxi is a linear or nonlinear function of xi, his the regression coefficient to be estimated, and z(xi) is a stochastic function with mean zero and variance 2. The spatial correlation function between stochastic functions is given by: corr z xi ? ?; z xj? ? R; xi; xj ? a m l1 exp ?lxi l ? x j l ?2 hi 6 The parameters h, 2, and lcan be estimated by maximizing the likelihood of samples. The likelihood function is: 1 2n=22n=2Rj j1=2 exp ? y ? fT ?TR?1 y ? fT ? 22 # 7 In practice, they can be obtained by maximizing the logarithm of the likelihood function, ignoring constant terms: ? n 2 log 2 ? ? 1 2 logRj j ? y ? fT ?TR?1 y ? fT ? 22 8 Let the derivatives of this expression with respect to 2and be equal to zero; then, we can obtain: b 2 y ? fTb ?T R?1y ? fTb ? n 9 b fTR?1y fTR?1f 10 Substituting Eqs. 9 and 10 into Eq. 8, we can obtain the so- called “concentrated log-likelihood” function: ? n 2 log b s2 ? ? 1 2 logRj j11 It depends on R only and, hence, on the correlation parameters ls. By maximizing the function we can obtain: b min ? Rj j1=m2 no 12 Then, the estimatesbb and b s2can be obtained from Eqs. 9 and 10. 3.2 Predictor The function value b y x? at a new point x* can be approximately estimated as a linear combination of the response values of samples Y: b y x? cTY13 The error is: b y x? ? y x? cTY ? y x?14 Substituting Eq. 1 into Eq. 14 gives: b y x? ? y x? cTF Z ?f x? T z ? cTZ ? z FTc ? f x? ?T 15 where Z z1; z2;.; zn? and F f1; f2;.; fn?: To make the predictor unbiased for x*, the mean error at this point should be zero, i.e.: E b y x? ? y x? 016 Then, we have: FTc x? f x?17 The mean squared error (MSE) of the predictor shown in Eq. 15 is: x? Eb y x? ? y x?2 hi EcTZ ? z ?2 hi 21 cTRc ? 2cTr ? 18 where: r x? R q; x1; x? ?;.; R q; xn; x? ? 19 Minimizing (x*) with the constraint shown in Eq. 17, we can obtain: c R?1r ? Fe ? ;e FTR?1F ?1 FTR?1r ? f ? 20 then: b y x? f x?b r x?T21 Int J Adv Manuf Technol (2008) 37:953960955 where: R?1Y ? Fb ? 22 Thus, we can predict the function value b y x? at every new point x* by using Eq. 21. Simpson et al. 17 suggested that the Kriging model is the best choice for deterministic and highly nonlinear in a moderate number of variables (less than 50). It has been applied early by a number of researchers in designing complex engineering 1820. Recently, Huang et al. 21 have used the Kriging model to minimize die wear for metal-forming process design improvement. In addition, Hawe and Sykulski 22 have showed an application of the Kriging model to electromagnetic device optimization. 4 Warpage optimization based on the Kriging model 4.1 Optimization model and optimization process A warpage minimum design problem can be stated as follows: findx1; x2;.; xm minimize warpage x1; x2;.; xm subject to xj? xj? xjj 1; 2;.; m 23 where x1, x2,., xmare the variables, representing process conditions, warpage(x1, x2,.xm) is a quantified warpage value, which will be replaced by an approximate function based on the Kriging model in optimization iterations, and xjand xjare the lower and upper limits of the jth design variable. The optimization algorithm based on the Kriging model is described as follows: 1.Get a set of samples with n points (each point corresponding to a group of process conditions) using the MRG approach and run the Moldflow program to obtain the warpage values for the sample points. Then, select a group of process conditions corresponding to the minimum warpage value as the initial design. 2.Model the approximate relationship between warpage and the process parameters using the Kriging model based on the trial samples obtained. 3.Minimize the warpage value to obtain a modified design by means of the Kriging approximate function. Then, compute the corresponding warpage value by the Moldflow program. 4.Check convergence: if convergence criteria of the next section are satisfied, then stop; else, add the modified design into the set of samples and go to step 2. Note that the initial design will be renewed if the modified design is better than the former initial design. 4.2 Convergence criteria The convergence criteria are used to satisfy the accuracies of both optimization and the Kriging approximation simultaneously, i.e.: warpagek? warpagek?1 ? ? ? ? 1 24 Fig. 2 Mid-plane model of a cellular phone cover 956Int J Adv Manuf Technol (2008) 37:953960 warpagek?b yk ? ? ? ? 2 25 where k is the optimization iteration index and b ykis the approximate warpage value from the Kriging model. 5 Warpage optimization for a cellular phone cover As an example, a cellular phone cover is investigated. Its length, width, height, and thickness are 130 mm, 55 mm, 11 mm, and 1 mm, respectively. The cover is discretized by 3,780 triangle elements, as shown in Fig. 2. It is made of PC/ABS and its material properties are given in Table 1. The design variables are the mold temperature (A), melt temperature (B), injection time (C), and packing pressure (D). The warpage is quantified by the out-of plane displacement, which is the sum of the maximum upward deformation and the maximum downward deformation with reference to the default plane in Moldflow. The ranges of the four variables are given in Table 2. We hope to find the optimal 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. The range of the mold temperature is based on the recommen- ded values in Moldflows Plastics Insight, which considers the property of the material. The range of the melt temperature is 10C higher than the minimum values that should be used in Moldflow, as a lower melt temperature may result in melt-short shot. Those of injection time and packing pressure are determined based on the experience of the manufacturer. Fifty-four process combinations are selected by the MRG approach. After FE simulations, the trial samples are obtained and then the Kriging model is constructed using the DACE toolbox. Under the condition of constant regression term and 1=2=1.0e-3, only five modifications were needed to obtain the optimal solution, and the result is given in Table 3. It takes 11 hours of CPU time (running Moldflow and executing optimization) on an Intel P4 processor PC and the net time consumed by the optimiza- tion process is only 2.3s. Figure 3 shows the iteration histories for the cellular phone cover optimization. The simulation values from the Kriging model approaches the analysis values in Moldflow gradually as the iterations increase. Figures 4 and 5 show the warpage before and after optimization, respectively. 6 Results and discussion 6.1 Analysis of optimization results In order to analyze the results in detail, each factors effect on the warpage will also be investigated by means of FE simulation under the condition that all other factors are kept at their optimum level. The results are shown in Fig. 6. In general, if the mold temperature is low, higher residual stress will occur because the melt in the cavity has a high cooling rate. Therefore, from the view of quality, the highest mold temperature is best in its range. But Fig. 6 shows that the mold temperature has very little effect on warpage when all other factors are kept at their optimum value. This phenomenon results in that the optimal mold temperature isnt the highest value in its range. Table 3 Optimization results ParameterA (C)B (C)C (s)D (%) Warpage (mm) Before optimization 752651.080.00.8111 After optimization 83.002299.320.25984600.134 0 0.05 0.1 0.15 0.2 0.25 0.3 12345 Iterations Wapage(mm) Kriging MoldFlow Fig. 3 Iteration histories of optimization Table 1 Material properties of PC/ABS Properties Melt density0.98258 g/cm3 Solid density1.1161 g/cm3 Eject temperature99C Maximum shear stress0.4 MPa Maximum shear rate40,000 (1/s) Thermal conductivity0.27 W/mC Elastic module2,780 MPa Poisson ratio0.23 Table 2 Ranges of the process parameters ParameterA (C)B (C)C (s)D (%) Lower limit502600.260 Upper limit923001.090 Int J Adv Manuf Technol (2008) 37:953960957 Figure 6 shows that the warpage value decreases non- linearly as the melt temperature changes from 260C to 300C. A lower melt temperature has bad liquidity, which can generate higher shear stress. If there isnt enough time to release the shear stress, the warpage will increase. The result shows that high melt temperature is desirable for minimizing warpage, which agrees with the optimization results. A short injection time can induce fast melt flow in the cavity, which has contributions to residual stress and molec- ular orientation. On the other hand, a long injection time will cause the ratio of the frozen skin layer to the molten core layer to rise. This will cause a higher shear stress and more molecular orientation in the material. Figure 6 shows that the latter effect may be more important in the chosen range. Fig. 5 Warpage of the cover after optimization Fig. 4 Warpage of the cover before optimization 958Int J Adv Manuf Technol (2008) 37:953960 The packing pressure affects warpage in two aspects. A low packing pressure cannot compact the plastic material in the cavity, which can form volume shrinkage and induce large warpage. On the other hand, a high packing pressure can generate higher residual-stress-induced flow and high pressure when transferring more melt into the cavity. Figure 6 shows that the latter effect is more important in the chosen range because the warpage almost increases linearly when the packing pressure gets higher and higher. 6.2 Result analysis of the Kriging model Two correlation functions for a design variable are shown in Fig. 7, corresponding to =1 and =5. The curve for =5 drops off more rapidly with the change in the design variable. This illustrates that the larger is, the more active the variable is. Therefore, the parameter can be interpreted as measuring the importance of the corresponding variable 15. For this example, the number of parameter ls is the same as the process parameters, so each element lreflects the effect of corresponding process parameters on the warpage. Table 4 shows that the lvalue corresponding to injection time is bigger than the others in the Kriging model after optimization, so the injection time has the most effect on warpage and it is also consistent with Figure 6. 7 Conclusions In this study, a modified rectangular grid (MRG) is proposed. Comparing with RG, MRG moves some points lying on the boundary into the internal design region, which will provide more useful information for the Kriging model. Moreover, it can ensure that the points have less replicated coordinate values. With the heritance of RG, it can avoid the case that the points are spaced close to each other. Based on the MRG, an effective optimization method for minimizing warpage in injection molding is presented. This Fig. 7 Correlation functions for a design variable Table 4 Corresponding lvalues for every parameter ParameterABCD l0.07870.39692.00.1575 Fig. 6 Each factors individual effect on the warpage Int J Adv Manuf Technol (2008) 37:953960959 method performs optimization based on an approximate function from the Kriging model instead of expensive warpage analysis by means of Moldflow. The optimization method has been used to minimize the warpage of a cellular phone cover and the results show that it has good accuracy and effectiveness for warpage optimization. The Kriging model can not only help to reduce the computational cost in optimization, but it is also conducive to analyzing the effect of process parameters on warpage, especially to reflect their nonlinear relationship. As far as a cellular phone cover is concerned, the injection time is the important factor of warpage in the chosen range. AcknowledgmentsThe authors gratefully acknowledge the finan- cial support for this work from the Major program (10590354) of the National Natural Science Foundation of China and wish to thank the Moldflow Corporation (Framingham, MA) for making their simula- tion software available for this study. References 1. Wang TJ, Yoon CK (2000) Shrinkage and warpage analysis of injection-molded parts. In: Proceedings of the SPE ANTEC Annual Technical Conference, Orlando, Florida, May 2000, pp 687692 2. Huang MC, Tai CC (2001) The effective factors in the warpage problem of an injection-molded part with a thin shell feature. J Mater Process Technol 110(1):19 3. Liao SJ, Chang DY, Chen HJ, Tsou LS, Ho JR, Yau HT, Hsieh WH, Wang JT, Su YC (2004) Optimal process conditions of shrinkage and warpage of thin-wall parts. Polym Eng Sci 44(5):917928 4. Wang TH, Young WB, Wang J (2002) Process design for reducing the warpage in thin-walled injection molding. Int Polym Process 17(2):146152 5. Dong B-B, Shen C-Y, Liu C-T (2005) The effect of injection process parameters on the shrinkage and warpage of PC/ABSs part. Polym Mater Sci Eng 21(4):232235 6. Lee BH, Kim BH (1995) Optimization of part wall thicknesses to reduce warpage of injection-molded parts based on the modified complex method. Polym Plast Technol Eng 34(5):793811 7. Lee BH, Kim BH (1996) Automated selection of gate location based on desired quality of injection molded part. Polym Plast Technol Eng 35(2):253269 8. Sahu R, Yao DG, Kim B (1997) Optimal mold design method- ology to minimize warpage in injection molded parts. Technical papers of the 55th SPE ANTEC Annual Technical Conference, Toronto, Canada, April/May 1997, vol 3, pp 33083312 9. Shen C-Y, Wang L-X, Zhang Q-X (2005) Process optimization of injection molding by the combining ANN/HGA method. Polym Mater Sci Eng 21(5):2327 10. Ozcelik B, Erzurumlu T (2005) Determination of effecting dimensional parameters on warpage of thin shell plastic parts using integrated response surface method and genetic algorithm. Int Commun Heat Mass Transfer 32(8):10851094 11.