Int J Adv Manuf Technol DOI 10.1007/s00170-015-8100-4

ORIGINAL ARTICLE

Multiobjective optimization of injection molding process parameters based on Opt LHD, EBFNN, and MOPSO Junhong Zhang 1,2 & Jian Wang 1 & Jiewei Lin 3 & Qian Guo 1 & Kongwu Chen 1 & Liang Ma 1,2

Received: 17 June 2015 / Accepted: 11 November 2015 # Springer-Verlag London 2015

Abstract The injection molding process parameters strongly affect plastic production quality, manufacturing cost, and molding efficiency. In this study, the effects of the process parameters, including the valve gate open timing, the molding temperature, the melt temperature, the injection time, the packing pressure, the packing time, and the cooling time, on the warpage of the plastic product and the clamping force during the injection molding process are analyzed using the analysis of variance method. A multiobjective optimization of the injection molding process parameters for a diesel engine oil cooler cover was carried out based on the optimal Latin hypercube design, ellipsoidal basis function neural network, and multiobjective particle swarm optimization. According to the calculated results using the optimal parameters, a structural optimization on the oil cooler cover cooling and a cooling channel improvement are proposed to further reduce the warpage. At last, a suite of overall tools are developed to treat the cooling deformation. As a result, the reduction on warpage is about 4 mm, the peak stress of the optimized plastic oil cooler cover is reduced by 60 MPa, and the stress distributes more evenly throughout the whole product. The peak clamping force is decreased from 760 to 470 t which

* Jiewei Lin [email protected] Jian Wang [email protected] 1

State Key Laboratory of Engines, Tianjin University, Tianjin 300072, People’s Republic of China

2

Department of Mechanical Engineering, Tianjin University Renai College, Tianjin 301636, People’s Republic of China

3

Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK

makes the machine selection more flexible and reduces the production cost. Keywords Plastic injection molding . Multiobjective optimization . Optimal Latin hypercube design . Ellipsoidal basis function neural network . Multiobjective particle swarm optimization . Diesel engine oil cooler cover . Lightweight

1 Introduction To achieve a lightweight vehicle, plastic components are increasingly used in contemporary automobile industry because of the benefits of cost, weight reduction, and fuel efficiency. The entire injection molding process can be divided into six stages including clamping, filling, packing, cooling, opening, and ejecting. Besides, the cycle of injection molding could be affected by many factors such as mold temperature, melt temperature, injection pressure and time, packing pressure and time, cooling time, and gate location. Such a complicated manufacturing process affected by a variety of factors makes it very challenging to attain a desired product with a high level of quality. In the early works, the finite element method was usually used to predict the residual stress and the warpage of plastic components [1–4]. Based on the prediction results, the researcher tried to improve the plastic product quality by adjusting injection parameters and structural geometry [5, 6]. With the help of the design of experiment (DOE) method [7], the effects of injection parameters on the warpage have been well investigated, and the product optimization efficiency has been improved. Using the Taguchi design, Huang and Tai [8] found that the packing pressure is the most significant factor affecting the warpage and the mold temperature, and the melt temperature and the packing time also have considerable effects on the warpage. Chen et al. [9] carried out a number of Moldflow analyses

Int J Adv Manuf Technol

by utilizing the combination of process parameters based on a three-level L18 orthogonal array table and used the DOE approach to determine the optimal parameter setting. Erzurumlu and Ozcelik [10] investigated the effects of process parameters on the warpage and the sink index with orthogonal arrays of Taguchi, the signal-to-noise ratio, and the analysis of variance (ANOVA). The optimal process parameters, rib cross-section types, and the rib layout angle for PC/ABS, POM, and PA66 were found, respectively. However, these methods have a common limitation that the best combination of the process parameters strongly depends on the sampling interval or the level of parameters used in the DOE. In other words, the optimal results found by the DOE methods may not be the global optimal solution. With the computer-aided engineering widely used, more and more optimization methods based on the fitting technology were proposed. Shi et al. [11] combined the backpropagation neural network and the non-binary genetic algorithm (GA) to optimize the plastic injection molding process parameters. As a result, the maximum shear stress of a butter container lid was significantly reduced by 24.9 %. Kurtaran et al. [12] integrated the three-level full factorial design, the response surface methodology (RSM), and the GA to obtain the efficient minimization of the warpage on a bus ceiling lamp base. In order to obtain a highly accurate response surface and find the global optimal solution, the Kriging method, the radial basis function neural network, and the sequential approximate optimization (SAO) were employed in some researches. Gao and Wang [13, 14] built an approximate function of the warpage and the process parameters with the Kriging method. Then, the SAO was performed to decrease the warpage of a cellular phone cover. Deng et al. [15] adopted the Kriging method combined with the mode-pursuing sampling method to optimize the process parameters to minimize the warpage of a food tray. Li et al. [16] introduced a step-by-step optimization method based on the radial basis function surrogate model for the packing profile optimization of a fan gate. Chiang et al. [17, 18] used the grey-fuzzy logic analysis on the optimal process design of an injection-molded part with a thin shell feature. Nevertheless, the injection molding process parameters are not independent with each other. Increasing melt temperature can decrease the melt viscosity as well as the shear stress, but the cooling time will be increased. The combination of a higher injection pressure and a shorter injection time can reduce the temperature difference between different areas of the production but may increase the temperature at the flow front. The sink marks decrease with increased packing time, while the flash will be increased at the same time. With the increasing amount of factors considered, the number of the optimization objects increases accordingly. Thus, the multiobjective optimization was applied to trade off these conflicting parameters. Cheng et al. [19] integrated the variable complexity methods, the constrained non-dominated sorted genetic algorithm and the back

propagation neural network to locate the Pareto optimal solutions of the constrained multiobjective optimization problem by approximating three objective functions (difference of the volume shrinkage, total volume of the runner system, and the cycle time). Xu et al. [20] combined the gray correlation analysis and the particle swarm optimization to find the Pareto solutions for a multiobjective optimization problem. Zhao et al. [21] adopted the non-dominated sorting-based genetic algorithm II to find a good spread of design solutions and a good convergence near the true Pareto front. Although a lot of previous studies have been conducted on the injection parameter optimization, the clamping force in the injection molding optimization has never been well discussed. In fact, the clamping force should be considered as an important factor affecting the choice of the injection molding machine. In practice, a smaller clamping force can make the machine choice more flexible which may reduce the cost greatly. In this study, a multiobjective optimization of the injection molding process parameters was proposed based on the optimal Latin hypercube design (Opt LHD), the ellipsoidal basis function neural network (EBFNN), and the multiobjective particle swarm optimization (MOPSO). The aim was to reduce the warpage, the clamping force, and the manufacture cost at the same time. Firstly, the valve gate timing was optimized individually. Secondly, based on the optimum timing, the sampling was carried out using the Opt LHD method and the correlations between the injection parameters, and the warpage and the clamping force were fitted using the EBFNN. Another two methods were employed to verify the predicting performance of the EBFNN including the RSM and the Kriging method. Thirdly, the final objects were determined according to the ANOVA results. Then, the optimal combination of the injection molding process parameters was found using the MOPSO method. Finally, a structural optimization of the product, an improvement on the cooling channel and a suit of overall tools were proposed to further reduce the warpage.

2 Optimization methodology To find out how the injection parameters affect the warpage, the clamping force and the melt temperature is the basis of the multiobjective optimization. The EBFNN is used to approach the relationships between the objectives and the injection parameters so that the optimization process can be conducted based on a reliable prediction system. The prediction precision of the EBFNN depends on the sample information. The EBFNN surrogate model can capture the variation tendency accurately as long as the samples distribute uniformly, so the Opt LHD sampling method is employed. Then, the MOPSO method is adopted to search for the optimal parameter combination.

Int J Adv Manuf Technol Finite element model development

Sampling by the Opt LHD

For calculating the warpage and the clamping force For predicting the relationship between the factors and the responses

Training the predicting system conducted by the EBFNN

Verification of the predicting system

Effect analysis of parameters by ANOVA

Comparing to the Kriging method and the RMS Standardized effect, main effect, and interaction

Final decision on the process parameters used in the optimization

Fig. 1 Sketch map of the Opt LHD sampling

2.1 Optimal Latin hypercube design

The multi-objective optimization based on MOPSO

Fig. 3 Flowchart

The Opt LHD [22] was proposed based on the LHD [23] in order to minimize the integrated mean squared error [24] and maximize the entropy [25]. The Opt LHD sampling improves the LHD in aspect of uniformity and stability. It can make the fitting between factors and responses more accurate. The sampling result of the Opt LHD is given in Fig. 1. It can be carried out as follows: STEP 1 The algorithm first selects several active pairs, which make the objective function value minimum. It finds the best components from the selected active pairs for minimizing the integrated mean squared error and maximizing the entropy. It

Fig. 2 Block diagram of EBF neural network

selects pairs randomly and iterates until there is no further improvement. STEP 2 The constrained quasi-Newton routine algorithm is adopted to optimize the midpoint Latin hypercube design. It searches the optimal location in the interval by varying the coordinate from 0 to 1.

2.2 Ellipsoidal basis function neural network EBFNN [26] is a forward-feedback neural network consisting of three layers: the input layer, the hidden layer, and the output layer (shown in Fig. 2). The core function is the EBF which can make the partition of the input space limitary and bounded. Each pair of the input node and the hidden node is connected by two weights representing the core and the axis of the ellipsoidal unit in the input space [27]. All the hidden nodes and the output nodes are connected. Let X(x1, x2, …, xn) be an input vector, the hidden nodes and the output nodes of the EBFNN are given as follows: 2 n  X xi −c ji in ð1Þ E j ðX Þ ¼ 1− b2ji i¼1

Fig. 4 The plastic oil cooler cover of diesel engine

Int J Adv Manuf Technol Table 1

Properties of PA66 (GF 30 %)

Material structure

Glass fiber

Solid density (g/cm3)

Melt density (g/cm3)

Elastic modulus (MPa)

Poisson’s ratio

Shear modulus (MPa)

Ejection temperature (°C)

Recommended mold temperature (°C)

Recommended melt temperature (°C)

Crystalline

30 %

1.36

1.16

8500

0.4

1910

185

90

300

E out j ðX Þ ¼

eα þ 1 1   eα 1 þ exp −αEin ðxÞ

ð2Þ

j

where cji and bji are the core and the long radius axis of the ellipsoidal unit function j in the input space i. Ein j (X) ranges (X) ranges from 0 to 1. The slope of from −∞ to 1, and Eout j curve can be adjusted by α in Eq. 2. 2.3 Multiobjective particle swarm optimization The PSO algorithm works by initializing a flock of particles randomly over the searching space. Each particle successively adjusts its position toward the global optimum through finding the two extreme values: the best position encountered by itself (Pi =(pi1,pi2,⋯,piD)) and the best position found so far by the whole swarm (Pg =(pg1,pg2,⋯,pgD)). Then, the velocity vid and the position xid of particle i at the next iteration are calculated according to the following equations:     t t t ¼ w⋅v þ c ⋅r ⋅ p −x ⋅r ⋅ p −x vtþ1 þ c ð3Þ 1 1d 2 2d i g id id id id t t xtþ1 id ¼ xid þ vid

STEP 3 According to the objects and the constraints, population ranking will be done and the non-dominated individuals of the population are stored in an external repository. STEP 4 The population will be evolved through the update of the velocities and the positions of particle s. STEP 5 The population combines with the nondominated individuals in external repository, and the sorting is calculated. STEP 6 All the non-dominated individuals of the population are stored in the external repository again. When the predetermined maximum iterations are obtained by circulation, the algorithm will be ended, and the Pareto optimal results will be achieved.

3 Optimization process 3.1 Injection molding process parameters

ð4Þ

where w is the inertia weight of the particle, c1 and c2 are the acceleration constants, and r1d and r2d are two random numbers in the range [0, 1] for the dth dimension separately. The main algorithm of MOPSO can be described as [28]: STEP 1 Initialize xtid of the population; STEP 2 Initialize vtid of each particle;

Fig. 5 PVT characteristics of PA66 (GF 30 %) at different temperatures

The seven injection process parameters studied in this paper are the valve gate timings, the mold temperature, the melt temperature, the injection time, the packing pressure, the packing time, and the cooling time. Among the above parameters, the valve gate timing is relatively independent in the molding injection process.

Fig. 6 Shear viscosities of PA66 (GF 30%) under different pressures

Int J Adv Manuf Technol

Fig. 7 The finite element model of the injection molding system of the oil cooler cover

So, the optimization of the valve gate timing will be carried out individually before the other six. Fig. 8 Predicted warpage with different gate timings

3.2 Objectives 3.3 Flowchart To improve the plastic product quality and to provide more options on the injection mold machine, the warpage and the clamping force are the two main objectives in this study. Besides, in order to reach smaller energy consumption and higher production efficiency, the melt temperature and the production period is considered as another two objectives.

The flowchart of the proposed optimization method based on the Opt LHD, the EBFNN, and the MOPSO is given as below (Fig. 3).

4 Application Table 2

Samples of the valve gate timings

The plastic oil cooler cover used in this study (shown in Fig. 4) is 872 × 308 × 4 mm, which is a part of the cooling system of a diesel engine and supposed to have

No.

t1 (s)

t2 (s)

t3 (s)

Warpage(mm)

1

0

1.11

2.37

5.547

2 3 4 5 6

0 0 0 0 0

−3 −2.05 −1.11 −2.68 −2.37

−1.11 −2.37 1.42 1.74 0.16

4.622 4.492 4.983 5.277 4.923

1

0

−3

−1.42

4.610

7 8 9 10

0 0 0 0

3 0.79 −0.79 −1.42

1.11 3 −0.47 −3

5.217 5.401 4.637 4.539

2 3 4 5

0 0 0 0

−2.84 −2.68 −2.53 −2.37

−0.95 −2.68 −1.74 −0.32

4.709 4.458 4.501 4.826

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1.42 0.16 1.74 0.47 2.37 2.68 2.05 −1.74 −0.16 −0.47 1.5 3 −1.5 0 −3

−1.74 −2.05 −2.68 0.47 −0.16 −1.42 0.79 2.68 −0.79 2.05 3 −1.5 −3 1.5 0

4.723 4.600 4.959 4.909 5.341 4.701 5.205 5.465 4.591 5.088 5.756 4.627 4.602 4.981 4.863

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

−2.21 −2.05 −1.89 −1.74 −1.58 −1.42 −1.26 −1.11 −0.95 −0.79 −0.63 −0.47 −0.32 −0.16 0

−0.63 −1.89 0 −3 −2.05 −0.47 −2.53 −1.58 −2.21 −2.84 −1.11 −0.79 −2.37 −0.16 −1.26

4.782 4.462 4.881 4.651 4.485 4.677 4.652 4.533 4.604 4.482 4.562 4.670 4.509 4.810 4.514

Table 3 No.

Additional samples of the valve gate timings t1 (s)

t2 (s)

t3 (s)

Warpage (mm)

Int J Adv Manuf Technol

4.2 The optimization process of the valve gate timings 4.2.1 The sampling of the valve gate timings The opening times of gate 1, gate 2, and gate 3 are defined as t1, t2, and t3 respectively, and t2 and t3 correlate to t1. In this section, the three opening times are used as the design variables, and the objective is to minimize the warpage of the plastic structure. The ranges of t1, t2, and t3 are all from −3 to 3 s. Twenty-five samples were obtained by the Opt LHD and listed in Table 2. 4.2.2 Warpage prediction with different gate opening timings

Fig. 9 Detailed warpage in the gate open timing range [−3, 0]s

high strength and high durability. The material used is a type of PA66 with 30 % fiber glass supplied by Lanxess Chemical Co. Ltd. The properties, the PVT characteristics, and the shear viscosities of the PA66 (GF 30 %) are listed in Table 1, Figs. 5 and 6, respectively.

4.1 The finite element model of the injection molding system The injection molding system used consists of the oil cooler cover, the cooling channels, and the runner system. The cooling channel diameter is 10 mm. A valve gated hot runner with three gates was employed to control the valve gate timing so as to carry out a sequential injection which can reduce the warpage in the injection molding process [29]. As shown in Fig. 7, a finite element (FE) model containing 48,725 elements was developed for simulating the injection molding process using Moldflow (version 2012). Table 4

0 0 0 0 0 0 0 0 0 0

4.2.3 Optimal valve gate timing A sample constrained single-objective optimization of the valve gate timing was carried out based on the refined prediction. The optimal valve gate timings are t1 =

Test results of warpage predictions

No. t1(s) t2(s) 1 2 3 4 5 6 7 8 9 10

Based on the samples, an EBFNN was then trained to predict warpage. The prediction results with different timings are shown in Fig. 8. It can be seen that the warpage is lower when t2 and t3 are in the range of −3 to 0 s. To refine the predicting performance in this range, another 20 samples were added (Table 3). With the 45 samples listed in Tables 2 and 3, the prediction was carried out by EBFNN. The smoothing filter is 0.005 to relax the requirement of the RBF approximation passing through every single data point. Its primary purpose is to smooth out noisy data. The maximum iteration is 5000. A more detailed distribution of warpage within the gate open timing range of −3 to 0 s is shown in Fig. 9. To test the accuracy of the EBFNN prediction system, ten samples not included in the prediction were used in the testing. The test results (shown in Table 4) show a good agreement between the predicted warpage and the numerical experiments. So, it is reasonable to use the proposed EBFNN prediction system instead of FE simulations to save computational cost.

−2.33 −3.00 0.33 1.67 3.00 −1.00 −1.67 −0.33 1.00 2.33

t3(s)

Simulated warpage (mm) First predicted warpage (mm) Deviation (%) Second predicted warpage (mm) Deviation (%)

−0.33 0.33 1.67 2.33 1.00 −3.00 3.00 −1.00 −1.67 −2.33

4.846 4.978 5.022 5.475 5.352 4.476 5.489 4.547 4.639 5.001

4.815 4.935 5.089 5.584 5.203 4.524 5.468 4.581 4.724 4.800

0.640 0.874 1.342 1.998 2.793 1.072 0.377 0.737 1.832 4.027

4.823 4.966 5.161 5.485 5.223 4.516 5.472 4.577 4.675 4.865

0.466 0.235 2.770 0.190 2.408 0.891 0.315 0.662 0.776 2.723

Int J Adv Manuf Technol Table 5

Variable ranges of the injection molding parameters

Parameters

A (°C)

B (°C)

C (s) D (MPa) E (s)

Variable range 60–120 270–320 5–10 20–80

4.3 Optimization process of the injection molding parameters

F (s)

10–30 20–60

3, t2 =0, and t3 =0 s. The corresponding warpage from the FE simulation and the EBFNN are 4.42 and 4.43 mm, respectively. The deviation is only 0.22 %. Table 6

4.3.1 Sampling of the injection molding parameters Since the valve gate timings have been optimized in Sect. 4.2, the optimization in this section contains six variables including the mold temperature (A), the melt temperature (B), the injection time (C), the packing pressure (D), the packing time (E), and the cooling time (F). The variable ranges are given in

Samples of the injection molding parameters

Training samples

Test samples

No.

A (°C)

B (°C)

C (s)

D (MPa)

E (s)

F (s)

Warpage (mm)

Clamping force (t)

1

87.69

316.15

5.64

30.77

25.38

42.56

4.255

455.9

2 3

98.46 83.08

308.46 275.13

9.62 9.49

73.85 60

18.72 20.26

52.82 26.15

5.406 3.687

1247.1 815.5

4

120

290.51

9.87

47.69

16.15

37.44

3.985

677.5

5 6 7 8 9

60 90.77 75.38 110.77 84.62

293.08 280.26 299.49 284.1 302.05

8.85 9.1 9.74 6.28 8.85

27.69 38.46 53.85 33.85 41.54

21.79 17.18 10 26.41 30

40.51 58.97 39.49 50.77 56.92

3.823 3.772 4.32 3.735 3.513

300.1 417.7 852 365.8 606.6

10 11 12 13 14

86.15 66.15 109.23 64.62 89.23

313.59 271.28 304.62 314.87 295.64

10 7.56 5 6.41 8.59

49.23 43.08 35.38 40 26.15

23.85 12.05 13.59 16.67 15.13

30.26 38.46 33.33 27.18 21.03

4.029 4.013 4.338 4.686 4.071

807.9 454.8 527.5 644.9 291.2

15 …… 100 1 2 3 4 5

96.92 …… 115.38 97.89 66.32 120 85.26 110.53

308.46 …… 317.44 283.16 293.68 304.21 277.89 280.53

5.26 …… 7.69 9.74 6.84 8.16 7.11 8.42

69.23 …… 58.46 42.63 70 51.05 67.89 32.11

22.31 …… 27.95 35.26 25.79 16.32 27.37 17.89

28.21 …… 44.62 26.32 53.68 24.21 20 51.58

5.109 …… 4.825 3.93 3.69 4.51 3.59 3.84

1180.3 …… 993.6 525.1 1112.1 823.9 1008.7 301.6

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

116.84 104.21 78.95 82.11 107.37 88.42 113.68 94.74 60 69.47 101.05 72.63 91.58 63.16 75.79

285.79 317.37 314.74 291.05 306.84 320 309.47 288.42 272.63 312.11 275.26 298.95 270 301.58 296.32

8.68 7.63 8.95 10 5 9.21 7.37 5.53 7.89 6.32 5.79 5.26 6.05 6.58 9.47

65.79 53.16 30 59.47 63.68 61.58 36.32 34.21 40.53 55.26 57.37 38.42 48.95 46.84 44.74

30.53 14.74 19.47 11.58 28.95 32.11 36.84 24.21 21.05 13.16 10 22.63 38.42 40 33.68

49.47 57.89 34.74 41.05 38.95 32.63 45.26 22.11 36.84 28.42 43.16 55.79 47.37 30.53 60

3.38 5.56 4.78 4.22 4.11 3.75 3.54 3.88 3.97 4.39 4.47 4.33 3.66 3.55 3.56

1004.6 892.7 443 918.7 1074.6 1056.8 538.8 425.8 416.3 927.6 788.9 555.9 571.6 719.2 635.6

Int J Adv Manuf Technol

where x is the sample point, β is the regression coefficient, f(x) is a polynomial function of x, and Z(x) is a stochastic function with mean zero, variance σ2, and nonzero covariance. The covariance matrix of Z(x) that dictates the local deviations is:       Cov Z ðxi Þ; Z x j ¼ σ2 R R xi ; x j ð6Þ where R is the correlation matrix, and R(xi,xj) is the correlation function between any two of the sampled data points xi and xj. In this study, the Gaussian function was selected. The fit type is anisotropic fit. The filter distance is 0.0005. The maximum iteration is 5000. 2   corr xi ; x j ¼ ∏eθk jxik −x jk j

Fig. 10 Comparison of predicted warpages of different methods

Table 5. One hundred combinations of variable values were sampled as the training set, and another 20 samples are generated as the testing set (listed in Table 6).

where θk is the unknown correlation parameters used to fit the model. The parameters β, σ2, and θk can be estimated by maximizing the likelihood of samples [13]. The RSM model can be represented by a polynomial of the following form: f ¼ a0 þ

n X

a i xi þ

i¼1

4.3.2 The prediction of the warpage and clamping force In this study, the clamping force is taken into account which could strongly affect the choice of injection machine and the power consumption. Therefore, both the warpage and the clamping force are predicted with the above 100 samples. To verify the predicting precision of the proposed EBFNN system, anther two methods are employed to compare with the EBFNN which are the Kriging method and the RSM. The parameter settings of the EBFNN are the same as Sect. 4.2.2. The Kriging model can be written as: yðxÞ ¼ F ðβ; xÞ þ Z ðxÞ ¼ f T ðxÞβ þ Z ðxÞ

ð5Þ

ð7Þ

n n X X i¼1

ai j xi x j þ ⋅⋅⋅

ð8Þ

j¼1

where a0, ai, and aij are the polynomial coefficients, xi is the set of model inputs, and n is the number of model inputs. Coefficients of the polynomial are determined by solving a linear system of equations (one equation for each analyzed design point). In this study, a quartic polynomial model was constructed. The predicted warpage and clamping force using the EBFNN, the Kriging method, and the RMS overlaid with the FE results are shown in Figs. 10 and 11. It can be seen that most of the EBFNN predicted values agree with the FE results better than the other two methods. Thus, it is appropriate to conduct the following analysis and optimization based on the EBFNN prediction system. 4.3.3 Effect analysis using ANOVA Using the samples obtained in Sect. 4.3.1, the multivariate quadratic regression model is created as the following [30]: X X X y ¼ β0 þ β i xi þ βii x2i þ β i j xi x j ð9Þ i≠ j

where y is the objective, xi and xj are the variables, β0, βi, βii, and βij are the regression coefficients. Before fitting the polynomial model, the input data was normalized into the range of –1 to 1 in order to compare the variable contributions fairly. The regression was performed using the least squares fitting method:

Fig. 11 Comparison of clamping forces of different methods

  2 xi −xmin i T xi ¼ max min −1 xi −xi

ð10Þ

Int J Adv Manuf Technol Fig. 12 Main effects of variables on warpage

Fig. 13 The effect of the parameters on the warpage

Fig. 14 The effect of the parameters on the clamping force

Int J Adv Manuf Technol Fig. 15 Interactions between mold temperature and significant factors in the warpage case

(a) Packing pressure

(b) Packing time

(c) Melt temperature Fig. 16 The interaction effect of mold temperature and highimpact parameters on the clamping force

(a) Packing pressure

(b) Packing time

(c) Melt temperature

Int J Adv Manuf Technol Fig. 17 The Pareto frontier achieved by MOPSO for EBFNN model

y ¼ λ0 þ

X

λ i T xi þ

X

λii T xi 2 þ

X

λ i j T xi T x j

ð11Þ

i≠ j

where Txi and Txj are the normalized variables, λ0, λi, λii, and λij are the corresponding regression coefficients. The effects of the variables on the objective can be calculated as: δxi ¼

jλi j  100% n X jλi j

ð12Þ

i¼1

The main effects of the injection molding parameters on the warpage are shown in Fig. 12. A main effect is the effect of an independent variable on the dependent variable averaging across the levels of any other independent variables. It ignores the effects of all other independent variables. The value of the main effect curve is the mean value of the responses Table 7 The optimal parameter combination with MOPSO and verification results

A(°C)

93.5

B(°C)

280.91

C(s)

7.5

D (MPa)

43.16

corresponding to all variable combinations at a certain level. From the results, it can be found that the relationship between the warpage and the mold temperature, the melt temperature, the injection time, the packing pressure, the packing time, and the cooling time are all nonlinear. Besides, the correlations between the affecting factors (except the packing time) and the response are not monotonic. However, in the given parameter ranges, only one minimum response value can be found in each case. Therefore, the optimal combination of the injection molding parameters with respect to the warpage can be found as 93.5 °C for the mold temperature, 283 °C for the melt temperature, 9.5 s for the injection time, 45 MPa for the packing pressure, 40 s for the packing time, and 40 s for the cooling time. The corresponding warpage and clamping force are 3.68 mm and 566.6 N, respectively. The standardized effects of the process parameters on the warpage and the clamping force are given in Figs. 13 and 14, respectively. It can be found that the packing time and the melt E (s)

19

F (s)

26

Warpage(mm)

Clamping force(t)

Predicted

Simulated

Predicted

Simulated

3.66

3.73

519.7

539.8

Int J Adv Manuf Technol

Fig. 20 The average temperature of plastic oil cooler cover Fig. 18 The simulation warpage based on MOPSO

temperature affect the warpage significantly, and the packing pressure and the melt temperature have significant influence on the clamping force. Overall, there are three significant factors affecting the responses of the warpage and the clamping force, which are the packing time, the melt temperature, and the packing pressure. It should be noticed that the mold temperature has very low impacts on both the warpage and the clamping force. The interaction between the mold temperature and the significant factors (the melt temperature, the packing pressure, and the packing time) on the warpage and the clamping force are shown in Figs. 15 and 16. An interaction graph shows the main effect of a selected factor on a response at each level of another factor. Here, the interaction graph includes the higher level and the lower level of the corresponding factor. When the two curves are parallel, it means that no or less interaction effect exists. Otherwise the interaction effect exists. For the warpage, the effects of significant factors are not correlated to the mold temperature. For the clamping force, the effect of packing time shows a relatively strong interaction with the mold temperature, but in this case, the packing time does not significantly influence the response (clamping force). As a result, in order to improve the efficiency of the multiobjective optimization, it is reasonable to set the

mold temperature as a constant. According to the main effect result given in Fig. 12, the mold temperature is fixed at 93.5 °C in the following optimization. 4.3.4 The multiobjective optimization based on MOPSO The multiobjective optimization problem in this study was defined as follows: x ¼ ðA; B; C; D; E; F Þ WarpageðxÞ Clamping forceðxÞ Minimize : Production periodðxÞ B A ¼ 93:5 270 ≤ B ≤ 320 5s≤ C ≤ 10s Subject to : 20Mpa≤ D ≤ 80Mpa 10s≤ E ≤ 40s 20s≤ F ≤ 60s Production periodðxÞ ¼ C þ E þ F Find

:

(a)The front side

(b)The back side Fig. 19 The simulation clamping force based on MOPSO

Fig. 21 The optimization structure of plastic oil cooler cover

ð13Þ

Int J Adv Manuf Technol

(a) Original structure Fig. 24 The conformal cooling channels

(b) Optimized structure Fig. 22 The stress comparison between original structure and optimized structure

where x is the set of design variables. The lower and upper limits of the design variables are provided by the manufacturer. The warpage (x) and the clamping force (x) were replaced by the EBFNN predicting system in optimization iterations. The maximum number of iterations is 1000. The Pareto frontier achieved by the MOPSO is shown in Fig. 17. Different combinations of optimal tradeoff solutions can be selected from these solutions, and the optimum combination is shown as red in Fig. 17. The detailed parameters of the optimal result are shown in Table 7, a comparison between the optimal and the simulated results of the peak warpage and the peak clamping force are given as well. The warpage distribution of the plastic oil cooler cover and the clamping force throughout the injection molding process are shown in Figs. 18 and 19, respectively. 4.4 Structural optimization and cooling channel improvement The average temperature distribution of the plastic oil cooler cover (shown in Fig. 20) was calculated based on the optimal parameter combination. It can be seen

Fig. 23 The temperature distribution of the cooling channels

that the temperature varies in location. The temperature at edges is higher than that at the center after the same cooling time. It is because a thicker part needs longer cooling time, and the thickness of the edge is much larger than the center. To solve this problem, several grooves were added on the edge, and some reinforcing ribs were also added onto the surface to improve the structural strength. The optimized structure of the plastic oil cooler cover is shown in Fig. 21. To compare the optimized structure to the original one, the stress distributions were calculated under a pressure (1 MPa) applied on the inner face, and the bolt holes were constrained. From the comparison shown in Fig. 22, it can be seen clearly that the stress at the cover center, the interface between the higher and lower surface, and the fillet near the edge is reduced. The reduction of the peak stress of the optimized structure is about 60 MPa. The temperature distribution of the cooling channels (shown in Fig. 23) shows a difference of about 7 °C between the inlet and the outlet. To improve the cooling efficiency, a conformal cooling channels design was proposed (shown in Fig. 24). Each cooling channel is divided into three sections with the diameter increased from 10 to 15 mm. The middle and the right sections are much closer to the oil cooler cover than the original design. With the new design (conformal cooling channels), the temperature difference between the inlet and the outlet is reduced to about 2 °C (shown in Fig. 25). Besides, the temperature at the edge is decreased, and the temperature distributes more evenly throughout the entire production as shown in Fig. 26. Under this condition, the warpage and the clamping force were recalculated and shown in Figs. 27 and 28.

Fig. 25 The temperature distribution of the conformal cooling channels

Int J Adv Manuf Technol

Fig. 26 The average temperature of optimized plastic oil cooler cover

As shown in Fig. 27, the average warpage is about 2.892 mm, and only the warpage at the upper right corner is a bit higher. On the whole, the warpage of the plastic oil cooler cover with an optimized structure and produced in an optimized approach is reduced significantly. In addition, the peak clamping force is 470 t for the optimized design which is much smaller than the original one. 4.5 Overall tools and warpage measurement As the temperature of the new design production was still at a high level, a suite of overall tools consisting of a board and a hold-down device was developed to help prevent the cooling deformation after the injection molding process (shown in Fig. 29). The warpage of a plastic oil cooler cover treated with the proposed tools was measured using a three-coordinate measuring machine (shown in Fig. 30) and compared with that without post processing (shown in Fig. 31). The results show that the warpage of the former one is 1.027 mm and that of the latter is 2.819 mm. In other words, the warpage has been reduced by more than 60 % (1.792 mm).

5 Conclusions

Fig. 28 Calculated clamping force with the optimized design

responses. An EBFNN-based predicting system was then founded and trained based on the Opt LHD sampling and validated by comparing to the Kriging method and the RSM. Then, the effects of process parameters on the warpage and the clamping force were analyzed using ANOVA, and the design parameters used in the optimization were determined accordingly. A multiobjective optimization was carried out based on the MOPSO method. After that, a structural optimization and a cooling channel improvement were conducted to further reduce the warpage and the clamping force. At last, a suite of overalls tools was developed and utilized to prevent the cooling deformation after the injection molding. Main conclusions can be drawn as follows: (1) The open times of the valve gates affect the warpage independently so that it can be optimized individually. The optimal valve gate timings are t1 =3, t2 =0, and t3 =0 s, and the corresponding warpage is reduced by 12 %.

In this study, a multiobjective optimization of the injection molding parameters was proposed to tradeoff these conflicting process parameters. An FE model of the injection molding system was developed to calculate the

Fig. 27 Calculated warpage with the optimized design

Fig. 29 Overalls tools and the optimized production

Int J Adv Manuf Technol

References 1.

2.

3.

4. Fig. 30 Warpage measurement 5.

(2) The packing time and the melt temperature affect the warpage significantly, and the packing pressure and the melt temperature have significant influence on the clamping force. The mold temperature has very low interactions with other process parameters which can be set as constant in the optimization. (3) The proposed multiobjective optimization method based on the Opt LHD, the EBFNN, and the MOPSO can predict the responses (the warpage and the clamping force) and reduce the responses precisely and efficiently. Besides, the manufacture cost and melt temperature can be reduced as well. As a result, the optimized warpage and clamping force are 3.73 mm and 579.8 t. (4) The structural optimization, cooling channel improvement, and the overalls tools can reduce the warpage and the clamping force further. The stress of the optimized structure is reduced to about 60 MPa, and the warpage is 2.89 mm. The warpage of the treated production is 1.027 mm, which is reduced for about 4 mm compared with the original one. The clamping force is further reduced to 470 t.

6.

7.

8.

9.

10.

11.

12.

13.

14. Acknowledgments This work is supported by a grant from the National High Technology Research and Development Program of China (863 Program) (No. 2014AA041501).

15.

16.

17.

18.

19.

20. Fig. 31 Comparison between the after-treated plastic oil cooler cover (left) and the untreated one (right)

Chiang HH, Hieber CA, Wang KK (1991) A unified simulation of the filling and postfilling stages in injection molding. Part I: formulation. Polym Eng Sci 31(2):116–124 Turng LS (1995) Development and application of CAE technology for the gas-assisted injection molding process. Adv Polym Technol 14(1):1–13 Shelesh-Nezhad K, Siores E (1997) An intelligent system for plastic injection molding process design. J Mater Process Technol 63(1): 458–462 Kabanemi KK, Vaillancourt H, Wang H, Salloum G (1998) Residual stresses, shrinkage, and warpage of complex injection molded products: numerical simulation and experimental validation. Polym Eng Sci 38(1):21–37 Seow LW, Lam YC (1997) Optimizing flow in plastic injection molding. J Mater Process Technol 72(3):333–341 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):793–811 Chang TC, Faison E III (1999) Optimization of weld line quality in injection molding using an experimental design approach. J Inject Mold Technol 3(2):61 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):1–9 Chen CP, Chuang MT, Hsiao YH, Yang YK, Tsai CH (2009) Simulation and experimental study in determining injection molding process parameters for thin-shell plastic parts via design of experiments analysis. Expert Syst Appl 36(7):10752–10759 Erzurumlu T, Ozcelik B (2006) Minimization of warpage and sink index in injection-molded thermoplastic parts using Taguchi optimization method. Mater Des 27(10):853–861 Shi F, Lou ZL, Zhang YQ, Lu JG (2003) Optimisation of plastic injection moulding process with soft computing. Int J Adv Manuf Technol 21(9):656–661 Kurtaran H, Erzurumlu T (2006) Efficient warpage optimization of thin shell plastic parts using response surface methodology and genetic algorithm. Int J Adv Manuf Technol 27(5-6):468–472 Gao Y, Wang X (2008) An effective warpage optimization method in injection molding based on the Kriging model. Int J Adv Manuf Technol 37(9-10):953–960 Gao Y, Wang X (2009) Surrogate-based process optimization for reducing warpage in injection molding. J Mater Process Technol 209(3):1302–1309 Deng YM, Zhang Y, Lam YC (2010) A hybrid of mode-pursuing sampling method and genetic algorithm for minimization of injection molding warpage. Mater Des 31(4):2118–2123 Li C, Wang FL, Chang YQ, Liu Y (2010) A modified global optimization method based on surrogate model and its application in packing profile optimization of injection molding process. Int J Adv Manuf Technol 48(5-8):505–511 Chiang KT, Chang FP (2006) Application of grey-fuzzy logic on the optimal process design of an injection-molded part with a thin shell feature. Int Commun Heat Mass Transfer 33(1):94–101 Chiang KT (2007) The optimal process conditions of an injectionmolded thermoplastic part with a thin shell feature using grey-fuzzy logic: a case study on machining the PC/ABS cell phone shell. Mater Des 28(6):1851–1860 Cheng J, Liu Z, Tan J (2013) Multiobjective optimization of injection molding parameters based on soft computing and variable complexity method. Int J Adv Manuf Technol 66(5-8):907–916 Xu G, Yang Z (2015) Multiobjective optimization of process parameters for plastic injection molding via soft computing and grey correlation analysis. Int J Adv Manuf Technol 78(1):525–536

Int J Adv Manuf Technol 21.

Zhao J, Cheng G, Ruan S, Li Z (2015) Multi-objective optimization design of injection molding process parameters based on the improved efficient global optimization algorithm and non-dominated sorting-based genetic algorithm. Int J Adv Manuf Technol 78(9): 1813–1826 22. Park JS (1994) Optimal Latin-hypercube designs for computer experiments. J Statistical Plann Inference 39(1):95–111 23. McKay MD, Beckman RJ, Conover WJ (1979) Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245 24. Sacks J, Schiller SB, Welch WJ (1989) Designs for computer experiments. Technometrics 31(1):41–47 25. Shewry MC, Wynn HP (1987) Maximum entropy sampling. J Appl Stat 14(2):165–170

26.

Jakubek S, Strasser T (2002) Fault-diagnosis using neural networks with ellipsoidal basis functions. Proc Am Control Conf 5(8-10): 3846–3851 27. Luo J, Lin S, Ni J, Lei M (2008) An improved fingerprint recognition algorithm using EBFNN. Porc IEEE Second International Conference on Genetic and Evolutionary Computing, Hubei, China, p 504–507 28. Reyes-Sierra M, Coello CAC (2006) Multi-objective particle swarm optimizers: a survey of the state-of-the-art. Int J Comput Intell Res 2(3):287–308 29. Chen XF, Yu SH, Zhou SQ (2012) Improvement of the plastic part quality with the sequential injection molding (SIM). J Wuhan Univ Technol 34(6):32–35 30. Wang X, Zhao G, Wang G (2013) Research on the reduction of sink mark and warpage of the molded part in rapid heat cycle molding process. Mater Des 47:779–792