Injection molding is the most widely used process in manufacturing polymer products. The warpage induced during injection molding process has an impor...

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ORIGINAL ARTICLE

Optimization of injection molding process parameters to improve the mechanical performance of polymer product against impact Yingjie Xu & QingWen Zhang & Weihong Zhang & Pan Zhang

Received: 16 June 2014 / Accepted: 25 September 2014 # Springer-Verlag London 2014

Abstract Injection molding is the most widely used process in manufacturing polymer products. The warpage induced during injection molding process has an important influence on the mechanical performance of injection molded products. Therefore, how to optimize process parameters becomes the key issue in improving the mechanical performance of the product towards the expected service conditions. In this paper, a combined artificial neural network and particle swarm optimization (PSO) algorithm method is proposed to optimize the injection molding process. An integrated finite element analysis of the injection molding process, the warpage-induced residual stresses during assembly, and mechanical performance of serviced product is firstly proposed. A back propagation neural network model is then developed to map the complex nonlinear relationship between process parameters and mechanical performance of the product. The PSO algorithm is interfaced with this predictive model to optimize process parameters and thereby significantly improve the mechanical performance. A case study of vehicle window made of polycarbonate (PC) is presented. Optimum values of process parameters are determined to minimize the maximum von Mises stress within the PC vehicle window under impact loading.

Keywords Injection molding . Warpage . Impact . BP neural network . PSO algorithm

Y. Xu (*) : Q. Zhang : W. Zhang (*) : P. Zhang Engineering Simulation and Aerospace Computing (ESAC), Northwestern Polytechnical University, P.O. Box 552, 710072 Xi’an, Shaanxi, China e-mail: [email protected] e-mail: [email protected]

1 Introduction Polymer materials offer a wide range of advantages such as high strength-to-weight ratio, high transparency, high flexibility, recyclability, corrosion resistance, and fast processing times, which make them very attractive materials. An important application of polymer refers to the transparent products in automobile and aerospace industries, for instance vehicle window, aircraft windshield, and astronaut viewing window. In these particular applications, the destruction of products caused by high-speed impacting is one of the most common threatens [1]. The structural safety and stability of the products depend on the mechanical performance of the products under high-speed impact loading. Considering the fact that the geometry of a product is usually determined at the initial design stage and cannot be easily changed, optimization of process parameters should be more feasible and reasonable. Injection molding is the most widely used process for manufacturing polymer products [2]. In this process, the melted polymer is injected into a mold cavity with a desired shape and then cooled down under a high packing pressure. Warpage is the most serious defect in injection molded parts, especially the thin-walled products [3, 4]. Residual stresses are often generated within the injection molded product due to the forced deformation of the warped geometry during the assembly process. In fact, the warpage-induced residual stresses have a remarkable influence on the mechanical properties of product and are highly related to molding process parameters. It is therefore essential to take into account the influence of molding process in the description of the mechanical behavior of the serviced product. If molding process parameters can be adjusted in an intelligent way, the mechanical performance of the product can be improved towards the expected service conditions. In recent years, numerous research studies have been carried out to optimize molding process parameters to reduce the

Int J Adv Manuf Technol

Fig. 1 Illustration of the integrated analysis

warpage of the injection molded products [5–11]. The warpage is a nonlinear, implicit function of the process parameters, which can be typically evaluated by the finite element method. However, this approach is cost, ineffective, and not suitable for large number of reanalyses that are often required in an optimization process. In order to reduce the computational cost in warpage optimization, some researchers introduced the surrogate models, such as polynomial regression, artificial neural network, and support vector regression. These surrogate models can be used to construct a mathematical approximation to replace expensive finite element analyses. Intelligent optimization algorithms, for instance the famous genetic algorithm, are further used for optimization of the process parameters. Zhou et al. used support vector regression and genetic algorithm to optimize injection molding process [5]. Shen et al. optimized the process parameters by using neural network model and genetic algorithm to reduce the maximum difference of volume shrinkage of the top cover of an industrial refrigerator [6]. Kurtaran et al. performed process optimization to

reduce the warpage for a bus ceiling lamp base using neural network model and genetic algorithm [7]. Kurtaran and Erzurumlu combined response surface methodology and generic algorithm for optimization of process parameters to reduce the warpage of thin shell plastic part [8]. Mao et al. developed a back propagation neural network model for warpage prediction and optimization of the automobile glove compartment [9]. Then, Mao et al. proposed a hybrid approach combining back propagation neural network and genetic algorithm for optimization of injection molding process parameters [10]. Warpage and clamp force during injection molding process were investigated as the optimization objectives. Ozcelik and Erzurumlu developed an efficient optimization methodology using artificial neural network and genetic algorithm in minimizing warpage of a thin shell part [11]. The above research studies demonstrated that the optimized process parameters could effectively reduce the warpage value of the products. However, the optimization of process parameters was rarely taken into consideration for the improvement of the mechanical performance of the products. Motivated by this situation, optimization of the injection molding process parameters to improve mechanical performance of the product under impact loading is proposed in this paper. In order to take into account the process-induced warpage and describe the mechanical behavior of product correctly, an integrated approach closing the gap between process simulation and mechanical simulation is introduced. On the basis Table 1 Material properties of PC [15]

Fig. 2 Finite element model with gate and cooling channel

Density (kg/m3) Elastic modulus (MPa) Poisson ratio Yield strength (MPa) Tangent modulus (MPa)

1,200.00 1,500.00 0.37 62.00 32.00

Int J Adv Manuf Technol

of integrated approach, a mathematical function mapping the relationship between mechanical behaviors and process parameters is acquired by using back propagation (BP) neural network and the data obtained from the finite element simulations. The particle swarm optimization (PSO) algorithm is interfaced with this prediction model to find the optimal process parameters and thereby significantly improve the mechanical performance. The optimization study is finally carried out on a vehicle window made of polycarbonate (PC). Mold temperature, melt temperature, injection velocity, compression distance, compression force, compression velocity, and compression waiting time are considered as design parameters. Optimum values of process parameters are determined to minimize the maximum von Mises stress within the PC vehicle window against impact.

2 Integrated analysis of the PC vehicle window The proposed integrated analysis incorporates effects of the injection molding process into the mechanical performance simulation of impacted vehicle window. This new approach is able to close the gap between process simulation and mechanical simulation. The integrated analysis is schematically illustrated in Fig. 1. Firstly, the finite element package MoldFlow [12] is used to simulate the injection molding process and obtain the warpage results induced during the injection molding process. Secondly, the warped window is clamped and forced to deform to its expected shape. The generated residual stresses are calculated by using finite element program ANSYS [13]. Finally, the residual stresses are imposed on the finite element model of serviced product as the prestress. The mechanical behavior of the impacted product is obtained by using ANSYS. 2.1 Finite element analysis of the injection molding process Geometrical model of the vehicle window utilized in this study is created according to the real structure [14] and meshed in finite element package ANSYS. The finite element model is then imported into MoldFlow for injection molding process simulation. Figure 2 shows the finite element model of the vehicle window with gate and cooling channel. Its length, width, and thickness are 828, 632, and 4 mm. The material is PC and the material properties are given in Table 1 [15]. Finite element analysis is carried out using the process parameters given in Table 2. The computed warpage results in x, y, and z directions are displayed in panels a–c of Fig. 3, respectively. The coordinates of each node of the warped window are exported and denoted as xi, yi, and zi (i=1, 2,…, N, N is the amount of nodes). In addition, warpage values at all the nodes are also exported and denoted as Wix, Wiy, and Wiz.

2.2 Residual stresses analysis During assembly process, since the warped window is clamped and forced to deform to its expected shape, as illustrated in Fig. 4, residual stresses are generated and have a remarkable influence on the mechanical behaviors of the window under impact loading. Hence, an accurate modeling of residual stresses within the window is essential for the impact analysis. In this study, a “reconstruction model” is proposed for the calculation of residual stresses. Firstly, N nodes are generated according to the coordinate values xi, yi, and zi (i=1, 2,…, N) and elements are created by using these nodes. A new finite element model defined as “reconstruction model” is sequentially obtained in ANSYS based on the generated nodes and elements. Then, a set of displacements with values of −Wix, − Wiy, and −Wiz are applied on each node of the “reconstruction model” to generate the forced deformation of the warped window and finite element analysis is executed for computing the residual stresses. The obtained residual stress results will be applied on the finite element model as the prestress in the following impact analysis. 2.3 Impact analysis In this study, the mechanical response of PC vehicle window under impact loading is simulated. Residual stresses are applied on the finite element model as the prestress. Suppose that a 30-mm-diameter spherical steel projectile impacts at the midpoint of window. Finite element model of PC vehicle window against the sphere impact is shown in Fig. 5. As listed in Table 1, the material model used for the PC window is a bilinear plastic kinemics model in ANSYS with an elastic modulus of 2,300 MPa and a Poisson’s ratio of 0.37. A tangent modulus of 32 MPa is used to incorporate the plastic deformation. The PC material density used is 1,200 kg/m3. The yield strength for PC is 65 MPa. Here, it should be noted that the failure or damage of the PC window is not considered. Thus, the failure criterion and related values of material parameters are not given in this paper. The steel ball projectile is modeled as a rigid material with a density of 7,800 kg/m3. The elastic modulus is 200 GPa and the Poisson’s ratio is 0.30. Table 2 Process parameters

Mold temperature (°C) Melt temperature (°C) Injection velocity (cm3/s) Compression distance (mm) Compression force (t) Compression velocity (cm/s) Compression waiting time (s)

100 300 250 2 1,600 2 15

Int J Adv Manuf Technol Fig. 3 Warpage results in a x direction; b y direction; c z direction

(a)

(b)

(c) The edges of the window are constrained to avoid any translational and rotational motion which exercises a fully clamped boundary condition. Impact velocity of projectile is set to be 100 m/s and impact duration is 0.1 ms. A

surface to surface friction-less contact is defined between the projectile and the target window. The von Mises stress distribution within the impacted window is shown in Fig. 6.

Int J Adv Manuf Technol Fig. 4 Illustration of the forced deformation of warped window

3 BP neural network model for mechanical response An integrated computing procedure links up together simulations of molding process, residual stress, and impact is developed here. Undoubtedly, the computationally costly finite element models are not suitable for a large number of repeated analyses which are required in an optimization process. Therefore, in this study, a more efficient predictive model created by BP neural network is used for mechanical response analysis. It is known that BP neural network [16] has the powerful ability of nonlinear interpolation to obtain the mathematical mapping reflecting the internal law of the experimental data. Therefore, BP neural network has been widely used in engineering applications for prediction and optimization [17–19]. It has already been proved that a three-layer BP neural network with enough hidden units can give any precision solution of any continuous function in a bounded area [16]. In this work, a three-layer BP neural network is developed to predict the function relationship between the molding process parameters and the maximum von Mises stress of PC vehicle window under impact loading. The BP neural network architecture used in this study is displayed in Fig. 7. The network contains three parts: one input layer which has 7 neurons related to mold temperature,

melt temperature, injection velocity, compression distance, compression force, compression velocity, and compression waiting time; two hidden layers with 20 neurons for each one; and one output layer having 1 neuron representing the mechanical response of impacted window, i.e., the maximum von Mises stress. In the network, each neuron receives total input from all of the neurons in the preceding layer as: net j ¼

N X

ωij xi

ð1Þ

j¼0

where netj is the total input and N is the number of inputs to the jth neuron in the hidden layer. wij is the weight of the connection from the ith neuron in the forward layer to the jth neuron in the hidden layer. xi is the input from the ith neuron in the preceding layer. A neuron in the network produces its output outj by processing the input through a transfer function f (the tangent hyperbolic function is used in this study) as below: 1−e−net j out j ¼ f net j ¼ 1 þ e−net j

ð2Þ

3.1 Generation of training data To calculate the weights of connection between each neuron, the designed BP neural network should be trained by a group of experimental data. A full factorial experiment design means that a large number of finite element computations are needed. In order to reduce the number of finite element analysis, the orthogonal experiment design method [20] is used in this study. Each experimental factor range is divided into three levels between the lowest and the highest values. The setting of the processing parameters is shown in Table 3. 3.2 Training of the neural network

Fig. 5 Finite element model of PC vehicle window against sphere impact

The network training is carried out for calculating the weights of connection which minimize the mean square error between network prediction and training data. The weights are given randomly at the beginning. Then, they are iteratively updated

Int J Adv Manuf Technol Fig. 6 The von Mises stress distribution within the impacted window

until the convergence to a certain value by using the gradient descent method. The connection weights are updated as: ωnew ¼ ωold ij ij þ Δωij ∂E Δωij ¼ −η out j ∂ωij

ð3Þ

where E is the mean square error and outj is the jth neuron output. η is the learning rate parameter controlling the stability and rate of convergence of the network. η is a constant between 0 and 1 and is chosen to be 0.01 in this study. The mean square error of the training data is set as 1×10−4. Training process takes about 600 s of CPU time on HP personal workstation for 4.5×105 training iterations and Fig. 8 illustrate

Fig. 7 BP neural network architecture used in this study

the training progress of the network. It can be observed that with the updating of the connection weights, the mean square error between the network prediction data and training data declines gradually and converges to 1×10−4 within 4.5×105 iterations. After training, the mathematic mapping between the processing parameters and the maximum von Mises stress is stored in the trained net. The mathematic function can be expressed as: S ¼ fl

X

w3 f s

X

w2 f s

X

w1 X

ð4Þ

where S is the maximum von Mises stress of the impacted window; X=[x1, x2, x3, x4, x5, x6, x7] is the matrix consisting of the values of seven different process parameters; f l is the liner transfer function between hidden layer 2 and output layer; f s is

Int J Adv Manuf Technol Table 3 Setting of the processing parameters Process parameters

Level 1

Level 2

Level 3

A: mold temperature (°C) B: melt temperature (°C) C: injection velocity (cm3/s) D: compression distance (mm) E: compression force (t) F: compression velocity (cm/s) G: compression waiting time (s)

80 280 200 1 1,400 1 14

100 300 250 2 1,600 2 15

120 320 300 3 1,800 3 16

the transfer function between input layer and hidden layer 1, as well as hidden layers 1 and 2; and w1, w2, and w3 represent the connection weights between input layer and hidden layer 1, hidden layer 1 and hidden layer 2, and hidden layer 2 and output layer, respectively.

window is minimized. Mathematically, the optimization problem can be formulated as: Find : A ; B ; C ; D ; E ; F ; G Minimize : Maximum σvm ðA; B; C; D; E; F; GÞ 80 C ≤ A : mold temperature ≤ 120 C 280 C ≤ B : melt temperature ≤ 320 C 200 cm3 = s ≤ C : injection velocity ≤ 300 cm3 = s 1 mm ≤ D : compression distance ≤ 3 mm 1 ; 400 t ≤ E : compression force ≤ 1 ; 800 t 1 cm = s ≤ F : compression velocity ≤ 3 cm = s 14 s ≤ G : compression waiting time ≤ 16 s

ð5Þ

To solve the optimization problem, the PSO algorithm is coupled with the BP neural network model to yield a global optimum. 4.2 Basis of PSO algorithm

3.3 Neural network testing To test the accuracy of the neural network prediction system, 10 groups of process parameters not used in the training process are used in the testing. A detailed comparison between the BP neural network prediction system and finite element simulations can be seen in Table 4. It is seen that BP neural network prediction is in good agreement with the FE results. This indicates that the developed BP neural network model has good interpolation capability and can be used as an efficient predictive tool for the maximum von Mises stress.

The PSO algorithm is a global optimization algorithm and described as sociologically inspired [21]. In PSO, each individual of the swarm is considered as a particle in a multidimensional space that has a position and a velocity. These particles fly through hyperspace and remember the best position that they have seen. Members of a swarm remember the location where they had their best success and communicate good positions to each other, then adjust their own position and velocity based on these good positions. Updating the position and velocity is done as follows V ikþ1 ¼ ωV ki þ c1 r1 Pi k −X i k þ c2 r2 Pg k −X i k ð6Þ

4 Process optimization by particle swarm optimization algorithm

X ikþ1 ¼ X ki þ V ikþ1

4.1 Optimization problem The objective of this study is to optimize process parameters such that the maximum von Mises stress within the impacted

Fig. 8 Training process of the network

ð7Þ

where Vi and Xi represent the current velocity and the position of the ith particle, respectively (note that the subscripts k and k+ 1 refer to the recent and the next iterations, respectively); Pi is the best previous position (pbest) of the ith particle and Pg is the best global position (gbest) among all the particles in the swarm; c1 and c2 represent “trust” parameters indicating how much confidence the current particle has in itself and how much confidence it has in the swarm; r1 and r2 are two random numbers between 0 and 1; and ω is the inertia weight. The acceleration constants c1 and c2 indicate the stochastic acceleration terms which pull each particle towards the best position attained by the particle or the best position attained by the swarm. Low values of c1 and c2 allow the particles to wander far away from the optimum regions before being tugged back, while the high values pull the particles toward the optimum or make the particles to pass through the optimum abruptly. In this work, c1 =2 and c2 =2 are chosen. The role of the inertia weight ω is considered important for the convergence behavior of PSO algorithm. The

Int J Adv Manuf Technol Table 4 Comparison of BPNN predictions with FE analyses

Nos.

Process parameters

Maximum von Mises stress (MPa)

A

B

C

D

E

F

G

FE analysis

BPNN model

1

80

280

200

1

1,800

3

16

52.5483

52.5880

2 3 4 5 6 7 8 9 10

80 80 80 100 100 100 120 120 120

300 300 320 280 300 320 280 300 320

250 250 300 250 300 200 250 300 200

1 1 2 1 2 3 1 2 3

1,600 1,400 1,800 1,600 1,400 1,800 1,800 1,400 1,600

1 2 3 3 2 1 1 2 14

15 14 14 16 15 14 16 15 14

52.5330 55.2990 54.0991 52.1297 51.0041 57.9800 57.9834 57.4458 52.5866

53.3044 54.6517 53.6957 52.3264 51.2704 56.5787 57.5685 56.9829 51.8862

inertia weight is employed to control the impact of the previous history of velocities on the current velocity. Thus, the parameter ω regulates the trade-off between the global (wide ranging) and the local (nearby) exploration abilities of the swarm. A proper value for the inertia weight ω provides balance between the global and local exploration ability of the swarm, and thus results in better solutions. Numerical tests imply that it is preferable to initially set the inertia to a large value, to promote global exploration of the search space, and gradually decrease it to obtain refined solutions [21]. Thus, a dynamic variation of inertia weight proposed in reference [22] is used in this paper. The inertia weight ω is decreased dynamically based on a fraction multiplier kω as is shown below ωkþ1 ¼ k ω ωk

Fig. 9 Illustration of the variable limits handling strategy

ð8Þ

4.3 Variable limits handling strategy Most optimization problems include the problem-specific constraints and the variable limits. According to Eq. (5), it can be known that the considered problem belongs to unconstrained problems. Hence, only the variables limits, i.e., the design bounds of the values of process parameters, are needed to be effectively handled to make sure that all of the particles fly inside the search space. A classical way consists in using variables values obtained by random generations that meet the variable limits. In the present study, an alternative method introduced by Li et al. [23] dealing with the particles that fly outside the variables boundary is used. This method is derived from the harmony search (HS) algorithm. HS algorithm is based on natural musical performance processes that occur when a musician searches for a

Int J Adv Manuf Technol Fig. 10 Optimization history with iterations for maximum von Mises stress

better state of harmony. The engineers seek for a global solution as determined by an objective function, just like the musicians seek to find musically pleasing harmony as determined by an aesthetic [23]. The HS algorithm includes a number of optimization operators, such as the harmony memory (HM), the harmony memory size (HMS), the harmony memory considering rate (HMCR), and the pitch adjusting rate (PAR). In the study of Li et al. [23], the harmony memory concept had been used in the PSO algorithm to avoid searching trapped in local solutions. The other operators have not been employed. How the HS algorithm generates a new vector from its harmony memory and how it is used to improve the PSO algorithm is discussed as follows. In the HS algorithm, the harmony memory (HM) stores the feasible vectors, which are all in the feasible space. The harmony memory size determines how many vectors can be stored. A new vector is generated by selecting the components of different vectors randomly in the harmony memory. Undoubtedly, the new vector does not violate the variables boundaries. When it is generated, the harmony memory will be updated by accepting this new vector if it gets a better solution and deleting the worst vector. Similarly, the PSO algorithm stores the feasible and “good” vectors (particles) in the pbest swarm, as does the harmony memory in the HS algorithm. Hence, the vector (particle) violating the variables boundaries can be generated randomly again by such a strategy: if any component of the current particle violates its corresponding boundary, then it will be replaced by selecting the corresponding component of the particle from pbest swarm randomly. Since the new particle is generated based Table 5 Injection molding process parameters before and after optimization

on the feasible and “good” vectors (particles) in the pbest swarm, the global solution can be reached in relatively less number of iterations compared with the classical method. To highlight the presentation, a schematic diagram is given in Fig. 9 to illustrate this strategy. 4.4 Optimization result The optimization problem introduced in Section 4.1 is solved by using the above PSO algorithm. A population of 50 particles is used. The stopping criterion can be defined based on the number of iterations without update in the best values of the swarm or the number of iterations the algorithm execute. Although the latter is not a real physical stopping criterion, it is quite easy in programming implementation and hence widely used in PSO algorithms. In this work, the maximum number of iterations is limited to 1,000 and adopted as the stopping criterion. Optimization history with iterations for the maximum stress is shown in Fig. 10. The value of maximum von Mises stress within the PC vehicle window against impact, which is 58.2884 MPa before optimization, is finally reduced to 50.72082 by 12.9 % after optimization. Injection molding process parameters recommended by the MoldFlow software and the optimized process parameters are listed in Table 5. Compared with the recommended process parameters, the optimized process parameters have a lower mold temperature and a higher melt temperature. Meanwhile, a larger compression distance and compression force and a longer compression waiting time are needed. In addition, the

Injection molding process parameters

Recommended parameters Optimized parameters

A

B

C

D

E

F

G

120 85.366

280 314.2

300 293.194

2 2.7002

1,400 1,753.6

3 1.899

15 15.998

Int J Adv Manuf Technol

compression velocity and injection velocity slightly increase and decrease, respectively.

5 Conclusions In this study, an efficient optimization methodology using BP neural network and PSO algorithm is proposed in optimizing the mechanical response of thin shell polymer product manufactured by injection molding process. A PC vehicle window under impact loading is considered as example. The appropriate process parameters are determined to minimize the maximum von Mises stress within the impacted window. Mold temperature, melt temperature, injection velocity, compression distance, compression force, compression velocity, and compression waiting time are considered as design parameters. In order to take into account the process-induced warpage and describe the mechanical behavior of product correctly, an integrated analysis is introduced to close the gap between process simulation and mechanical simulation. An efficient predictive model for the maximum von Mises stress is created using BP neural network exploiting integrated analysis results. Neural network model is integrated with an effective PSO algorithm to find the optimum process parameter values. The optimization reduces the maximum von Mises stress of the initial model by 12.9 %. Most previous applications are focused on the optimization of process parameters for improving manufacturing performance. The presented study indicates that the optimization of process parameters can also be employed to improve the mechanical performances of polymer products. Acknowledgments This work is supported by 973 Program (2012CB025904), National Natural Science Foundation of China (11302174, 11432011) and the Fundamental Research Funds for the Central Universities (3102014JCS05003).

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