Int J Adv Manuf Technol (2013) 69:2605–2612 DOI 10.1007/s00170-013-5233-1

ORIGINAL ARTICLE

Study of water penetration length and processing parameters optimization in water-assisted injection molding Jian Gen Yang & Xiong Hui Zhou & Gu Ping Luo

Received: 25 March 2013 / Accepted: 29 July 2013 / Published online: 14 August 2013 # Springer-Verlag London 2013

Abstract Water penetration length is one of the most important indexes of the water-assisted injection molding parts, the maximization of which is a particularly significant optimization objective. The effects of processing parameters, such as the short shot size, melt temperature, water pressure, and delay time, on water penetration length were exploited by using single factor experiment method and computational fluid dynamics analysis. In addition, the maximization of water penetration length on dimensional transition and curved-section parts by integrating the Taguchi orthogonal array design, radial basis function neural network, and particle swarm optimization was investigated. The research results showed that the two primary parameters affecting the water penetration length were the short shot size and water pressure, and that the effects of the melt temperature and delay time were little. Furthermore, the maximum water penetration length after optimization was slightly bigger than that of the confirmation experiment, which indicated that the optimization methodology was reliable and effective.

Keywords Water-assisted injection molding . Water penetration length . Computational fluid dynamics . Design of experiments . Radial basis function . Particle swarm optimization J. G. Yang : X. H. Zhou (*) Institute of Forming Technology and Equipment, National Die and Mold CAD Engineering Research Center, Shanghai Jiao Tong University, Shanghai 200030, China e-mail: [email protected] J. G. Yang Mechanical and Electronic Engineering Department, Jingdezhen Ceramic Institute, Jingdezhen 333403, China G. P. Luo Ningbo Sunny Mould Co., Ltd, Ningbo 351400, China

1 Introduction Besides gas-assisted injection molding, water-assisted injection molding (WAIM) [1] is another new fluid-assisted injection molding technology used to mold hollow or partly hollow parts. The process of WAIM can be briefly described as follows: firstly, the mold cavity is partly filled with melt; and then, the high-pressure water is injected through dedicated hydraulic equipments; finally, the hollow parts are formed. Because of water instead of gas as molding medium, WAIM is more competitive for products with thin wall thickness and smooth inner surface. By far, WAIM technology has been developed to solve the problems that conventional injection molding process isn't able to, and is suitable in the fields of furniture and building, household items, and automotive industry. Since Germany developed WAIM technology in 1998, many research results have been achieved. Liu et al. [2] found that the water penetration was unstable in molded symmetric channels, however, it was more stable than that of gas; Liu and Lin [3] studied the effects of materials, processing parameters, and geometries of water channels on water fingering; Liu et al. [4] discussed the effects of processing parameters on surface gloss difference for thermoplastic parts; Lin and Liu [5] examined the morphology of fluid-assisted injection molded high density polyethylene/polycarbonate blends; Sannen et al. [6] investigated the influence of melt and process parameters on the residual wall thickness and part weight, and detected correlations in the occurrence of part defects. On the other hand, the studies for the water penetration length and its optimization are still comparatively rare. Ahmadzai and Behravesh [7] experimentally investigated the effects of three processing parameters on water penetration of branched pipe, and got the longest water penetration length; to optimize the WAIM process, Liu and Chen [8] conducted an experimental study, based on the Taguchi orthogonal array design, to characterize the effects of seven different processing

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parameters on WAIM of thermoplastic composites; Huang and Deng [9] experimentally studied the effects of four processing parameters on the water penetration length of curved pipe by single factor experiment method. Meanwhile, the optimal processing parameters for maximum water penetration length were obtained by Taguchi method. Up to now, the existing researches on water penetration length have been mainly focused on experiments, while the numerical simulation is fewer. Furthermore, the optimization method is the Taguchi orthogonal experimental design only which can't fully represent the design domain, and isn't quite applicable for the nonlinear optimization problem. In the first part of this study, the effects of processing parameters, namely the short shot size, melt temperature, water pressure, and delay time, on the water penetration length were determined using computational fluid dynamics (CFD) analysis results based on single factor experimental method. The second part of this study presented an integrated optimization strategy, Taguchi orthogonal array design, radial basis function (RBF) neural networks, and particle swarm optimization (PSO) algorithm, to find the optimal processing parameters resulting in maximum water penetration length. The research results indicated that the proposed approach could effectively achieve competitive advantages of product quality and computational efficiency.

2 Optimization model and methods 2.1 Finite element analysis The authors had already published data on the residual wall thickness of tubes in WAIM [10]. Based on the prior studies, the continued work for the water penetration length of polypropylene (PP) is investigated. As shown in Fig. 1, the part geometry used by the authors is taken as a case study, which is a tube with dimensional transitions and curved sections. It can also be seen from Fig. 1 that the computational domain of the cavity comprises three phases, namely, the water, melt, and air. The water injection inlet is at AB, and the air outlets are at GH and IJ. When the water begins to penetrate into the core of melt along the path of the least resistance, the water front CD and melt front EF will be formed. The water penetration length is computed by CFD method, which is proved to be excellent in handling the water penetration behavior in WAIM. Generally speaking, the CFD special pre-processing software, Gambit, is used to build up geometry models and divide meshes, by which the calculation domain is discretized with paved mesh of quadrilaterals of 15790 elements. Moreover, the water penetration length is calculated by CFD software, Fluent. For a more stable solution, the under-

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Fig. 1 The part geometry and computational domain

relaxation factors for all the samples are set to 0.2, and the water injection pressure is gradually increased from 0 to the desired pressure during the time 0.1 s of the water beginning injection. In WAIM, there are many processing parameters that will have effect on the water penetration behavior. Based on the water penetration length studies of Liu and Chen [8], and Huang and Deng [9] by experiments, the short shot, melt temperature, water pressure, and delay time are selected to investigate the water penetration length. 2.2 Design of experiment and RBF model Before performing processing parameters optimization, it is crucial to find a good model effectively mapping the relationship between the inputs and outputs. In this paper, the four input processing parameters are the short shot, melt temperature, water pressure, and delay time, and the output parameter is the water penetration length. RBF model [11] is an effective mapping tool and has been extensively applied to approximate the nonlinear spaces. Shie [12] analyzed the contour distortions of PP composite components by RBF, and determined an optimal parameter setting of the injection molding process; Li et al. [13] introduced a modified global optimization method based on RBF surrogate model and its application in packing profile optimization of injection molding process; Srinivasa Pai et al. [14] presented an estimation of flank wear in face milling operations using RBF networks; Mollah and Pratihar [15] used RBF networks to

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determine the input–output relationships of tungsten inert gas welding and abrasive flow machining. The studies stated that RBF networks had the advantages of reasonably fast training, reasonably compact networks, and relatively accurate approximation. As shown in Fig. 2, RBF model has a forward structure of three layers, which consists of one input, one hidden, and one output layers. The input layer has four neurons that are short shot size As, melt temperature Tm, water pressure Pw, and delay time td, respectively. The output layer corresponds to water penetration length. Once the hidden layer obtains the data from the input layer, the input vector will be transformed by nonlinear transformation. And then, the data of the output layer is constructed by linearly combined function responses. To keep theradial function as the basis function, RBF is constructed by linear superposition. Given inputs and outputs, a RBF predictor in design space is described as follows: f ð xÞ ¼

n X i¼1

n     X ωi ϕ r i ¼ ωi ϕ x−xi 

ð1Þ

i¼1

Where ωi is weight coefficient; ri is Euclidean distance, and ϕ(ri) is nonlinear radial basis function. For the current problem, the basis function is Gauss function, which is expressed as follows:  .  ϕðrÞ ¼ exp −r2 c2 ð2Þ

On condition that the samples of data are mismatching and ϕ(ri) is a positive definite function, there is a unique solution for Eq. (3): −1  ω ¼ Φ xi −x j  ⋅Y ð4Þ For the initialization of RBF model, it requires at least 2n+1 sets of data to be evaluated, where n is the number of inputs. The component being approximated can be executed multiple times to collect the required data. Therefore, to ensure the accumulation of RBF model, 32 sets of training data conforming to Taguchi L32(44) design of experiment (DOE) are employed to train RBF model. The Taguchi DOE is based on an orthogonal array table, and performs a fractional factorial experiment to maintain orthogonality among the various factors and interactions. In much of the studies [12, 16–18], the orthogonal DOE was chosen to create the training data, because it avoided a costly full-factorial experiment, greatly reduced the number of designs, and still obtained a meaningful factor-effect information. For orthogonal experiment, the four factors and four levels are shown in Table 1, and the orthogonal arrays are shown in Table 2. Furthermore, for the purpose of comparison, an additional Kriging model is constructed, which is an interpolation technology based on the theory of statistics. The expression between the input variables and response values is as follows: yð xÞ ¼ f ð xÞ þ μ ð xÞ

Where c is a constant that is greater than 0. According to the interpolation condition f(x j) = y j, the following equation set can be obtained   ð3Þ Φ xi −x j  ⋅ω ¼ Y

ð5Þ

Where f (x) is a known polynomial function of x, which is taken to be a constant term. μ(x) is a random function that reflects the approximation of partial deviations with mean zero, variance σμ2, and non-zero covariance, of which the covariance matrix is given by: Cov½μðvÞ; μðwÞ ¼ σ2μ ½Rðθ; V ; W Þ

ð6Þ

Table 1 Factors and levels in orthogonal experiment

Fig. 2 Radial basis function neural network

Factors

Level 1

Level 2

Level 3

Level 4

A: Short shot size (%) B: Melt temperature (K) C: Water pressure (Mpa) D: Delay time (s)

56 468 5.5 0.5a

58 488a 6.5 1

60a 508 7.5a 1.5

62 528 8.5 2

a

Standard for single factor experimental method.

2608 Table 2 L32(44) orthogonal array used in design of experiment

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Exp. number

A

B

C

D

1 2 3 4 5

1 1 1 1 1

1 2 3 4 1

1 2 3 4 1

1 2 3 4 2

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3

2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3

2 3 4 2 1 4 3 2 1 4 3 4 3 2 1 4 3 2

1 4 3 3 4 1 2 4 3 2 1 1 2 3 4 2 1 4

24 25 26 27 28 29 30 31 32

3 4 4 4 4 4 4 4 4

4 1 2 3 4 1 2 3 4

1 3 4 1 2 3 4 1 2

3 3 4 1 2 4 3 2 1

Where R(θ, V, W) is the relevant function of the sample points V and W with parameter θ. Kriging model [19] can assure the fitting accuracy in small samples, and its degree of exactitude is relatively high for nonlinear problem forecast. Besides, it has the character of part estimation under the action of correlation function.

2.3 PSO algorithm In this study, the water penetration length is determined by four independent processing conditions, namely, As, Tm, Pw, and td. Therefore, the mathematical format of the optimization problem with the maximum water penetration length can be defined as follows:

Find: x ¼ ½As ; T m ; Pw ; t d 

ð7aÞ

Maximize : Water penetration lengthðxÞ

ð7bÞ

Subjected to constraint : Water penetration length≤ 305:6 mm

ð7cÞ

Within ranges: 55 % ≤ As ≤ 63 % 460K ≤ T m ≤ 530K 5Mpa≤ Pw ≤ 9Mpa 0 s ≤ t d ≤ 2:5 s Liu and Hsieh [20] studied the residual wall thickness distribution at the transition and curved sections of WAIM tubes. By reference to the experimental data, the ranges of variables are obtained. To solve the above optimization problem, an effective PSO is coupled with the RBF model for the water penetration length to yield a global optimum. PSO is an evolutionary optimization algorithm initially suggested by Kennedy and Eberhart [21] in 1995, in which a simple velocity-position mode is adopted. Although being relatively new, PSO had already been successfully applied in some researches in injection molding area, which included the optimization studies of injection molded part warpage, weld lines, and air traps [22]; product and mold cost estimation [23]; product weight, flash, and volumetric shrinkage [24]; and parts cost estimation [25]. As a result, this paper introduces the combination of the PSO with the RBF model to effectively improve the network solving efficiency and quality. The basic principles of PSO are as follows: PSO mimics the social behavior of animal groups such as flocks of birds or fish shoals. The process of finding an optimal design point is compared to the food-foraging activity of these organisms. It is assumed that a swarm consisting of m particles flies with a certain velocity in dimension D. As probing, each particle adjusts its position and velocity by its own best value (pbest) and the global best value (gbest). The position of the ith particle is represented as: xi ¼ ðxi1 ; xi2 ; ⋅⋅⋅; xiD Þ

ð8Þ

The velocity of the ith particle is represented as: vi ¼ ðvi1 ; vi2 ; ⋅⋅⋅; viD Þ

ð9Þ

Where 1≤i≤m. At first, the initial position and velocity of each particle are determined. According to the objective function evaluation of

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each particle, the pbest and gbest are also identified. Then the position and velocity of each particle will be changed by the pbest and gbest, and accordingly updated by the following equation: xtþ1 ¼ xt þ vtþ1

ð10Þ

vtþ1 ¼ ωvt þ c1 r1 ðpbest−xt Þ þ c2 r2 ðgbest−xt Þ

ð11Þ

Where, ω is inertia weight; c1 and c2 are positive acceleration constants; and r1 and r2 are random numbers uniformly distributed between 0 and 1.

Fig. 4 Effect of the melt temperature on the water penetration length

Taking the example by the early study of the effects of processing parameters on the residual wall thickness [26], single factor experimental method is also adopted to study the effects of processing parameters on the water penetration length. As shown in Fig. 3, the water penetration length obviously decreases with the short shot size increasing. This is due to the fact that the more the short shot is, the less the space for the water penetration is. In WAIM, the reasonable short shot size is very important. If the short shot size is too small, it is easy for the water to blow out the melt. However, if the short shot size is too large, the water penetration length will dramatically decrease, and the water channel that is not penetrated by water will cause various part defects. It can be seen from Fig. 4 that the melt temperature has little effect on the water penetration length. This can be explained as follows. Two contradictory phenomena occur when increasing the melt temperature: one is that the viscosity of the melt becomes smaller, which is easier to make the water penetrate into the melt; the other is that the hollowed core rate

is bigger [10], which makes more water move to the mold wall. The effect of the delay time on the water penetration length is also small, as shown in Fig. 5. This is because increasing the delay time leads to the increase of the solidified layer of the melt, which decreases the void area at the beginning of the water channel and helps the water penetrate further into the melt core. On the other hand, increasing the delay time will increase the viscosity of the melt, which makes the water penetration difficult. Figure 6 illustrates the effect of the water pressure on the water penetration length. The results clearly indicate that increasing the water pressure will significantly increase the water penetration length. This is so that the higher water pressure will help the water overcome the high viscosity of the melt, which makes the water penetrate further into the melt and push the melt forward. The water pressure is a key processing parameter that affects the water penetration length. If the water pressure is too small, the water penetration length is noticeably less. On the contrary, if the water pressure is too large, the water will easily penetrate into the thin area, instead of the core of the melt. To sum up, in this study, the short shot and water pressure are found to be the two principal parameters affecting the water penetration length. Liu and Chang [27], investigating the influences of processing conditions on the gas penetration

Fig. 3 Effect of the short shot on the water penetration length

Fig. 5 Effect of the delay time on the water penetration length

3 Results and discussion 3.1 Effects of processing parameters on water penetration length

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Fig. 6 Effect of the water pressure on the water penetration length

Fig. 7 Comparison between the experimental and predicted values via two models

length, concluded that the key parameters are also the short shot and water pressure. It seems that the finding is in accordance with the previous research results. Besides, the melt temperature and delay time have little impacts on the water penetration length. Huang and Deng [9], studying the effects of processing parameters on the water penetration length, stated that increasing the melt temperature decreased the water penetration length and increasing the delay time increased the water penetration length slightly. For both our study and the literature, the effects of the melt temperature and delay time on the water penetration length are very small. The reasons for the difference might be the different mold structure and dimension. To this end, the research results only come from the special case study, and may not be correct for other cases.

3.3 Water penetration length optimization by PSO

3.2 Model validation As mentioned above, the RBF neural networks are firstly trained based on 32 samples of training data. However, the RBF algorithm doesn't assure a fulfilling generalization for the available data. Therefore, it is necessary to make predictions by verifying data. In this study, 11 samples of testing data that are different from the training data are used to verify the RBF model. The testing method is universally acknowledged in the studies of neural networks [24, 28–30]. Figure 7 illustrates the comparison between the experimental and predicted values via RBF and Kriging model. In general, it shows that the predicted values correspond with the experimental values, and the prediction accuracy is high for the two models. Whereas, it also clearly indicates that the predicted values of the samples 2 and 4 for the Kriging model deviate from the experimental values relatively much. Distinctly, the performance of the RBF quality predictor is better than that of the Kriging quality predictor. As a result, the predictive ability of the RBF model is sufficient enough to take the place of the computer-aided engineering (CAE) analysis and further be optimized with PSO.

To apply the PSO for the RBF model, the parameters are defined as follows: the maximum iterations, 50; the number of particles, 50; the inertia, 0.9; the global increment, 0.9; the particle increment, 0.9; and the maximum velocity, 0.1. After optimization by PSO, the result shows that the maximum water penetration length is 301.4 mm, and the corresponding processing parameters are: the short shot, 55 %; the melt temperature, 460 K; the water pressure, 9 Mpa; and the delay time, 0 s. However, in all the 32 training samples and 11 testing samples, the maximum water penetration length is 284.7 mm. Compared to all the samples, the optimized water penetration length by PSO is the biggest. To verify the optimum water penetration length that is proposed by PSO, a confirmation experiment of CAE under the corresponding optimal processing parameters is performed. The confirmation experiment is very important in parameter design, especially when DOE are small fractional factorial experiments. By the confirmation experiment, the water penetration length is 296.9 mm, which is slightly smaller than the optimum solution of 301.4 mm by PSO and the relative error is only 1.5 %. The confirmation experiment demonstrates that the proposed optimization scheme is effective and reliable. For comparison, a conventional Taguchi method is also used to optimize the water penetration length. Taguchi method performs a fractional factorial experiment to maintain orthogonality among the various factors and interactions. The result shows that the optimum length by Taguchi method is 285.2 mm, which is far less than that of the PSO coupled with RBF model. Despite the fact that the Taguchi method is easy to perform, it is only the best combination of factor levels and can't investigate the global optimum solution in the design space. In addition, the optimum water penetration length optimized by the PSO coupled with Kriging model is 286 mm. Accordingly, it is 282.5 mm by the confirmation experiment. Although the relative error is only 1.2 %, the optimum length by the PSO coupled with Kriging model is

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also much smaller than that of the PSO coupled with RBF model. Furthermore, the PSO optimization based on the RBF model can significantly improve computational efficiency while still achieving reasonable results. By the methodology, only 32 samples of training data are needed, for each of which the computational time is 1.5 h with the CAE analysis. Thus, the whole optimization time is 48 h. On the other hand, for the full-factorial experiment design the needed samples of training data are 44 =256. Accordingly, the computational time is 384 h. Although the PSO methodology at first needs a relatively long time to execute experiments for the samples of training data, it will afterwards bring a lot of benefits.

4 Conclusions In this paper, the water penetration length of tubes with dimensional transitions and curved sections was simulated by CFD method. Based on single factor experiment method, the short shot size and water pressure were found to be the most critical processing parameters influencing the water penetration length. Increasing the short shot size would obviously decrease the water penetration length, while the water penetration length would significantly increase with the water pressure increasing. Moreover, the melt temperature and delay time had little influences on the water penetration length. An efficient optimization methodology using Taguchi orthogonal array design, RBF neural networks, and PSO algorithm was introduced in maximizing the water penetration length. The predicted values for RBF agreed well with the experimental values, which meant that the RBF had good interpolation capability and could be used as an efficient predictive tool for the water penetration length. In addition, the maximum water penetration length by PSO was slightly smaller than that of the confirmation experiment. While still achieving reliable results, PSO could significantly improve computational efficiency. The proposed methodology has suggested the potential of PSO coupled to RBF in the optimization of the water penetration length. However, the effects on the quality of parts are various in WAIM. Therefore, further researches will be addressed to multi-objective optimization, including the water penetration length, residual wall thickness, difference of wall thickness, and so on.

Acknowledgments This work was supported by the Specialized Research Fund (No. 2010B10018) from Science and Technology Bureau of Ningbo City, ZheJiang, China. Conflict of interest The authors declare that they have no conflict of interest.

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