Residual wall thickness is an important indicator for water-assisted injection molding (WAIM) parts, especially the maximization of hollowed core ratio and minimization of wall thickness difference which are significant optimization objectives. Residual wall thickness was calculated by the computational fluid dynamics (CFD) method. The response surface methodology (RSM) model, radial basis function (RBF) neural network, and Kriging model were employed to map the relationship between process parameters and hollowed core ratio, and wall thickness difference. Based on the comparison assessments of the three surrogate models, multiobjective optimization of hollowed core ratio and wall thickness difference for cooling water pipe by integrating design of experiment (DOE) of optimized Latin hypercubes (Opt LHS), RBF neural network, and particle swarm optimization (PSO) algorithm was studied. The research results showed that short shot size, water pressure, and melt temperature were the most important process parameters affecting hollowed core ratio, while the effects of delay time and mold temperature were little. By the confirmation experiments for the best solution resulted from the Pareto frontier, the relative errors of hollowed core ratio and wall thickness are 2.2% and 3.0%, respectively. It demonstrated that the proposed hybrid optimization methodology could increase hollowed core ratio and decrease wall thickness difference during the WAIM process.
WAIM, one of the innovations of plastic injection molding technology, is the newest way to mold hollow parts. The development of this molding technique is a variant of gas-assisted injection molding, which has the main strength of reducing part costs and improving part characteristics. However, because of water instead of gas as a molding medium, there are some unique advantages in WAIM: thin residual wall thickness, smooth inner surface, and short cycle. By far, some automotive plastic products, such as foot pedals, handrails, and engine cooling water pipes, are made by WAIM.
Residual wall thickness is one of the most important indexes of part quality, which significantly affects the strength. Some scholars have conducted studies on how to improve the residual wall thickness from the aspects of water needle structure, process parameters, auxiliary media, and cavity cross-sectional shape. Park et al. [
However, the relationship between molding parameters and residual wall thickness is highly complex. In particular, the optimization of process parameters can be more feasible and reasonable to meet the requirement of product quality with the advantages of saving on materials and energy [
In this paper, the main research was devoted to present an integrated optimization strategy, DOE of Opt LHS, surrogate model, and PSO algorithm, to find the optimal process parameters resulting in maximizing hollowed core ratio and minimizing wall thickness difference. This paper studied the following: (1) the DOE of Opt LHS, and calculating of residual wall thickness by CFD; (2) the building of RSM, RBF, and Kriging models, and cross-validation; (3) the effects of process parameters on hollowed core ratio; and (4) the multiobjective optimization for maximizing hollowed core ratio and minimizing wall thickness difference, and verification. Now, we are working together with Xunyu Mould Co. Ltd of Ningbo to develop WAIM technology. For engine cooling water pipe, we have set up experimental equipment of WAIM including an injection-molding machine, a mold, and water auxiliary injection devices. The integrated optimization strategy that aims for better molding quality of plastic product is helpful to accelerate the development of optimization technology in WAIM and will lay a good foundation for future industrial applications.
The basic process of WAIM is shown in Figure
WAIM process.
In WAIM, the filling process is non-Newtonian laminar of polymer melt and turbulence of water with high Reynolds number. What makes WAIM different from traditional injection molding is that the injection of water is turbulence, and the interchange of heat between water and melt is obvious. Based on the generalized Hele-Shaw model, the improved Reynolds time-averaged motion equation including Reynolds stresses is
According to Boussinesq eddy viscosity assumption, Reynolds stresses is solved by the following equation:
Simulated by the standard
More details on the mathematical model of WAIM can be found in the author’s early study [
The surrogate model is based on experimental design and statistical analysis and is an alternative model method for reflecting real problems. There are some advantages for the surrogate model, such as small calculation amount, and high accuracy, which can ensure that the optimization algorithm searches for the optimal solution in the continuous space of design variables. At present, in the field of plastic injection molding, RSM, RBF, and Kriging models are frequently used [
The basic idea of the RSM model is to fit a response surface through a series of deterministic experiments, thereby expressing the complex relationship between multifactor input and output in a system. The model is essentially a polynomial function, which can be fitted by four orders. In general, the higher the polynomial order is, the more accurate the fitted response surface model will be. Moreover, it combines experiment design and mathematical modeling with the advantages of less test times and good predictive performance and can fit some complex response relationship in plastic injection molding [
The usual first-order and second-order polynomial equations are expressed by
The term selection method can be used to choose some important polynomial terms and discard some less important polynomial terms for improving the reliability of the model. It includes sequential replacement, stepwise, two-at-a-time replacement, and exhaustive search, which take the smallest residual sum of squares (RSS) as the goal to select the best item. The formula for RSS is as follows:
RBF neural network is an effective fitting model, which is widely used in fitting highly complex nonlinear problems. Li et al. [
The basis function of the RBF model is a radial function, and the construction method is linear superposition. Assuming input X and output Y, RBF is expressed as follows:
According to the interpolation
There are many basis functions, of which the Gaussian function is the most common one, namely,
Geologist Krige first proposes the Kriging model, which is based on structural analysis and variation function theory and can unbiasedly optimize the design of regionalized variables [
Under the condition of small samples, a certain fitting accuracy can be ensured by the Kriging model, which has been well applied to the part stiffness predictions [
PSO is an evolutionary optimization algorithm developed by Kennedy and Eberhart, which has been successfully applied in the optimization design of plastic injection molding, including the optimization studies of volumetric shrinkage of biaspheric lens [
The PSO algorithm is based on a group of particles. Assuming that a group of
The velocity of the
During each iteration of the particle swarm, each particle must not only find its own historical best point (
The global extreme of
After determining
By the hybrid DOE, surrogate model, and optimization technique, the optimization design process of this problem is shown in Figure
Flow chart of hybrid optimization design.
The experimental product is a brand of automobile cooling water pipe, of which the cross-section is round and precise, the outer diameter is 29 mm, the average wall thickness is 2 mm, and the length is about 400 mm, as shown in Figure
Model of cooling water pipe.
Because the cooling water pipe is connected to the automobile engine, the material of product must meet the requirements of high temperature, shock, and fatigue resistance. Through the contrast of engineering plastic properties, the material is selected as PA66 + 30% GF. It is a rare material that can meet the stringent requirements of automotive engine parts.
Considering the simplification of the model and calculation efficiency, the two brackets of product are removed. At first, Gambit is used to divide the tetrahedral mesh unit, and then, the 3D mesh model is imported into Fluent for simulation calculation. For CAE simulation, the multifluid volume of fluid is adopted. Boundary conditions are specified as follows: the pressure and temperature of the inlet derive from the values of water injection; the pressure of the outlet is atmospheric pressure; no-slip boundary condition is used at the wall, and fixed temperature boundary condition is employed for the method of heat exchange; no-slip condition is applied at the solid-melt interface; Dirichlet boundary condition is specified at the water-melt interface. In addition, the numerical calculation of the flow field is the PISO algorithm, the underrelaxation factors are set to 0.2, and the time step of the iterative calculation is set to 10-5 s.
The molding method is short shot, and the main process parameters are short shot size
In this paper, DOE of Opt LHS is applied to get the training samples, the process of which is as follows. First, the numbers of design varies
DOE of Opt LHS.
Sample | |||||||
---|---|---|---|---|---|---|---|
1 | 61.9 | 593.0 | 8.5 | 2.3 | 318.7 | 55.9 | 27.9 |
2 | 66.1 | 578.7 | 10.0 | 2.9 | 332.1 | 55.3 | 23.3 |
3 | 67.6 | 574.0 | 7.9 | 2.5 | 351.1 | 52.4 | 26.1 |
4 | 61.1 | 576.3 | 7.5 | 0.0 | 330.1 | 55.8 | 26.6 |
5 | 65.7 | 552.5 | 6.4 | 3.2 | 334.0 | 49.2 | 26.5 |
6 | 60.8 | 571.6 | 6.2 | 2.7 | 322.5 | 52.0 | 26.7 |
7 | 63.1 | 590.6 | 9.1 | 1.3 | 349.2 | 56.9 | 32.3 |
8 | 63.8 | 569.2 | 9.8 | 0.8 | 314.9 | 56.1 | 28.7 |
9 | 65.3 | 564.4 | 6.8 | 1.0 | 313.0 | 50.0 | 27.6 |
10 | 64.2 | 547.8 | 9.2 | 3.1 | 347.3 | 54.7 | 22.6 |
11 | 66.5 | 554.9 | 6.6 | 0.4 | 341.6 | 48.6 | 33.0 |
12 | 67.2 | 585.9 | 8.3 | 0.6 | 328.2 | 54.7 | 25.8 |
13 | 62.3 | 543.0 | 8.1 | 1.1 | 326.3 | 53.3 | 25.8 |
14 | 65.0 | 559.7 | 9.4 | 0.2 | 345.4 | 54.9 | 20.8 |
15 | 61.5 | 562.1 | 7.0 | 1.7 | 353.0 | 52.8 | 24.5 |
16 | 60.0 | 566.8 | 9.6 | 2.1 | 335.9 | 58.3 | 23.0 |
17 | 64.6 | 588.2 | 6.0 | 1.5 | 339.7 | 57.9 | 21.2 |
18 | 66.9 | 583.5 | 7.1 | 3.4 | 320.6 | 50.7 | 20.5 |
19 | 60.4 | 545.4 | 7.3 | 3.8 | 337.8 | 55.9 | 26.3 |
20 | 68.0 | 550.1 | 8.9 | 1.9 | 324.4 | 52.8 | 21.7 |
21 | 63.4 | 557.3 | 8.7 | 3.6 | 316.8 | 55.8 | 18.8 |
22 | 62.7 | 581.1 | 7.7 | 4.0 | 343.5 | 51.7 | 18.7 |
Training samples were used to construct RSM, RBF, and Kriging surrogate models, so as to establish the nonlinear relationship between the process parameters and hollowed core ratio and wall thickness difference. For the RSM model, the order of the polynomial depends mainly on the number of design variables and sample points. In this study, there are five design variables and twenty-two sample points. Therefore, second-order polynomial functions are selected. In order to improve the precision and quality of the model, the term selection method is an exhaustive search. Although it requires the highest computational amount, the quality is the best. For the RBF model, the nonlinear radial basis function is selected as the Gauss function, of which the prediction curve is smooth. The smoothing filter is 0, and the maximum iterations to fit is 50. Furthermore, for the Kriging model, the fit type is anisotropic, the correlation function is Gaussian, and the maximum iterations to fit is 1000. Cross-validation is used to test the accuracy of the three models to determine whether the fitted surrogate model meets the accuracy requirements (see Section
In order to ensure that the flow rate of water is sufficiently large, the hollowed core ratio should be as large as possible. In addition, the residual wall thickness distribution is also needed to be uniform, and the wall thickness difference should be controlled as small as possible. Therefore, the essence of this optimization problem is to find the process parameter combination with the largest hollowed core ratio and the smallest wall thickness difference in the feasible process space. This is a multiobjective optimization problem, and the mathematical model is as follows:
Find:
Within ranges:
The PSO algorithm flow is shown in Figure
PSO algorithm flow.
The optimization process is a combination strategy including the DOE, surrogate model, and optimization algorithm, and errors are inevitable in the calculation process. Therefore, it is necessary to verify the final optimization results and analyze the accuracy of the optimization method.
In WAIM, high-pressure water is injected into the mold cavity after the melt filling stage. As the rapid cooling effect of water on melt, a highly viscous membrane will be formed at the leading edge of the water. As shown in Figure
Simulation result of water filling.
As shown in Figure
Effect of short shot size on hollowed core ratio.
It can be seen from Figure
Effect of delay time on hollowed core ratio.
As shown in Figure
Effect of melt temperature on hollowed core ratio.
Figure
Effect of water pressure on hollowed core ratio.
As shown in Figure
Effect of mold temperature on hollowed core ratio.
In this paper, cross-validation is adopted, which is an effective test method widely used in metamodeling techniques in support of engineering design optimization [
RE and RP of hollowed core ratio for the three models are shown in Table
The actual value, predicted value, RE, and RP of hollowed core ratio of the three models.
Sample | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Mean |
---|---|---|---|---|---|---|---|---|---|---|---|
Actual value (%) | 53.8 | 54.0 | 54.6 | 54.3 | 55.7 | 54.0 | 51.9 | 54.6 | 53.1 | 54.0 | 54.0 |
RSM predicted value (%) | 54.2 | 54.7 | 55.3 | 55.1 | 56.7 | 54.2 | 52.8 | 55.1 | 54.8 | 54.9 | 54.9 |
RBF predicted value (%) | 54.5 | 54.6 | 54.7 | 53.9 | 56.5 | 53.8 | 52.9 | 53.9 | 54.7 | 54.4 | 54.4 |
Kriging predicted value (%) | 55.1 | 56.0 | 56.1 | 55.6 | 57.7 | 55.2 | 51.2 | 55.6 | 55.6 | 55.6 | 55.4 |
RSM | |||||||||||
RE (%) | 0.7 | 1.3 | 1.3 | 1.5 | 1.8 | 0.4 | 1.7 | 0.9 | 3.2 | 1.7 | 1.5 |
RP (%) | 99.3 | 98.7 | 98.7 | 98.5 | 98.2 | 99.6 | 98.3 | 99.1 | 96.8 | 98.3 | 98.5 |
RBF | |||||||||||
RE (%) | 1.3 | 1.1 | 0.2 | 0.7 | 1.4 | 0.4 | 1.9 | 1.3 | 3.0 | 0.7 | 1.2 |
RP (%) | 98.7 | 98.9 | 99.8 | 99.3 | 98.6 | 99.6 | 98.1 | 98.7 | 97.0 | 99.3 | 98.8 |
Kriging | |||||||||||
RE (%) | 2.4 | 3.7 | 2.7 | 2.4 | 3.6 | 2.2 | 1.3 | 1.8 | 4.7 | 3.0 | 2.8 |
RP (%) | 97.6 | 96.3 | 97.3 | 97.6 | 96.4 | 97.8 | 98.7 | 98.2 | 95.3 | 97.0 | 97.2 |
Table
The actual value, predicted value, RE, and RP of wall thickness difference of the three models.
Sample | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Mean |
---|---|---|---|---|---|---|---|---|---|---|---|
Actual value (%) | 25.7 | 23.4 | 24.6 | 25.0 | 25.8 | 27.1 | 26.7 | 20.3 | 23.1 | 20.3 | 24.2 |
RSM predicted value (%) | 24.0 | 24.1 | 24.1 | 22.6 | 27.5 | 23.8 | 25.2 | 22.6 | 22.5 | 22.6 | 24.1 |
RBF predicted value (%) | 23.0 | 22.8 | 23.3 | 20.9 | 26.2 | 23.4 | 23.4 | 20.9 | 21.3 | 20.9 | 22.6 |
Kriging predicted value (%) | 20.8 | 20.5 | 25.5 | 23.8 | 28.5 | 25.5 | 26.3 | 23.8 | 23.8 | 23.8 | 24.2 |
RSM | |||||||||||
RE (%) | 6.6 | 3.0 | 2.0 | 9.6 | 6.9 | 12.2 | 5.6 | 11.3 | 2.6 | 11.3 | 7.1 |
RP (%) | 93.4 | 97.0 | 98.0 | 90.4 | 93.1 | 87.8 | 94.4 | 88.7 | 97.4 | 88.7 | 92.9 |
RBF | |||||||||||
RE (%) | 10.5 | 2.6 | 5.3 | 16.4 | 1.6 | 13.7 | 12.4 | 3.0 | 7.8 | 3.0 | 7.6 |
RP (%) | 89.5 | 97.4 | 94.7 | 83.6 | 98.4 | 86.3 | 87.6 | 97.0 | 92.2 | 97.0 | 92.4 |
Kriging | |||||||||||
RE (%) | 19.0 | 12.4 | 3.6 | 4.8 | 10.5 | 5.9 | 1.5 | 17.2 | 3.0 | 17.2 | 9.5 |
RP (%) | 81.0 | 87.6 | 96.4 | 95.2 | 89.5 | 94.1 | 98.5 | 82.8 | 97.0 | 82.8 | 90.5 |
The comprehensive RP of RSM, RBF, and Kriging models for hollowed core ratio and wall thickness difference are 95.7%, 95.6%, and 93.9%, respectively, among which the prediction capabilities of RSM and RBF models are high and very close. Nevertheless, the Kriging model is significantly worse. The main reason is that the correlation function used in the Kriging model is a Gaussian, which is characterized by local interpolation. Compared with wall thickness difference, the hollowed core ratio of the cooling water pipe is a more important optimization object. Therefore, the RBF neural network with a stronger prediction ability of hollowed core ratio is used to couple the PSO algorithm for the multiobjective optimization problem.
For this problem, the optimal design requires the hollowed core ratio as large as possible, and wall thickness difference as small as possible. However, in actual WAIM optimization, the two objects are difficult to meet at the same time. Namely, as the hollowed core ratio increases, wall thickness difference also increases. Considering that the improvement of any one object is at the expense of other objects, the optimal design of this problem is actually a Pareto solution that compromises hollowed core ratio and wall thickness difference. With the above combined optimization strategy of DOE of Opt LHS, RBF model, and PSO algorithm, multiobjective optimization on hollowed core ratio and wall thickness difference is performed. For the problems of traditional multiobjective optimization, multiobjective is usually transformed into a single objective. Nevertheless, there is no need to set the weights and proportionality coefficients of each target artificially with multiobjective PSO, which will automatically calculate Pareto optimal solutions. The number of simulations is confined in a predetermined value of 511 runs for this case study due to the computing cost of each simulation and the budget of time. Based on the results of 511 runs, the Pareto frontier for hollowed core ratio and wall thickness difference is shown in Figure
Pareto frontier.
After determining the Pareto frontier, the next step is to select the appropriate trade-off solution according to the actual engineering problem. In this problem, the hollowed core ratio of engine cooling water pipe needs to be large enough to provide more flow rate for cooling, and wall thickness difference should not be too large. According to the Pareto frontier, the hollowed core ratio of 58.4% at point
To verify the best solution of the hollowed core ratio and wall thickness difference obtained by hybrid optimization strategy, a confirmation experiment is absolutely necessary. As shown in Table
Comparison between the best solution and the verification value.
RE | RE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
% | K | MPa | s | K | Optimal | Verification | % | Optimal | Verification | % | |
Point | 60 | 563 | 10 | 2.4 | 330 | 58.4 | 59.7 | 2.2 | 22.9 | 23.6 | 3.0 |
In this paper, the residual wall thickness of the cooling water pipe was simulated by the CFD method. It was concluded that water pressure, short shot, and melt temperature were the most critical process parameters influencing residual wall thickness. Rising water pressure and short shot both would obviously decrease residual wall thickness, or rather, hollowed core ratio increased. On the contrary, the hollowed core ratio increased with melt temperature rising. Furthermore, delay time and mold temperature had little influence on the hollowed core ratio.
DOE of Opt LHS improved balance and spatial filling of sample points, and it was suitable for sample acquisition in plastic injection molding. As for the RP comparison of RSM, RBF, and Kriging models, the RBF model has the highest RP for hollowed core ratio. The comprehensive RP for hollowed core ratio and wall thickness difference of RSM and RBF models is very close and very high, while the RP of the Kriging model is relatively low. As a result, RBF was an effective model to fit the relationship between process parameters and residual wall thickness.
An efficient optimization methodology coupled with DOE of Opt LHS, RBF neural networks, and multiobjective PSO algorithm was adopted in maximizing hollowed core ratio and minimizing wall thickness difference. The optimization design requires that the hollowed core ratio is as large as possible, and the wall thickness difference is as small as possible. Through the confirmation experiment for the best solution, it shows that the RE for hollowed core ratio is 2.2%, and the RE for wall thickness difference is 3%. In conclusion, the combined optimization strategy can find the optimal solution of hollowed core ratio and wall thickness difference in the entire design space, and the accuracy also meets the requirements for WAIM.
All data used during the study appear in the submitted article, and no additional data are available.
The authors declare that they have no conflict of interest.
This work was supported by the National Natural Science Foundation of China (Grant No. 51741505) and Dr. Start-up Project of Shanghai Ocean University (Grant No. A2-2006-00-200370), for which the authors are very grateful.