Abstract: Using volume shrinkage rate (Y1) and maximum warpage deformation in Z-direction (assembly direction) (Y2) of a large plastic truck wheel arch crossbeam as response objectives, melt temperature, mold temperature, first-stage holding time, second-stage holding time, first-stage holding pressure, second-stage holding pressure were selected as experimental variables. One hundred sets of samples were designed using an optimal Latin hypercube experimental design, simulations were performed using Moldex3D mold flow analysis software. Regression prediction models for Y1 and Y2 were established using a Bayesian regularized neural network (BRNN), with determination coefficients (R²) of 0.991 and 0.989, respectively. Models were then optimized using a non-dominated sorting genetic algorithm II (NSGA-II) to obtain optimal experimental variable parameters. Optimal experimental variable parameters were simulated in Moldex3D and applied in actual field. It was found that for Y1 and Y2, errors between simulation results and optimal results predicted by BRNN-NSGA-II were 0.14% and 7.28%, respectively, which were reduced by 3.16% and 64.42% compared with initial simulation results. Actual molded parts had good molding quality and met production requirements. These results indicate that proposed BRNN combined with NSGA-II method can effectively solve multi-objective optimization problem of injection molding process for large and complex plastic parts.
Injection molding, due to its high efficiency and high precision in molding complex structures, has become core process for manufacturing lightweight automotive body parts. Most lightweight automotive body parts must simultaneously meet requirements of high strength and high precision. Use of fiber-reinforced composite materials can significantly improve tensile strength of plastic parts, but warping deformation caused by non-uniform shrinkage still seriously restricts molding quality of plastic parts. In order to achieve requirements of high molding quality, it is necessary to perform multi-objective collaborative optimization of injection molding process parameters (such as melt temperature, mold temperature, holding pressure, etc.). Traditional optimization of injection molding process parameters mainly relies on operators to repeatedly adjust injection molding machine parameters through trial and error, but this method has disadvantages of strong experience dependence, difficulty in cost control, and long optimization cycle. With continuous improvement of computer-aided simulation technology, injection molding process can now be numerically simulated by mold flow analysis software such as Moldex3D and Moldflow, thereby predicting key quality indicators such as weld lines, volume shrinkage rate and warping deformation of plastic parts. Based on simulation results, molding process parameters of plastic parts are optimized, thereby improving efficiency of mold trial and reducing cost of mold trial. In cutting-edge research on injection molding process parameter optimization, most of them use surrogate models combined with multi-objective optimization algorithms to optimize injection molding process. Liu et al. took glass fiber reinforced plastic product-pump fixing bracket as research object, and took warpage deformation as optimization target. They used Moldflow for mold flow analysis, adopted response surface methodology (RSM) for experimental design, established RSM, backpropagation neural network (BPNN), genetic algorithm optimized backpropagation neural network (GA-BPNN) and adaptive enhancement genetic algorithm optimized backpropagation neural network (AdaBoost-GA-BP) models. By comparing model accuracy, AdaBoost-GA-BP model was finally selected, and optimal parameter combination was obtained by particle swarm optimization algorithm. Optimal combination was simulated, and relative error was 0.6%. Compared with before optimization, maximum warpage deformation was reduced by 8.41%. Li et al. used warpage deformation, volume shrinkage rate and residual stress of short fiber reinforced composite injection molded parts as optimization targets, used fiber content, fiber width ratio, melt temperature, injection pressure and cooling time as experimental variables. Based on orthogonal experimental design, variance analysis was performed to obtain importance of each parameter to three targets. Then, injection molding process was optimized by response surface methodology and non-dominated sorting genetic algorithm II (NSGA-Ⅱ) algorithm to obtain optimal process parameters. Yin Lei et al. used warpage deformation and volume shrinkage rate of automobile B-pillar exterior parts as research targets. Data was sampled by optimal Latin hypercube experiment and a Kriging model was established. Surrogate model was optimized by neighborhood cultivation genetic algorithm (NCGA) algorithm. Optimized warpage deformation was reduced by 18.56% and volume shrinkage rate was reduced by 50.94% compared with original process. Hong et al. proposed an optimization method based on BP neural network and NSGA-II algorithm, taking volume shrinkage rate and warpage deformation of junction box shell as research object. Optimized volume shrinkage rate and warpage deformation decreased by 33.2% and 3.8% respectively, optimization effect was significant. In existing research on injection molding process parameter optimization, most of them are small plastic parts, there are few studies on large and complex plastic parts. Moreover, most of them adopt discrete combination methods such as orthogonal experiment and response surface when designing experiments. Parameter levels of each experimental variable are small, which cannot completely cover parameter space and easily leads to omission of key parameters. Optimal Latin hypercube experimental design can ensure that all intervals of each variable factor have samples through stratified sampling, and parameter space coverage is more uniform. Therefore, this paper focuses on volume shrinkage rate and Z-direction (assembly direction) warpage of a large plastic part (truck wheel arch beam). 100 sets of samples were designed using optimal Latin hypercube experimental design, Pearson correlation analysis was used to obtain magnitude and ranking of influence of each experimental variable on research object. To ensure accuracy of model's predictions, a Bayesian regularized neural network was selected to establish a regression prediction model. Finally, multi-objective optimization was performed using NSGA-II, parameters were adjusted by simulating an actual injection molding machine during optimization iteration process, with all variable parameters constrained to integers. Optimization process is shown in Figure 1. This method has a clear process and significant effects, can provide a certain reference for optimizing injection molding process parameters for similar complex plastic parts.

 

Figure 1 Optimization flowchart for injection molding process parameters

This paper uses truck wheel arch beam as research object and performs simulation using Moldex3D software. Material used for plastic part is 30% by mass long glass fiber (E glass fiber) reinforced polypropylene (manufacturer: Polystar, Canada, model: 80GL6-BK772), and material properties are shown in Table 1. The overall size of truck wheel arch beam is large (1459 mm * 772 mm * 303 mm) and material is fiber composite material. In order to avoid molding defects caused by excessive flow length ratio, a gating system of "9-point valve hot runner (Ø22 mm) + U-shaped cold runner (12 mm * 8 mm) + overlapping gate (16 mm * 1.2 mm) / direct gate (Ø6 mm)" is adopted. Cooling system adopts 13 sets of "straight-through (Ø15 mm) + baffle (Ø24 mm)" combined circulating water channels to enhance cooling effect. Two-dimensional dimension drawing and finite element model of plastic part are shown in Figure 2 and Figure 3.

Table 1 Material properties

 

Fig. 2 2D dimensional drawing of truck wheel cover cross beam

 

Fig. 3 Finite element model of truck wheel cover cross beam

Appearance of plastic part requires high molding quality, so opening of valve hot runner is controlled by flow front (mesh node) method. Opening node position is set at about 80 mm along flow direction of gate. This control method can make melt present a "relay" flow and avoid conflict with front melt. Melt temperature and mold temperature adopt initial parameters recommended by material manufacturer. Holding pressure and holding time adopt segmented holding pressure. This holding pressure method can effectively reduce residual stress inside plastic part and reduce molding defects. Molding process parameters and opening time of valve hot runner are shown in Table 2.

Table 2 Initial molding process parameters and opening times of valved hot runners

Initial analysis of truck wheel arch beam was carried out using Moldex3D software, as shown in Figure 4. As shown in Figures 4a and 4b, plastic part was completely filled in 4.432 s, and the overall filling contour lines were relatively uniform, indicating that there was no short shot or retention phenomenon in plastic part; As shown in Figure 4c, due to large number of anisotropic structures in plastic part, melt flow direction was varied, and fiber orientation was concentrated in range of 0.6~0.8, and fiber orientation in some structures could reach 0.9. As shown in Figure 4d, there were no obvious weld lines on the surface of plastic part, and weld lines in other positions were mainly caused by hole structure, which could not be avoided; gating system was reasonably designed and met appearance requirements of plastic part. As shown in Figures 4e and 4f, maximum Z-direction warpage, which primarily affects assembly accuracy of plastic part, is 5.051 mm, failing to meet assembly requirement (less than 5 mm). Maximum volume shrinkage rate, a secondary factor, is 10.307%, but most of it falls within the 5%–7% range, with maximum value concentrated at sprue. To achieve higher assembly accuracy and improve success rate of first-time molding trials, it is necessary to minimize maximum values of Z-direction warpage and volume shrinkage rate, which negatively impact molding quality of plastic part.

 

Fig. 4 Initial analysis results

Reasonable selection of injection molding process parameters is key to improving molding quality of plastic parts. Based on previous production experience and relevant literature, we selected melt temperature (A), mold temperature (B), first stage holding time (C), second stage holding time (D), first stage holding pressure (E), and second stage holding pressure (F) as experimental variables, volume shrinkage rate (Y1) and maximum warping deformation in Z direction (Y2) as response targets. Parameter range of each experimental variable was set by material properties and initial analysis results, as shown in Table 3. There are 6 factors (A-F) in experiment, and they are nonlinear. In order to avoid stacking of sample points and to cover all experimental variable parameter spaces, as well as to improve generalization ability of neural network model, optimal Latin hypercube experimental design method was adopted to design experiment. To avoid underfitting in neural network model, number of sample points should be much larger than number of experimental variables, satisfying N>10K (N is number of sample points, K is the total number of experimental variables), sample data should be divided into a training set of 75% and a test set of 25%. After comprehensive consideration, number of sample points was designed to be 100 groups. Figure 5 is a schematic diagram of three-dimensional spatial distribution of sampling points. It can be seen that distribution of samples in three-dimensional space is relatively uniform, without any stacking phenomenon. 100 groups of samples were simulated using Moldex3D software, and simulation results are shown in Table 4. Pearson correlation analysis was performed on 100 groups of samples to analyze correlation between each experimental variable and response target. When Pearson correlation coefficient is in range of [0, 1], variable and response target are positively correlated, that is, when variable increases, corresponding target increases; when Pearson correlation coefficient is in range of [0, -1], variable and response target are negatively correlated, that is, when variable increases, corresponding target decreases. Figure 6 shows Pearson correlation heatmap for two response objectives Y1 and Y2. It can be seen that experimental variables A (0.50) and D (0.12) are positively correlated with Y1, while experimental variables B (-0.17), C (-0.03), E (-0.32), and F (-0.76) are negatively correlated with Y1. Order of influence of each experimental variable on Y1 is F>A>E>B>D>C. Similarly, experimental variables A (0.09) and B (0.73) are positively correlated with Y2, while experimental variables C (-0.14), D (-0.22), E (-0.04), and F (-0.13) are negatively correlated with Y2. Order of influence of each experimental variable on Y2 is B>D>C>F>A>E.

Table 3 Parameter ranges for test variables

 

Fig. 5 Schematic of three-dimensional spatial distribution of sampling points

 

Table 4 Simulation results

 

Fig. 6 Pearson correlation heatmap of Y1 and Y2

To accurately construct nonlinear mapping relationship between test variables and response target, a neural network is used to establish a regression prediction model. Classic training algorithms for this model include Levenberg-Marquardt (LM), Bayesian regularization (BR), and quantized conjugate gradient (SCG). LM is based on second derivative information of Gauss-Newton and gradient descent, balances second-order optimization accuracy and first-order optimization stability by dynamically adjusting damping factor. Its advantage is that it has a fast convergence speed, is suitable for small and medium-sized networks and nonlinear least squares problems. SCG avoids time-consuming linear search steps in traditional methods by using a dynamic step size adjustment mechanism and quantization parameters. Its advantage is that it is suitable for large-scale networks and classification tasks with low real-time requirements. Principle of BR is to introduce a weight decay term (L2 regularization) into loss function, automatically optimize regularization parameters α and β through Bayesian evidence framework to achieve a balance between model complexity and data fitting, making it suitable for classification or regression tasks with small datasets.
Given that dataset studied by us contains only 100 samples, which is a typical small dataset, BR was chosen to train neural network to improve model's generalization ability. Training procedure for BR neural network is as follows.
Step 1: Data preprocessing. Min-Max normalization is used to map experimental variables and response targets to interval [-1, 1] to eliminate dimensional differences. Mathematical expression for Min-Max normalization is shown in equation (1).

 

Where: X is original data;  is minimum value in dataset;  is maximum value in dataset; is result after normalization.
Step 2: Network architecture design. Network adopts a single hidden layer, and number of neurons is selected through 1-3 iteration tests. Hidden layer activation function is hyperbolic tangent sigmoid function (tansig), and output layer activation function is a linear function (purelin), as shown in equations (2) to (4). Model training algorithm is Bayesian regularization (trainbr), as shown in equations (5) to (8), with a maximum iteration count of 1000.

 

 

 

Where:  is weight;  is output of previous layer; b is bias.

 

 

 

 

Where: E is the total error after regularization; β is data error weight; α is weight decay weight;  is data error term;  is true value of sample;  is model predicted value;  is weight decay term; J is Jacobian matrix; μis damping factor; I is identity matrix; e is error vector (difference between predicted value and true value of each sample).
Step 3: Train network, calculate test set determination coefficient (R²) and root mean square error (RMSE) for cyclic testing (1-3 neurons), mathematical expressions are shown in equations (9) and (10), and select number of neurons with the largest R² and smallest RMSE as final number of neurons used in model. Model training is completed, final model is saved and encapsulated as a network function.

 

 

Where:  is true value of i-th sample;  is predicted value of i-th sample;  is mean of true values; n is number of samples; ERMS is root mean square error.

Cyclic testing results for 1-3 neurons are shown in Table 5. Y1 model ultimately selected 3 neurons, with an R² of 0.991 and an RMSE of 0.02; Y2 model ultimately selected 2 neurons, with an R² of 0.989 and an RMSE of 0.097. Test set error curves for Y1 and Y2 are shown in Figure 7. Both models fit requirements (R² > 0.9), providing good predictive ability for further optimization.

 

Table 5 R² and RMSE for 1-3 neuron cycling test

 

Figure 7 Test set error curves for Y1 and Y2

To avoid getting trapped in local optima during optimization and to meet constraint that experimental variables must be integers, NSGA-II was chosen to solve optimal solution for Bayesian regularized neural network. NSGA-II is a multi-objective global optimization algorithm based on evolutionary principles. It evaluates individual priority through non-dominated sorting and crowding distance, and iteratively optimizes using selection, crossover, mutation operations to ultimately generate Pareto optimal solution set. NSGA-II algorithm flowchart is shown in Figure 8, where g is current iteration generation, G is number of iterations, and Gmax is maximum number of evolution generations.

 

Figure 8 NSGA-II flowchart
Two network functions Y1 and Y2 are combined into one function for multi-objective optimization. Different weights are set according to influence of Y1 and Y2 on assembly accuracy of plastic part; weight for Y1 is 0.4, and weight for Y2 is 0.6. Population size of NSGA-II is 200, maximum number of iterations is 500, function tolerance is set to 1*10⁻¹⁰, and a constraint is added that experimental variables are integers in each iteration. After 102 iterations, NSGA-II reached set termination condition (1*10⁻¹⁰). Pareto optimal solution optimized by NSGA-II is shown in Figure 9. Optimal combination of experimental variables is: melt temperature 245 ℃, mold temperature 105 ℃, first holding time 10 s, second holding time 8 s, first holding pressure 55 MPa, second holding pressure 40 MPa. Predicted volume shrinkage rate under optimal combination of process parameters is 9.995%, and maximum warpage in Z direction is 1.675 mm.

 

Figure 9 Pareto frontier solution

Optimal combination of experimental variables obtained from NSGA-II optimization was substituted into Moldex3D for simulation. Simulation results are shown in Figure 10. Maximum warpage in Z direction is 1.797 mm, and maximum volume shrinkage rate is 9.981%. Table 6 compares initial simulation results, Bayesian Regularized Neural Network (BRNN)-NSGA-II optimization results, and Moldex3D simulation results. Table 6 shows errors between Moldex3D simulation results and BRNN-NSGA-II optimization results: error in maximum warpage deformation in Z direction is 7.28%, and error in volume shrinkage rate is 0.14%, both within 10%, meeting requirements. Compared with initial analysis results (maximum warpage deformation in Z direction 5.051 mm, volume shrinkage rate 10.307%), two results from Moldex3D simulation decreased by 64.42% and 3.16%, respectively. Actual mold testing was conducted based on optimal combination of experimental variables. Actual sample is shown in Figure 11. It can be found that sample has no appearance defects such as short shots, flow marks, or weld lines. Gap between sample and fixture, checked by go gauge (Ø4 mm) and no-go gauge (Ø6 mm), both meet design requirements (5 mm). All holes, checked by locating pins, also meet requirements, achieving a successful mold test on first attempt.

 

Fig. 10 Simulation results of optimum process parameters

 

Table. 6 Comparison of initial simulation, BRNN-NSGA-II optimization, and Moldex3D simulation results

 

Fig. 11 Actual sample of truck wheel cover cross beam

(1) Based on optimal Latin hypercube experimental design, 100 sets of experimental samples were designed, and Pearson correlation analysis showed that: experimental variables A and D are positively correlated with Y1, and other experimental variables are negatively correlated with Y1. Influence of each experimental variable on Y1 is ranked as F>A>E>B>D>C; experimental variables A and B are positively correlated with Y2, and other experimental variables are negatively correlated with Y2. Influence of each experimental variable on Y2 is ranked as B>D>C>F>A>E.
(2) Bayesian regularized neural network models for Y1 and Y2 were established respectively, and cyclic testing with 1 to 3 neurons was performed. Optimal number of neurons was 3 and 2 respectively, with R² values of 0.991 and 0.989, RMSE values of 0.02 and 0.097 respectively. Optimal process parameters obtained through NSGA-II optimization were: melt temperature 245 ℃, mold temperature 105 ℃, first-stage holding time 10 s, second-stage holding time 8 s, first-stage holding pressure 55 MPa, second-stage holding pressure 40 MPa. Under this optimal injection molding process parameter combination, volume shrinkage rate was 9.995%, and maximum warpage deformation in Z direction was 1.675 mm.
(3) Optimal injection molding process parameters were verified by Moldex3D simulation and trial molding. Moldex3D simulation results showed a volume shrinkage rate of 9.981% and a maximum warpage deformation in Z direction of 1.797 mm, with errors of 0.14% and 7.28% respectively compared to optimal results from NSGA-II. Compared to initial simulation analysis results, volume shrinkage rate and maximum warpage deformation in Z direction obtained by Moldex3D simulation were reduced by 3.16% and 64.42% respectively. After trial molding, actual sample had no appearance defects and met assembly requirements. Therefore, it is proven that Bayesian regularized neural network model combined with NSGA-II can effectively solve multi-objective optimization problem of large and complex plastic parts in injection molding, can provide some reference for injection molding process design of this type of material and product.