Abstract: To address problems of large warping deformation and residual stress in aircraft navigation light covers during injection molding, leading to cracking during secondary processing, assembly, and use, a regression analysis-based method is proposed to optimize warping deformation and residual stress of light cover using multiple objectives. Melt temperature, mold temperature, injection time, injection pressure, holding time, holding pressure, and cooling time were selected as experimental factors. With product warpage and residual stress as optimization objectives, a 7-factor, 21-level uniform design experiment was established. Multi-objective optimization problem was transformed into a comprehensive objective value evaluation using entropy weight method and improved compromise programming, and a multiple regression equation with comprehensive objective value as response value was established. Particle swarm optimization algorithm was used to iteratively optimize regression equation, obtaining a minimum response value of 0.096. Corresponding optimal process parameter combination was: melt temperature 306 ℃, mold temperature 82 ℃, injection time 3s, injection pressure 85 MPa, holding time 24 s, holding pressure 138 MPa, and cooling time 54 s. Simulation of plastic part according to optimal process parameter combination showed a maximum warpage of 0.812 mm, a maximum residual stress of 1.256 MPa, and a comprehensive objective value of 0.103. Simulation results of optimized scheme showed little difference from predicted values, indicating that regression model had high prediction accuracy. Optimized process parameter combination was applied to actual trial production. Measured results of product were consistent with optimized simulation trend, meeting assembly and usage requirements.
Aircraft navigation light covers need to operate in extreme aerial environments such as large temperature differences, low air pressure, and strong ultraviolet radiation, requiring high structural strength. Navigation light covers are generally manufactured using sheet bending or injection molding. Sheet bending is only suitable for simple, single-curvature products, has a long molding cycle and high cost. Injection molding, on the other hand, can achieve high-precision molding of complex structures, with a short cycle and stable, excellent quality, making it main development method for current aircraft navigation light covers. Large-size, thin-walled navigation light covers have an injection flow length ratio exceeding 80, making them prone to warping and residual stress during molding. This leads to localized stress concentration during installation and use, especially in peripheral connection areas of light cover, which can easily cause cracking and other problems, threatening effective operation of signal lights and thus affecting flight safety. Warpage and residual stress under injection molding are affected by many factors such as materials, processes, molds and post-processing. Among them, process parameters (such as time, temperature, pressure, etc.) directly determine quality status of product. Optimizing process parameters has become an effective way to control above two properties. Orthogonal experimental design is a typical experimental optimization method. For multi-index optimization problems, multiple objectives can be transformed into a comprehensive objective by weighting methods such as comprehensive weighted scoring, entropy weight method, and Critic weight method. Then, range and variance analysis are performed on comprehensive objective results to determine optimal process parameter combination. Domestic and foreign scholars have also combined optimization algorithms such as NSGA-II algorithm, multi-objective differential evolution (MODE) algorithm, multi-objective particle swarm optimization (MOPSO) algorithm, improved particle swarm optimization algorithm with orthogonal experimental design to fit regression models and predict optimal process parameters for injection molded products. When number of experimental factors is large, orthogonal experimental method requires a large number of experiments. Uniform design can select fewer experimental points as representatives from full experimental range, and results can still reflect main characteristics of analysis system, which has obvious advantages. Deng Kexin et al. used uniform design experiments and genetic algorithms to optimize warping deformation of automotive headlight covers and obtained minimum warping deformation of product. Liu Bangyu et al. used optimal Latin hypercube sampling method to design experimental scheme and optimized warping deformation of automotive taillight dual-color covers through multi-island genetic algorithms, and maximum warping deformation decreased by 20.35%. Above studies mainly target conventional products such as automotive lamp covers. These products are mostly installed by adhesive bonding, and stress concentration during assembly is small, which is quite different from aircraft navigation lamp covers. At the same time, existing studies only optimize warping deformation and do not consider structural strength requirements of aircraft navigation lamp covers under special working conditions, nor do they optimize residual stress during injection molding process. This paper takes an aircraft navigation light cover as research object, adopts a uniform experimental design to optimize warpage deformation and residual stress during injection molding process of light cover through multiple objectives. Maximum warpage deformation and maximum residual stress of plastic part are transformed into comprehensive objective values using entropy weight method and an improved compromise programming method. A multiple regression equation with comprehensive objective value as response value is fitted, particle swarm optimization algorithm is used to iteratively optimize regression equation to find minimum comprehensive objective value and its corresponding process parameters. Injection molding simulation is performed according to optimal process parameter combination. Combining regression analysis and simulation, actual injection molding production of navigation light cover is guided.

Structure of aircraft navigation light cover is shown in Figure 1a. Its basic dimensions are 301 mm * 220 mm * 217.5 mm, and its thickness is 4 mm. In actual use, navigation light cover requires secondary machining and hole making around its perimeter. Hole distribution is shown in Figure 1a. Light cover is fastened to fuselage by bolts, and assembly diagram is shown in Figure 1b. When significant warping occurs along X-direction, product edges will experience substantial deformation under pressure of machine body and frame during lampshade assembly. This deformation is called assembly deformation. If residual stress at lampshade edges is high, it can lead to silver streaks or even cracks around holes during secondary processing, causing product failure and scrapping. High warping deformation and residual stress along X-direction increase risk of cracking during assembly and use. Therefore, maximum warping deformation and maximum residual stress in X-direction are selected as optimization targets.

 

Fig. 1 Structure and assembly diagram of product

Lampshade mold cavity arrangement is a two-cavity mold, using a cold runner and fan-shaped gate for gating. As shown in Figure 2, main runner length is 100 mm, upper diameter is 5 mm, and lower diameter is 11.8 mm. Length of fan-shaped gate is 40 mm; width of gate's starting end is 32.2 mm, and thickness is 4.5 mm; shape of injection end is arc-shaped, with a chord length of 81.5 mm, an arc length of 84.8 mm, and a thickness of 1.4 mm.

 

Fig. 2 Design of gating system
Cooling system of lampshade mold is shown in Figure 3. To ensure uniform cooling, cavity side is designed with approximately conformal cooling, as shown in Figure 3a. It consists of 10 series-connected cooling loops, each with a water pipe diameter of 12 mm. Distance between centerline of water loop and product surface is 20 mm, and spacing between water loops is 40 mm. Core side is designed with 7 series-connected cooling loops, with baffle-type water wells to enhance cooling effect. Water well diameter is 18 mm, spacing between water wells is 38 mm, spacing between water loops within same loop is 40 mm, and distance between loops is 60 mm.

 

Figure 3. Cooling System Design

Lampshade model and gating system model were imported into Moldex3D. Lampshade mesh size was 4 mm, and gating system mesh size was 1 mm. Both meshes were 3D solid hybrid meshes, with a 3-layer triangular prism mesh on the surface and a tetrahedral mesh inside. The total number of solid meshes was 8,490,554. A linear model of water channel was established based on cooling system design, with a mesh size of 5 mm and a hexahedral mesh type. In Moldex3D material library, TEIJIN-manufactured polycarbonate (PC) material Panlite L-1250 was selected. Recommended melt temperature range is 280~320 ℃, recommended mold temperature range is 80~120 ℃, ejection temperature is 129 ℃, and curing temperature is 149 ℃.
Based on product's geometric characteristics and material properties of simulation software, initial process parameters were set as follows: melt temperature 300 ℃, mold temperature 100 ℃, injection time 3.8 s, injection pressure 160 MPa, holding time 22 s, holding pressure 160 MPa, cooling time 30 s, and V/P switching point at 98% volume filling. Simulation results of initial scheme are shown in Figure 4. Lampshade was completely filled. Positive values in warping deformation along X-axis represent deformation along positive X-axis, negative values represent deformation along negative X-axis. Maximum warping deformation was taken as maximum absolute value of warping deformation along both directions. Therefore, under initial scheme, maximum warping deformation along X-axis was 3.556 mm, manifested as inward shrinkage on both sides. Maximum residual stress was 7.163 MPa, mainly distributed at filling end on inner side of product. Initial design resulted in significant warpage and residual stress in product. During subsequent drilling and assembly processes, these factors easily led to cracks due to assembly deformation and residual stress, failing to meet usage requirements.

 

Fig. 4 Simulation results of initial scheme

To improve injection molding quality and production efficiency of navigation light cover, relationship between process parameters and optimal target values was predicted by combining uniform design experiments and regression analysis. Optimal target value and its corresponding optimal process parameter combination were then found using a particle swarm optimization algorithm. Optimization process is shown in Fig. 5, with specific steps as follows:

 

Fig. 5 Flowchart of experimental optimization method
Step 1: Based on simulation analysis results of initial scheme, determine indicators to be optimized. Determine value range of each factor level based on recommended process parameter range for material and actual production requirements.
Step 2: Design a uniform experiment with 7 factors and 21 levels, perform injection molding simulation in Moldex3D to analyze and calculate simulated values of two optimization objectives.
Step 3: Calculate weights of two optimization objectives using entropy weight method, and calculate comprehensive objective value.
Step 4: Perform a quadratic multiple regression analysis on comprehensive objective value using backward method in SPSS software. Perform regression analysis on each fitted model to verify fitting accuracy of each model and derive a regression equation with high prediction accuracy.
Step 5: Develop a particle swarm optimization algorithm and use final determined regression equation as fitness function to predict optimal combination of process parameters.

Uniform experimental design involves conducting only one experiment for each level of each factor. When any two experimental points are on a planar grid, there is exactly one experimental point in each row and column, ensuring that experimental points are evenly distributed within experimental range. Neither uniformity nor comparability is considered. Therefore, compared to orthogonal design, it can significantly reduce number of experiments when there are many factors and levels, obtaining the most information with fewer experiments, ensuring that results still reflect main characteristics of system. We selected melt temperature (A), mold temperature (B), injection time (C), injection pressure (D), holding time (E), holding pressure (F), and cooling time (G) as experimental factors, with maximum warpage deformation (Y1) and maximum residual stress (Y2) in X direction as target values. Using initial process parameters as intermediate values, ranges of each factor's level values were determined based on experience with actual process parameter settings: 280 ℃≤A≤320 ℃, 80 ℃≤B≤120 ℃, 2 s≤C≤6 s, 80 MPa≤D≤160 MPa, 20 s≤E≤60 s, 80 MPa≤F≤160 MPa, and 20 s≤G≤60 s. A uniform design experiment with 7 factors and 21 levels was established using Latin hypercube sampling method (see Table 1). Moldex3D was used for filling-holding-cooling-warpage simulation analysis.

 

Table 1 Experiment table of uniform design
Notes: A is melt temperature; B is mold temperature; C is injection time; D is injection pressure; E is holding time; F is holding pressure; G is cooling time.

To avoid errors in fitted regression simulation due to large multiple differences between level values of different factors, level values of each factor were all logarithmic with base 10, i.e., {ai, bi, ci, di, ei, fi, gi}={lg (Ai), lg (Bi), lg (Ci), lg (Di), lg (Ei), lg (Fi), lg (Gi)}, i= 1⁓21. Entropy weight method was used to determine weights of two optimization objectives, maximum warping deformation (Y1) and maximum residual stress (Y2), in comprehensive evaluation. Since two optimization objectives are different in type and order of magnitude, data must first be standardized. In this experiment, optimization objectives for warpage deformation and residual stress were both set to minimize target value (i.e., the smaller target value, the better). Standardization matrix was calculated using standardization formula (1) for negative indices.

 

Where: is standardization matrix, and  is value of j-th index for i-th sample, i = 1~21, j = 1, 2. Weight of i-th sample under j-th index was calculated using formula (2).

 

Where: n is number of samples, n = 21. Information entropy was calculated using formula (3).

 

Information entropy was substituted into formula (4) to calculate difference coefficient of each index. The larger difference coefficient, the higher importance of index.

 

Difference coefficient was substituted into formula (5) to determine weight of each index.

 

Where: m is number of indices, m = 2. Weights of Y1 and Y2 were determined to be 40.82% and 59.18% respectively, based on entropy weight method. Combined target value of two optimization objectives was calculated using improved compromise programming method proposed by Wu Peng et al., as shown in equation (6).

 

Where: is combined target value;  is objective function of j-th sub-objective; is maximum value corresponding to objective function of j-th sub-objective; p is penalty factor, generally taken as 2. Uniform test results and combined target value calculated based on two target values are shown in Table 2. Combined target value of Experiment 2 was the smallest, at 0.133, with a maximum warpage deformation of 2.397 mm and a maximum residual stress of 1.303 MPa.

 

Table 2 Test results and comprehensive target values

Regression analysis is application of statistical methods to establish a regression relationship function expression between dependent and independent variables. Through regression analysis, main factors affecting dependent variable can be obtained, regression equation can be used to predict and control practical problems. Experimental data were analyzed using quadratic multiple regression analysis, and regression model is shown in equation (7).

 

Where: Y is response value; m is the total number of factors; is linear term of factor; is quadratic term of factor;  is interaction term of factor;  is constant term; , , , are all regression coefficients; is random error.
In SPSS software, backward method was used to perform quadratic multiple regression analysis on experimental results. Principle is to first fit a regression equation including all variables that affect response value, then delete independent variables with no significant influence according to importance of factors (F-value) until all remaining variables have a significant influence on response value. In Model 1, independent variables are X67, X34, X15, X1, X14, X12, X66, X11, X33, X35, X37, X57, X3, X77, X45, X36, X47, X26, X25. X35 has the smallest F-value (0.0049) and the largest significance level (P-value) (0.955), indicating its least significant influence; therefore, X35 is removed. Based on F-values and P-values, X35, X33, X66, X77, X36, X34, X14, X57 are removed sequentially from Models 2 to 9. In Model 9, X15 has the least significant influence compared to the other coefficients. Model 9 has a total degrees of freedom of 20 and a residual degrees of freedom of 9. Consulting F-distribution table, we find F_0.05(1,9) = 5.12. F-value for X15 is 11.35 > 5.12; P-value for X15 is 0.008 < 0.05, indicating that X15 has a significant impact on response value, there are no factors with insignificant influence in this model.
Table 3 compares regression models. The larger absolute value of correlation coefficient (R), the stronger correlation between independent variable and response value. Modified R² is an adjustment to R², taking into account number of independent variables in model. It is more suitable for comparing models with different numbers of independent variables. The larger modified R², the stronger model's explanatory power. Standard estimation error reflects model's predictive accuracy; the smaller value, the higher model's predictive accuracy. Comparing R², corrected R², and standard estimation error, Model 7 has a corrected R² of 0.981, indicating strong interpretability of experimental results and reliable model performance. Its standard estimation error is 0.022353, the smallest among nine models, indicating the highest predictive accuracy. Therefore, Model 7 was selected as final fitted model.

 

Table 3: Comparison of regression models in backward method
Analysis of variance for Model 7 is shown in Table 4. F-value is used to characterize significance of model; a larger F-value indicates stronger significance. With a significance level of α = 0.05, table shows F_0.05(13, 7) = 3.550. F-value of regression model is 90.200, significantly greater than 3.550, therefore model can be considered significant.

Table 4. Analysis of variance of regression model 7
Regression equation fitted to the overall target value calculated in SPSS based on results of 21 experimental groups is shown in Equation (8).
Y=17.024-13.839X1-6.562X3+5.216X11-7.279X12+0.773X14+1.621X15+4.354X25+5.56X26+4.5X37-6.651X45+5.569X47+0.58X57-7.162X67
Regression coefficient analysis was performed on each term of regression equation, as shown in Table 5. Absolute value of standardized coefficient indicates degree of influence of corresponding term on response value. Interaction factors with greater influence are X67, X45, X47, X25, X26, namely, interaction between holding pressure and cooling time, injection pressure and holding time, cooling time, mold temperature and holding time, holding pressure, and other process parameters. Except for X14 and X57, significance level (P value) of other terms is less than 0.05.

 

Table 5. Analysis of regression coefficient
Process parameter combinations from 21 experiments were input into regression equation to calculate predicted values, which were then compared with simulated values of each experiment, as shown in Figure 6. Experiment 2 had the largest deviation, with a value of 0.075. Average deviation of all experiments was 0.029, approximately 5.6% of average of simulated results. Small error between simulated and predicted values verifies that regression model has high accuracy, can predict optimal response value and optimal process parameter combination well.

 

Figure 6. Comparison of simulated values and predicted values

A standard particle swarm optimization (PSO) algorithm was written using MATLAB. Regression fitting equation was used as fitness function, constraint range of independent variables was range of process parameter values, minimum value of fitness function was iterative optimization objective of PSO. Search space is set to have a dimension of 7, a particle count of 30, an iteration count of 100, an inertia weight of 0.5, an individual learning factor and a swarm learning factor of 2, and a maximum velocity of 0.1. Basic framework of algorithm is shown in Figure 7. First, position and velocity of each particle are randomly determined. Position and velocity of each particle are continuously updated, historical best position and fitness of individual and swarm are recorded. Termination condition is reaching maximum number of iterations or minimizing difference in fitness values between two adjacent iterations. Once termination condition is met, optimal value is output and analyzed.

 

Figure 7. Basic framework of particle swarm optimization algorithm.
Algorithm reached its minimum value and began to converge on 11th iteration, as shown in Figure 8. Minimum fitness function value was 0.096, which is minimum comprehensive target value predicted by regression fitting equation. Corresponding process parameter combination is: melt temperature (A), 306 ℃; mold temperature (B), 82 ℃; injection time (C), 3 s; injection pressure (D), 85 MPa; holding time (E), 24 s; holding pressure (F), 138 MPa; cooling time (G), 54 s.

 

Figure 8. Iterative optimization process of particle swarm optimization algorithm.

Filling-holding-cooling-warpage analysis was performed in Moldex3D according to optimal process parameter combination. Simulation results of lampshade's warpage deformation and residual stress are shown in Figure 9. Maximum warpage deformation is 0.812 mm, which is 2.744 mm less than initial scheme, a reduction of 77.17%. Maximum residual stress is 1.256 MPa, which is 5.907 MPa less than initial scheme, a reduction of 82.47%. Comprehensive target value is 0.103, which differs from predicted optimal value by 0.007, indicating a small error. This demonstrates that regression model proposed in this study has high prediction accuracy and can be used as a reference for actual production.

 

Figure 9 Simulation results of optimization scheme

Lampshade was verified by actual injection molding using optimal process parameter combination. Injection molding equipment was Haitian JU24000IIs/14 600. Molded product is shown in Figure 10. Plastic part has a complete structure, without shrinkage marks, flow marks, bubbles, or other appearance defects. Light transmittance reaches over 89%, and haze is less than 1%, meeting product design requirements. Meanwhile, measured performance of products from initial and optimized processes was compared (see Table 6). Warpage deformation was measured using a steel ruler along span in X direction. Residual stress was tested using a SINT MTS3000-Restan blind hole stress analyzer, with a drilling depth of 1 mm at the end of lampshade filling. It can be seen that optimized product showed significant improvements in both warpage deformation and residual stress, with a deformation reduction of 81.81% and a residual stress reduction of 46.15%. Comparison of experimental and simulation results revealed that trends in warpage deformation and residual stress of navigation lampshade were basically consistent with simulation values. However, actual warpage value was greater than simulation value, while actual reduction in residual stress was lower than simulation. This was mainly due to a certain deviation between boundary settings of simulation analysis and actual working conditions, and use of a viscous constitutive equation, which neglected elastic effect of material. This led to certain deviations in deformation and stress analysis of product. Further research will be conducted on viscoelastic constitutive relationship of material to further improve accuracy of simulation analysis. Meanwhile, drilling, assembly, and use of lampshade manufactured under optimized process were completed. After more than one year of service, no problems such as edge cracking have occurred, showing a significant improvement in crack resistance compared to previous models.

 

Fig. 10 Sample of mold trial

Table. 6 Properties of lamp by process optimization

(1) To address requirements for structural strength and assembly connections of aircraft navigation lampshades under service conditions, multi-objective optimization was conducted on warpage deformation and residual stress during injection molding process of lampshade. A uniform test with 7 factors and 21 levels was designed. Combined target value of two optimization objectives was calculated using entropy weight method and improved compromise programming method. A backward regression analysis was performed on test results to compare interpretability and prediction accuracy of each model, final regression model was determined.
(2) A multiple regression equation with comprehensive target value as response value was fitted using SPSS software. Fitted regression model was iteratively optimized using a particle swarm optimization algorithm, yielding an optimal comprehensive target value of 0.096. Corresponding process parameters were: melt temperature 306 ℃, mold temperature 82 ℃, injection time 3 s, injection pressure 85 MPa, holding time 24 s, holding pressure 138 MPa, and cooling time 54 s.
(3) Injection molding simulation was performed using optimized process parameters. Maximum warpage deformation of lampshade was 0.812 mm, a reduction of 77.17% compared to initial scheme; maximum residual stress was 1.256 MPa, a reduction of approximately 82.47%. Comprehensive target value was 0.103, differing from predicted optimal value by 0.007, indicating a small prediction error in regression model. Actual production verification confirmed that product quality met usage requirements.