An Effective Shrinkage Control Method for Tooth Profile Accuracy Improvement of Micro-Injection-Molded Small-Module Plastic Gears

An effective method to control the non-linear shrinkage of micro-injection molded small-module plastic gears by combining multi-objective optimization with Moldflow simulation is proposed. The accuracy of the simulation model was verified in a micro-injection molding experiment using reference process parameters. The maximum shrinkage (Y1), volume shrinkage (Y2), addendum diameter shrinkage (Y3), and root circle diameter shrinkage (Y4) were utilized as optimization objectives to characterize the non-linear shrinkage of the studied gear. An analysis of the relationship between key process parameters and the optimization objectives was undertaken using a second-order response surface model (RSM-Quadratic). Finally, multi-objective optimization was carried out using the non-dominated sorting genetic algorithm-II (NSGA-II). The error rates for the key shrinkage dimensions were all below 2%. The simulation results showed that the gear shrinkage variables, Y1, Y2, Y3, and Y4, were reduced by 5.60%, 8.23%, 11.71%, and 11.39%, respectively. Moreover, the tooth profile inclination deviation (fHαT), the profile deviation (ffαT), and the total tooth profile deviation (FαT) were reduced by 47.57%, 23.43%, and 49.96%, respectively. Consequently, the proposed method has considerable potential for application in the high-precision and high-efficiency manufacture of small-module plastic gears.


Introduction
In recent decades, with the rapid development of high-performance engineering plastics [1,2], small-module plastic gear has been widely applied in cutting-edge fields, such as aerospace [3], medical devices [4], and automobiles [5]. Compared with small-module metal gear, small-module plastic gear has obvious advantages, such as self-lubrication, strong shock and vibration resistance, and low noise [6,7]. Micro-injection molding is likely to become the main processing method for small-module plastic gear in the future because it enables production in large quantities [8,9]. However, because of the high shrinkage of engineering plastics and the irregular surface shape of gear teeth, plastic gear can undergo significant non-linear shrinkage during micro-injection molding [10][11][12][13]. Moreover, the specific surface area of small module gear teeth is small, which leads to complex and changeable interface heat transfer events during the cooling process [14,15]. Hence, the non-linear shrinkage of micro-injection molded small-module plastic gears is difficult to control.
Because of the small size of the small-module gear itself, the non-linear shrinkage phenomenon during the micro-injection molding greatly influences the tooth profile accuracy of small-module plastic gears [16]. In addition to the rational design of the mold between the simulation and experimental tests were less than 2%. Moreover, tooth profile inclination deviation , tooth profile deviation and total tooth profile deviation were reduced by 47.57%, 23.43%, and 49.96%, respectively. Consequently, the proposed method can inform the high-precision and high-efficiency manufacture of smallmodule plastic gears.

3D model and Material of Studied Gear
In this study, small-module involute cylindrical gear (module = 0.5 mm) was studied. A three-dimensional (3D) model and the dimension parameters of the studied gear are shown in Figure 2. Polyformaldehyde (POM) is most commonly used for production of plastic gears. It is a thermoplastic crystalline polymer with high strength and good wear resistance. The studied gear was made from POM-100 P, produced by DuPont,US, which is the most used high-performance crystalline engineering plastic for small-module plastic gears [31][32][33]. The physical properties of the materials used are shown in Table 1.

3D model and Material of Studied Gear
In this study, small-module involute cylindrical gear (module = 0.5 mm) was studied. A three-dimensional (3D) model and the dimension parameters of the studied gear are shown in Figure 2. Polyformaldehyde (POM) is most commonly used for production of plastic gears. It is a thermoplastic crystalline polymer with high strength and good wear resistance. The studied gear was made from POM-100 P, produced by DuPont, US, which is the most used high-performance crystalline engineering plastic for small-module plastic gears [31][32][33]. The physical properties of the materials used are shown in Table 1.
between the simulation and experimental tests were less than 2%. Moreover, tooth profile inclination deviation , tooth profile deviation and total tooth profile deviation were reduced by 47.57%, 23.43%, and 49.96%, respectively. Consequently, the proposed method can inform the high-precision and high-efficiency manufacture of smallmodule plastic gears.

3D model and Material of Studied Gear
In this study, small-module involute cylindrical gear (module = 0.5 mm) was studied. A three-dimensional (3D) model and the dimension parameters of the studied gear are shown in Figure 2. Polyformaldehyde (POM) is most commonly used for production of plastic gears. It is a thermoplastic crystalline polymer with high strength and good wear resistance. The studied gear was made from POM-100 P, produced by DuPont,US, which is the most used high-performance crystalline engineering plastic for small-module plastic gears [31][32][33]. The physical properties of the materials used are shown in Table 1.

Moldflow Simulation
Autodesk Moldflow Insight (AMI) 2021 software, US, [34,35] was used for the simulation of small-module plastic gear micro-injection molding. For small-module gears, the polymer filling process is extremely short, and the process cannot be observed visually, so it is difficult to characterize experimentally. However, the polymer flow molding process can be studied visually using Moldflow. To facilitate later extraction of the 3D shrinkage gear model, a built 3D model of the studied gear was imported into the Moldflow software using an .stp extension. The gating system of the simulation model was established according to the actual forming gating system. A double-layer mesh was used for meshing, and the global edge length was 0.15 mm. The mesh chord angle was 20 degrees. The minimum curvature size was 10% of the global size. The mesh model was optimized by reducing the aspect ratio through operations such as merging nodes and integrating nodes. The injection gate position was then set according to the actual micro-injection molding. The gear simulation model is shown in Figure 3.

Moldflow Simulation
Autodesk Moldflow Insight (AMI) 2021 software, US, [34,35] was used for the simulation of small-module plastic gear micro-injection molding. For small-module gears, the polymer filling process is extremely short, and the process cannot be observed visually, so it is difficult to characterize experimentally. However, the polymer flow molding process can be studied visually using Moldflow. To facilitate later extraction of the 3D shrinkage gear model, a built 3D model of the studied gear was imported into the Moldflow software using an .stp extension. The gating system of the simulation model was established according to the actual forming gating system. A double-layer mesh was used for meshing, and the global edge length was 0.15 mm. The mesh chord angle was 20 degrees. The minimum curvature size was 10% of the global size. The mesh model was optimized by reducing the aspect ratio through operations such as merging nodes and integrating nodes. The injection gate position was then set according to the actual micro-injection molding. The gear simulation model is shown in Figure 3.  The analysis type chosen for this study was "thermoplastic injection molding". The analysis sequence was "Filling + Packing + Warpage". Then, POM 100-P from DuPont was selected in the software material library. Parallel computing was specified to improve the computing speed. The analysis type chosen for this study was "thermoplastic injection molding". The analysis sequence was "Filling + Packing + Warpage". Then, POM 100-P from DuPont was selected in the software material library. Parallel computing was specified to improve the computing speed.

Experimental Design
Studies have found that the local shrinkage, overall shrinkage and key dimension shrinkage of plastic gears directly affect the tooth accuracy [25]. For this reason, the maximum shrinkage (Y 1 ), volume shrinkage (Y 2 ), addendum diameter shrinkage (Y 3 ) and root circle diameter shrinkage (Y 4 ) were taken to be optimization objectives for multi-objective optimization. The maximum shrinkage (Y 1 ) and volume shrinkage (Y 2 ) were obtained directly from the simulation results. The key dimension shrinkage was also assessed to verify the accuracy of the simulation model. The 3D model of the gear after shrinkage was extracted through Moldflow simulation. The diameter of the addendum and root circle were measured by CAD software. The addendum diameter shrinkage (Y 3 ) and the root circle diameter shrinkage (Y 4 ) were calculated using Equations (1) and (2).
where d a and d a are the addendum circle diameters of the studied gear and the shrinkage gear, and d f and d f are the root circle diameters of the studied gear and the shrinkage gear. Based on previous research and long-term practical experience in the production of small-module plastic gears, the mold temperature, melt temperature, packing pressure, packing time, and cooling time were selected as input factors. Compared to traditional single-factor experimental designs and orthogonal experimental designs, the Box-Behnken design (BBD) method facilitates the analysis of individual and interactional effects of parameters on responses with fewer experiments and lower computational cost [36,37]. Thus, the BBD method was utilized to quantitatively analyze the influence of key process parameters on the shrinkage of small-module plastic gear. The micro-injection molding process levels were selected according to the range of process parameters recommended by the Moldflow material library, and production experience, as shown in Table 2. The 46 groups of BBD simulation results are shown in Table 3. It was found that the optimal results for Y 1 , Y 2 , Y 3 and Y 4 appeared in 39, 13, 12 and 41 groups of trials, respectively. Therefore, the interaction between the objectives was substantial, and the four objectives could not be minimized at the same time through conventional single-objective optimization.

Approximate Surrogate Model
An approximate surrogate model can effectively save time required for CAE simulation. If the accuracy of the established model is reliable, the model can be applied to predict the properties of unknown points in space. In this study, Kriging models [38], radial basis function (RBF) models [39], linear response surface models (RSM) [40], RSM-Quadratic, RSM-Cubic and RSM-Quartet models between process parameters, and optimization objectives, were established using the Isight platform, US [41,42]. Table 3. Box-Behnken experimental design and simulation results.

Run#
Factor Response of Optimize Objectives To determine if a model is representative, it is necessary to verify the convergence accuracy of the model through error analysis. The method used to verify the accuracy of the surrogate model involves comparison of the approximation of the predicted value and the test sample. This study mainly used R-square (R 2 ) to evaluate the accuracy of the built surrogate model. The most suitable surrogate model was selected through error analysis to predict the shrinkage of the gear more accurately. Moreover, the approximate surrogate model was suitable for further multi-objective optimization to minimize the shrinkage of the small-module plastic gear.

Multi-Objective Optimization
Since practical engineering problems are complex and involve uncertain factors, most problems can be studied with multi-objective optimization strategies. Multi-objective optimization algorithms provide more satisfactory solutions than single-objective optimization methods [43,44]. Multi-objective optimization methods have attracted the attention of researchers in different disciplines and fields. Many population-based algorithms have been proposed which have been used to solve the multi-objective optimization problem (MOP) in engineering applications.
The improved non-dominated sorting genetic algorithm, NSGA-II [45,46], has good exploration performance and an efficient sorting process. The principle of the NSGA-II algorithm is shown in Figure 4. The algorithm not only uses crowding degree and a crowding degree comparison operator to replace the fitness sharing strategy under the control of sharing radius, but also introduces an elite strategy to ensure that the best individual is not lost. Therefore, in this study, the NSGA-II multi-objective optimization algorithm was selected to optimize gear shrinkage. For the algorithm settings, the population side was set to 12, the number of generations was set to 20, and the number of samples was 240, which meets the recommended value of between 20 and 200. The crossover probability crossover rate was set to 0.9, the recommended value was 0.6~1, and the remaining settings were set by default.
the surrogate model involves comparison of the approximation of the predicted value and the test sample. This study mainly used R-square ( ) to evaluate the accuracy of the built surrogate model. The most suitable surrogate model was selected through error analysis to predict the shrinkage of the gear more accurately. Moreover, the approximate surrogate model was suitable for further multi-objective optimization to minimize the shrinkage of the small-module plastic gear.

Multi-Objective Optimization
Since practical engineering problems are complex and involve uncertain factors, most problems can be studied with multi-objective optimization strategies. Multi-objective optimization algorithms provide more satisfactory solutions than single-objective optimization methods [43,44]. Multi-objective optimization methods have attracted the attention of researchers in different disciplines and fields. Many population-based algorithms have been proposed which have been used to solve the multi-objective optimization problem (MOP) in engineering applications.
The improved non-dominated sorting genetic algorithm, NSGA-Ⅱ [45,46], has good exploration performance and an efficient sorting process. The principle of the NSGA-II algorithm is shown in Figure 4. The algorithm not only uses crowding degree and a crowding degree comparison operator to replace the fitness sharing strategy under the control of sharing radius, but also introduces an elite strategy to ensure that the best individual is not lost. Therefore, in this study, the NSGA-II multi-objective optimization algorithm was selected to optimize gear shrinkage. For the algorithm settings, the population side was set to 12, the number of generations was set to 20, and the number of samples was 240, which meets the recommended value of between 20 and 200. The crossover probability crossover rate was set to 0.9, the recommended value was 0.6~1, and the remaining settings were set by default. The mathematical model of the process optimization problem of gear shrinkage in this study was expressed as Equation (3).
230 ° 60 80 1 5 10 30 (3) The mathematical model of the process optimization problem of gear shrinkage in this study was expressed as Equation (3).

Experimental Verification of Simulation Model and Optimization Method
First, the reliability of the gear simulation model required to be verified through micro-injection molding pre-testing with reference process parameters. The self-built microinjection molding experiment platform is shown in Figure 5a, and the mold structure is shown in Figure 5b. Figure 5c is the actual mold of the studied gear. According to the recommended process for assessing materials, the reference process parameters are shown in Table 4. The diameter of the addendum circle and root circle were measured using a  (1) and (2), respectively. The reliability of the simulation model was verified by comparing the key dimensions shrinkage between the simulation and experiment under reference process parameters.
First, the reliability of the gear simulation model required to be verified through micro-injection molding pre-testing with reference process parameters. The self-built microinjection molding experiment platform is shown in Figure 5a, and the mold structure is shown in Figure 5b. Figure 5c is the actual mold of the studied gear. According to the recommended process for assessing materials, the reference process parameters are shown in Table 4. The diameter of the addendum circle and root circle were measured using a digital super-depth microscope (KEYENCE VHX-1000C, US). The addendum diameter shrinkage ( ) and the root circle diameter shrinkage of the gear samples were calculated using Equations (1) and (2), respectively. The reliability of the simulation model was verified by comparing the key dimensions shrinkage between the simulation and experiment under reference process parameters.  The results of multi-objective optimization require to be verified, and injection molding experiments were carried out under the obtained optimal process parameters. The  The results of multi-objective optimization require to be verified, and injection molding experiments were carried out under the obtained optimal process parameters. The optimization results of the NSGA-II algorithm were verified by Moldflow simulation. The accuracy of the optimized method was verified by comparing the key dimensions shrinkage and the tooth profile accuracy of the gear from the micro-injection molding experiments under reference and optimal process parameters. The tooth profile accuracy was automat-ically measured using an FPG 3002B small-module gear measuring machine [47]. Tooth profile accuracy [48,49] includes the deviation of tooth profile inclination ( f HαT ), tooth profile deviation ( f f αT ) and total tooth profile deviation (F αT ). The gear clamping and specific clamping method for the small-module gear are shown in Figure 6.
Polymers 2022, 14, 3114 9 of 21 optimization results of the NSGA-II algorithm were verified by Moldflow simulation. The accuracy of the optimized method was verified by comparing the key dimensions shrinkage and the tooth profile accuracy of the gear from the micro-injection molding experiments under reference and optimal process parameters. The tooth profile accuracy was automatically measured using an FPG 3002B small-module gear measuring machine [47]. Tooth profile accuracy [48,49] includes the deviation of tooth profile inclination ( ), tooth profile deviation ( ) and total tooth profile deviation ( ). The gear clamping and specific clamping method for the small-module gear are shown in Figure 6.

Simulation Model Accuracy
Since the flow channel was injected symmetrically, the shrinkage of the two gears was basically the same. For this reason, we only selected a single gear for analysis. The shrinkage results of the studied gear obtained by Moldflow analysis are shown in Figure  7. From Figure 7a, it can be seen that the shrinkage of the gear was different at different positions. The tooth part of the gear shrank the most. During the cooling process, the cooling rate indicated by the gear teeth was very high due to the large temperature difference between the mold and the polymer melt. Accordingly, the gear teeth measurements indicated that the volume of the polymer molecular chain changed greatly. The degree of shrinkage near the addendum and the root circle of the studied gear was the largest, which is consistent with previous research findings [15,25]. In addition, the shrinkage of the part where the gear shrank the most was 0.4183 mm. The test results for the key dimensions of the gear obtained by simulation are shown in Figure 7b. The volume shrinkage of the gear obtained by simulation was 13.49% (Figure 7c).

Simulation Model Accuracy
Since the flow channel was injected symmetrically, the shrinkage of the two gears was basically the same. For this reason, we only selected a single gear for analysis. The shrinkage results of the studied gear obtained by Moldflow analysis are shown in Figure 7. From Figure 7a, it can be seen that the shrinkage of the gear was different at different positions. The tooth part of the gear shrank the most. During the cooling process, the cooling rate indicated by the gear teeth was very high due to the large temperature difference between the mold and the polymer melt. Accordingly, the gear teeth measurements indicated that the volume of the polymer molecular chain changed greatly. The degree of shrinkage near the addendum and the root circle of the studied gear was the largest, which is consistent with previous research findings [15,25]. In addition, the shrinkage of the part where the gear shrank the most was 0.4183 mm. The test results for the key dimensions of the gear obtained by simulation are shown in Figure 7b. The volume shrinkage of the gear obtained by simulation was 13.49% (Figure 7c).
The teeth morphology of the gear simple under reference process parameters was observed using a 200-× microscope, as shown in Figure 8a. It can be seen that the forming quality of the gear teeth was intact, so the gear samples could be adopted to verify the simulation model. The addendum and root circle diameters of the gear samples measured under the 50-× microscope are shown in Figure 8b. The key dimension shrinkage of the gear derived through simulation and experiment was calculated using Equations (1) and (2), as shown in Table 5. It was found that the errors between the experiment and the simulation were all less than 2 %. Therefore, the simulation model was able to accurately simulate the shrinkage of the studied gear. The teeth morphology of the gear simple under reference process parameters was observed using a 200-× microscope, as shown in Figure 8a. It can be seen that the forming quality of the gear teeth was intact, so the gear samples could be adopted to verify the simulation model. The addendum and root circle diameters of the gear samples measured under the 50-× microscope are shown in Figure 8b. The key dimension shrinkage of the gear derived through simulation and experiment was calculated using Equations (1) and (2), as shown in Table 5. It was found that the errors between the experiment and the simulation were all less than 2 %. Therefore, the simulation model was able to accurately simulate the shrinkage of the studied gear.

Approximate Surrogate Models and Effect Analysis of Process Parameters
The data in Table 2 was input into Isight software. The Approximate module was used to build the approximate surrogate models. Since the input factors are different types of physical quantities, the adaptation type was selected as anisotropic. To select the approximate surrogate model with the highest accuracy, the accuracy of the established five typical approximate surrogate models was compared; the results are shown in Table 6. It was found that the accuracy of the RSM-Linear model was clearly lower, and the accuracy of the optimization objectives was less than 85%. This indicated that there was a strong non-linear relationship between the optimization objectives and the process parameters. It was found that that the accuracy of volume shrinkage (Y 2 ) in the Kriging models and the RBF models was below 90%. The RSM-Quadratic, RSM-Cubic, and RSM-Quartet models all achieved the required accuracy. However, compared with the RSM-Quadratic model, the accuracy improvement using the RSM-Cubic and RSM-Quartet models was not very significant; however, the computational complexity increased substantially. Accordingly, the RSM-Quadratic model was applied to represent the response relationship between the process parameters and the optimization objectives of the studied gear. The RSM-Quadratic coefficients of the optimization objectives are shown in Table 6. Table 6. The accuracy of the Kriging models, RBF models, and the RSM-Linear, RSM-Quadratic, RSM-Cubic and RSM-Quartet models.

Approximation Surrogate Model
Accuracy (%) Therefore, the following RSM-Quadratic model was employed to represent the relationship between the input factors and the responses: where f is the output response, a i = 0, 1, . . . , k are coefficients of regression, and x i , i = 0, 1, . . . , k are input factors. The RSM-Quadratic coefficients of the optimization objectives are shown in Table 7.
The significance of the RSM-Quadratic coefficients was evaluated, as shown in Table 8. The p-value for Y 1 , Y 2 , Y 3 , and Y 4 , as a function of D 2 , was 0.000. Therefore, besides A, B, C and D, there was a strong interaction between D and D. To simplify the RSM-Quadratic, only A, B, C, D, E and D 2 were considered in the regression equations for the optimization objectives. The regression equations for Y 1 , Y 2 , Y 3 , and Y 4 are Equations (5)- (8).   Figure 9 shows the error analysis results of the RSM-Quadratic of the Gaussian spatial correlation function. The abscissa is the predicted value of the RSM-Quadratic and the ordinate is the actual measured value. The blue horizontal line is the average response value of the model, and the white background indicates that the model has high accuracy. The black line with a slope of 1 represents the ideal model. The closer the red dot is to the black slash, the more accurate and representative the surrogate model is. The red dots in the figure are evenly distributed on both sides of the black line and are very close. This indicates that the fitting accuracy of the RSM-Quadratic established in this study was very high. Therefore, the RSM-Quadratic established in this study can be used to predict the shrinkage of the studied gear micro-injection molding. It will save time spent on multiple Moldflow simulation attempts.  Table 9 is the variance analysis table of the maximum shrinkage in which the contribution ratio of the mold temperature and the dwell time is shown to be 5.05% and 81.61%. Table 10 is the variance analysis table of volume shrinkage in which the contribution ratio of the melt temperature and packing time is shown to be 13.68% and 73.71%. Table 11 is the variance analysis table of the addendum diameter shrinkage in which the contribution ratio of the mold temperature and the packing time is shown to be 7.99% and 80.59%. Table 12 is the variance analysis table of root circle diameter shrinkage in which the contribution ratio of the mold temperature and the packing time is shown to be 9.81% and 75.44%. Therefore, it can be concluded that the packing pressure had the greatest influence on the non-linear shrinkage of the small-module plastic gears.
Then, the interactive effects of the micro-injection molding process parameters on the shrinkage of the small-module plastic gears was further investigated. The interaction effect plots for the maximum shrinkage ( ), volume shrinkage ( ), addendum diameter shrinkage ( ) and root circle diameter shrinkage ( ) were obtained, as shown in Figure  11. If the two straight lines are not parallel or intersect, there is an interaction effect between the process parameters; the greater the degree of non-parallelism, the stronger the Then, the main effect diagram and variance analysis were used to analyze the influence of the process parameters on the optimization objectives. The main effect diagram is shown in Figure 10. Figure 10a,c,d are the main effect diagrams of maximum shrinkage, addendum diameter shrinkage and root circle diameter shrinkage, respectively. It was found that there was a positive correlation between the mold temperature and the maximum shrinkage, the addendum diameter shrinkage and the root circle diameter shrinkage. However, there was a negative correlation between the melt temperature, packing pressure and packing pressure and the maximum shrinkage, addendum diameter shrinkage and root circle diameter shrinkage. The maximum shrinkage, addendum diameter shrinkage and root circle diameter shrinkage increased first and then decreased with increase in cooling time. Figure 10b is the main effect diagram of the volume shrinkage. It can be seen that the volume shrinkage of the studied gear first increased and then decreased with increase in the packing pressure.    Table 9 is the variance analysis table of the maximum shrinkage in which the contribution ratio of the mold temperature and the dwell time is shown to be 5.05% and 81.61%. Table 10 is the variance analysis table of volume shrinkage in which the contribution ratio of the melt temperature and packing time is shown to be 13.68% and 73.71%. Table 11 is the variance analysis table of the addendum diameter shrinkage in which the contribution ratio of the mold temperature and the packing time is shown to be 7.99% and 80.59%. Table 12 is the variance analysis table of root circle diameter shrinkage in which the contribution ratio of the mold temperature and the packing time is shown to be 9.81% and 75.44%. Therefore, it can be concluded that the packing pressure had the greatest influence on the non-linear shrinkage of the small-module plastic gears. Then, the interactive effects of the micro-injection molding process parameters on the shrinkage of the small-module plastic gears was further investigated. The interaction effect plots for the maximum shrinkage (Y 1 ), volume shrinkage (Y 2 ), addendum diameter shrinkage (Y 3 ) and root circle diameter shrinkage (Y 4 ) were obtained, as shown in Figure 11. If the two straight lines are not parallel or intersect, there is an interaction effect between the process parameters; the greater the degree of non-parallelism, the stronger the interaction effect. It can be seen that there was a large interaction between the process parameters, so the optimal combination of process parameters cannot be determined using a traditional single factor design.

NSGA-II Algorithm Optimization Results
The optimal process parameters obtained by NSGA-II were a mold temperature of 61.44 • C, a melt temperature of 193.64 • C, a packing pressure of 79.84 MPa, a packing time of 4.48 s and a cooling time of 20 s. Figure 12 shows the simulation results under the optimal parameters. The maximum shrinkage of the studied gear was 0.1400 mm (Figure 12a), which was reduced by 5.60% compared to the shrinkage of the gear under the reference process parameters. The volume shrinkage of the studied gear was 12.38% (Figure 12b), which was reduced by 8.23% compared to the shrinkage of gear under the reference process parameters. The addendum diameter shrinkage (Y 3 ) and root circle diameter shrinkage (Y 4 ) calculated according to Figure 12c were 5.618% and 5.762%, which were reduced by 11.71% and 11.39%, respectively. Compared with the optimization objectives predicted by the approximate surrogate models, the relative errors were all less than 5%, as shown in Table 13.

NSGA-II Algorithm Optimization Results
The optimal process parameters obtained by NSGA-II were a mold temperature of 61.44 °C, a melt temperature of 193.64 °C, a packing pressure of 79.84 MPa, a packing time of 4.48 s and a cooling time of 20 s. Figure 12 shows the simulation results under the optimal parameters. The maximum shrinkage of the studied gear was 0.1400 mm ( Figure  12a), which was reduced by 5.60% compared to the shrinkage of the gear under the reference process parameters. The volume shrinkage of the studied gear was 12.38% ( Figure  12b), which was reduced by 8.23% compared to the shrinkage of gear under the reference process parameters. The addendum diameter shrinkage ( ) and root circle diameter shrinkage ( ) calculated according to Figure 12c were 5.618% and 5.762%, which were reduced by 11.71% and 11.39%, respectively. Compared with the optimization objectives predicted by the approximate surrogate models, the relative errors were all less than 5%, as shown in Table 13.

Optimization Method Validation
The measurement results for the key dimension shrinkage of the gear samples in the micro-injection molding experiment under reference and optimal process parameters are shown in Figure 13a. It was found that the addendum diameter shrinkage (Y 3 ) and the root circle diameter shrinkage (Y 4 ) decreased by 9.44% and 9.91%, respectively. Moreover, the tooth profile inclination deviation ( f HαT ), tooth profile deviation ( f f αT ) and total tooth profile deviation (F αT ) were reduced by 47.57%, 23.43%, and 49.96%, respectively (Figure 13b). Figure 13c shows the tooth profile deviation curve of the gear sample with the highest tooth profile accuracy under optimal process parameters. It can be seen that the deviation near the addendum circle and the root circle was larger, while the deviation near the graduation circle was smaller, which was also close to the simulation results. Consequently, the tooth profile accuracy of the small-module plastic gears could be significantly improved using the proposed method. profile deviation ( ) were reduced by 47.57%, 23.43%, and 49.96%, respectively ( Figure  13b). Figure 13c shows the tooth profile deviation curve of the gear sample with the highest tooth profile accuracy under optimal process parameters. It can be seen that the deviation near the addendum circle and the root circle was larger, while the deviation near the graduation circle was smaller, which was also close to the simulation results. Consequently, the tooth profile accuracy of the small-module plastic gears could be significantly improved using the proposed method.

Conclusions
In summary, this paper has proposed an effective method to improve the tooth profile accuracy of micro-injection molded small-module plastic gears. First, based on the Box-Behnken design method, a simulation of the micro-injection molding of the studied

Conclusions
In summary, this paper has proposed an effective method to improve the tooth profile accuracy of micro-injection molded small-module plastic gears. First, based on the Box-Behnken design method, a simulation of the micro-injection molding of the studied gear was carried out. Next, the response relationship between the process parameters and the optimization objectives was analyzed using the RSM-Quadratic. The optimization objectives were non-normalized optimized using the NSGA-II algorithm. Finally, an experimental analysis of the micro-injection molding of the studied gear was carried out under reference and optimal process parameters to verify the optimization method. The main conclusions of this study are as follows.

1.
The shrinkage of the gear was different at different positions. The tooth part of the gear shrank the most. Moreover, the shrinkage degree near the addendum and the root circle of the small-module plastic gear was the largest. The relative errors of the key dimension shrinkage between the micro-injection molding experiment and simulation were all less than 2%.

2.
The factor that had the greatest influence on the shrinkage of the small-module plastic gears was the packing time, and there was a complex interaction between the process parameters. The accuracy of the RSM-Quadratic for the optimization objectives reached more than 92 %. 3.