## 4 Response-Surface Modeling  
Response-surface methodology (RSM) is a common multi-objective optimization model, which is widely used in materials-related fields and has also been applied to motor optimization in recent years. In order to obtain the relationships between the optimization parameters and the optimization objectives, we tested the modeling accuracy of different regression equations to build a response surface in the software Design Expert 13, through which it was concluded that quadratic polynomial response-surface modeling had the highest accuracy, with variance above 0.9. Therefore, in this study, the quadratic polynomial method was chosen to establish the response-surface model, the formula for which is as follows [29]:  
\[f(x)=\beta_{0}+\beta_{i}\underset{i=1}{\sum}x_{i}+\beta_{ii}\underset{i=1}{ \sum}x_{i}^{2}+\beta_{ij}\underset{i<j}{\sum}x_{i}x_{j}+\varepsilon \tag{5}\]  
Figure 5: Sensitivity analysis.  
Figure 6: Integrated sensitivity analysis.  
where \(\beta_{0}\) is the constant term; \(\beta_{i}\) is the first-order coefficient; \(\beta_{ii}\) is the second-order coefficient; \(\beta_{\bar{y}}\) is the interaction term coefficient; and \(\epsilon\) is the error constant.  
In order to reduce the experimental volume, we adopted the Box-Behnken Design (BBD) test method to establish the response-surface model. The BBD method classifies individual parameters into three levels (\(-1\), \(0\), and \(1\)), with \(0\) representing the center point and \(-1\) and \(1\) representing the lowest and highest levels, effectively reducing the number of finite element methods [30]. First, a 3-level, 4-factor experimental model was established using the BBD experimental method in Design Expert 13 software, and then Ansys Maxwell 2022r2 software was used to complete the simulation calculation for 29 groups of experiments; finally, the second-order regression equation was chosen to establish the response-surface model and analyze the model's accuracy. The parameters and significance levels are shown in Table 4.  
The results of the target model evaluation are shown in Table 5, in which the signal-to-noise ratios of the output torque, back EMF amplitude, and torque ripple quadratic models are all greater than 4 dB, the variances are all greater than 0.9, and the CVs are all less than 10%, thus proving that the tests are highly credible and accurate, and suggesting that the response-surface model can adequately describe the relationships between the variables and the target variables. In Figure 7, which illustrates the response-surface model plots of optimization parameters and objectives, it can be seen that all three response-surface models have extreme values, where the optimization parameters have optimal ranges, but they are different for each model. Therefore, in order to avoid conflicts between the optimization objectives, it was necessary to use the multi-objective optimization algorithm to obtain the optimal solution set.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter/Level** & \(-1\) & \(0\) & \(1\) \\ \hline RW1 & 4 mm & 5.5 mm & 7 mm \\ WM1 & 25 mm & 27.5 mm & 30 mm \\ RW2 & 11 mm & 13 mm & 15 mm \\ WM2 & 11 mm & 13 mm & 15 mm \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameters and significance levels.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Optimization Goal** & **Model** & \(\mathbf{R^{2}}\) & **CV (\%)** &
\begin{tabular}{c} **Signal-to-Noise** \\ **Ratio (dB)** \\ \end{tabular} \\ \hline Output torque & Quadratic & 0.99 & 0.1 & 242.12 \\ Torque ripple & Quadratic & 0.92 & 6.3 & 13.46 \\ Back EMF amplitude & Quadratic & 0.99 & 0.38 & 59.45 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Assessment results.  
Figure 7: Response-surface model: (**a**) output torque; (**b**) torque ripple; and (**c**) back EMF amplitude.