## 4 Optimal Design Process  
The design process followed the parameters of the basic WFSM, maintaining consistency in the stator's outer diameter and other design variables. However, only the parameters related to the assisted magnet were considered design variables in this study. Moreover, to achieve comparable output performance, an equality constraint aiming for a rated torque of 32 Nm was established. To ensure optimal performance, a constraint was implemented to keep the torque ripple below 6%. Furthermore, in line with cost-effectiveness and resource conservation objectives, the optimization objective function aimed to minimize the cross-sectional area of the permanent magnets. In total, eight design variables were selected, taking into account feasibility. Their respective contributions to the alterations in output performance were thoroughly evaluated through sensitivity analysis. The main variables are marked with yellow boxes in the table to the right of Figure 9. Subsequently, based on the results of this analysis, four major design variables, which exhibited substantial impacts on torque and torque ripple, were further refined and chosen for subsequent optimization, as shown in Figure 10.  
To ensure optimal projective properties and space filling across the designated design range, a sampling approach utilizing OLHD was employed. Initially, a comprehensive set of 80 sampling points was strategically chosen, a number calculated based on the inclusion of four essential design variables. These training data points were designated for use in data learning purposes, and their number was selected to be 80, where \(nTR\) represents the number of training data points and \(nDV\) denotes the number of design variables [19]. \[nTR>1.5(nDV\ +1)(nDV\ +2)/2 \tag{6}\]  
The initial 80 training points were calculated using FEM analysis, and a surrogate model was generated based on these data. To generate a surrogate model, two primary types of methods are available: one is regression methods and the other is interpolation methods. Interpolation methods unlike regression methods do not rely on correlation coefficients to predict distributed training data. Instead, interpolation techniques like kriging offer a continuous and smooth estimation of the underlying function throughout the entire design space. This detailed estimation fosters a deeper understanding of the relationship between input variables and output performance. Furthermore, interpolation enables the estimation of output performance at any point within the design domain, even in cases where training data points are scarce or absent. This versatility enhances the applicability of the surrogate model, particularly in situations where exhaustive experimental testing  
Figure 8: The flux density of the PMa-WFSM: (**a**) \(I_{dc}\) = 5 A; (**b**) \(I_{dc}\) = 3.5 A.  
is impractical or overly costly. Unlike regression, interpolation does not introduce errors between experimental data and predicted values, as it relies solely on the spatial distance between each data point. To quantify the error between the FEM calculation results and the predicted values, 9 test points, which represent approximately 10% of the 80 training data points, were selected, where \(nTE\) represents the number of test points. The error between the training points and FEM analysis results was then computed using the root mean square error (RMSE), where \(y_{1}(X_{i})\) is the value of the FEM analysis results and \(y_{2}(X_{i})\) is the value of the surrogate model. Figure 10: Design variables of PMa-WFSM. Figure 9: Optimal process of PMa-WFSM. \[nTE=0.1\cdot nTR \tag{7}\] \[\text{RMSE}=\sqrt{\frac{1}{nTR}{\sum_{i=1}^{nTR}}[y_{1}(X_{i})-y_{2}(X_{i}) ]}^{2} \tag{8}\]  
Utilizing the data obtained from FEM calculations, we constructed a surrogate model employing kriging interpolation to encompass the entire spectrum of variables.