## 4 Optimal Design Process
With this surrogate model in place, we can effectively predict output performance values across the designated range without the substantial time investment typically associated with FEM computations. Having defined our objective functions, constraints, and target functions, we allowed for the MGA to derive the optimal model. The MGA offers expedited computational time compared with normal genetic algorithms, aligning well with the goal of efficient optimization. Additionally, if applicable, we could discuss the trade-offs between conflicting objectives (e.g., maximizing efficiency while minimizing torque ripple) and how these were addressed during the optimization process. The previously set constraint conditions included an average torque of 32 Nm, a torque ripple of less than 6%, and the objective function of minimizing the cross-sectional area of the permanent magnet. Additionally, weights were assigned between the conditions. In this study, the primary objective was to match the target torque while minimizing the number of permanent magnets used. Therefore, the weight for the objective function was set to 0.8, and the remaining constraint conditions were assigned a weight of 0.2. The iterative process for evaluating the objective function and constraints was conducted over 200 iterations, taking into account convergence criteria. This iterative process of validation and refinement not only bolstered the reliability of the training approach, laying a solid foundation for a highly reliable data-driven design optimization process. The optimized model obtained through the MGA revealed a reduction in magnet width by \(-49.21\%\) and a decrease in magnet length by \(-4\%\) compared with the initial model, as illustrated in Figure 11a. Figure 11: MGA algorithm convergence results: (**a**) variation in design variables; (**b**) target function; (**c**) constraint condition; (**d**) objective function. Moreover, there were notable adjustments in other variables, such as a considerable increase of 93.77% in barrier width and an even more substantial increase of 12\({}^{\circ}\) in barrier angle. The significant increase in barrier width and angle suggests a more concentrated magnetic flux within the rotor core, potentially leading to enhanced efficiency and power density. In the initial model, the reduction in \(I_{dc}\) led to a failure to meet the target torque. The torque of the surrogate model achieved 31.91 Nm. However, the optimized model successfully achieved this target, delivering a torque of 31.61 Nm within the acceptable error range. Furthermore, there was a small improvement in torque ripple, which decreased by 0.03%. Used surrogate models serve as substitutes for intricate original models, thereby facilitating time and cost savings in the design and optimization phases of electric motors. Whereas FEM computations demand significant time, the adoption of surrogate models obviates the necessity for such computations, thereby expediting results. Additionally, surrogate models facilitate swift exploration of a wide array of design scenarios, augmenting the efficiency of the design process by providing instantaneous access to outputs for various input conditions. The comparison between the surrogate model and FEM analysis results revealed minimal discrepancies, as shown in Table 3, with torque exhibiting a mere 0.94% error and torque ripple showing only a 0.17% deviation. These results underscore the effectiveness of the optimization strategy in not only meeting the performance objectives but also in enhancing the overall performance of the WFSM.