## 3 Design Optimization
### Optimization Results  
The MOGA is a global optimization method suitable for multi-objective function problems. It uses the non-dominant sorting and crowding distance and is based on a genetic algorithm. This algorithm can be used without encoding the design variables if actual values are used, making calculations simpler and allowing continuous and discrete design variables to be processed simultaneously [18]. Most previous studies on the optimal design with multi-objective functions found optimal solutions with fixed weights, making it difficult to determine the influence of the individual weights on the optimal result. This influence can be easily determined with the MOGA. In this study, the metamodels of the multi-objective functions and constraints with the best accuracy were combined with the MOGA to find optimal solutions. We used a workstation with an Intel Xeon Silver 4110 processor (11 M cache, 2.1 GHz) and 64 GB of RAM. The number of function calls was 25,002, and the optimization time was 6 min and 44 s. Figure 8 shows the optimization results for the demagnetization rate and average torque. The blue dot represents the initial model value, and the gray dots represent 25,002 analysis solutions in the optimization process; the red dots represent 1902 optimal solutions obtained through optimization. From the figure, it can be seen that the 1902 solutions can be used depending on the user's needs or application. Because there are 1902 optimal solutions, the characteristic change was examined while changing the weight of the multi-objective function, as shown in Figure 9. The multi-objective function set in this study was to minimize the demagnetization rate and maximize the torque simultaneously. The weight of the demagnetization rate to the torque was determined to be 0.1:0.9. Because the optimal design method using metamodeling uses an approximation function, it is necessary to verify the optimal design result through FEA. Table 5 presents the verification results. This table reveals that the optimization result predicted using the metamodel and the result verified by FEA are similar. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Metamodel** & **Demanetizanion Rate** & **Torque** & **Torque Ripple** & **Efficiency** \\ \hline Krigging & 0.299578388 & 0.480370464 & 0.661102263 & 0.068150470 \\ RBF (Interpolation) & 0.162832835 & 0.316088471 & 1.684960769 & 0.054181548 \\ RBF (Regression) & 0.184083130 & 0.185910892 & 1.660534066 & 0.049919916 \\ PRG (Forward step) & 0.219662647 & 0.199215669 & 1.649310197 & 0.054889271 \\ PRG (Backward step) & 0.245157992 & 0.365451547 & 1.344382800 & 0.041693768 \\ PRG (Linear) & 0.281644165 & 0.811042784 & 1.465072536 & 0.105565885 \\ PRG (Simple quadratic) & 0.249909024 & 0.907408134 & 1.366168145 & 0.126457557 \\ PRG (Full quadratic) & 0.194996813 & 0.197564341 & 1.84740458 & 0.037366471 \\ PRG (Simple cubic) & 0.195295038 & 0.221251507 & 1.524009121 & 0.034044144 \\ EDT (Hybrid) & 0.563395063 & 0.855283102 & 1.071512674 & 0.116880624 \\ EDT (Hybrid / opt.) & 0.486268030 & 0.767061634 & 1.265285952 & 0.094801174 \\ MLP & 0.252095717 & 1.8330424299 & 1.687341801 & 0.168041319 \\ MLP (Custom) & 0.111563863 & 0.159207851 & 1.861550458 & 0.114412812 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Root mean squared error (RMSE) test results of metamodels. Figure 8: Pareto front of the multi-objective functions. Figure 10a shows the torque waveforms of the initial and optimal models. It can be seen that the average torque of the optimal model was 63.648 Nm, which is 4.45% higher than that of the initial model. The torque ripple of the optimal model was 7.01%, which is slightly higher than that of the initial model but satisfies the constraint within 10%. Figure 10b shows the demagnetization rate of the initial and the optimal models. It can be seen that the demagnetization rate improved by 2.7%. Figure 10: Torque and back EMF waveforms: (**a**) torque; (**b**) back EMF.