# A Modified Shuffled Frog Leaping Algorithm for the Topology Optimization of Electromagnet Devices
## 4 Problem Formulation and Numerical Experiments
However, because there is a lack of escape mechanism, the optimized result found by the SFLA has a relative low objective value, with an average torque of 100.8 Nm, torque ripple of 28.2%, and weighted objective value of 0.539. The population size is 185, and the number of cycles to find the optimum is two and does not go through any evolvement for the remaining optimization process, making the result of the algorithm highly dependent on the initial guess of the optimization process. On the other hand, the LS-GA algorithm has the longest optimization time, as well as the best optimization results. It can successfully locate the solution with the largest average torque value of 130.8 Nm, as well as the best weighted objective values of 0.91. In view of the material consumptions, the ratios of the volumes of the material to that of the entire design domain under the optimized topologies of the conventional SFLA, the LS-GA, and the proposed SFLA are, respectively, 67.7%, 60%, and 63.5%. However, the average number of FEM evaluations for LS-GA is about 1800, which is three times the optimization time of the conventional SFLA algorithm. The performance of the proposed algorithm falls in between the conventional SFLA algorithm and the LS-GA algorithm. The average time cost of the proposed algorithm is 950 FEM evaluations, which is 1.5 times of the conventional SFLA and almost half of the LS-GA, and the optimized result found by the proposed method is 0.88, with an average torque of 126.8 Nm and torque ripple of 19.8%. The tradeoff between the optimization speed and the result performance of the proposed algorithm is the most efficient among all three EAs. Comparing with the LS-GA algorithm, the proposed algorithm sacrifices only a small degradation in the quality of the optimization result (96.7% objective value) in return for a shorter optimization time (50% optimization time). On the other hand, the local search step in the proposed algorithm enhances the algorithm's ability for exploration, and the number of cycles to find the optimum is found is 6, which is the last iteration. Because the design space is more extensively explored, the optimization time is relatively longer (1.5 times of the conventional SFLA), but with a much better optimization results (1.27 times the objective value of the conventional SFLA). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Conventional SFLA** & **LS-GA** & **Proposed SFLA** \\ \hline Average number of FEM evaluations & 661 times & 1800 times & 905 times \\ \hline Standard deviation in FEM evaluations & 60 & 246 & 141 \\ \hline Optimized average torque & 100.8 Nm & 130.8 Nm & 126.8 Nm \\ \hline Optimized torque ripple & 28.2\% & 20.0\% & 19.8\% \\ \hline Weighted sum objective & 0.539 & 0.91 & 0.88 \\ \hline No. of cycles to find the Optimum & 2 & 7 & 6 \\ \hline Ratio of the volume of the material to & 67.7\% & 60.0\% & 63.5\% \\ that of the entire design domain & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of optimization performances of all three EAs.