## 4 Tested Cases
### Engineering Application
\begin{table}
\begin{tabular}{l c} \hline \hline
**Fixed Parameters** & **Values** \\ \hline Elastic modulus \(E\) & \(2.09\times 10^{5}\) MPa \\ Density \(\rho\) & 7850 kg/m\({}^{3}\) \\ Poisson’s ratio \(\mu\) & 0.3 \\ The length of the Hull \(L\) & 12,000 mm \\ The radius of the Hull \(R\) & 3300 mm \\ Rib space \(l\) & 600 mm \\ Size of the ribs’ & \(14\times 224/26\times 80\) mm \\ The radius of the base web opening \(r\) & 75 mm \\ Width of the base web opening \(d\) & 210 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The values of the fixed structural and material parameters. Figure 6: The structural profile of the base. As shown in Figure 9, the red line denotes the real impedance curve, which fluctuates significantly under different frequencies. The black line is the impedance curve after the pre- processing operation, which is smooth and the minimum value of the original curve remains the same. The blue line is the required impedance curve under different frequencies. Moreover, the allowance weight for this case is 3.027t and the maximum iteration of this case is set to be 50. To eliminate the randomness of the initial samples and the genetic algorithm, all methods use the same 60 initial LHS samples and the same setups of the genetic algorithm. Moreover, all the methods are repeated 20 times to avoid randomness occurring even though the setups are the same. The statistical optimization results of this problem with all compared methods of under 20 runs are summarized in Table 9. Furthermore, the detailed design variables, optimal, and weights of all runs are listed in Table A1 in the Appendix A. As illustrated in Table 9, the best value of the proposed method is \(4.062\times 10^{5}Nm/s\), which is the maximum optimal value among all methods. Moreover, the proposed method has the maximum mean value on the objective among all the listed methods. It indicates the effectiveness of the proposed approach. Regarding robustness, the proposed method also performs best among all these methods because the proposed method obtains the minimum standard deviation. It is worth mentioning that the LCB and PLCB methods obtain some infeasible solutions. In detail, there are 15 and 9 runs that have failed for the LCB and PLCB methods respectively. The results show that the proposed method is a stable and effective method to solve this engineering optimization problem. Figure 10 shows the impedance curves of the optimal scheme of the proposed approach and the original scheme. As illustrated in Figure 10, the impedance curve of the optimal scheme is better than that of the \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{4}{c}{\(f(x)\)**(\(\times\)10\({}^{5}\) Ns/m)**} \\ \cline{2-5}  & **Max** & **Mean** & **Std** & **Succeeded** \\ \hline EI & 4.031 & 3.998 & 0.03060 & **20/20** \\ WEI & 4.042 & 3.983 & 0.04316 & **20/20** \\ LCB & 3.857 & 3.733 & 0.08629 & 5/20 \\ PLCB & 3.892 & 3.723 & 0.12230 & 11/20 \\ EW-LCB & **4.062** & **4.027** & **0.01953** & **20/20** \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The statistical optimization results of the engineering case with different methods. Figure 9: The impedance curves before and after pre-processing. original scheme because the impedance curve has larger impedance values in the frequencies which are critical to the performance of the base as shown in the sub-figure of Figure 9. On the other side, in the frequencies which are not critical to the performance of the base, the optimal scheme has smaller values than those of the original scheme. Therefore, the proposed approach is an effective method for this engineering case.