## 5 Optimization Results  
Based on the weighted B-Spline FCM performed in Matlab script and GCMMA implemented in the Boss-Quattro optimization platform, the shape optimization results for the four cases are obtained. The final design solutions of the four cases provide us with the minimum mass that removes material to the maximum extent. Meanwhile, the allowable von-Mises stresses are all still satisfied. The final design solutions of the four cases of the lifting lug and the von-Mises stress distributions are depicted in Figure 12.  
The corresponding iteration histories of the von-Mises stress and the mass of the four cases are shown in Figure 13.  
Figure 12: Initial and final von-Mises stress distributions of the four cases (left: initial, right: final). Figure 14a,b compare the maximum von-Mises stress and the mass between the initial and the final design solutions of the four cases, respectively. The results are listed in Table 1.  
In Table 1 and Figures 12-14, it can be seen that in contrast to the initial configurations, although the maximum von-Mises stresses of the four cases all increased, they still meet the strength requirements of the lifting lug, i.e., \(\sigma_{\max}\leq 400\)MPa. The mass reduction by shape optimization in the four cases is very obvious. Below, the four cases are discussed in detail. Case 1: Compared with the initial configuration, the maximum stress change is very small; however, by the optimization design for the outer contour of the lifting lug, the final mass was reduced \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Maximum Von-Mises Stress/MPa**} & \multicolumn{2}{c}{**Mass/kg**} \\ \hline  & **Initial** & **Final** & **Initial** & **Final** \\ \hline Case 1 & 396.95 & 398.25 & 987.51 & 915.20 \\ Case 2 & 390.03 & 399.32 & 909.86 & 819.76 \\ Case 3 & 394.68 & 399.68 & 921.60 & 827.51 \\ Case 4 & 390.34 & 398.72 & 874.57 & 781.82 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of the initial and the final maximum von-Mises stress and mass of the four cases. Figure 14: Comparison of the maximum von-Mises stress and mass between the initial and final design. (**a**) The von-Mises stress (**b**) Mass. Figure 13: Iteration histories of the von-Mises stress and the mass. (**a**) Iterations of the von-Mises stress (**b**) Iterations of the mass. by 72.31 kg, i.e., 7.3%, which is still the largest among the four cases. This indicates that although the original single web configuration can achieve the goal of weight reduction by shape optimization on the premise of meeting the strength requirements, the effect of weight reduction is limited. Case 2: With the maximum stress increasing by 9 MPa based on the shape optimization, the mass decreases by 90.1 kg from 909.86 kg to 819.76 kg, i.e., by 10%. Compared with the optimized single web, the weight loss of the porous dual web is about 96 kg, which shows the configuration of case 2 is better than that of case 1. Case 3: The maximum stress in the optimized configuration does not change much, but as seen in Figure 12, the stress distribution changes greatly. The maximum stress in the initial configuration occurs at the connection hole with the mounting base plate, while that in the final configuration occurs in multiple places. Additionally, the final mass declines by 94.09 kg based on shape optimization, i.e., by 10%. In contrast to the optimized single web in case 1, the final open deep groove dual web in case 3 reduces 87.69 kg, which is less than that of case 2. Case 4: Compared with the initial configuration, the maximum stress and stress distribution in the optimized configuration do not change substantially, but the final mass drops by 92.75 kg, i.e., by 10.6%. Compared with the optimized single web, the weight loss of the connected closed dual web is 133.38 kg, which indicates that the final configuration of case 4 is the best among the four cases. The above results indicate that while satisfying the stress requirement, the final mass of case 4 is the least.