## 3 Results
### Study of a Frame-Type Structure
#### 3.2.3 Case Study 3--Frame-Type Structure with Different Square Cross-Sections  
In the third case study, it is considered that the structure is built with members with different cross-section dimensions, although maintaining the square configuration. This approach took into account the structure characteristics, so, edge b1 was considered for vertical elements, b2 for horizontal elements, and b3 for the inclined ones. The results of this study are shown in Table 11.  
In opposition to previous case studies, in this case as the number of design variables increased, the optimal solutions obtained during the ten runs show a greater dispersion, which is an expected trend. Although the minimum maximum displacement is fairly close \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Run** & \begin{tabular}{c} **Best Solution** \\ **[mm]** \\ \end{tabular} & \
\begin{tabular}{c} **Displacement** \\ **[mm]** \\ \end{tabular} &
\begin{tabular}{c} **Fundamental** \\ **Frequency [Hz]** \\ \end{tabular} & **Mass [kg]** \\ \hline
1 & 68.1797 & 21.6346 & 52.2594 & 999.9921 \\
2 & 68.1798 & 21.6344 & 52.2595 & 999.9959 \\
3 & 68.1784 & 21.6361 & 52.2583 & 999.9555 \\
4 & 68.1794 & 21.6350 & 52.2590 & 999.9850 \\
5 & 68.1799 & 21.6349 & 52.2594 & 999.9975 \\
6 & 68.1783 & 21.6364 & 52.2583 & 999.9504 \\
7 & 68.1799 & 21.6343 & 52.2594 & 999.9977 \\
8 & 68.1799 & 21.6343 & 52.2594 & 999.9979 \\
9 & 68.1790 & 21.6356 & 52.2586 & 999.9721 \\
10 & 68.1799 & 21.6345 & 52.2595 & 999.9991 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Optimization of the structure with equal constant square cross-section in all members with mass restriction. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Run** & \begin{tabular}{c} **Best Solution** \\ **(b1) [mm]** \\ \end{tabular}  & \begin{tabular}{c} **Best Solution** \\ **(b2) [mm]** \\ \end{tabular}  & \begin{tabular}{c} **Best Solution** \\ **(b3) [mm]** \\ \end{tabular}  & \ \begin{tabular}{c} **Displacement** \\ **[mm]** \\ \end{tabular}  & \begin{tabular}{c} **Fundamental** \\ **Frequency [Hz]** \\ \end{tabular}  & **Mass [kg]** \\ \hline
1 & 72.3722 & 104.5734 & 89.4797 & 15.0004 & 64.3588 & 1530.3103 \\
2 & 73.4485 & 163.9175 & 21.4854 & 15.0070 & 18.7650 & 1187.2763 \\
3 & 71.4657 & 167.461  
among the different runs, the corresponding structures' masses and fundamental frequencies present, in some cases, significant differences. This is an aspect for which analysis is relevant to better characterize the different optimal design variables' configurations. To this purpose, the K-means algorithm was used [41] for clustering, while the Silhouette score was considered to determine the number of clusters that should be used in each case. The global average Silhouette scores for the other number of clusters is given in Table 12.  
According to the scores in Table 12, we selected three clusters as this number of clusters corresponds to the highest Silhouette score (0.516). The three clusters can be observed in the scatter plot presented in Figure 5, using different colors for the three regions. In Cluster 1, we have high values for \(b3\) and moderate values for \(b1\) and \(b2\) variables yielding heavier structures with higher fundamental frequencies. In Cluster 2, varied values for \(b1\), very high values of \(b2\), and very low values of \(b3\) are found. This inverse correlation between \(b2\) and \(b3\) leads to very low frequency and mass values. Finally, for Cluster 3, we find a low value for \(b2\) and moderate values for \(b1\) and \(b3\). The centroids and intervals of each cluster considering the original scale of each variable are presented in Table 13. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Clusters** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline Silhouette score & 0.500 & 0.516 & 0.385 & 0.440 & 0.405 & 0.318 & 0.262 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Silhouette score for 2 to 8 clusters in the optimization process using 30 individuals and 100 iterations in each run.