## 3 Results
### Study of a Frame-Type Structure
#### 3.2.3 Case Study 3--Frame-Type Structure with Different Square Cross-Sections
Regarding Cluster 4, it includes moderate values of \(b3\) and \(b1\) and very high values of \(b2\). Finally, Cluster 5 includes instances with very high values of \(b3\), high values of \(b2\), and moderate \(b1\) values. Figure 6 presents a scatter plot where these clusters are represented. According to the achieved clustering, it is possible to summarize that the configurations in Cluster 1 provide the lightest structures with lower fundamental frequencies while the ones in Cluster 5 provide heavier structures with higher frequencies followed by Cluster 3. The remaining configurations demonstrate intermediate values, with opposite range values regarding the \(b2\) and \(b3\) variables. The design variables of the achieved optimal configurations show less discrepant standard deviations. Now, we have 30.91, 45.73, and 63.89 for the features \(b1\), \(b2\), and \(b3\), respectively. Although \(b3\) continues to be the most influential on clustering, \(b2\) lost some importance while the opposite happened in the case of \(b1\). Again, the influence of \(b3\) from the mechanical perspective supports the results arising from the optimal solutions analysis. As a final scenario in this case study, we have considered a mass constraint of 1000 kg. The optimal solutions obtained in ten runs, maintaining the latter number of individuals and iterations, are presented in Table 17. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cluster** & **Runs** & **b1 [mm]** & **b2 [mm]** & **b3 [mm]** \\ \hline
1 & 6, 7 & 76.1528 & 94.4798 & 17.8596 \\  & & (76.10-76.21) & (87.50-101.46) & (16.88–18.84) \\
2 & 1, 4, 5 & 72.3373 & 105.1059 & 133.9703 \\  & & (72.01–72.93) & (90.33–113.50) & (129.95–138.27) \\
3 & 3 & 175.6964 & 19.9670 & 13.8323 \\
4 & 8, 9 & 71.5970 & 170.6276 & 73.2114 \\  & & (71.52–71.68) & (152.54–188.71) & (69.05–77.37) \\
5 & 2, 10 & 71.4913 & 157.1890 & 185.6313 \\  & & (71.48–71.50) & (156.23–158.15) & (172.14–199.13) \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Cluster centroids for the optimization process using 20 individuals and 150 iterations in each run. \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 & 85.6639 & 86.6051 & 27.9362 & 9.5307 & 24.3466 & 944.8848 \\
2 & 99.3422 & 26.6380 & 20.1129 & 8.9263 & 16.9451 & 986.8938 \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Optimization of the structure considering mass constraint, using 20 individuals and 150 iterations in each run. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Clusters** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline Silhouette score & 0.559 & 0.366 & 0.488 & 0.628 & 0.520 & 0.436 & 0.303 \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Silhouette score for 2 to 8 clusters in the optimization process using 20 individuals and 150 iterations in each run. Regarding these optimal configurations, the Silhouette score pointed to a two-clusters structure, for which the scatterplot is presented in Figure 7.  
The centroids of the two clusters are characterized in Table 18, being presented also with the corresponding design variables' intervals and the runs in which solutions were assigned to the clusters. The imposition of a mass constraint produced a clear effect in narrowing the domain of the previously most influential design variable, \(b3\), significantly reducing its standard