## 3 Results
### Study of a Frame-Type Structure
#### 3.2.3 Case Study 3--Frame-Type Structure with Different Square Cross-Sections
Figure 5: Clusters for the optimal solutions dataset in the optimization process using 30 individuals and 100 iterations in each run. Note that all the instances are effective optimal solutions, so the unique point in the third cluster does not constitute an outlier, but a valid instance as well as the others. Summarizing the key aspects for this optimization process, we can say that high values of \(b3\) consistently correlate with high frequencies as shown in the first and third cluster. Low values of \(b3\) lead to low frequencies. The structure mass follows a similar trend. These conclusions are supported by the role that the inclined elements play in the analyzed structure. Regarding \(b2\), when these values are very high, Cluster 2 forces \(b3\) values to be lower. However, for moderate \(b2\) values, high values of \(b3\) are found, yielding to high masses and frequencies. Overall, it is seen that \(b3\) has a greater influence on clustering followed by \(b2\) as they possess greater standard deviations, namely 58.25 and 50.81 against 8.19 for \(b1\). Within the same case study, it was intended to allow the optimization algorithm to progress in a higher number of iterations, although maintaining the same number of function evaluations, so we have further considered the following optimization parameters: 10 runs, where in each run a population of 20 individuals is allowed to progress along 150 iterations. The results obtained in this optimization process are presented in Table 14.  
As expected, and similar to the results obtained previously, Table 13 shows that the results obtained for the objective function are very close, although the design variables' configurations achieved in each run show clear differences. Thus, as shown previously, we have proceeded to a clustering analysis of those configurations, for a more systematic characterization. In this set of optimal solutions, five clusters were found to be the best clustering arrangement, according to the Silhouette score (0.628). The global average Silhouette scores for other numbers of clusters are given in Table 15. \begin{table}
\begin{tabular}{c c c c c} \hline
**Cluster** & **Runs** & **b1 [mm]** & **b2 [mm]** & **b3 [mm]** \\ \hline
1 & 1, 3, 5, 6, 8, 9 & 72.2717 & 113.2629 & 136.4777 \\  & & (71.47–72.49) & (86.23–167.46) & (89.48–178.94) \\
2 & 2, 7, 10 & 75.8940 & 181.1797 & 20.9110 \\  & & (73.45–81.89) & (163.92–193.24) & (10.81–30.44) \\
3 & 4 & 99.0440 & 19.7390 & 82.8014 \\ \hline \end{tabular}
\end{table} 
Table 13: Cluster centroids for the optimization process using 30 individuals and 100 iterations. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Run** & **Best Solution** & **Best Solution** & **Best Solution** & **Displacement** & **Fundamental** & **Mass [kg]** \\
**(b1) [mm]** & **(b2) [mm]** & **(b3) [mm]** & **(mm)** & **Frequency [Hz]** & & \\ \hline
1 & 72.0103 & 113.4950 & 138.2667 & 15.0026 & 65.0105 & 2651.5073 \\
2 & 71.4793 & 158.1470 & 199.1264 & 15.0045 & 65.7291 & 4926.6675 \\
3 & 175.6964 & 19.9670 & 13.8323 & 15.0049 & 11.7764 & 2939.5891 \\
4 & 72.0715 & 111.4927 & 129.9525 & 15.0018 & 64.9146 & 2424.9065 \\
5 & 72.9301 & 90.3299 & 133.6916 & 15.0015 & 63.8515 & 2431.7774 \\
6 & 76.0955 & 101.4636 & 16.8798 & 15.0039 & 14.7473 & 816.6170 \\
7 & 76.2100 & 87.4959 & 18.8374 & 15.0011 & 16.4516 & 762.8236 \\
8 & 71.5161 & 188.7119 & 77.3680 & 15.0012 & 65.2092 & 1904.0974 \\
9 & 71.6778 & 152.5432 & 69.0547 & 15.0092 & 59.3841 & 1496.8936 \\
10 & 71.5032 & 156.2309 & 172.1362 & 15.0008 & 65.6971 & 3938.5718 \\ \hline \end{tabular}
\end{table} 
Table 14: Optimization of the structure considering different square cross-section values, using 20 individuals and 150 iterations in each run. From Table 14, we can see that Cluster 1 is particularly characterized by very low values of \(b3\), while \(b1\) and \(b2\) present moderate values. Cluster 2 is characterized by high \(b3\) values and moderate \(b1\) and \(b2\) values. Cluster 3 contains a low value of \(b3\) and \(b2\), and a very high \(b1\) value.