## 5 Optimization of the Three-Layer One-Segment Tube
### Maximization of the Fundamental Natural Frequency \(f_{1}\)  
The maximization (optimization) of \(f_{1}\) was performed by applying the procedure shown in Figure 2, with the following assumptions:  
* the lamination angle increment is expressed either in real numbers (in the tables with the results coded as '0'), integer numbers ('1'), or integers divisible by 2, 5, or 15 ('2', '5', or '15', respectively);
* the usefulness of all networks listed in Table 4 is verified;
* the GA optimization procedure for each lamination angle increment ('0', '1', '2', '5', or '15') and for each network (from ANN\({}_{13}\) to ANN\({}_{245}\)) is repeated 50 times with different random initial populations.  
Altogether, the optimization procedure was repeated \(5\times 6\times 50=1500\) times. The results obtained are illustrated in Figure 5.  
Figure 5 presents the so-called _box-and-whisker diagram_ with the following metrics used to describe the results of 250 runs (five different lamination angle increments and 50 repetitions in each case) of \(f_{1}\) maximization for a particular ANN applied: minimum and maximum values of the obtained results (whiskers); the interquartile range, representing the results between the 75 th and 25 th percentiles (the colored boxes); and the mean (the white lined inside the colored boxes). The box-and-whisker diagram was created using the \(f_{1}^{\max}=g^{FEM}\) results.  
\begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**Training** & **Testing** & **ANN** & **Training** & \multicolumn{3}{c}{**Testing Error**} & **No. of FE** & **Time** \\
**Pattern Sets** & **Pattern Sets** & **Symbol** & **Patterns** & **MSE** & \(\boldsymbol{\sigma}\) & **ARE\({}_{1}\)** & **Calls** & **Consumption** \\ \hline \(\mathcal{P}_{1}+\mathcal{P}_{3}\) & \(\mathcal{P}_{2}+\mathcal{P}_{5}\) & ANN\({}_{13}\) & 559 & 3.51 & 1.87 & 2.64 & 1775 & 459 min \\ \(\mathcal{P}_{1}+\mathcal{P}_{2}+\mathcal{P}_{3}\) & \(\mathcal{P}_{5}\) & ANN\({}_{123}\) & 775 & 1.97 & 1.40  
Of six verified networks, only one, ANN\({}_{13}\), gave a mean result of \(f_{1}^{\text{max}}=\text{g}^{FEM}\) that is lower than 36.97 Hz (the maximum value of \(f_{1}\) available in the data sets). Four of the six networks gave a minimum of \(f_{1}^{\text{max}}=\text{g}^{FEM}\) of almost 36.97 Hz, with the mean and the maximum clearly higher than 36.97 Hz. The best results are from applying optimization procedure using the ANN\({}_{1235}\) network: the value \(f_{1}=37.77\) Hz is almost 1 Hz higher than the value that can be found in rather dense sets of patterns.  
The proposed optimization approach can be adopted for both real values of lamination angles and integer values with a user-defined angle increment. The rather narrow spread of results, e.g., those from ANN\({}_{4}\), indicates the robustness of the proposed procedure. The time consumption of one optimization case (for example, using the network ANN\({}_{4}\), a lamination angle increment of '0', and 50 repetitions was one of the most time-consuming cases investigated) is 802 min for pattern generation, 15 s for GA+ANN optimization, and 22 min for FEM verification of the obtained results for a total time of slightly less than 14 h. The number of FE calls is 3413 during pattern generation and 50 during FE verification of GA+ANN results for a total of 3463 FE calls. For the network ANN\({}_{1235}\), whose application generates the best-obtained results (for the increment '0') of \(f_{1}=37.67\) Hz and \(\boldsymbol{\Lambda}^{*}=[51.5/-6.33/77.5]\), the overall time consumption is 9 min less, and the number of FE calls remains constant. In a reference case (classical optimization involving FEM), the best-obtained results are \(f_{1}=37.786\) Hz and \(\boldsymbol{\Lambda}^{*}=[50.9/-4.5/77.7]\), and the overall time and the number of FE calls are 21 h and 9061, respectively.