## 7 Optimization of the Four-Layer Three-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 3, with the following assumptions:  
* the lamination angle increment is expressed only in real numbers, and other approaches are not tested,
* the usefulness of the two networks (ANN\({}_{3}^{1}\) and ANN\({}_{123}^{1}\), see Table 10) is verified,
* different numbers of generations of GA optimization are applied to prevent GA from achieving a sharp minimum during intermediate optimization loops,
* the GA optimization procedure for each network and each considered number of generations is repeated 50 times, with different random initial populations. Altogether, the optimization procedure was repeated \(19\times 50=950\) times. The results obtained are shown in Figure 8 and Table 11.  
The total number of GA generations (see Table 11, the third column) was not only \(G=100\) (as it was for every case of MODEL1 optimization) but also \(G=50\) and \(G=75\). The number of generations was increased to \(G=200\) in only one case (the last row of Table 11) in order to check whether such \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Pattern** & & & & **Number** & **Time Consumption** \\
**Set** & \(\boldsymbol{\lambda^{\text{min}}}\) & \(\boldsymbol{\lambda^{\text{max}}}\) & \(\boldsymbol{\Delta\lambda}\) & **of Patterns** & \(\boldsymbol{f_{1}}\) & \(\boldsymbol{f_{1}}\) through \(\boldsymbol{f_{10}}\) \\ \hline \hline \multicolumn{8}{c}{Training sets} \\ \hline \(\mathcal{P}_{1}\) & \(-60\) & \(+60\) & \(60\) & \(729\) & \(44\) min & \(97\) min \\ \(\mathcal{P}_{2}\) & \(-75\) & \(+75\) & \(60/30/60\) (*) & \(4096\) & \(248\) min & \(546\) min \\ \(\mathcal{P}_{3}\) & \(-90\) & \(+90\) & \(60\) & \(4096\) & \(248\) min & \(546\) min \\ \(\mathcal{P}_{4}\) (**) & \(-90\) & \(+90\) & randomly & \(1000\) & \(61\) min & \(133\) min \\ \hline \multicolumn{8}{c}{Testing sets} \\ \hline \(\mathcal{P}_{5}\) & \(-75\) & \(+75\) & \(30\) & \(4000\) (***) & \(243\) min & \(533\) min \\ \(\mathcal{P}_{6}\) & \(-87\) & \(+88\) & \(35\) & \(4000\) (***) & \(243\) min & \(533\) min \\ \hline \hline \end{tabular} (*) \(\Delta\lambda\) was not constant, and the considered values are \(\lambda_{i}=(-75,-15,+15,+75)\); (***) \(\mathcal{P}_{4}\) is not used in \(f_{1}\) maximization; (***) only 4000 randomly selected patterns out of 46,656 patterns were generated.  
\end{table} 
Table 9: Pattern sets (three-segment MODEL2). an extension of the procedure would be expedient. Since there are no observable improvements with this increase, \(G=100\) is treated as the appropriate value. Figure 8 shows a clear improvement in the results, especially for ANN\({}_{3}^{1}\), which was trained on the data set \(\mathcal{P}_{3}\), which is half the size of the training set \(\mathcal{P}_{1}+\mathcal{P}_{2}+\mathcal{P}_{3}\) used for the second network, ANN\({}_{123}^{1}\). The final results obtained from both networks are comparable. However, the time required to generate the necessary pattern set, train the networks, and perform the optimization procedure is significantly different: 792 min (13.2 h) for ANN\({}_{3}^{1}\) and 1073 min (17.9 h) for ANN\({}_{123}^{1}\). The mechanism of improvement during three CL loops in the application of ANN\({}_{3}^{1}\) is clearly visible in Figure 9. The results obtained from ANN\({}_{3}^{1}\) and ANN\({}_{123}^{1}\) are visualized in a manner that is typical for the presentation of neural network accuracy: \(f_{1}^{\rm ANN}\) versus \(f_{1}^{\rm FEM}\) (values predicted