## 7 Optimization of the Four-Layer Three-Segment Tube  
### Pattern Generation and ANN Training  
The proposed CL optimization improvement (see Figure 3) was tested on the MODEL2 structure (see Figure 1c), with four symmetric layers of composite material and sparse FE mesh applied during FEM computations. The number of different lamination angles equals six, i.e., \(\boldsymbol{\Lambda}=[\lambda_{1}/\lambda_{2}|\lambda_{3}/\lambda_{4}|\lambda _{5}/\lambda_{6}]_{s}\in\mathbb{L}^{6}\). The lamination angles in the first segment (the one with the highest diameter) are \([\lambda_{1}/\lambda_{2}]_{s}=[\lambda_{1}/\lambda_{2}/\lambda_{2}/\lambda_{1}]\); the angles in the second one (the one with non-constant diameter) are \([\lambda_{3}/\lambda_{4}]_{s}=[\lambda_{3}/\lambda_{4}/\lambda_{4}/\lambda_{3}]\); and the angles in the last one are \([\lambda_{5}/\lambda_{6}]_{s}=[\lambda_{5}/\lambda_{6}/\lambda_{6}/\lambda_{ 5}]\) \begin{table}
\begin{tabular}{c c c} \hline \hline
**Fiber** & **Optimal** & **Optimal Solution [Hz]** \\
**Angle** & **Lamination** & \(f_{1}^{\text{max}}\) & \(f_{1}^{\text{max}}\) \\
**Increment** & **Angles \(\boldsymbol{\Lambda}^{*}\)** & \(\mathbf{g}_{\text{(FEM)}}^{\text{(ANN)}}\) & \(\mathbf{g}^{FEM}\) \\ \hline \multicolumn{3}{c}{3 layers} \\ \hline
0 & \([51.5/-6.33/77.5]\) & \(37.33\) & \(37.67\) \\
1 & \([51/-6/77]\) & \(37.32\) & \(37.73\) \\
5 & \([-55/5/-75]\) & \(37.27\) & \(37.77\) \\ \hline \multicolumn{3}{c}{6 layers} \\ \hline
0 & \([73.7/-13.3/-10.7/16.1/10.8/-89.9]\) & \(43.88\) & \(44.17\) \\
1 & \([75/5/-17/8/6/-90]\) & \(44.02\) & \(43.76\) \\
5 & \([-70/-5/20/-5/0/-90]\) & \(43.79\) & \(43.79\) \\ \hline \multicolumn{3}{c}{8 layers} \\ \hline
0 & \([78.7/-46/5.94/4.64/5.43/-0.637/-3.15/89.9]\) & \(44.94\) & \(45.24\) \\
1 & \([82/-37/16/-2/6/-2/-7/90]\) & \(45.01\) & \(44.99\) \\
5 & \([85/-35/20/0/5/-5/-5/90]\) & \(44.95\) & \(45.08\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Numerical results of \(f_{1}\) optimization using MODEL1 with a different number of layers. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Fiber** & **Optimal** & **Optimal Solution [Hz]** \\
**Angle** & **Lamination** & \(\min\lvert f_{1}(\boldsymbol{\Lambda}^{*})-p\rvert\) \\
**Increment** & **Angles \(\boldsymbol{\Lambda}^{*}\)** & \(\mathbf{g}_{\text{(FEM)}}^{\text{ANN}}\) & \(\mathbf{g}_{\text{(FEM)}}^{FEM}\) \\ \hline \multicolumn{3}{c}{3 layers} \\ \hline
0 & \([62.8/73.6/-71.4]\) & \(21.62\) & \(21.36\) \\
1 & \([66/81/-69]\) & \(21.91\) & \(21.09\) \\
5 & \([-55/-70/85]\) & \(22.11\) & \(20.72\) \\ \hline \multicolumn{3}{c}{6 layers} \\ \hline
0 & \([76.4/-22.9/7.82/17.6/16.8/-60.9]\) & \(23.75\) & \(23.44\) \\
1 & \([90/-11/-13/23/26/-61]\) & \(25.19\) & \(23.20\) \\
5 & \([-70/-10/20/-5/0/-90]\) & \(23.80\) & \(20.36\) \\ \hline \multicolumn{3}{c}{8 layers} \\ \hline
0 & \([70.6/-43.3/13.9/-3.89/-2.19/-13.7/18.8/72.5]\) & \(24.35\) & \(24.32\) \\
1 & \([72/-46/21/9/-3/-5/-16/70]\) & \(24.54\) & \(24.55\) \\
5 & \([75/-40/25/0/5/-5/-20/70]\) & \(24.67\) & \(24.39\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Optimization of the distance between natural frequencies and \(p=70\) Hz using MODEL1 with a different number of layers. Two groups of pattern sets for ANN training were calculated using the FE ADINA code [40]. Each was composed of lamination angles \(\boldsymbol{\Lambda}=[\lambda_{1}/\lambda_{2}|\lambda_{3}/\lambda_{4}|\lambda_{5}/ \lambda_{6}]_{s}\) and the corresponding fundamental frequency of the structure \(\mathbf{f}=\{f_{1}\}\) or ten corresponding natural frequencies of the structure \(\mathbf{f}=\{f_{1},f_{2},\ldots,f_{10}\}\):  
\[\mathcal{P}=\{(\mathbf{f},\boldsymbol{\Lambda})^{p}\}_{p=1}^{p}, \tag{16}\]  
where the fundamental frequencies were obtained through FEM computations.