## 4 Verification Example  
The optimization procedure described above, with GA as a tool to solve the optimization problem and ANN as a tool that substitutes time-consuming FE calculations, was verified using an example described in [41]. All of the following assumptions (regarding the FE model, material parameters, lamination angles, and so on) are based on this article. The structure (see Figure 4a) is a cantilever tube. Its model is a thin-walled cylindrical shell with a circular cross-section, and the diameter of the middle-surface of the cross-section is \(R=0.1\) m. The thickness of the shell is \(t=0.002\) m, and the length of the cantilever is \(l=0.2\) m. The shell web consists of 64 layers of composite material of equal thickness but with different fiber orientations. The boundary conditions are defined on the cylinder axis and correspond to the support conditions of a simply supported beam. The material properties are \(E_{1}=275.0\) GPa, \(E_{2}=11.0\) GPa, \(\
u_{12}=0.25\), \(G_{12}=G_{13}=5.50\) GPa, and \(\rho=1500\) kg/m\({}^{3}\). The lamination angles considered are \(0\), \(\pm 45\), and \(90\). The laminates are considered symmetric, and neighboring layers are grouped into pairs, with lamination angles equal to \([0/0]\) (herein called '\(0\)'), \([+45/-45]\) (called '\(\pm 45\)'), or \([90/90]\) (called '\(90\)'). For example, the stacking sequence described as \([0_{4}/\pm 45_{8}/90_{4}]_{s}\) should be read as four pairs of layers \([0/0]\), eight pairs of layers \([+45/-45]\), and four pairs of layers \([90/90]\). Assuming that the stacking sequence is symmetric, the overall number of layers is equal to 64, while the number of varying lamination angles equals 16. The fundamental natural frequency reported herein was obtained by using the commercial FE code ADINA [40]. The FE model (called MODEL3) was built using multi-layered shell 4-node MITC4 elements (first-order shear theory), the number of which equals 72 along the circumferential direction and 20 in the axial direction. The overall number of FE elements, nodes, and degrees of freedom equals 1440, 1514, and 9084, respectively. In order to verify the compatibility of MODEL3 with the FE model used by [PERSON] and [PERSON] in [41], the fundamental natural frequency was calculated for three different stacking sequences described in [41] and compared with the values in that paper. The results, shown in Table 1, prove that the proposed FE model and the FE model in [41] can be considered compatible. Figure 4: MODEL3: Finite element model (**a**) and the fundamental mode shape obtained either for constant lamination angles \([90_{16}]_{s}\) (**b**) or for [PERSON] and [PERSON]’s optimal solution \([90_{2}/\pm 45_{11}/0/90/\pm 45]_{s}\) (**c**). The optimization of the lamination angles in order to obtain the maximum value of the fundamental natural frequency \(f_{1}\) was performed according to the scheme shown in Figure 2. Four pattern sets were generated. The first one, \(\mathcal{P}_{1}\), consists of 5184 patterns, each composed of 16 angles of neighboring laminate pairs (ANN inputs) and one corresponding fundamental frequency (ANN output). The fundamental natural frequency corresponding to the particular set of lamination angles was computed using FEM. The patterns gathered in \(\mathcal{P}_{1}\) were generated according to two scenarios:  
1. For the first three pairs of lamination angles, all three possible cases (\([0/0]\), \([+45/-45]\), and \([90/90]\)) are considered. Starting from the fourth lamination angle pair to the 16 th (the last one), two cases (\([0/0]\) and \([90/90]\)) are considered for even lamination pairs (the 4 th, the 6 th, the 8 th, and so on), and only one case (\([+45/-45]\)) is considered for odd lamination angle pairs. 2. For the first three pairs of lamination angles, all three possible cases (\([0/0]\), \([+45/-45]\), and \([90/90]\)) are considered.