## 3 The GA+ANN Optimization Procedure  
### One-Step Optimization  
As mentioned above, in order to make the optimization procedure less time-consuming, the values of natural frequencies (gathered in vector \(\mathbf{f}=\{f_{1}/f_{2},\ldots,f_{10}\}\)) for the given lamination angles  
Figure 1: The considered models: (**a**) MODEL1 (laminate tube); (**b**) MODEL1 cross-section; (**c**) MODEL2 (three-section shell structure). (gathered in vector \(\boldsymbol{\Lambda}\)) reported herein were calculated using neural networks \(\mathbf{f}=\text{ANN}(\boldsymbol{\Lambda})\) (instead of the usual FEM calculations) and the GA was applied as a tool to find the lamination angles that yield the maximum value of the fundamental natural frequency \(f_{1}\). The scheme of the one-step optimization procedure proposed herein is shown in Figure 2 and is explained in the following:  
1. **Initial solution population**: Random generation of 100 vectors \(\boldsymbol{\Lambda}\in\mathbb{L}^{n}\), where \(\lambda^{\text{min}}=-90\) and \(\lambda^{\text{max}}=+90\). In the jargon of GAs, each vector \(\boldsymbol{\Lambda}\) is called a chromosome, and a set of \(\boldsymbol{\Lambda}\) vectors is called a population. The dimension \(n\) of \(\mathbb{L}^{n}\) space is determined by the number of lamination angles in \(\boldsymbol{\Lambda}\). 2. **Objective function evaluation**: For each chromosome (\(\boldsymbol{\Lambda}\) vector of \(\lambda_{i}\) lamination angles), the value of the objective function \(g\left(\boldsymbol{\Lambda}\right)\) being minimized (see Equations (7) and (8)) is calculated by a trained deep network: \(g^{\text{ANN}}(\boldsymbol{\Lambda})=\text{ANN}(\boldsymbol{\Lambda})\). 3. **Stop criteria**: All stop criteria are verified, e.g., the number of iterations is compared with the arbitrarily selected maximum, or the value of the decrease in the objective function in the subsequent iteration steps is compared with the required minimum. 4. **GA: selection, crossover, mutation**: If the stop criteria are not fulfilled, the usual GA procedure involving selection, crossover, and mutation will produce a new population (a set of new \(\boldsymbol{\Lambda}\) vectors), and the next iteration step begins. 5. **Optimal solution**: When the stop criteria are fulfilled, the chromosome (\(\boldsymbol{\Lambda}^{*}\) vector) giving the smallest value of the objective function \(\boldsymbol{\Lambda}^{*}=\arg\min\left\{g(\boldsymbol{\Lambda})\right\}\) is treated as an optimal solution; the value of the objective function \(g\left(\boldsymbol{\Lambda}^{*}\right)\) for the solutions obtained during multiple repetitions of the GA+ANN procedure with random initiation (denoted as \(g^{\text{ANN}}(\boldsymbol{\Lambda}^{*})\) or \(g^{\text{ANN}}\) for short) are verified by FEM (\(g^{\text{FEM}}(\boldsymbol{\Lambda}^{*})\) or \(g^{\text{FEM}}\) for short). FEM is undoubtedly a more accurate method than ANN; therefore, as an optimal solution, the best \(g^{\text{ANN}}\) resulting from multiple GA+ANN runs is not being chosen. Instead, the maximum value from among all obtained \(g^{\text{FEM}}\) is chosen (the corresponding ANN prediction for the chosen \(g^{\text{FEM}}\) is called \(g^{\text{FEM}}_{(\text{ANN})}\)). The procedure described above lacks one important step shown in Figure 2: the generation of patterns for ANN training and the training itself (in the figure, the step is labeled **FEM: examples for ANN training**). FE simulations obtain the patterns (or examples) that consist of already known values of the parameters to be obtained (i.e., the natural frequencies gathered in vector \(\mathbf{f}\)), together with varying parameters (lamination angles gathered in vector \(\boldsymbol{\Lambda}\)). This is the most time-consuming part of the whole procedure. In order to increase time efficiency in comparison with the procedure without ANN application, the number of FE calls during the generation of patterns for ANN training should be as low as possible; otherwise, this approach is pointless.