## 4 Discussion  
Soft computing methods applied in mechanical engineering problems may greatly reduce the size of the problems considered without losing the accuracy of the solution. In this paper, it was proven that the artificial neural network applied to the engineering problem may be an interesting alternative to consider by the designers instead of using design standards. First, in Section 3.1, the homogenization method implemented was validated with the scientific paper. The example considered was replicated from the literature after [10] and it proved a faultless implementation of the method in the Python environment as presented in Table 2. In the table, the last column shows the errors computed as shown in Section 2.5. The values achieved are very low, and in most cases they are even equal to 0 within the assumed precision. Secondly, in Section 3.2, the configuration of the ANN with the best performance was sought. More than one hundred and fifty configurations were analyzed; selected cases are presented in Figures 5 and 6. From the presented configurations, the best ones were selected on the basis of the error measurement according to the MSE measure from Section 2.5. As shown in Figure 6a, the best results were obtained by the model with two hidden layers with 1024 and 8192 neurons (400 epochs), in which the accuracy was 0.00017. This model was used to compare the results from ANN and homogenization algorithm in next section, i.e., Section 3.3. Moreover, based on the results obtained it was observed that the ANN with \begin{table}
\begin{tabular}{c|c c c c c} \hline \hline
**Effective** & & & **h [m]** & & \\
**Stiffnesses** & **0.300** & **0.325** & **0.350** & **0.375** & **0.400** \\ \hline \(A_{11}\) [kPa \(\cdot\) m] & 4.81 & 5.63 & 6.47 & 7.30 & 8.14 \\ \(A_{22}\) [kPa \(\cdot\) m] & 6.67 & 7.51 & 8.33 & 9.16 & 9.97 \\ \(A_{12}\) [kPa \(\cdot\) m] & 1.08 & 1.27 & 1.47 & 1.68 & 1.88 \\ \(A_{33}\) [kPa \(\cdot\) m] & 2.04 & 2.35 & 2.66 & 2.98 & 3.29 \\ \(D_{11}\) [Pa \(\cdot\) m\({}^{3}\)] & 0.0627 & 0.0828 & 0.1057 & 0.1318 & 0.1623 \\ \(D_{22}\) [Pa \(\cdot\) m\({}^{3}\)] & 0.0735 & 0.0947 & 0.1196 & 0.1480 & 0.1808 \\ \(D_{12}\) [Pa \(\cdot\) m\({}^{3}\)] & 0.0135 & 0.0181 & 0.0234 & 0.0294 & 0.0366 \\ \(D_{33}\) [Pa \(\cdot\) m\({}^{3}\)] & 0.0248 & 0.0326 & 0.0416 & 0.0516 & 0.0634 \\ \(R_{44}\) [Pa \(\cdot\) m] & 1.33 & 1.64 & 1.96 & 2.26 & 2.57 \\ \(R_{55}\) [Pa \(\cdot\) m] & 1.75 & 2.06 & 2.36 & 2.67 & 2.97 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Effect of increasing the height of a precast concrete slab model on the effective stiffnesses of a representative volume element. Figure 7: The plots of effects of increasing the height of a precast concrete slab model on the effective stiffness of a representative volume element in: (**a**) compression/tension (**b**) bending and (**c**) transverse shearing. one layer of neurons always was characterized with high MSE. Those ANN configurations are not recommended to be used. In general, the more neurons, the better the accuracy. For example, if the total number of neurons was greater than 8000, the MSE obtained was no higher than 0.0005. However, also for a smaller total number of neurons, low MSE could be obtained, e.g., for the \"4096, 512\" ANN configuration, the MSE was 0.00018, which was one of the best results. Also, the \"1024, 2048\" ANN configuration gave a low MSE, i.e., 0.0002, while \"1024, 1024\" gave a MSE equal to 0.00025. These four values can be thought of as a Pareto front when considering the number of neurons and the MSE accuracy. In this study, the influence of a number of epochs was also verified. It was observed that increasing the number of epochs significantly affects the MSE if a small number of total neurons is considered, see Figure 5. MSE can be decreased even several times, like in \"128, 64\", in which the MSE for 2200 epochs was almost five times lower than for 200 epochs. Similar features may be observed for the \"512, 128, 64\" configuration.