# Stress Analysis of 2D-FG Rectangular Plates with Multi-Gene Genetic Programming
## 3 Application in GP Programming
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Equivalent Stress** & **Criteria** & **Model 1** & **Model 2** \\ \hline \multirow{4}{*}{\(\left(\sigma_{\text{eqv}}\right)_{1}\)} & Node & 22 & 42 \\  & Complexity & 56 & 129 \\  & Tree Depth & 4 & 4 \\  & Gen & 4 & 4 \\ \hline \multirow{4}{*}{\(\left(\sigma_{\text{eqv}}\right)_{2}\)} & Node & 29 & 21 \\  & Complexity & 82 & 53 \\  & Tree Depth & 4 & 4 \\  & Gen & 4 & 4 \\ \hline \multirow{4}{*}{\(\left(\sigma_{\text{eqv}}\right)_{3}\)} & Node & 18 & 19 \\  & Complexity & 44 & 50 \\  & Tree Depth & 4 & 4 \\  & Gen & 4 & 4 \\ \hline \multirow{4}{*}{\(\left(\sigma_{\text{eqv}}\right)_{4}\)} & Node & 24 & 17 \\  & Complexity & 64 & 43 \\ \cline{1-1}  & Tree Depth & 4 & 4 \\ \cline{1-1}  & Gen & 4 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Information about trees of the best models. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Dataset** & **Equations** \\ \hline \multirow{4}{*}{\(\left(\sigma_{\text{eqv}}\right)_{1}\)} & \(24.2\text{tanh}\left(m_{1}\hat{1}\right)-64.2\text{tanh}\left(m_{1}\hat{1} \right)-3.74n_{1}+\) \\  & \(5.16\text{ tanh}\left(m_{1}\right)^{2}\left(n_{1}+m_{1}\hat{1}\right)-7.47m_{1}\hat{1} +427\) \\ \cline{2-3}  & \(\left(\sigma_{\text{eqv}}\right)_{2}\) & \(6.56n_{1}+8.97m_{1}+54.3\exp\left(-2m_{1}\right)-\) \\  & \(162\exp\left(-1.26m_{1}\right)\exp\left(-n_{1}m_{1}\right)\left|n_{1}\right|m_{1} \right|+116\) \\ \cline{2-3}  & \(\left(\sigma_{\text{eqv}}\right)_{3}\) & \(760\text{tanh}\left(\text{tanh}\left(m_{1}\hat{1}\right)\right)+707\exp\left(-m _{1}\hat{1}\right)+36.9m_{1}\text{tanh}\left(\exp(n_{1})\right)+31.9n_{1}\text{ tanh}\left(n_{1}m_{1}\right)-613\) \\ \cline{2-3}  & \(\left(\sigma_{\text{eqv}}\right)_{4}\) & \(10.1\text{tanh}\left(n_{1}\right)+28.6\text{tanh}\left(m_{1}\right)\text{tanh} \left(n_{1}\right)-5.76m_{1}\hat{1}^{3/2}\text{tanh}\left(m_{1}\right)^{3}+\) \\  & \(1.28\exp\left(m_{1}\right)\text{tanh}\left(n_{1}\right)^{2}+0.655\) \\ \hline \multirow{4}{*}{\(\left(\sigma_{\text{eqv}}\right)_{1}\)} & \(100\exp\left(-\frac{\left(2m_{1}\right)}{n_{1}+0.10557}\right)+58.4n_{1}\hat{3}m _{1}\exp\left(-m_{1}\right)-\) \\  & \(5.78n_{1}\left(m_{1}+0.116\right)\left(n_{1}^{2}-9.29\right)+\frac{66.4n_{1}}{ \left(n_{1}+0.116\right)\left(n_{1}+m_{1}+0.117\right)}+326\) \\ \cline{2-3}  & \(\left(\sigma_{\text{eqv}}\right)_{2}\) & \(399-252\exp\left(-n_{1}-m_{1}\hat{1}\right)-29.3\exp\left(-2m_{1}\right)-\) \\  & \(153\text{tanh}\left(m_{1}\hat{1}\right)\exp\left(-\text{tanh}\left(n_{1}\right) \right)-29.1\exp\left(\text{tanh}\left(n_{1}\right)^{\frac{1}{2}}\right)\) \\ \cline{2-3}  & \(\left(\sigma_{\text{eqv}}\right)_{3}\) & \(602m_{1}^{3/4}-59.4\text{tanh}\left(m_{1}\hat{1}^{\frac{1}{2}}\exp\left(-m_{1} \right)\right)-\) \\  & \(\tanh\left(\text{tanh}\left(n_{1}\right)\right)\left(m_{1}-m_{1}\hat{1}\right)-380 m_{1}+93.1\) \\ \cline{2-3}  & \(\left(\sigma_{\text{eqv}}\right)_{4}\) & \(107\text{tanh}\left(m_{1}-4.29\exp\left(-n_{1}\right)+54\exp\left(-2m_{1}\exp \left(-n_{1}\right)\right)-\) \\  & \(14.6n_{1}\text{tanh}\left(n_{1}m_{1}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Equations of best models. The results obtained from the numerical analysis results are compared with the values obtained in genetic programming. Considering the order of the input data, graphs are created with the equivalent stress value, which is our output data. It is important that the figures in the graphics are similar to each other in order to check whether the program is working correctly. When we look at all the graphics, it is remarkable that they are similar to each other. Figure 6 shows the graph of the actual and predicted values of the training simulation results of all equivalent stresses of Model 1, and Figure 7 has test simulation results. In error graphs, it is desired that the actual value and the predicted value be very close to each other. For both training and test simulation results, the actual and predicted values in the graphs converged.