## 3 Results
### Structure 1
#### 3.1.2 Nonlinear Analysis by Employing AHS  
For nonlinear cases, the nodal displacement and minimum total potential energy are presented in Table 3. The example has only 8 non-zero displacements. That is why the method is presented for a higher number of freedoms than structure 1 in other numerical examples. The nonlinear constitutive equations used in the paper show that the nonlinear stress-strain relationship in case 2 is very close to one in the linear case, while the other \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{4}{c}{**TopMA**} \\ \cline{2-13}  & \multicolumn{2}{c}{**Node 1**} & \multicolumn{2}{c}{**Node 2**} & \multicolumn{2}{c}{**Node 3**} & \multicolumn{2}{c}{**Node 4**} & \multicolumn{2}{c}{**Node 5**} & \multicolumn{2}{c}{**Node 6**} & \multicolumn{2}{c}{**\(\Pi\)**p**} \\ \cline{2-13}  & **Ax** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** \\ \hline
**FEM** & 0.472 & 0.000 & 0.424 & 0.000 & 0.370 & 0.370 & 0.297 & 0.297 & 0.000 & 0.472 & 0.000 & 0.424 & \(-\)7611.581 \\ \hline
**GA** & 2.208 & 0.000 & 2.239 & 0.000 & 1.521 & 0.054 & 1.168 & 0.131 & 0.000 & \(-\)0.698 & 0.000 & \(-\)0.980 & 34,413.583 \\ \hline
**DE** & 0.472 & 0.000 & 0.424 & 0.000 & 0.370 & 0.370 & 0.297 & 0.297 & 0.000 & 0.472 & 0.000 & 0.424 & \(-\)7611.581 \\ \hline
**PSO** & 0.393 & 0.000 & 0.335 & 0.000 & 0.329 & 0.430 & 0.272 & 0.336 & 0.000 & 0.577 & 0.000 & 0.529 & \(-\)7511.990 \\ \hline
**HS** & 0.471 & 0.000 & 0.424 & 0.000 & 0.369 & 0.362 & 0.294 & 0.290 & 0.000 & 0.474 & 0.000 & 0.424 & \(-\)7607.300 \\ \hline
**AHS** & 0.472 & 0.000 & 0.424 & 0.000 & 0.370 & 0.370 & 0.297 & 0.297 & 0.000 & 0.472 & 0.000 & 0.424 & \(-\)7611.581 \\ \hline
**ABC** & \(-\)0.389 & 0.000 & \(-\)0.486 & 0.000 & \(-\)0.185 & 0.325 & \(-\)0.039 & 0.320 & 0.000 & 0.492 & 0.000 & 0.461 & 470.741 \\ \hline
**TLBO** & 0.472 & 0.000 & 0.424 & 0.000 & 0.370 & 0.370 & 0.297 & 0.297 & 0.000 & 0.472 & 0.000 & 0.424 & \(-\)7611.581 \\ \hline
**FPA** & 0.472 & 0.000 & 0.424 & 0.000 & 0.370 & 0.370 & 0.297 & 0.297 & 0.000 & 0.472 & 0.000 & 0.424 & \(-\)7611.581 \\ \hline
**GWO** & \(-\)0.656 & 0.000 & 0.037 & 0.000 & \(-\)0.400 & \(-\)2.337 & 0.816 & \(-\)2.336 & 0.000 & 0.048 & 0.000 & \(-\)0.553 & 359,953.164 \\ \hline
**JA** & 0.472 & 0.000 & 0.424 & 0.000 & 0.370 & 0.370 & 0.297 & 0.297 & 0.000 & 0.472 & 0.000 & 0.424 & \(-\)7611.581 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimization results for the structural system with 4 members-6 nodes (linear). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Best** & **\(\Pi\)p** & **Mean of \(\Pi\)p** & **Standard Deviation of** & **Best** \\
**Methods** & **(Nmm)** & **(Nmm)** & **(Nmm)** & **(\PiP (Nmm))** & **Iteration** \\ \hline DE & \(-\)7611.581 & \(-\)7611.581 & 0.00000000002 & 93,515 \\ TLBO & \(-\)7611.581 & \(-\)7611.581 & 0.000000000002 & 4882 \\ FPA & \(-\)7611.581 & \(-\)7611.581 & 0.000000000002 & 2824 \\ JA & \(-\)7611.581 & \(-\)7611.581 & 0.000000000002 & 10,698 \\ AHS & \(-\)7611.581 & \(-\)7611.581 & 0.000000000002 & 2023 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Evaluation of best iteration steps among maximum iteration numbers to reach the minimum total energy. 2 relationships corresponding to case 1 and case 3 are very different from the one in the linear case. Cases 1 and 3 involve cubic powers of the strain on the right-hand side of the equations, while the only difference between nonlinear case 2 and linear relationships is the shifting of the value of the elasticity modulus according to the values of strains. In numerical examples, the value of elasticity modulus for the nonlinear part of case 2 is half of the linear case. It seems that the strains generally tend to the nonlinear part with the half elasticity modulus value that is resulted in the doubling of displacements compared to the linear case.