## 3 Results
### Structure 2  
The ensured optimum analysis results are shown in Table 7. As expected, linear and nonlinear results are different from each other. The linear results of FEM show a good match with those of TPO/MA. Both displacement and total potential energy of the second nonlinear case are the same as those of the linear case since all deformations are within the limitation of the linearity. Different from the other example, the strain values of nonlinear case 2 are the same as those of the linear case because the linearity limit used in nonlinear case 2 was not exceeded. The same structure was analyzed for different Poisson's ratio values, and the minimum total potential energy values are shown in Table 8. From the results, the changes of Poisson's ratio have a small effect on the values for the cases with a small absolute value of the total potential energy. Although the results of nonlinear case 1 show a linear and expected reduction by the increase in the Poisson's ratio, the effects are significant due to big energy values in absolute value. The different percentages of maximum and minimum energy values are 0.52%, 63.90%, 0.52%, and 3.86% for the linear case, nonlinear case1, nonlinear case2, and nonlinear case3, respectively.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**FEM**} & \multicolumn{2}{c}{**TPOMA**} & \multicolumn{2}{c}{**TPOMA**} & \multicolumn{2}{c}{**TPOMA**} \\  & \multicolumn{2}{c}{**(Linear)**} & \multicolumn{2}{c}{**(Linear)**} & \multicolumn{2}{c}{**(Nonlinear 1)**} & \multicolumn{2}{c}{**(Nonlinear 2)**} & \multicolumn{2}{c}{**(Nonlinear 3)**} \\ \hline \multirow{2}{*}{**Node**} & **Ax** & **Ax** & **Ax** & **xy** & **Ax** & **xy** & **Ax** & **xy** & **Ax** & **xy** \\  & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** \\ \hline
1 & 0.479 & 0.000 & 0.479 & 0.000 & 29.882 & 0.000 & 0.959 & 0.000 & 27.652 & 0.000 \\
2 & 0.422 & 0.000 & 0.422 & 0.000 & \(-\)0.219 & 0.000 & 0.845 & 0.000 & 20.305 & 0.000 \\
3 & 0.480 & 0.131 & 0.480 & 0.131 & 40.580 & 25.3407 & 0.961 & 0.261 & 28.044 & 9.018 \\
4 & 0.404 & 0.104 & 0.404 & 0.104 & 8.190 & 18.872 & 0.807 & 0.208 & 18.110 & 6.868 \\
5 & 0.442 & 0.254 & 0.442 & 0.254 & 59.811 & 53.252 & 0.883 & 0.508 & 28.741 & 18.130 \\
6 & 0.368 & 0.210 & 0.368 & 0.210 & 40.500 & 48.925 & 0.736 & 0.420 & 21.545 & 16.379 \\
7 & 0.373 & 0.373 & 0.373 & 0.373 & 86.303 & 86.303 & 0.746 & 0.746 & 24.585 & 24.585 \\
8 & 0.294 & 0.294 & 0.294 & 0.294 & 84.501 & 84.501 & 0.588 & 0.588 & 25.376 & 25.376 \\
9 & 0.254 & 0.442 & 0.254 & 0.442 & 53.252 & 59.811 & 0.508 & 0.683 & 18.130 & 28.741 \\
10 & 0.210 & 0.368 & 0.210 & 0.368 & 48.925 & 40.500 & 0.420 & 0.736 & 16.379 & 21.545 \\
11 & 0.131 & 0.480 & 0.131 & 0.480 & 25.341 & 40.580 & 0.261 & 0.961 & 9.018 & 28.044 \\
12 & 0.104 & 0.404 & 0.104 & 0.404 & 18.872 & 8.190 & 0.208 & 0.807 & 6.868 & 18.110 \\
13 & 0.000 & 0.479 & 0.000 & 0.479 & 0 & 29.882 & 0.000 & 0.959 & 0.000 & 27.652 \\
14 & 0.000 & 0.422 & 0.000 & 0.422 & 0 & \(-\)0.219 & 0.000 & 0.845 & 0.000 & 20.305 \\ \hline \(\Pi_{\rm P}\) (Nmm) & & & \(-\)7887.851 & \(-\)1.09356257515 \(\times\) 10\({}^{8}\) & \(-\)15,775.702 & \(-\)740,764.295 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Optimum results for the system with 12 members-14 nodes (AHS).  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**FEM**} & \multicolumn{2}{c}{**TPOMA**} & \multicolumn{2}{c}{**TPOMA**} & \multicolumn{2}{c}{**TPOMA**} \\  & **(Linear)** & & \multicolumn{2}{c}{**(Linear)**} & \multicolumn{2}{c}{**(Nonlinear 1)**} & \multicolumn{2}{c}{**(Nonlinear 2)**} & \multicolumn{2}{c}{**(Nonlinear 3)**} \\ \hline \multirow{2}{*}{**Node**} & **Ax** & **xy** & **Ax** & **xy** & **Ax** & **xy** & **Ax** & **xy** & **Ax** & **xy** \\  & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** \\ \hline
1 & 0.000000 & 0.000000 & 0.000000 & 0.0000000 & 0.000000 & 0