## 3 Results and Discussion
### Direct Optimization by Running the Parameterized Finite Element Model  
The main goal of this section is to run directly the local stochastic search algorithm trained on the parameterized finite element model created for shape optimization. For adjustments of the algorithm based on cases of it being run in the previous section, see Table 8.  
Table 9 includes better function values and their geometry variable values accepted by the algorithm, while Figure 7 visualizes the path done by the algorithm. Based on the results of it being run, we managed to find an appropriate environment for the global optimum when the function was run 3000 times. Based on the results, the objective function values were equal until three decimals were reached, even when the variables were within one decimal to the known optimum value. The effect of tightening search space coincided with what was experienced with the surrogate model. The optimum found by the algorithm occurred in the \(m=2373\) iteration, which involves the fact that the random number generated with the \(\sigma=0.5\) mm standard deviation meant a search space that was too large while on the way to a global optimum. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**P-V**} & \multicolumn{2}{c}{**P-VI**} & \multicolumn{2}{c}{**P-VII**} & \multicolumn{2}{c}{**P-VIII**} \\ \hline
**Test** & \multirow{2}{*}{\(k\)} & \multirow{2}{*}{\(m\)} & \(\begin{array}{c}E(D_{opt})_{SVM}\\ (\mathbf{k}\mathbf{N})^{2}\end{array}\) & \multirow{2}{*}{\(k\)} & \multirow{2}{*}{\(m\)} & \(\begin{array}{c}E(D_{opt})_{SVM}\\ (\mathbf{k}\mathbf{N})^{2}\end{array}\) & & & \(\begin{array}{c}E(D_{opt})_{SVM}\\ (\mathbf{k}\mathbf{N})^{2}\end{array}\) & & & \(\begin{array}{c}E(D_{opt})_{SVM}\\ (\mathbf{k}\mathbf{N})^{2}\end{array}\) & & & \(\begin{array}{c}E(D_{opt})_{SVM}\\ (\mathbf{k}\mathbf{N})^{2}\end{array}\) \\ \hline
1 & 18 & 3566 & \(-\)772.50 & 13 & 4752 & \(-\)707.86 & 13 & 2454 & \(-\)772.82 & 18 & 3800 & \(-\)772.84 \\
2 & 20 & 4884 & \(-\)772.67 & 10 & 3379 & \(-\)772.87 & 16 & 3729 & \(-\)772.84 & 14 & 4615 & \(-\)772.52 \\
3 & 15 & 2212 & \(-\)772.29 & 22 & 2378 & \(-\)772.32 & 14 & 2319 & \(-\)772.80 & 17 & 1266 & \(-\)772.89 \\
4 & 9 & 2022 & \(-\)771.61 & 17 & 3601 & \(-\)707.86 & 11 & 3097 & \(-\)772.49 & 11 & 3270 & \(-\)707.86 \\
5 & 15 & 2549 & \(-\)772.78 & 8 & 4125 & \(-\)707.86 & 19 & 3892 & \(-\)707.86 & 13 & 3004 & \(-\)707.86 \\
6 & 9 & 3682 & \(-\)707.86 & 13 & 2810 & \(-\)772.63 & 11 & 4643 & \(-\)772.86 & 18 & 1144 & \(-\)772.72 \\
7 & 19 & 3851 & \(-\)772.67 & 10 & 4120 & \(-\)707.86 & 21 & 2600 & \(-\)772.77 & 16 & 2462 & \(-\)772.72 \\
8 & 20 & 2825 & \(-\)772.76 & 13 & 1770 & \(-\)772.67 & 16 & 3268 & \(-\)772.73 & 23 & 3227 & \(-\)707.86 \\
9 & 15 & 3756 & \(-\)772.80 & 16 & 4896 & \(-\)772.39 & 21 & 3889 & \(-\)772.60 & 13 & 2591 & \(-\)772.89 \\
10 & 17 & 3768 & \(-\)772.36 & 16 & 4844 & \(-\)772.01 & 11 & 2173 & \(-\)772.74 & 19 & 1758 & \(-\)772.85 \\ \hline avg. & 16 & 3312 & \(-\)766.03 & 14 & 3668 & \(-\)746.64 & 15 & 3206 & \(-\)766.25 & 16 & 2714 & \(-\)753.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The values of the optimal objective function with algorithms run by different adjustments. \begin{table}
\begin{tabular}{c c c} \hline  & \multicolumn{2}{c}{**P-IX**} \\ \hline iteration number start, \(m_{start}\) & 1 & 2001 \\ iteration number stop, \(m_{stop}\) & 2000 & 3000 \\ standard deviation, \(\sigma(1)\) (mm) & 30 & 0.5 \\ standard deviation, \(\sigma(2)\) (mm) & 30 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Adjustment of local stochastic search algorithm by tightening search space. Optimal variable values found by the basic adjustment local search methods were compared with ones found by the local search method trained on surrogate models, and they were related to optimal design variable values (Table 10).