## 5 Results and Discussion
### Tuning Simulated Annealing Algorithm for Shape Optimization  
To terminate the SA algorithm with a high probability near the global optimum, the cooling strategy must be chosen case-by-case for the shape optimization task. While  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Test** \\ **Function** \\ \end{tabular} } & \multirow{2}{*}{\(\overline{E}(\mathbf{p}_{opt})_{SA}\)} & \multirow{2}{*}{**Repeatability**} & \multicolumn{2}{c}{**S(\(\mathbf{T}_{0}\),\(\mathbf{A}\),\(\mathbf{N}\)), Annealing Schedule**} & \multirow{2}{*}{\(\beta\), Step Size**} \\ \cline{3-3} \cline{5-8}  & & & \multirow{2}{*}{**Mean**} & & \multirow{2}{*}{**SD**} & \(\mathbf{\Lambda}\)**, Cooling** & \(\mathbf{\alpha}_{1}\)**,** & \(\mathbf{\alpha}_{2}\)**, Linear** & \(\mathbf{T}_{0}\)**, Initial** & **Adaptation Factor** \\ \hline Rosenbrock & 0.74659 & 0.9291 & 0.27690 & 1 & 0.70 & - & 436,335 & 0.925 \\ [PERSON] & 0.07565 & 0.0752 & 0.00023 & 1 & 0.70 & - & 1.065 & 0.625 \\ Six-hump & 0.14214 & 0.1426 & 0.00045 & 1 & 0.70 & - & 72.731 & 0.625 \\ [PERSON] & 0.12670 & 0.1274 & 0.00040 & 1 & 0.70 & - & 26.894 & 0.625 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Optimal parameter settings for different mathematical test functions.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Test Function** \\ \end{tabular} } & \multicolumn{2}{c}{**Variable Domain**} & \(\mathbf{x_{opt}}\) & \(\mathbf{f_{opt}}\) & \(\mathbf{f_{limit}}\) \\ \hline [PERSON] & \(x_{1}\in[-5,10]\) & \(x_{2}\in[-5,10]\) & \((1,1)\) & 0 & 0.01 \\ [PERSON] & \(x_{1}\in[0,\pi]\) & \(x_{2}\in[0,\pi]\) & \((2.20,1.57)\) & \(-1.8013\) & \(-1.8\) \\ Six-hump camel & \(x_{1}\in[-3,3]\) & \(x_{2}\in[-2,2]\) & \((0.0898,-0.7126)\) & \(-1.0316\) & \(-1.03\) \\ [PERSON] & \(x_{1}\in[-1.5,4]\) & \(x_{2}\in[-3,4]\) & \((-0.54719,-1.54719)\) & \(-1.9133\) & \(-1.91\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Mathematical optimization test functions.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Test Function** \\ \end{tabular} } & \multicolumn{2}{c}{**Initial Temperature**} & \multicolumn{2}{c}{**Step Size Adaptation**} & \multicolumn{2}{c}{**Convergence**} \\ \cline{2-7}  & \(\mathbf{m}_{0}\) & \(\mathbf{\chi}(\mathbf{T}_{0})\) & \(\mathbf{\rho}_{0}\) & \(\mathbf{N}\) & \(\mathbf{q}\) & \(\mathbf{\varepsilon}\) & \(\mathbf{m}_{C}\) & \(\mathbf{m}_{max}\) \\ \hline [PERSON] & 100 & 0.85 & \([7.5,7.5]\) & 20 & 0.2 & \([0.1,0.1,0.001]\) & 100 & 6000 \\ [PERSON] & 100 & 0.85 & \([\pi/2,\pi/2]\) & 20 & 0.2 & \([0.1,0.1,0.001]\) & 100 & 6000 \\ Six-hump camel & 100 & 0.85 & \([3,2]\) & 20 & 0.2 & \([0.1,0.1,0.001]\) & 100 & 6000 \\ [PERSON] & 100 & 0.85 & \([3.25,3.5]\) & 20 & 0.2 & \([0.1,0.1,0.001]\) & 100 & 6000 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Parameters of the simulated annealing algorithm for the different test functions.  
the accuracy of the optimum approach is affected by the step size adaptation factor, \(\beta\). This parameter also greatly affects the number of function evaluations of the algorithm, which results in an increased length for the simulation-based design process. Thus, the determination of the optimal parameters was performed by running the trained SVR surrogate model, following the method used on the test functions. The dimensional tolerances of the rubber product were \(\pm 0.1\)\([\)mm\(]\); thus, the \(f_{limit}\) value was determined (see Table 7) on the following discrete domain:  
\[f_{limit}=\min_{d\in\Omega}E(d)_{SVR}, \tag{30}\]  
subject to  
\[\begin{array}{l}d_{1}\in\{70:0.1:130\}\ [\text{mm}]\\ d_{2}\in\{10:0.1:60\}\ [\text{mm}]\\ x_{1}-d_{2}/2\geq 15\text{ where }x_{1}=d_{1}/2-d_{3}\tan(d_{4})\end{array} \tag{31}\]  
The constant parameters were chosen empirically according to Table 8, and then the parameters of the simulated annealing algorithm were tuned. Table 9 shows the settings that performed the best on the SVR surrogate model.