## 5 Results and Discussion
### Shape Optimization of the Rubber Bumper Using the Tuned Simulated Annealing Algorithm  
The current section aims to evaluate the performance of the SA algorithm with the SVR tuned parameter setting by solving the finite element-based shape optimization problem directly. The search was repeated \(11\) times from the initial design \(\mathbf{d}_{initial}\). Table 10 shows the found \(\mathbf{d}_{opt,\ SA}\) optimal construction relating to the median \(E\left(\mathbf{p}_{opt}\right)_{SA}\) value run. The algorithm approached the known optimal construction \(\mathbf{d}_{opt}\) within the dimensional tolerances of the rubber product, meaning that the optimum variables' value \(\mathbf{d}_{opt,\ SA}\) is accurate from a technical point of view. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Point** & \(\mathbf{d}_{1}[\text{mm}]\) & \(\mathbf{d}_{2}[\text{mm}]\) & \(\mathbf{E(d)_{FEA}[\text{kN}^{2}]}\) \\ \hline \(\mathbf{d}_{initial}\) & 75 & 20 & 9666.118 \\ \(\mathbf{d}_{opt}\) & 108 & 33 & 0 \\ \(\mathbf{d}_{opt,\ SA}\) & 108.034 & 33.071 & 0.00005 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The optimal design variables found by the simulated annealing algorithm. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Objective Function** & \(\mathbf{d}_{limit}[\text{mm}]\) & \(\mathbf{f}_{limit}[\text{kN}^{2}]\) \\ \hline Cubic SVR & \((106.7,32.2)\) & \(-840.487\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The found optimum of the fitted SVR metamodel for the discrete variable domain. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Surrogate** & \multirow{2}{*}{\begin{tabular}{c} \(-\) \\ **Model** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **\(\mathbf{E(p_{opt})_{SA}}\)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **\(\mathbf{Mean}\)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **\(\mathbf{SD}\)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} \(\mathbf{\Lambda}\)**, Cooling** \\ **Function** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} \(\mathbf{\alpha}_{1}\), \\ **Cooling sp.** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} \(\mathbf{\alpha}_{2}\), **Linear** \\ **Cooling sp.** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} \(\mathbf{\gamma}_{0}\), **Initial** \\ **Temp.** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} \(\mathbf{\beta}\)**, **Step Size** \\ **Adaptation** \\ **Factor** \\ \end{tabular} } \\ \hline Cubic SVR & 0.07787 & 0.07811 & 0.00032 & 2 & - & 64 & 20430 & 0.625 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Tuned SA parameters for the SVR surrogate model. The working characteristic found by the tuned SA algorithm can be seen in Figure 8. It approaches the desired compressive force values within a 0.1% relative error, as seen in Table 11.  
Figure 9a shows the search path taken by the SA algorithm. Using the initial temperature value calculated on the SVR model, the Metropolis criterion also accepts cost-increasing function values at the beginning of the search process. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{**Compressive Extension [mm]**} \\ \cline{2-10}  & **1.2** & **2.4** & **3.6** & **4.8** & **6** & **7.2** & **8.4** & **9.6** & **10.8** & **12** \\ \hline \(F_{i,\mathcal{A}_{pre}}[N]\) & 4805 & 10,266 & 16,424 & 23,445 & 31,462 & 40,627 & 51,226 & 63,479 & 77,933 & 95,028 \\ \(F_{i,\mathcal{A}_{pre,5\times 4}}[N]\) & 4805 & 10,267 & 16,426 & 23,447 & 31,464 & 40,629 & 51,228 & 63,480 & 77,932 & 95,024 \\ \(RE[\%]\) & 0.013 & 0.012 & 0.010 & 0.009 & 0.007 & 0.006 & 0.003 & 0.001 & \(-0.002\) & \(-0.005\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: The working characteristic found by the tuned SA algorithm and its relative error when compared to the desired characteristic. Figure 8: Working characteristics found by the tuned simulated annealing algorithm. Figure 9: (a) Candidate designs accepted by the SA algorithm’s Metropolis criterion. (b) Steps accepted with cost function reduction.