# Optimization Design Method of Pipe-Insulating Joints Based on Surrogate Model and Genetic Algorithm
## 4 Conclusions  
This study established a surrogate model for calculating the maximum stress of the right flange of the insulation joint through parameterized finite element results. Based on this surrogate model and combined with the genetic algorithm, the dimensions of the right flange of the insulating joint were optimized. The main conclusions are as follows:  
1. The overall finite element analysis of the insulated joints before optimization reveals that the maximum stress in the joint, under the most adverse condition of four-point bending, is located at the right flange. The overall average stress of the insulating joint is relatively low, indicating a significant space for optimization. 2. Based on the parametric computational results of the right flange dimensions, a surrogate model for calculating the maximum stress of the right flange can be constructed. The one established using the ridge regression algorithm demonstrated the best predictive performance, with an \(R^{2}\) value of 0.928 for the random test set and relative errors all below 10%. 3. The optimization algorithm established by integrating the surrogate model with genetic algorithms can optimize the dimensions of the right flange under various optimization objectives, demonstrating exact optimization performance. Among all the dimensional parameters, \(r_{\text{BI}}\) exhibits a strong correlation with the maximum stress, with its optimal value increasing as the stress limit decreases, indicating that the chamfer radius has a greater influence on improving the stress concentration effect in the right flange. The maximum stress obtained from the finite element analysis based \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Optimization Targets** \\ **(MPa)** \\ \end{tabular} } & \begin{tabular}{c} **Stresses** \\ **Calculated by** \\ **Finite Element** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Relative Error in** \\ **Genetic** \\ **Algorithm** \\ **Optimization (\%)** \\ \end{tabular} & \begin{tabular}{c} **Stresses** \\ **Calculated by the** \\ **Surrogate Model** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Relative Error in** \\ **Surrogate Model** \\ **(\%)** \\ \end{tabular} & \
\begin{tabular}{c} **Cross-** \\ **Sectional Area** \\ **(mm\({}^{2}\))** \\ \end{tabular} &
\begin{tabular}{c} **Optimization** \\ **Rate** \\ **(\%)** \\ \end{tabular} \\ \hline \(\sigma\) & \(\sigma_{\text{f}}\) & \(\left\lfloor\sigma_{\text{f}}-\sigma\right\rceil/\sigma\) & \(\sigma_{\text{r}}\) & \(\left\lfloor\sigma_{\text{r}}-\sigma_{\text{f}}\right\rceil/\sigma_{\text{f}}\) & \(S\) & \((S-S_{0})/S_{0}\) \\ \hline \(\sigma\leq 0.7\sigma_{\text{max}}=339.5\) & 352.09 & 3.71 & 336.41 & 5.77 & 32,802 & \(-16.25\) \\ \hline \(\sigma\leq 0.8\sigma_{\text{max}}=388\) & 371.33 & 4.30 & 387.22 & 4.28 & 30,538 & \(-8.23\) \\ \hline \(\sigma\leq 0.9\sigma_{\text{max}}=436.5\) & 402.56 & 7.78 & 432.25 & 7.37 & 25,661 & 9.06 \\ \hline \(\sigma\leq\sigma_{\text{max}}=485\) & 490.54 & 1.14 & 482.51 & 1.64 & 9906 & 64.89 \\ \hline \(\sigma\leq 1.1\sigma_{\text{max}}=533.5\) & 545.85 & 1.93 & 527.95 & 3.28 & 8451 & 70.05 \\ \hline \(\sigma\leq 1.2\sigma_{\text{max}}=582\) & 634.26 & 8.98 & 573.07 & 9.65 & 7942 & 71.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Relative error of stresses. on the optimized dimension parameters for the right flange has a maximum relative error of 8.98% and an average relative error of 4.63% compared to the optimization target. Based on the optimized dimensions, the predicted maximum stress of the right flange by the surrogate model has a maximum relative error of 9.65% and an average relative error of 5.33% compared to finite element calculations. Compared to improving material strength, optimizing cross-sectional dimensions is more efficient for stress optimization of the right flange. The variety of insulating joint models is extensive, and this study focuses solely on the DN1400 model as the optimization target, which does not have strong universality.