# Optimization Design Method of Pipe-Insulating Joints Based on Surrogate Model and Genetic Algorithm
## 3 Results
### Surrogate Model-Genetic Algorithm Optimization Results  
To optimize the dimensions of the right flange, the surrogate model established in Section 2 is utilized as the fitness function in a genetic algorithm. Different optimization targets are set to find the optimal dimensions to verify the capability of the genetic algorithm in solving optimization problems of varying degrees. The current design's maximum material yield strength of the right flange, \(\sigma_{\text{max}}=485\) MPa, is taken as the reference for optimization targets. The optimization target range for the right flange is set between \(0.7\sigma_{\text{max}}\) and \(1.2\sigma_{\text{max}}\), with intervals of \(0.1\sigma_{\text{max}}\). The main hyperparameters of the genetic algorithm are as follows: The initial population size is 300, the maximum number of iterations is 3000, and the mutation probability is 0.001. The upper and lower bounds for the optimization parameters are the same as those of the training dataset from the surrogate model, as shown in Table 6. The optimized design dimensions of the right flange by the genetic algorithm with different optimization targets are shown in Table 7.  
As seen from Table 7, the dimension \(b_{\text{B2}}\) of the right flange consistently adopts its lower limit value regardless of the stress limit, indicating that \(b_{\text{B2}}\) has minimal effectiveness in reducing the maximum stress of the right flange. When the stress is less than 485 MPa, a notable increasing trend is observed in \(b_{\text{B1}}\), \(t_{\text{B1}}\), \(t_{\text{B2}}\), and \(r_{\text{B1}}\); this shows that these dimensions have a role in reducing the maximum stress in the right flange. Among all the dimensional parameters, \(r_{\text{B1}}\) strongly correlates with the maximum stress, with its optimal value increasing as the stress limit decreases. This indicates that the chamfer radius has a significant impact on improving the stress concentration in the right flange. The optimal dimensions exhibit strong nonlinearity with the optimized stress objective. Specifically, as \begin{table}
\begin{tabular}{c c c c c c} \hline
**Dimension Type** & \(b_{\text{B1}}\) & \(b_{\text{B2}}\) & \(f_{\text{B1}}\) & \(f_{\text{B2}}\) & \(r_{\text{B1}}\) \\ \hline Range of values (mm) & 76–228 & 66.4–199.2 & 28–44 & 28–84 & 5–25 \\ \hline \end{tabular}
\end{table} 
Table 6: Range of values for the optimized size of the genetic algorithm. \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Optimized Targets**} & \multicolumn{5}{c}{**Optimized Dimensions**} \\ \cline{2-6}  & \multicolumn{5}{c}{**(mm)**} \\ \cline{2-6}  & \(b_{\text{B1}}\) & \(b_{\text{B2}}\) & \(f_{\text{B1}}\) & \(f_{\text{B2}}\) & \(r_{\text{B1}}\) \\ \hline \(\sigma\leq 0.7\sigma_{\text{max}}=339.5\) & 228 & 66.4 & 41.8 & 84 & 14.4 \\ \hline \(\sigma\leq 0.8\sigma_{\text{max}}=388\) & 212.3 & 66.4 & 40.9 & 84 & 13.3 \\ \hline \(\sigma\leq 0.9\sigma_{\text{max}}=436.5\) & 183.3 & 66.4 & 36.3 & 84 & 13.7 \\ \hline \(\sigma\leq\sigma_{\text{max}}=485\) & 76 & 66.4 & 28 & 70.4 & 9.2 \\ \hline \(\sigma\leq 1.1\sigma_{\text{max}}=533.5\) & 76 & 66.4 & 28 & 53 & 6.3 \\ \hline \(\sigma\leq 1.2\sigma_{\text{max}}=582\) & 76 & 66.8 & 28 & 47.5 & 5 \\ \hline \end{tabular}
\end{table} 
Table 7: Optimized dimensions of the right flange for different targets. the stress limit varies, the optimal dimensions undergo an abrupt change near a particular limit, which corresponds to a stress limit of 485 MPa in this study. The optimized right flange dimensions are simulated and analyzed using finite elements, with the loading and boundary conditions identical to those described in Section 2.1. The stress of the right flange under different optimization targets is shown in Figure 11.  
As can be observed from Figure 11, the stress concentration of the right flange primarily occurs at two locations, namely, the upper and lower edges of the chamfer. Moreover, the position of the maximum stress gradually shifts upward with an increase in the stress limit.