# Optimization Design Method of Pipe-Insulating Joints Based on Surrogate Model and Genetic Algorithm
## 2 Methods
### Surrogate Model  
#### 2.2.1 Construction of the Training Dataset  
From the analysis in Section 1, it is evident that the maximum stress in the insulating joint under the most adverse conditions occurs at the right flange on the lower edge of the insulating pipe. Moreover, the dimensions of the insulating joint are primarily determined by the right flange, with the dimensions of other components being correlated with it. Consequently, when the maximum stress in the right flange is optimized, it can be determined that the stress in the entire insulating joint is also optimized. Based on the optimized dimensions of the right flange, the design dimensions of the whole insulating joint's cross-section can be established. Based on the loading conditions of the right flange, it can be equivalently modeled, as shown in Figure 7, for calculations. The moment load can be equivalently transformed into tensile and compressive stresses on the transverse section. Given that the connected pipes to the right flange are thin-walled structures, the variation of tensile and compressive stresses due to the moment in the y-direction, as depicted in Figure 5, can be neglected, simplifying the forces on the thin wall to a uniformly distributed load \(P^{\prime}\). To account for the most adverse condition, the magnitude of the uniformly distributed load is taken as the yield strength of the pipe material. In this study, the DN1400 insulating joint has an external pipe thickness of 26 mm, with a yield strength of 485 MPa. Therefore, the load \(P^{\prime}\) is calculated as (26 mm \(\times\) 485 MPa)/\(t_{\text{B1}}\). The boundary conditions for the right flange primarily consider the interactions in the y-direction. The interface between the right flange and the insulating filler on the right side is predominantly subject to mutual compression with negligible relative displacement. Thus, it is simplified to a fixed boundary condition. On the left side of the right flange, where it contacts the sealing ring, since this is on the tensile side of the insulating joint, there is no interaction other than possible compression between the two; therefore, the left boundary condition is set as a compression-only support. The independently optimizable dimensional parameters for the right flange are \(b_{\text{B1}}\), \(b_{\text{B2}}\), \(t_{\text{B1}}\), \(t_{\text{B2}}\), and \(r_{\text{B1}}\), as shown in Figure 8. The optimization objective is to minimize the weight of the insulating joint while ensuring that the maximum stress does not exceed the material's stress limit. Given that the current design parameters for the insulating joint are overly conservative, the optimized parameters are expected to be less than the current design values. The closer the data range in the training dataset is to the final optimized  
Figure 7: Right flange force model. design values, the more likely it is to yield the optimal solution, and the more accurate the prediction of the optimal solution will be. Therefore, the value ranges for \(b_{\text{B1}}\), \(b_{\text{B2}}\), and \(t_{\text{B2}}\) are set to be 40% to 120% of the current design values. Since the right side of the right flange needs to be welded to the pipe, its dimension \(t_{\text{B1}}\) must not be smaller than the thickness of the pipe. Considering the common pipe thicknesses connected to DN1400, its value is taken as 70% to 110% of the current design value. Based on design experience, the larger the chamfer dimension \(t_{\text{B1}}\), the smaller the stress concentration effect, so its value ranges from 70% to 250% of the current design value. Each dimensional parameter is uniformly sampled with five values within its range, and the final parameters are shown in Table 2.  
In order to consider the coupling effect of different parameters on the maximum stress, the parameters in Table 2 were cross-combined to generate the final \(5^{5}=3125\) sets of size combinations.