## 3 Numerical Simulation
### Numerical Calculation
#### 3.4.2 Variable Temperature Conditions  
With the gauge pressure at 0 Pa in the duct, the stress distribution contours of the duct system at various characteristic temperatures are shown in Figures 10-12.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Temperature** & \begin{tabular}{c} **Yield** \\ **Strength** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Young’s** \\ **Modulus** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} & \
\begin{tabular}{c} **Expansion** \\ **Coefficient** \\ **(\(\times\)10\({}^{-6/5}\)C)** \\ \end{tabular} &
\begin{tabular}{c} **Density** \\ **(kg/m\({}^{3}\))** \\ \end{tabular} \\ \hline
21.11 & 655.0 & 205,471 & 0.28 & 12.744 & 8442.37 \\
37.78 & 654.8 & 199,955 & 0.28 & 12.751 & 8442.37 \\
93.33  
According to Figure 10, under atmospheric pressure conditions, stress concentration was observed at the tees and bends of the duct system due to the high temperature. The main reason for the phenomenon mentioned above is the thermal expansion deformation of the duct, and the internal force distribution was more concentrated in the locations where the spatial curvature or gradient of the tees and bends had changed noticeably. The thermal expansion displacement caused by a high temperature promoted the expansion deformation of the entire annular duct system towards the outer side of the envelope surface. Meanwhile, since the tee is located in the middle of the duct system, the effects produced by the three branches were superimposed, leading to a very obvious concentration of secondary stress. Moreover, as shown in Figures 10-12 with the temperature changing from 315.56 \({}^{\circ}\)C to 537.78 \({}^{\circ}\)C, the stress concentration phenomenon became more prominent at the position of structural discontinuity, and the maximum stress increase occurred in the tee area of region A at 56.572 MPa.  
Furthermore, it is noteworthy that the stress at the connection between the tie rod lug and the duct system increased significantly with temperature variation, as shown in Figure 13.  
Figure 11: Duct stress distribution at 426.67 \({}^{\circ}\)C: (**a**) region A; (**b**) region B; (**c**) region C; (**d**) region D.  
Figure 12: Duct stress distribution at 537.78 \({}^{\circ}\)C: (**a**) region A; (**b**) region B; (**c**) region C; (**d**) region D.  
Figure 10: Duct stress distribution at 315.56 \({}^{\circ}\)C: (**a**) region A; (**b**) region B; (**c**) region C; (**d**) region D.  
According to Figure 13, the stress at points E and G on the tie rod lug base was significantly higher than that at points F and H, and the main reason for this phenomenon is the rotational movement trend of the tie rod lug base around the FH axis, which caused a squeezed state at point E and a stretched state at point H. The maximum stress of the base increased from 375.62 MPa to 771.25 MPa, with an increase of 395.63 MPa, which exceeded the yield limit of 565.1 MPa at this temperature. Specifically, at a temperature of 537.78 \({}^{\circ}\)C, the stress in the duct system not only exceeded the limit but also caused separation and failure of the structure.