## 2 Finite Element Analysis of the Original Structure
### Theoretical Model and Simulations of Temperature Field and External Heat Flow in Orbit
\(\varphi_{S}\), serving as an indicator describing the sun's position, is obtained through calculations based on the date used to determine the sun's location:  
\[\varphi_{S}=\frac{Date-Date0}{365.25}\times 360 \tag{6}\]  
\[q_{x}=-k\frac{\partial T}{\partial x} \tag{2}\]0  
Equation (6) [22] describes the solar position using the ecliptic longitude \(\varphi_{S}\), where \(Date\) represents the calculation date, and \(Date0\) represents the vernal equinox date. In Equation (7) [22], \(\beta\) represents the angle between the sun and the orbital plane, \(I\) represent the inclination of the celestial body's orbit relative to the ecliptic. \(\Omega\) represents the right ascension of the ascending node of the sun, and \(i\) represents the orbital inclination. \(\beta\) servs as a primary determinant of the illumination conditions for different surfaces of the satellite. Based on the satellite's orbital parameters, a simulation was performed of the external heat flux reaching the satellite from various directions when considering the maximum and minimum values of the beta angle, which are shown in Figures 4 and 5. As shown in Figure 6, +x represents the direction of the antenna's in-orbit flight, +z is the direction toward the Earth, and +y is the rear direction exposed to solar radiation. \[q_{x}=-k\frac{\partial T}{\partial x} \tag{2}\]1 The external heat flux curves at the minimum value of the beta angle. \[q_{x}=-k\frac{\partial T}{\partial x} \tag{2}\]2 The external heat flux curves at the maximum value of the beta angle. The external heat flow in Figures 4 and 5 is the sum of solar irradiation, Earth's reflected radiation, and Earth's infrared radiation. Because the \(-\)y surface is the sun-illuminated surface, the magnitude of the external heat flux reaching the \(-\)y surface should be the greatest among the six directions. This is confirmed in both Figures 4 and 5. As for the measurement points, based on the previous simulations, extreme temperatures will occur at the upper ends of the three reflector surfaces, and nine temperature measurement points were set, as shown in Figure 7a and corresponding to a discrete grid, as illustrated in the Figure 7b:  
Based on the external heat flux data in orbit, the temperature variation over time for temperature measurement points on the antenna reflector surface under extreme low-temperature conditions could be obtained, as shown in \[q_{x}=-k\frac{\partial T}{\partial x} \tag{2}\]3  
Figure 8 shows the temperature variation of the temperature measurement points over time. As shown in Figure 8, the temperature variation of the measuring points over time followed a regular pattern. This aligns with our assumption of the external heat flux being a periodic function. It can be observed that during the in-orbit operation of the antenna, the lowest temperature under extreme low-temperature conditions is \(-113.65\)\({}^{\circ}\)C.  
\[q_{x}=-k\frac{\partial T}{\partial x} \tag{2}\]4 The flight coordinate system during in-orbit operation. \[q_{x}=-k\frac{\partial T}{\partial x} \tag{2}\]5 (**a**) Setting the measuring points; (**b**) setting measuring points of the discrete grid.