## 2 Finite Element Analysis of the Original Structure
### Theoretical Model and Simulations of Temperature Field and External Heat Flow in Orbit  
The main types of heat transfer for the antenna during in-orbit operation are thermal radiation and heat conduction, and the equations for thermal radiation and heat conduction are as follows:  
\[Q_{rad}=\varepsilon A\sigma_{0}T^{4} \tag{1}\]  
\[q_{x}=-k\frac{\partial T}{\partial x} \tag{2}\]  
Figure 1: The dimensions of the generatrix. Figure 2: Structural concept diagram of the parabolic cylindrical antenna. where \(Q_{rad}\) represents the radiation heat of the antenna. \(\varepsilon\) is the emissivity of the antenna, whose value is less than 1. \(A\) represents the surface area of the antenna, measured in square meters. \(\sigma_{0}\) is the Stefan-Boltzmann constant, its value equals \(5.67\times 10^{-8}\) W/\(\left(\text{m}^{2}\text{.K}^{-4}\right)\). \(T\) denotes the absolute temperature of the blackbody, measured in Kelvin. When calculating the heat flux density, it needs to specify the direction of temperature change. Equation (2) takes the x-direction as an example. \(q_{x}\) represents the heat flux density in the x-direction, measured in W/m\({}^{2}\). \(k\) is the thermal conductivity, measured in W/m\({}^{-}\). \(\frac{\partial T}{\partial x}\) represents the temperature gradient in the x-direction, with units in K/m. The thermal equilibrium relationship during the in-orbit operation of the antenna can be expressed as  
\[Q_{1}+Q_{2}+Q_{3}+Q_{4}=Q_{5}+Q_{6} \tag{3}\]  
where \(Q_{1}\) is the heat absorbed by the antenna from solar irradiation. \(Q_{2}\) is the heat absorbed by the antenna from Earth's reflected radiation. \(Q_{3}\) is the heat absorbed by the antenna from Earth's infrared radiation. \(Q_{4}\) is the heat generated by the antenna during in-orbit operation. \(Q_{5}\) is the heat radiated by the antenna into outer space. \(Q_{6}\) is the change in internal energy of the antenna. The thermal equilibrium of the satellite in orbit is shown in Figure 3.  
Equation (3) considers the satellite as a whole, representing the overall thermal equilibrium relationship. When this is discretized, there are temperature differences between different nodes, which lead to heat conduction and thermal radiation between nodes. The equation of the thermal balance for discretized nodes can be obtained as in [22]:  
\[m_{jcj}\frac{dT_{j}}{dt}=Q_{external\_j}+q_{j}+\sum_{k=1}^{N}E_{kj}\big{(}T_{k} -T_{j}\big{)}+\sum_{k=1}^{N}G_{kj}\Big{(}T_{k}^{4}-T_{j}^{4}\Big{)} \tag{4}\]  
where subscripts \(j\) and \(k\) represent grid nodes, \(m\) is mass, \(c\) is the heat capacity of the antenna material, \(T\) is temperature, and \(t\) is time. \(Q_{external}\) is the rate of external heat flux. \(q\) is the power of the internal heat source in the antenna. \(E_{kj}\) and \(G_{kj}\) represent the linear heat transfer coefficient and radiation coefficient between node \(j\) and node \(k\). Therefore, the left side of the equation represents the rate of change in internal energy for node \(j\) of the antenna. On the right side of the equation, \(Q_{external\_j}\) is the rate of external heat flux heating at node \(j\), and \(q_{j}\) is the self-generated heating power of the antenna at node \(j\). \(\sum_{k=1}^{N}E_{kj}\big{(}T_{k}-T_{j}\big{)}\) represents the sum of all heat conduction rates flowing into node \(j\), while \(\sum_{k=1}^{N}G_{kj}\Big{(}T_{k}^{4}-T_{j}^{4}\Big{)}\) denotes the sum of all radiation heat transfer rates flowing into node \(j\). Figure 3: Schematic diagram of satellite thermal equilibrium. In this study, Equation (4) is considered a periodic transient equation, which means that the external heat flux is a periodic variable over cycles. This is defined as follows:  
\[Q_{external\_j}(t+\tau_{0})=Q_{external\_j}(t) \tag{5}\]  
The magnitude of external heat flux, such as solar radiation, depends on the angle between the sun and the satellite's orbital plane.