## 3 Topology Optimization and Parametric Analysis  
### Topology Optimization Mathematical Model  
This study employs the solid isotropic material with penalization (SIMP) model for topology optimization calculations. In this approach, the 'element density' of each element in the designed region of the finite element model serves as the design variable. The element density takes continuous values between 0 and 1. After the optimization process, an element density value of 1 or close to 1 indicates that the material at that element's position is crucial and should be retained in the design. Conversely, an element density value of 0 or close to 0 signifies that the material at that element's position is not essential and can be removed, thus achieving efficient material utilization. The selection of the SIMP method involves choosing porous materials during the original material selection, with a theoretical model as follows:  
\[E_{i}=\left(1-\left(\rho_{i}\right)^{p}\right)E_{min}+\left(\rho_{i}\right)^{p }E_{0} \tag{11}\]  
where \(E_{i}\) is the interpolated elastic modulus, \(\rho_{i}\) is the relative density value of the element (a numerical value between 0 and 1, where 0 represents a void at that element, and 1 represents the presence of material at that element), \(E_{min}\) is the elastic modulus of the porous portion, \(P\) denotes the penalty factor, and \(E_{0}\) is the elastic modulus of the solid portion.  
This study chose the maximum thermal deformation in each iteration as the optimization objective, aiming for displacement minimization. The specific mathematical model is as follows:  
\[\rho=\left[\rho_{1},\rho_{2},\rho_{3},\ldots,\rho_{n}\right]^{T} \tag{12}\]  
\[Objective=minimax(f(\rho)) \tag{13}\]  
\[f(\rho)=\left(\frac{\mathit{Disp}_{1}(\rho_{1})}{r_{1}},\frac{\mathit{Disp}_{2 }(\rho_{2})}{r_{2}},\frac{\mathit{Disp}_{3}(\rho_{3})}{r_{3}},\ldots,\frac{ \mathit{Disp}_{n}(\rho_{n})}{r_{n}}\right) \tag{14}\]  
\[\mathit{Disp}_{n}=\sqrt{\left(x_{n}^{\prime}-x_{n}\right)^{2}+\left(y_{n}^{ \prime}-y_{n}\right)^{2}+\left(z_{n}^{\prime}-z_{n}\right)^{2}} \tag{15}\]  
\[s.t.\ \frac{\sum_{t=1}^{n}\rho_{i}V_{t}}{V_{0}}\leq V_{fraction}\qquad 0\leq\rho_{i}\leq 1 \tag{16}\]  
where Equation (12) is the optimization objective equation, aiming at minimizing the displacement of the maximum thermal deformation within the design area in each iteration calculation. Equation (13) is the objective function, which is the absolute value of the displacement of all nodes within the optimization target area. \(r_{n}\) is the reference value for the response, determined based on the response type. Regarding this setting, \(r_{n}\) equals 1, which means \(f(\rho)\) is the displacements of all nodes on the reflector surface. Equation (14) represents the absolute value of the displacement for the nth node. In Equation (15), \(V_{fraction}\) is the volume fraction of the region requiring the design of reinforced ribs relative to the  
Figure 14: Conceptual diagram of the deformation of the parabolic antenna.  
original reflector surface volume. Upper or lower limits can be independently set, and for this optimization, an upper limit of 0.3 was imposed.  
Topology optimization calculations can be performed using finite element method software. Many commercial software packages such as Hypermesh, Ansys, Creo, etc., can perform optimization calculations based on mathematical models and settings, achieving topology optimization objectives and enabling further parametric analysis. This study chose Hypermesh and Optistruct to carry out the optimization calculations.