# Structural Optimization Design of Spaceborne Microwave Probe Antenna
## 3 Topology Optimization of the Planar Antenna Reinforcement Structure  
### Topology Optimization Mathematical Model  
The structural design needs to reduce the magnitude of the rotational inertia while ensuring dynamic stiffness. When the topology optimization aims to reduce the moment of inertia of the structure, the reinforcement of the new structure will be concentrated near the rotating axis, and the dynamic stiffness of this structure is very low. Therefore, in this paper, the topology optimization is aimed at increasing the intrinsic frequency of the structure, and the size optimization is aimed at reducing the moment of inertia. In this paper, the solid isotropic material with penalization (SIMP) method is used to solve the topology optimization problem of intrinsic frequency. Compared to other methods, SIMP is a more mature and widely used topology optimization method that has been extensively applied in engineering design. The structural design variable (element density) is directly linked to the optimization problem, meaning that the topology structure is explicitly dependent on the design variables. The optimization algorithm converges well, and the sensitivity is simple and easy to calculate.  
Considering the topology optimization of the intrinsic frequency, i.e., obtaining a set of design variables in the design domain to maximize the intrinsic frequency of the structure, the mathematical model can be expressed as follows:  
Find  
\[x=[x_{1},x_{2},\cdots,x_{n}]^{T} \tag{1}\]  
\[\max\omega_{1} \tag{2}\]  
\[s.t.\left\{\begin{array}{l}K\Phi_{1}=\omega_{1}^{2}M\Phi_{1}\\ 0.2V_{0}\leq V^{\prime}\leq 0.4V_{0}\,,\\ 0<x_{\min}\leq x_{i}\leq 1\end{array}\right. \tag{3}\]  
where: \(x\) represents the column vector of design variable density, the variable \(n\) represents the number of elements in the designable domain, and the relative density values of the \(n\) elements in the designable domain form \(x\); \(x_{j}\) is the relative density of the material in the jth cell; and n is the number of finite elements. In Equation (2), \(\omega_{1}\) is the first-order intrinsic frequency of the structure and is the objective function. The first equation in Equation (3) represents the characteristic equation of structural vibration; \(K\) and \(M\) are the overall stiffness and mass matrices of the structure, respectively; and \(\Phi_{1}\) is the corresponding vibration mode. The second equation in Equation (3) represents the constraints on the volume of the reinforcement; \(V_{0}\) is the volume of the designable region and \(V^{\prime}\) is the volume of the retained reinforcement, i.e., the volume of the portion of the reinforcing bars retained in the topology optimization results. When the retained volume is below 20% of the designable domain, the dynamic stiffness of the structure deteriorates, while exceeding 40% causes the structure's moment of inertia to become too large. Therefore, the range of 20% to 40% is defined. The third equation in Equation (3) represents the setting of a minimum value of the design variable \(x_{\min}\) to avoid singularities in the calculation.