# Structural Optimization Design of Spaceborne Microwave Probe Antenna
## 4 Size Optimization of the Planar Antenna Reinforcement Structure
### Optimization of the Size of the Planar Antenna Reinforcement Structure  
Once the layout of the stiffeners has been determined, the choice of the size of the stiffeners also determines their effectiveness in reducing the rotational inertia of the planar antenna. In existing designs, the choice of the height and thickness of the stiffeners depends heavily on the experience of the designer. Sizing optimization is usually achieved by setting an optimization range based on experience and simulating it with finite element software to obtain the best size within that range. The variable settings for size optimization are shown in Table 2. There are 12 variables for size optimization. As shown in Figure 8, they are the thickness of the center reinforcement indicated by the red arrow, the heights and thicknesses of the three reinforcement bars on both sides of the center reinforcement, the heights and thicknesses of the two reinforcement bars perpendicular to the center reinforcement, and the thicknesses of the planar antenna base plate. The step size of each variable is 0.5 mm. After the sensitivity analysis calculation, the sensitivity results of the design variables are finally obtained. The sensitivity analysis results of each design variable on the moment of inertia are shown in Figure 9 and the sensitivity analysis results of the frequency are shown in Figure 10. The variable fl denotes the thickness of the base plate, which is a design variable. As can be seen from the sensitivity analysis results, the thickness of the base plate has the greatest influence on the moment of inertia and frequency, and the influence of the \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model Order**} & \multirow{2}{*}{
\begin{tabular}{c} **Shell-Beam Model** \\ **Frequency Value** \\ **(Hz)** \\ \end{tabular} } & \multicolumn{2}{c}{**Solid Model**} & \multirow{2}{*}{**Error\%**} \\  & & **Frequency Values** & & \\  & & **(Hz)** & & \\ \hline
1 & 260.5 & 265.8 & 2.03\% \\
2 & 286.8 & 285.4 & \(-\)0.49\% \\
3 & 362.0 & 371.7 & 2.61\% \\
4 & 379.1 & 398.9 & 4.96\% \\
5 & 410.0 & 413.5 & 0.07\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of modal frequency values between two modeling methods for planar antennas. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Variable Name**} & \multirow{2}{*}{**Initial Value (mm)**} & **Variable Underline** & **Variable Upline** \\  & & **Value (mm)** & **Value (mm)** \\ \hline Center & & & \\ reinforcement & 6 & 4 & 8 \\ thickness & & & \\ Thickness of the & & & \\ rest of the & 6 & 4 & 8 \\ reinforcement & & & \\ Height of the rest of & 13 & 8 & 18 \\ the reinforcement & & & \\ Thickness of the & 1.5 & 1 & 3 \\ base plate & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Size Optimization Variable Settings. remaining variables on the moment of inertia and frequency is slightly lower; however, each variable is very important, and the combined influence will be larger. The constraints of size optimization are the RMS of the maximum stress value of all elements < 50 Mpa and first-order frequencies > 240 Hz, and the objective is to minimize the rotational inertia around the central axis. The mathematical model for size optimization is shown below:  
Find  
\[t=[t_{1},t_{2},\cdots,t_{12}]^{T} \tag{4}\]  
\[\min I \tag{5}\]  
\[s.t.\left\{ \begin{array}{c}(RMS\ \ \mathrm{Stress})_{\max}<50\mathrm{Mpa}\\ \omega_{1}>240\mathrm{Hz}\\ t_{\min}\leq t_{i}\leq t_{\max}\end{array} \right., \tag{6}\]  
where \(t\) represents the column vector of 12 design variables and \(t_{j}\) is the thickness of the jth variable. \(I\) in Equation (5) is the rotational inertia of the structure around the central axis, which is the objective function of optimization. \(\omega_{1}\) in Equation (6) is the first-order intrinsic frequency of the structure. \(t_{\min}\) is the lower limit of the ith thickness and \(t_{\max}\) is the upper limit of the ith thickness.