# Structural Optimization Design of Spaceborne Microwave Probe Antenna
## 4 Size Optimization of the Planar Antenna Reinforcement Structure  
In order to obtain the detailed size of the antenna reinforcement structure, such as width, thickness, height, and other design parameters, size optimization is carried out on the basis of topology optimization results. The optimization algorithm for size optimization is the method of feasible directions (MFD).  
Figure 6: Topology Optimization Results.  
Figure 7: New Layout Form of Reinforcement Ribs.  
The size optimization needs to use the shell-beam element model, so it is necessary to verify the consistency between the simplified model of the shell-beam element and the refined model of the solid element first. Next, the antenna reinforcement structure is size-optimized. After sensitivity analysis of the 12 design variables for the moment of inertia and frequency, it is concluded that each variable is important. The constraints for the size optimization are RMS < 50 MPa for all cell maximum stress values and a first-order frequency > 240 Hz. The objective is to minimize the rotational inertia around the central axis. The new reinforcing structure is built based on the iteration results and compared with the original structure.  
Verifying the Consistency of the Shell-Beam Cell Mesh Used for Size Optimization with the Solid Cell Mesh  
Topology optimization and sizing optimization are two key stages in structural optimization, and they are usually applied at different stages of the optimal design. Topology optimization focuses on the presence or absence of material, while size optimization targets further adjustments to the thickness and width of the reinforcement in order to control the structure more finely. Size optimization targeting the thickness and width of the reinforcement requires the use of a mesh of shell-beam cells, so the error between the shell-beam cells and the solid cells is examined.  
As shown in Figure 8 below, the top figure shows the shell-beam cell model used for size optimization, and the bottom figure shows a schematic of the refined solid cell model. The different colors in figure (a) represent the reinforcing ribs at different locations. The shell-beam element model is simulated with shell elements except for the center reinforcement, and the rest of the reinforcement is simulated with beam elements. The red arrow in the middle indicates the center reinforcement. The solid element models all use hexahedral or pentahedral elements. The overall weight of the shell element model is 425.0 g, and the overall weight of the solid element model is 424.5 g. The error between the two is 0.12%, which is within the acceptable range.  
The two modeling methods of the finite element model are a modal analysis of the plane antenna to obtain the first five orders of frequency values and a vibration pattern and comparison. The frequency value of each order comparison results are shown in Table 1. The obtained modal frequency value errors of the first five orders are within 5%, and the largest gap between the fourth-order frequency errors only reached 4.96%. The model built by the shell-beam element can guarantee the accuracy of the size optimization, and the results are credible. The first five orders of the shell-beam element model and the body element model are consistent. This suggests that both the shell-beam element model and  
Figure 8: (**a**) Schematic diagram of the shell and beam element models; (**b**) Schematic diagram of the solid element model.  
the solid element model provide similar results for the first five modes, ensuring reliability and consistency in the simulation. The fact that the shell-beam model is able to closely replicate the solid element model's results indicates that it is a valid simplification for subsequent design steps. It can be considered that the simplified frame model using shell-beam elements can replace the solid element to complete the subsequent size optimization of the width and thickness of the reinforcement.