# Lightweight Design and Evaluation of Square Reflector
## 2 Square Reflector Lightweight Design
### Design of Mirror Support Point Position  
The stress point of the mirror is related to the support position, and the support position is designed according to the size and structure of the mirror. For mirrors with a caliber of 300 mm-700 mm, the number of support points, according to [PERSON] [14], is calculated by the following formula:  
\[N=\frac{1.5r^{2}}{t}\sqrt{\frac{\rho g}{\delta E}} \tag{2}\]  
In the formula, \(r\) is the radius, \(t\) is the thickness, \(\rho\) is the material density, \(g\) is the acceleration of gravity, \(E\) is the elastic modulus, and \(\delta\) is the maximum self-weight deformation. According to the requirement that the gravity deformation of the mirror should be less than \(\lambda/100\) (\(\lambda\) = 632.8 nm), it is calculated by the Formula (2) that the number of support points of the \(\Phi\)707 mm circular mirror \(N\) = 3.6. For a mirror in space, each support point has six degrees of freedom, that is, the translation along the \(X\), \(Y\), and \(Z\) directions and the rotation around the \(X\), \(Y\), and \(Z\) coordinate axes; if we restrict the six degrees of freedom entirely, the object can be fixed. For the space shape, by leveraging the stability of the triangle, the three-point support becomes suitable for application in small and medium-sized aperture (\(\leq\)\(\Phi\)500 mm) optical elements. Regarding large aperture (\(\geq\)\(\Phi\)500 mm) optical elements, three points or a multiple of three are also used to complete the design of  
Figure 1: Schematic diagram of mirror blank. the supporting system [15]. Given the number of support points is calculated as 3.6, we select the three-point support design. Shown in Figure 2.  
### Design of Mirror Support Point Position  
(1) Simulation of Sandwich Structures  
The structure of the interlayer is the key part for releasing the stress, and it is also the core to adapt to the environment of large-span temperature change. The structure of the interlayer mainly includes the thickness of the outer ring, the width of the reinforcing rib, and its length and thickness. Common lightweight forms of sandwich structures include the triangular rib structure, square rib structure, and hexagonal rib structure [16] (see Figure 3). The structural form of the rib should consider not only the weight of the reflector but also the rigidity, manufacturing process, and processing process. Modal analysis can characterize the rigidity of the designed structure; the larger the modal value, the stronger the rigidity. In this paper, aiming to obtain ribs with the same thickness (tentatively 4 mm) in the three structural forms, the three forms of mirrors were modeled and modal analysis was performed, as shown in Table 2, to obtain the weight and fundamental frequency of the three styles of mirrors. Among them, the mirror with a triangular rib structure has the lightest weight and the highest fundamental frequency, indicating that the triangular rib structure is more stable than the other two forms. Therefore, this paper chooses the triangular rib structure. Figure 3: Honeycomb grid structure form. Figure 2: Mirror gravity load. The size of the ribs of the honeycomb structure determines the weight and rigidity of the sandwich structure. When the width and thickness are constant, the longer the ribs are, the worse the overall rigidity will be. In addition, under the action of gravity, the deformation of the mirror surface supported by the ribs is small, and the amount of deformation without rib support is so large that the lattice effect is obvious. At the same time, the removal amount of optical processing increases, and the difficulty increases. On the contrary, the use of smaller ribs can obtain better structural rigidity, but it will increase the weight of the mirror. (2) Stress simulation of mirrors  
Compared with other structures, the sandwich structure has a certain stress-release ability. At the same time, the honeycomb structure with a triangular sandwich is also less stressed than other structures.