# Lightweight Design of Multi-Objective Topology for a Large-Aperture Space Mirror
## 4 Performance Evaluation of the New Configuration  
### Comparison of Optical Performance of Three Naked Mirrors  
In order to confirm the optical performance of the optimized mirror, we construct two classic lightweight forms of mirrors with hexagonal lightening holes and triangular lightening holes, respectively, which are commonly used and shown in Figure 8a,b[14; 15; 16; 17]. The triangular configuration lightweight mirror is generally considered to have a high lightweight ratio, and the hexagonal configuration lightweight mirror is generally considered to possess a high stiffness [18; 19]. To prove the effectiveness of the new lightweight configuration, we make the two classic designs have the same thickness in front plate, outer ring, and inner ring with the optimized mirror. The diameter of the inscribed circle of the lightening hole is 60 mm, which is 15 times the thickness of the front surface [20]. And the optimal thickness of the ribs is obtained by size optimization. The influences of the ribs on the RMS under lateral gravity (\(RMS_{L}\)), RMS under lateral gravity (\(RMS_{A}\)), and total mass (\(m\)) of the mirror are equivalent. Each of the influence is normalized by their \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Variable** & **Lower Bound** & **Upper Bound** & **Optimum Value** \\ \hline DV1 & 4 mm & 8 mm & 4 mm \\ DV2 & 4 mm & 8 mm & 5.6 mm \\ DV4 & 12 mm & 18 mm & 16.4 mm \\ DV5 & 4 mm & 10 mm & 5.2 mm \\ DV6 & 4 mm & 10 mm & 4 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Lower bound, upper bound, and optimum value of the DVs for size optimization. Figure 8: Two classic lightweight mirrors. (**a**) Triangular configuration lightweight mirror; (**b**) Hexagonal configuration lightweight mirror. Figure 7: Sensitivity analysis using the FEM on primary mirror. maximum value and the objective function is obtained by using unification-object method [21], it can be expressed as follows  
\[\text{Minimize }F(t)=\frac{RMS_{A}}{RMS_{1}}\bullet\frac{RMS_{L}}{RMS_{2}} \bullet\frac{m}{M}. \tag{13}\]  
where \(RMS_{1}\), \(RMS_{2}\), and \(M\) are the maximum values of \(RMS_{A}\), \(RMS_{L}\), and \(m\), respectively. The relationship between objective function and the thickness of the rib is shown in Figure 9, and the optimal thickness of rib of the two classic lightweight mirrors both are 4 mm. Then the FEM is used to analyze the deformation of the mirror under lateral gravity, axial gravity, and polishing pressure, respectively. Figure 10a-c illustrates the axial gravity induced deformation of these three mirrors. The simulated deformation of the mirror surface is transferred into Zernike polynomials through optomechanical codes written in MATLAB, the peak valley (PV), and RMS deviation can be calculated after removing the bias, tilt, and defocus [22; 23]. Then we compare the new configuration design mirror with two classic mirrors. The PV values, RMS values, fundamental frequencies, mass and lightweight ratios of these three mirrors  
Figure 10: Deformation of the mirror surface under axial gravity. **(a)** Triangular configuration design; **(b)** Hexagonal configuration design; **(c)** New configuration design. Figure 9: The objective functions considered \(RMS_{L}\), \(RMS_{A}\), and \(m\) of the triangular and hexagonal configuration lightweight mirror with respect to the thickness of the ribs \(t\). are listed in Table 5. The RMS values and PV values under axial gravity are much larger than that under the lateral gravity, which is consistent with the iterative curve characteristics of topology optimization in Figure 3. The hexagonal configuration mirror has a high lightweight ratio, and the triangular configuration mirror has superiority in optical performance. The new design configuration lightweight mirror combines the advantages of the two classical lightweight mirrors, and has a high fundamental frequency.