# Topology Optimization Based Parametric Design of Balloon Borne Telescope's Primary Mirror
## 2 Optimization Design of the Primary Mirror
### Topology Optimization of Primary Mirror  
Topology optimization can optimize the material distribution or obtain the best force transmission path in the design domain, which can play a strong basis for the classification of the primary mirror ribs. Compared with the traditional optimization based on the location and experience, it can improve the RMS values of the surface shape error of the primary mirror and reduce the mass. In this study, a topology optimization method with a multi-objective function is introduced for the primary mirror design using Optistruct. To reduce the calculation time, improve the optimization efficiency, and increase the flexibility of variables, the primary mirror finite-element model is established with linear shell elements. As shown in Figure 3, the topology optimization model comprises a design domain and a non-design domain. The mirror face, the six back support holes, and the bonding surface of the peripheral support are assigned to the non-design domain, and the rest to the design domain. The total number of elements is 41,946, and the total number of nodes is 38,706. The distribution of the material is determined based on SIMP to find the best force transmission path [21; 22]. The basic idea is to assume a material element that has variable relative density \(\rho\) between 0 and 1. The red part represents the elements with a density of 1, and the blue part represents elements with a density near zero. The nearer the density to 1, the higher the material importance is [23]. To avoid more intermediate density units, the mathematical model introduces penalty factor r, and the assumed function relationship between density and the elastic modulus of the material is as follows:  
\[E_{i}=(\rho_{i})^{r}E_{0}\quad i=1,2,\ldots,n. \tag{3}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Primary Mirror** & **RMSx (nm)** & **RMSz (nm)** & **M (kg)** & **f1 (Hz)** \\ \hline Index & \(\leq\)\(\lambda\)/60 & \(\leq\)\(\lambda\)/60 & \(\leq\)32.5 & \(\geq\)2100 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design requirements of primary mirror. Figure 3: Design domain and non-design domain. Here, \(n\) is the total number of the discrete finite elements. \(E_{i}\) is the material characteristic of the design unit, that is, the elastic modulus; \(\rho_{i}\) is the density of the material; \(E_{0}\) is the initial value of the elastic modulus of the material; \(r\) is the penalty factor (in order to obtain a clear 0-1 design, take 3); and the topology optimization model can be expressed as follows:  
\[Find\qquad\qquad X{=(\rho_{1},\rho_{2},.....,\rho_{n})^{T}} \tag{4}\]  
Optimization objective  
\[Min\qquad C_{w}=\alpha C_{0^{\circ}}+\beta C_{30^{\circ}}+\chi C_{60^{\circ}} +\gamma C_{90^{\circ}} \tag{5}\]  
Subject to  
\[\delta\leq\overline{U} \tag{6}\]  
\[mass\leq M \tag{7}\]  
\[\mathrm{G}_{k}(X){=}0 \tag{8}\]  
\[KU=F \tag{9}\]  
\[0<\rho_{\min}\leq\rho_{i}\leq 1\quad(i=1,2\ldots,n) \tag{10}\]  
The optimization goal is that the weighted compliance of the telescope is the minimum at the observation angles of \(0^{\circ}\), \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\), that is, the stiffness is the maximum. (The stiffness under each observation angle is very important, so take \(\alpha=\beta=\chi=\gamma=0.25\)), which can improve the RMS values of the surface shape error of the primary mirror. \(\delta\) is the displacement of the node of mirror face (\(\overline{U}\): the upper line of the displacement of the primary mirror node along the optical axis is 30 nm). According to [PERSON]'s research [24], constrained mirror face deformation can obtain better surface figure RMS.