# Topology Optimization Based Parametric Design of Balloon Borne Telescope's Primary Mirror
## 3 Parametric Design of Primary Mirror
### Parametric Design of Primary Mirror  
Given the automatic parametric design, this study adopts the Isight software, which combines Hypermesh, Optistruct, and SigFit. The parametric design can find the global optimal solution, while automatic update iterations improve the optimization efficiency. The flow chart of the parametric design is shown in Figure 7. On the basis of the sensitivity analysis, 22 variables were selected for the parametric design. Considering the manufacturing capability of the mirror, the minimum thickness of primary mirror ribs is 3 mm, so the domain of rib26 and rib29 variable is 6-10 mm, and the domain of the rest of ribs variable is 3-8 mm. The parameterized model aims to obtain the minimum F(t) while constraining the mass of the primary mirror and the fundamental frequency. The parametric design mathematical model is shown in formula (19). Figure 6: Contributions of the design parameters to the F(t). Figure 7: Flow chart of the parametric design of the primary mirror. \[\begin{split} Find\ t=(t_{1},t_{2},t_{3},t_{4},t_{5},t_{6},t_{7},t_{8},t_{9},t_{10},t_{11},t_{12},t_{13},t_{14},t_{16},t_{17},t_{18},t_{20},t_{21},t _{25},t_{26},t_{29})^{T}\\ \min F(t)=\left\{ \begin{array}{l}0.2^{2}\left[\frac{R_{z}(t)-1.64 \sigma e^{-6}}{2.5977e^{-6}-1.64\sigma e^{-6}}\right]^{2}+0.3^{2}\left[\frac{R_{z }(t)-4.884e^{-6}}{7.234e^{-6}-4.684e^{-6}}\right]^{2}\\ +0.3^{2}\left[\frac{m(t)-0.029}{0.04069-0.029}\right]^{2}+0.2^{2}\left[\frac{f(t )-20024}{2457-20024}\right]^{2}\\ s.t.\\ mass\leq M\\ f_{1}\geq\overline{f_{1}}\\ t_{i,low}\leq t_{i}\leq t_{i,up}\end{array} \right.\quad i=1,2,\ldots,n \tag{19}\]  
The multi-island genetic algorithm is used to optimize the design of the compromise objective function. The multi-island genetic algorithm has better global solving ability and computational efficiency than the traditional genetic algorithm [31]. The optimized results of each variable are shown in Table 3. The flow chart of the parametric design of the primary mirror is shown in Figure 7. Figure 8 shows the optimization results of the primary mirror using the multi-island genetic algorithm. After obtaining the set of Pareto-optimal solutions, we selected the optimal value. Table 4 shows a comparison of the results between the initial design and the optimum design. The optimal mirror surface accuracy is shown in Figure 9. In the mirror optimization, mirror optimization variables are mostly grouped according to the position of the ribs. Since the stiffness, mirror surface deformation, and mass are considered in the topology optimization of the mirror, the overall performance of the mirror can be further improved by grouping according to the topology optimization. The single objective optimization cannot improve the overall performance of the mirror, and multi-objective optimization will pose a huge challenge to optimization efficiency. Therefore, parametric design using compromise planning can not only consider multiple contradictory objectives, but also improve optimization efficiency, so the method in this paper has a certain reference value for mirror optimization. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Variables** & **Domains (mm)** & **Initial Value (mm)** &
\begin{tabular}{c} **Optimum Value** \\ **(mm)** \\ \end{tabular} \\ \hline ribs1 & [3; 8] & 5 & 6.4 \\ ribs2 & [3; 8] & 5 & 3.0 \\ ribs3 & [3; 8] & 5 & 3.2 \\ ribs4 & [3; 8] & 5 & 3.3 \\ ribs5 & [3; 8] & 5 & 7.4 \\ ribs6 & [3; 8] & 5 & 4.8 \\ ribs7 & [3; 8] & 5 & 3.4 \\ ribs8 & [3; 8] & 5 & 3.3 \\ ribs9 & [3; 8] & 5 & 3.6 \\ ribs10 & [3; 8] & 5 & 4.9 \\ ribs11 & [3; 8] & 5 & 3.1 \\ ribs12 & [3; 8] & 5 & 7.2 \\ ribs13 & [3; 8] & 5 & 8.0 \\ ribs14 & [3; 8] & 5 & 4.6 \\ ribs16 & [3; 8] & 5 & 6.9 \\ ribs17 & [3; 8] & 5 & 8.0 \\ ribs18 & [3; 8] & 5 & 6.8 \\ ribs20 & [3; 8] & 5 & 4.0 \\ ribs21 & [3; 8] & 5 & 5.2 \\ ribs25 & [3; 8] & 6 & 4.5 \\ ribs26 & [6; 10] & 8 & 7.7 \\ ribs29 & [6; 10] & 8 & 6.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal thicknesses of the lightweight ribs.