# Topology Optimization Based Parametric Design of Balloon Borne Telescope's Primary Mirror
## 2 Optimization Design of the Primary Mirror
### Initial Model Design of Primary Design  
The balloon-borne telescope is affected by gravity. The premise to ensure the RMS of the primary mirror is that the mirror body has sufficient stiffness to resist gravity deformation. Meanwhile, referring to the design of GHAPS, the composite support mode of the back and peripheral supports is adopted to ensure the RMS during the observation process, so the whiffletree and A-frame structure are used to realize kinematic spatial positioning [19]. The thickness ratio, the support points, and the lightweight structure are usually considered in the design of the primary mirror. According to [PERSON] et al. 's research, empirical Formula (1) is given; the minimum number of back support points can be calculated by empirical Formula (2) given by [PERSON] [20]. \[\delta=\frac{3\varrho gr^{4}}{16 Et^{2}}=\frac{3\varrho gdr^{2}D^{2}}{256E} \tag{1}\]  
\[N=\left(\frac{15r^{2}}{t}\right)\left(\frac{\rho}{E\delta^{\prime}}\right)^{0.5} \tag{2}\]  
where \(\delta\) is the maximum self-weight deformation of the mirror surface, \(\mu\)m; \(\rho\) is the density of the primary mirror, kg/m\({}^{3}\); \(g\): the acceleration of gravity, g/s\({}^{2}\); \(r\) is the radius of primary mirror (r = D/2), m; \(E\) is the elastic modulus of the primary mirror, Gpa; \(t\) is the thickness of the primary mirror, m; and \(\delta^{\prime}\) is the surface PV value, nm. According to the error distribution and the relationship between the PV and the RMS values, the PV value is no more than 30 nm. The balloon-borne telescope primary mirror assembly adopts passive support, which consists of a 6-point axial support and three A-frame lateral supports. The passive support employs a 6-point whiffletree for axial constraint (piston, tip, and tilt DOFs) and a set of  
Figure 1: Optical system. three A-frame supports for lateral support (two in-plane and one clocking DOFs). The common lightweight cells of the mirror include triangular, fan, square, and hexagonal structures. The triangle lightweight structure can provide better specific stiffness, stability, and optical quality than the other structures. Considering that the distribution of ribs very near the center hole will increase the difficulty of processing, the lightweight structure of the triangle and local central sector is adopted to design the primary mirror. A SiC primary mirror with a partially closed back was designed. The advantages of the partially closed back design include high mirror flexural rigidity relative to open-back designs and fabrication simplicity relative to built-up configurations. However, a partially closed back structure tends to have a much slower thermal response time than an open-back structure. To reduce the deformation of the edge of the primary mirror, radial ribs are arranged at the edge. The aperture of the primary mirror is 800 mm. Considering the edge effect during processing, the aperture of the primary mirror is 816 mm, the inner ring diameter is 722 mm, the diameter of the back support hole distribution circle is 519 mm, the diameter of the central hole is 160 mm, the diameter of six back support is 30 mm, the thickness of the mirror face is 6 mm, the thickness of the back face is 8 mm, and the thickness of the other ribs is 5 mm. Considering the commonly used materials of the primary mirror, the comprehensive performance of third-generation mirror material RB-SiC is better than that of glass-ceramic and Be, and Changchun Institute of Optics, Fine Mechanics and Physics has an independent SiC mirror-manufacturing technology, which is in the leading position in China. Thus, the RB-SiC material is used for the primary mirror. The material properties are shown in Table 1. The mass of the initial primary mirror is 33.8 kg, and a 75.8% lightweight ratio is achieved. Figure 2 shows primary mirror assembly showing the lateral support, axial support, mirror body, the back support holes, and basic dimensions. The primary mirror of the balloon-borne telescope must ensure good imaging quality.