# Topology Optimization Based Parametric Design of Balloon Borne Telescope's Primary Mirror
## 1 Introduction  
Near-space, also called suborbital space, is the airspace 20-100 km away from the Earth's surface [1, 2]. With the increasingly fierce competition in space exploration, scientists have turned their attention to near space. Near-space exploration has a bright future in the fields of military strategy and scientific observation. European and American technologies are relatively mature and have numerous successful payload observation missions for balloon-borne telescopes, and Japan developed the FUJIN telescope for planetary observations. These tasks are mainly for multidisciplinary observations (astrophysics, heliophysics, and planetary and earth science), such as the ball-borne large-aperture submillimeter telescope [3, 4], Terahertz Observatory (STO) [5], Sunrise [6], and the balloon-borne telescope [7], which have obtained observational data in planetary physics and solar physics, providing some effective bases for scientists to study planets and solar systems. Near-space exploration has more possibilities than space exploration, has the advantages of low cost and fast response, and can be recyclable multiple times. Compared with the ground-based large-diameter telescope, developing the balloon-borne telescope has an important scientific significance because reducing the atmospheric impact can detect the ultraviolet, gamma ray, and other wave bands absorbed by the atmosphere. As the key optical element of the telescope, the performance of the primary mirror affects the imaging quality to a certain extent. Therefore, the design of a primary mirror with high stiffness, low mass, andhigh stability is the near-space objective. The design of a balloon-borne telescope's primary mirror faces two main challenges: the root mean square (RMS) (RMS with piston, tilt, and focus Zernike terms removed) must satisfy the requirements under the effect of changing gravity, and the mass of the primary mirror is restricted by the balloon limitation. The primary mirror is the most important and sensitive element of an optical telescope. The design of the primary mirror mainly considers surface shape accuracy, mass and fundamental frequency. Significant research on the design and optimization of the primary mirror has been reported. [PERSON] et al. presented a study of the Sunrise telescope for observing the sun. This telescope has a 1000 mm Zerodur primary mirror, which has a three-point support at the back, an open-back triangular lightweight cell (lightweight cell means the space between ribs) [8]. [PERSON] et al. presented a study of the Gondola for High Altitude Planetary Science (GHAPS) that adopted triangular lightweight cells and an open-back structure. The mass of the primary mirror is 40 kg, and its supporting structure has a composite support form of lateral rods and an 18-point whiffletree structure at the back. The components meet the design requirements [9]. In addition to the traditional empirical design of the primary mirror, topology optimization methods are also widely used in the primary mirror design process. [PERSON] et al. made a topology optimization design for the primary mirror by considering the gravity in the optical axis direction and the polishing stress, achieving an RMS with a better performance than that of the traditional hexagonal lightweight structure [10]. [PERSON] et al. took the minimum compliance of the mirror as the objective to constrain the mass of the topological optimization design of the mirror. Compared with that of the traditional hexagonal lightweight mirror, the RMS decreased by 62.5% [11]. [PERSON] et al. presented a method that combines topology optimization with a multi-objective function with parametric optimization. The new configuration design shows obvious superiority over two types of traditional lightweight configurations [12]. [PERSON] et al. designed a \(\varphi\)760 mm lightweight mirror and a flexible support structure. Under the condition of constraining its mass, topology optimization maximizes the stiffness of the mirror.