# Optimization Design of the Spaceborne Connecting Structure for a Lightweight Space Camera
## 3 Optimization Design of the Spaceborne Connecting Structure  
### Topology Optimization of Connecting Structures  
The finite element model of the camera was established by using Hypermesh software, as shown in Figure 4, with 145,225 elements and 16,642 nodes, including 81,894 hexahedral eight-node elements \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Components** & **Materials** & \begin{tabular}{c} **Elastic** \\ **(10\({}^{9}\) Pa)** \\ \end{tabular} & \begin{tabular}{c} **Poisson** \\ **Ratio** \\ \end{tabular} & \
\begin{tabular}{c} **Density** \\ **(g/cm\({}^{3}\))** \\ \end{tabular} &
\begin{tabular}{c} **Thermal Expansion** \\ **Coefficient (10\({}^{-6}\)/K)** \\ \end{tabular} \\ \hline Camera backplane & SiC & 420 & 0.17 & 3.16 & 2.5 \\ Connecting structure & Ti alloy & 109 & 0.34 & 4.44 & 9.1 \\ Satellite platform & Carbon fiber composite & 90 & 0.30 & 180 & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of the components. and 63,551 tetrahedral six-node elements. We fixed threaded holes at the bottom of the spaceborne connecting structures. In subcase 1, we applied the X-direction self-weight load of the camera. In subcase 2, we applied a 4 \({}^{\circ}\)C temperature rise load to all nodes. The top and bottom faces of the connecting structure were functional faces, which were connected to the camera and the satellite platform, respectively. Therefore, the top and bottom 5 mm thickness areas were set as a non-design domain, and the rest were set as a design domain. The mass of the initial connecting structure is 2.4 kg. The support form of the optomechanical structure is a single pole support, and positions A, B, and C are the positions where the connecting structures are installed. The relative material densities of design domain elements are defined between zero and one. The elastic modulus of each element is introduced by using the Solid Isotropic Material with Penalty (SIMP) method [16]. Therefore, the topology optimization model is described as taking the relative material density of each element as a design variable, constraining the first-order nature frequency of the camera and the relative displacement of mirrors within design requirements, and the objective is to minimize the mass of the structure. A mathematical description of the topology optimization model is as shown in Equation (4). \[ \begin{split} X&=(\rho_{1},\rho_{2},\cdots,\rho_{n }),\min(MASS)\\ s.t.&\left\{\begin{array}{c}f_{1}\geq 100\text{Hz}\\ R_{i}^{\text{G}}\leq\overline{R},\quad i=1,\ldots,6\\ R_{i}^{\text{T}}\leq\overline{R},\quad i=1,\ldots,6\end{array}\right.\end{split}  \tag{4}\]  
in which \(R_{i}^{\text{G}}\) and \(R_{i}^{\text{T}}\) represent the relative displacement of the \(i\)th degree of freedom of mirrors under gravity load and temperature rise loads, respectively. When \(i\) is between 1 and 3, \(\overline{R}=10\mu m\); when \(i\) is between 4 and 6, \(\overline{R}=10^{\prime\prime}\). The topological optimization model was solved by using the Method of Moving Asymptotes (MMA) [17]. The optimal material distribution results are shown in Figure 5. The connecting structures presented typical bipod structures. The structure of position A is symmetrical, and some materials are removed from the bottom half of the bipod, which increases compliance greatly. The structures of position B and C are not symmetrical, and one end is thick, while the other end is thin. The connecting structures are in the form of mirror symmetry, which is consistent with the camera. Compared with the initial structure, a lot of materials are removed. The compliance of the structure is improved, and external loads transmitted into the camera from the bottom can be attenuated to a certain extent. Figure 4: Finite element model of the camera and the initial connecting structure.