# Lightweight Design of Multi-Objective Topology for a Large-Aperture Space Mirror
## 2 Lightweight Design of the Primary Mirror
### Formulation of the Optimization Problem  
The purpose of this research is to obtain the optimum configuration of the mirror, which is to minimize the mirror surface distortion while reducing the weight of the mirror. The mirror surface deviation is mainly caused by the self-weight of the mirror. In addition, the polishing pressure applied in the grinding manufacturing process causes serious local deformation and has a great impact on the optical performance of the mirror. In order to reach the minimum values of mirror surface shape error caused by lateral gravity, axial gravity, and polishing pressure at the same time, the multi-objective optimization problem is selected for the design method. The practical way to deal with the multi-objective optimization problem is to combine the multiple objectives into a single objective function, which is easy to be solved by assigning weighting factors to the objectives, or choosing the most important one as the objective function and treating the other objectives as the constraint conditions [12]. The wavefront RMS aberration is a common way to evaluate the quality of the mirror surface, which is defined as the root mean square of the distances between the deformed mirror surface nodes and the fitted surface. It can be expressed as  
\[\text{RMS}=\sqrt{\sum_{i=1}^{n}\frac{1}{n}(x_{i}-\overline{x})^{2}} \tag{2}\]  
where \(\text{x}_{i}\) is the distance from the i-th node to the fitting surface after deformation, and \(\overline{x}\) is the average distance between all nodes and the fitting surface. However, the value of RMS cannot be used as the optimization objective directly in the existing commercial finite element software. Considering time consumption and computational efficiency, the RMS deformation error under each loading cases is replaced by the corresponding structural compliance. The minimum structural weighted compliance is selected as the objective function to minimize the displacements of the surface nodes, which has similar effects to decrease the RMS value [10]. The weighted structural compliance can be obtained by the formula as follows  
\[C_{W}=\sum W_{j}C_{j}=\sum W_{j}u_{j}^{T}f_{j}, \tag{3}\]  
where \(C_{W}\) is the optimization objective weighted by \(C_{j}\), which represents the compliance function of the mirror surface under the \(j\)-th load subcase, \(W_{j}\) is weighting factors, \(u_{j}\) is the displacement vector, and \(f_{j}\) is the load vector. The structural configuration can be described by the material relative density \(\rho_{i}\), and the optimization model can be expressed as follows:  
\[\text{Find }X=(\rho_{1},\rho_{2},...,\rho_{N})^{T}, \tag{4}\]  
optimization objective  
\[\text{Minimize }C_{W}=\gamma C_{L}+\eta C_{A}+\mu C_{P}, \tag{5}\]  
subject to  
\[g(X)=\sum_{i=1}^{n}V_{i}-\alpha V^{*}\leq 0, \tag{6}\]  
\[f_{L}=ku_{L},\ f_{A}=ku_{A},\ f_{P}=ku_{P}, \tag{7}\]  
\[D_{L}-D_{1}\leq 0,D_{A}-D_{2}\leq 0, \tag{8}\]  
\[\rho_{0}\leq\rho_{i}\leq\rho_{1}. \tag{9}\]In this model, \(C_{W}\) is the weighted compliance function, and \(C_{P}\) is the compliance function under polishing pressure, \(C_{L}\) and \(C_{A}\) are the compliance functions of the mirror face under lateral gravity and axial gravity, respectively. \(\gamma\), \(\eta\), and \(\mu\) are weighting factors. \(V^{\star}\) is the volume of the initial structure, \(V_{i}\) is the \(i\)th element volume, and \(\alpha\) is the upper bound of the allowed volume fraction. \(f_{L}\), \(f_{A}\) and \(f_{P}\) are the lateral gravity, axial gravity and polishing pressure load vectors. \(u_{L}\), \(u_{A}\), and \(u_{P}\) are the corresponding displacement vectors. \(D_{L}\) is the nodal displacement of the mirror surface under lateral gravity, and \(D_{1}\) is the upper bound of lateral nodal displacement. \(D_{A}\) is the nodal displacement of the mirror under axial gravity, and \(D_{2}\) is the upper bound of axial nodal displacement.