# Equivalent Modeling and Verification of a High-Steepness and Lightweight Elliptical Aluminum Mirror
## 4 Optomechanical Design and Optimization of Mirror
### Mirror Shape Optimization  
In the previous section, we described the completed construction of the initial structure of the solid mirror, which was unable to meet lightweight design requirements.  
In order to reduce the weight of the mirror, large-aperture mirrors are usually designed to be single-arch, double-arch, flat back and other special shapes. Therefore, it is necessary to select a reasonable lightweight structure based on the processing characteristics of the mirror.  
We chose a topology optimization design combined with the characteristics of mirror processing to complete the shape optimization of the mirror, allowing us to carry out conceptual design. Topology optimization is a mathematical method to optimize material distribution under given load conditions, boundary conditions and performance constraints [22; 23; 24], as shown in Figure 7.  
Based on the constraints of the single-point diamond turning process, the design requirements of turning force on the lightweight support stiffness of the mirror back were analyzed. Machining turning force is a static optimization problem, which usually takes the minimization of structural compliance (or the minimization of strain energy and the  
Figure 6: Elliptical initial structure converted by equivalent circular plate: (**a**) circular equivalent plate; (**b**) high-steepness elliptic mirror.  
Figure 7: Establishment of initial finite element model for topology shape optimization: (**a**) division of optimization areas; (**b**) fixed support constraint.  
maximization of stiffness) as the objective function and the structural volume ratio as the constraint function. Its mathematical model can be expressed in Equation (9):  
\[\begin{split}\min:C_{SPDT}=F_{SPDT}^{T}U_{SPDT}=U_{SPDT}^{T}K_{SPDT}U _{SPDT}\\ s.t.\left\{\begin{array}{l}KU=F\\ V(\rho_{i})/V_{0}-f\leq 0\\ 0<\rho_{\min}\leq\rho_{i}\leq 1\end{array}\right.\end{split} \tag{9}\]  
where \(C\) is the compliance of the mirror, \(\rho_{i}\) is the unit density variable, \(K\) is the global stiffness matrix, \(U\) is the global displacement vector, \(F\) is global load vector is the design domain volume constraints, \(f\) is volume fraction, and \(V_{0}\) is design domain volume.  
Topology optimization design can provide a good reference to guide the design of a lightweight structure for the mirror back according to the density distribution. We use the optimization module of HyperMesh software (Altair, Troy, MI, USA) to complete the topology optimization analysis under given boundary constraints. After 25 iterations, the optimization results converge to a stable value. The density ranges from 0 to 1. The closer the density is to 1, the more important the structure is, which can be retained according to the actual design. The closer the density is to 0, the less important this part is [25]. This can be appropriately removed according to the actual design. In order to ensure the continuity of materials, we choose the result of threshold 0.1 as the reference result of shape optimization. As shown in Figure 8, it can be seen from the iteration results that a better lightweight density distribution of the mirror design is that less material is left in the long axis and more material is left in the short axis. The back shape of the mirror is a single arch. Unfortunately, the existing topology optimization results still failed to meet our expected requirements, but provided a direction for the subsequent lightweight design.