# Equivalent Modeling and Verification of a High-Steepness and Lightweight Elliptical Aluminum Mirror
###### Abstract  
In order to realize the compact layout of aerospace payloads, the design and manufacture of high-steepness lightweight aluminum alloy mirrors is a key technology to be explored. For high-steepness mirrors, the traditional method is to establish the initial thickness that satisfies the bending stiffness through finite element optimization iteration, which cannot achieve fast modeling and performance estimation. In this paper, firstly, the equivalent modeling method of the mirror with high steepness is proposed to achieve the equivalent of the elliptic mirror with a diameter of 410 \(\times\) 310 mm and F# less than 0.7. Based on the mathematical model, topology shape optimization was used to build a highly lightweight mirror structure that could be quickly assembled, and the equivalent area-mass density of the mirror is less than 34 kg/mm2. Next, the rationality of design feasibility was verified by simulation analysis. Finally, by using single point diamond turning combined with post polishing process, the high-precision manufacturing of conventional aluminum alloy mirror was realized. The results show that the mirror shape accuracy is 1/10 \(\lambda\) (\(\lambda\) = 632.8 nm), and the surface roughness Ra is 3.342 nm. This research provides strong theoretical support and application prospects for the low-cost and rapid manufacturing of high-steepness lightweight aluminum alloy mirrors. aluminum alloy; elliptic mirror; steepness; lightweight; plate equivalent modeling; single-point diamond turning +
Footnote †: journal: Applied Sciences  
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Footnote 0: _Applied Sciences_  
In order to achieve an ultra-compact layout and small moment of inertia, the most effective method is to increase the curvature (or steepness) of the mirror. This means that an ultra-lightweight design becomes more difficult to achieve, especially for the specific stiffness (\(E/\rho\)). It is still a technical challenge to realize a high steepness and ultra-lightweight design with low-specific-stiffness aluminum alloy materials [9]. The bending stiffness of the mirror is positively correlated with the thickness of the mirror. The suitable selection of an initial mirror thickness can improve the iterative efficiency of a lightweight design with high steepness. Many scholars have conducted outstanding research on the construction of initial mirror models. One method is based on an analytical deformation solution of a thin elastic plate with a circular thickness distribution under easily supported boundary constraints [10; 11]. The other method is to establish a parametric topology shape optimization model, using an optimization algorithm to determine an optimal solution in the global solution [12; 13]. Both methods require a lot of calculation and data processing, and it is impossible to quickly plan the initial mirror thickness suitable for an accurate mirror surface, which means that it is difficult to achieve fast modeling and performance estimation. The thin plate hypothesis theory can be used for estimating the self-deformation of a mirror with a small curvature aperture under the condition that the mirror diameter thickness ratio (diameter/thickness) is fixed [14; 15; 16]. For mirrors with large curvatures, especially elliptic mirrors, it is impossible to directly use the thin plate theory to select the appropriate mirror diameter thickness ratio. Further research is needed in order to quickly select the mirror diameter thickness ratio using an analytical approach. Topology optimization is an effective way to achieve a very lightweight design. Topology optimization and additive manufacturing are combined to reduce the weight of the mirror by about 20% under the same mechanical properties, which has been applied in designing small-aperture metal mirrors [17; 18]. However, for medium- and large-aperture metal mirrors, the traditional material reduction manufacturing method remains the main design choice at this stage [19].