2.1. Fundamental Concepts
Topology optimization is to find the best material distribution or force transmission path in a given design space, so that the lightweight design can be obtained under various constraints. Topology optimization of the primary mirror structure based on solid isotropic material with penalization (SIMP) was applied in this study using the software OptiStruct [
11]. Its basic idea is to assume a material element that has variable relative density
ρ between 0 and 1. When
ρ is 1 or near 1, it means the element is required and need to be retained; when
ρ is 0 or near 0, it says the element is not filled with material and needs to be removed. In order to avoid too many middle density elements, introducing penalty factor to enable continuous variables approach to 0 or 1 approximately, and the material properties of the element can be defined by
where
ρi is the material relative density and
Ei is the relative elasticity modulus of the
i-th element, respectively. In the commercial software OptiStruct, the lower bound
ρ0 takes the value of 0.01, that is to avoid the singularity of stiffness matrix in the process of finite element analysis, and the upper bound
ρ1 takes the value of 1.
E0 is the inherent elasticity module of a given isotropic material, and
P is the penalization power.
2.2. 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
where x
i is the distance from the i-th node to the fitting surface after deformation, and
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
where
CW is the optimization objective weighted by
Cj, which represents the compliance function of the mirror surface under the
j-th load subcase,
Wj is weighting factors,
uj is the displacement vector, and
fj is the load vector. The structural configuration can be described by the material relative density
ρi, and the optimization model can be expressed as follows:
optimization objective
subject to
In this model, CW is the weighted compliance function, and CP is the compliance function under polishing pressure, CL and CA are the compliance functions of the mirror face under lateral gravity and axial gravity, respectively. γ, η, and μ are weighting factors. V* is the volume of the initial structure, Vi is the ith element volume, and α is the upper bound of the allowed volume fraction. fL, fA and fP are the lateral gravity, axial gravity and polishing pressure load vectors. uL, uA, and uP are the corresponding displacement vectors. DL is the nodal displacement of the mirror surface under lateral gravity, and D1 is the upper bound of lateral nodal displacement. DA is the nodal displacement of the mirror under axial gravity, and D2 is the upper bound of axial nodal displacement.
The sensitivities of the objective function
CW with respect
ρi can be obtained by the adjoint method
The optimal criterion adopted in this paper is based on the Kuhn-Tuker (KT) condition, and the topology-optimized Lagrange function is constructed by introducing the KT condition to be satisfied in mathematics. The Lagrange multiplier
λ0,
λ1j,
λ2i,
λ3i is introduced to derive the Lagrange function of the continuum topology optimization problem with minimized compliance under volume constraints:
The condition that the design variable takes the optimal value is
In this paper, the importance of the
CL,
CA, and
CP are equivalent, so the weighting factors
γ,
η, and
μ take the same value of 0.33. The initial mass of the mirror is 42.6 kg, we take a more stringent lightweight ratio of 70% compared with the demand ratio of 67.1%, so the volume fraction
α takes 0.3. The design specification requires upper limits 25 nm for
D1 and 160 nm for
D2 [
2]. The shape of the polishing plate is like pentagram and its outer diameter is 160 mm, and the static polishing load on the whole surface of the mirror is 0.2 kPa [
9,
10]. Design parameters in topology optimization are shown in
Table 1.