# Three-Band Spectral Camera Structure Design Based on the Topology Optimization Method
## 2 Design of Three-Band Structural Components
### Optimization Method of the Structure  
#### 2.3.1 Topology Optimization Method  
In the design of the spectral camera, the optimization of the housing and the components carrying the optical lenses is crucial, particularly in terms of the fundamental frequency. By increasing the natural frequency of the housing and supports, resonance with external vibration frequencies can be avoided, ensuring system stability. Therefore, the structural design should focus on material selection and structural reinforcement to improve the rigidity, minimize deformation, and ensure the stability and reliability of the optical system in complex operating environments.  
This study adopts the continuum topology optimization method based on a variable density, aiming to achieve a lightweight design for the supporting structure while meeting practical requirements. The optimization design variable is the relative density of the elements [25]. When handling discrete optimization problems, the design variable values for the elements are either 0 or 1. However, by introducing the concept of the relative density, the design variable values for grid elements become continuous within the range of 0 to 1, thereby transforming the problem from discrete to continuous. The variable density method assumes a constant material density and defines the design variable as the element density, converting the problem into a material distribution optimization problem. The mathematical formulation is expressed as  
\[\rho=x\rho_{0} \tag{1}\]  
where \(x\) represents the relative density of each element, \(\rho_{0}\) is the inherent density of each element, and \(\rho\) is the topology design variable. When applying the variable density method to optimize the supporting structure, commonly used interpolation models include the RAMP interpolation model and the SIMP interpolation model, with the SIMP model being the most widely used. The mathematical expression for the SIMP model is  
\[E=\rho^{p}E_{0} \tag{2}\]  
In the equation, \(p\) represents the penalty factor, \(E\) is the elastic modulus, and \(E_{0}\) is the inherent elastic modulus of the material. For 3D structures, the limitation for the penalty factor is expressed as  
\[p\geq\text{max15}\bigg{\{}\frac{1-v^{0}}{7-5v^{0}},\frac{3}{2}\frac{1-v^{0}}{1 -2v^{0}}\bigg{\}} \tag{3}\]  
In the equation, \(v\) is the Poisson ratio of the material. Topology optimization based on the SIMP method can be widely applied to various objective function constraint scenarios, such as the maximum eigenvalue problem, minimum compliance problem, and minimum weight problem [26]. Based on the SIMP model, the mathematical formulation of the continuous topology optimization can be expressed as follows:Find  
\[x=\left\{x_{1},x_{2},x_{3},\ldots x_{n}^{T}\right\}\in R \tag{4}\]  
and minimize  
\[C=F^{T}U \tag{5}\]  
\(C\) represents the structural compliance, \(F\) is the force vector matrix, and \(U\) is the displacement matrix.  
\[Q=yQ_{0}=\sum_{i=1}^{n}x_{i}Q_{i}\leq Q^{*} \tag{6}\]  
\[F=KU \tag{7}\]  
\[0<x_{\min}\leq x_{i}\leq x_{\max}\leq 1 \tag{8}\]  
\(K\) represents the total stiffness matrix of the structure, \(Q\) is the optimized structural volume, \(Q_{0}\) is the initial total volume, and \(Q*\) is the maximum allowable material volume in the design. \(Y\) represents the optimized volume ratio, while \(x_{min}\) and \(x_{max}\) are the minimum and maximum limits of the relative density of the elements, respectively.