### Material Interpolation Format of SIMP and Structural Analysis
For a linear elastic material, the element stiffness matrix is typically calculated using:  
\[\mathbf{K}_{i}^{e}=\int_{\Omega^{i}}\mathbf{B}^{\mathrm{T}}\mathbf{D}(x_{i}) \mathbf{B}d\Omega^{i} \tag{3}\]  
where \(\mathbf{B}\) is the strain-displacement matrix; \(\Omega^{i}\) is the volume of the \(i-th\) element. The global stiffness matrix \(\mathbf{K}\) is constructed by assembling the stiffness matrices of all the individual elements. The global stiffness matrix represents the stiffness properties of the entire structure. \[\mathbf{K}=\mathbf{A}_{i=1}^{NaCl}\mathbf{K}_{i}^{e} \tag{4}\]  
where \(\mathrm{N}^{element}\) is the total number of elements in the discretized design domain. Structural analysis in the context of SIMP involves solving the equilibrium equations of the structure, where the stiffness matrix is a function of the interpolated material properties. The design domain is discretized into finite elements. Each element is assigned a density variable \(x_{i}\)The global stiffness matrix \(\mathbf{K}\) is assembled using the element stiffness matrices \(\mathbf{K}_{i}^{e}\). \(\mathbf{A}\) is the assembly matrix. The equilibrium equation for the structure is given by:  
\[\mathbf{K}\mathbf{U}=\mathbf{F} \tag{5}\]  
In static structural optimization, minimizing the compliance of the structure is a common objective. Compliance is a measure of the deformation of a structure under a given load and reflects the response characteristics of the structure to external forces. Specifically, \(\mathbf{C}\) is defined as the ratio of the work undertaken by external forces to the displacements (or equivalently, the dot product of the force vector \(\mathbf{F}\) and the vector \(\mathbf{U}\)), but more commonly and physically meaningful is its expression as a quadratic form of the displacement vector \(\mathbf{U}\):  
\[C=\mathbf{F}^{\mathrm{T}}\mathbf{U}=\mathbf{U}^{\mathrm{T}}\mathbf{K}\mathbf{U} \tag{6}\]  
Therefore, minimizing compliance means maximizing the stiffness of the structure under the given conditions. Mathematical Model for Structural Topology Optimization of the APTS Mechanical Support Structure Bracket  
Structural topology optimization aims to determine the optimal material distribution within a design domain to achieve specific performance objectives while adhering to defined constraints. The Solid Isotropic Material with Penalization (SIMP) method is one of the most widely adopted approaches for this purpose. The SIMP method formulates the problem as one of optimizing material layout, where the goal is to identify the best distribution of material and voids within a continuous region. This ensures that the structure meets performance criteria such as stress resistance and flexibility under loading conditions. The primary goal in the topological optimization of continuum structures is to identify a material subregion within the design domain \(\Omega\), with a specified volume, such that the objective function achieves an extremum. Objectives may include maximizing stiffness under applied loads, minimizing weight to achieve a lightweight design, controlling deformations by minimizing weight with displacement constraints, or maximizing natural frequencies to avoid resonance. Additionally, compliance minimization ensures structural safety under stress constraints, while reliability-based optimization accounts for uncertainties in material properties or loading conditions. The following is the mathematical model of the topology optimization of the APTS mechanical support structure in this study, which is mathematically solved as a constraint optimization problem. \[findX=\{x_{i}\} \tag{7}\] \[minC=\mathbf{U}^{\mathrm{T}}\mathbf{K}(x_{i})\mathbf{U}\] \[sub\;to:\left\{ \begin{array}{l}K(\mathbf{D}(x_{i}))U=F\\ V=\sum\limits_{i=1}^{N^{element}}x_{i}v_{i}\\ V\leq\mathrm{V}_{f}\times\mathrm{V}_{sum}\\ S\leq S_{max}\\ |\sigma_{i}|\leq\sigma_{max}\\ 0\leq x_{min}\leq x_{i}\leq 1\end{array}