### Material Interpolation Format of SIMP and Structural Analysis  
Structural topology optimization aims to determine the optimal material distribution within a given design domain to achieve desired performance objectives while satisfying various constraints. The Solid Isotropic Material with Penalization (SIMP) method, pioneered by [PERSON] and [PERSON] [32], revolutionized the field by formulating the optimization problem as a continuous material distribution problem, which enables efficient exploration of design spaces. The SIMP method is a widely used approach in structural topology optimization. It seeks to find the optimal material distribution within a given design domain to minimize structural compliance while satisfying various constraints. The basic idea of the SIMP material interpolation method is to use the artificial material density as a design variable. Through material interpolation models such as SIMP or RAMP [33], the relationship between artificial material density and the material elastic tensor is established. By varying the artificial density between 0 and 1, the optimization of two-phase material distribution is achieved. The SIMP method builds upon the concept of density-based optimization, where the material density of each finite element is used as a design variable. The method's foundation lies in formulating an objective function to minimize structural compliance or maximize stiffness, subject to volume and other constraints. For the topology optimization model presented in this paper, the SIMP interpolation model is used. In SIMP, the relationship between material density and material properties is described using a material interpolation function. A commonly used interpolation function is:  
\[E(x_{i})=E_{min}+x_{i}^{p}(E_{0}-E_{min}) \tag{1}\]  
where \(E(x)\) represents the elastic modulus of the material, \(E_{min}\) denotes the minimum allowable elastic modulus, ensuring numerical stability and avoiding singularities, usually \(E_{min}\) is a small value (often chosen as a fraction of \(E_{0}\), e.g., \(10^{-9}\)\(E_{0}\)) to avoid singular stiffness matrices in finite element analysis, \(E_{0}\) is the maximum elastic modulus, \(x_{i}^{p}\) denotes the material density design variable for the \(i-th\) element, ranging from 0 (void) to 1, where \(x_{i}^{p}=1\) represents solid material and \(x_{i}^{p}=0\) represents void, and \(p\) is the penalization factor influencing the distribution of material densities, typically \(p\geq 3\), promoting a binary material distribution. For numerical stability, typically, let \(E_{min}=E_{0}/1000\) in the present paper. Under the effect of \(p\), intermediate density values gradually approach 0 or 1. A larger value of \(p\) results in a stronger penalty effect on intermediate densities. However, \(p\) should not be too large, as an excessively large \(p\) can also lead to singularity in the matrix, making it impossible to find feasible solutions to the optimization problem. When \(x_{i}^{p}=1\) or \(x_{i}^{p}=0\), it represents the design point as solid material or void material, corresponding to a \"black\" or \"white\" design in topology optimization. The parameter \(p\) represents the penalization exponent. By employing a continuous penalization strategy on the artificial density \(x_{i}\), design variables tend toward 0 or 1 as much as possible, facilitating clear material selection and achieving structural topology optimization design. For isotropic materials, the elastic constitutive matrix \(\mathbf{D}(x_{i})\) of the element is calculated as follows:  
\[\mathbf{D}(x_{i})=\frac{E(x_{i})}{1-\
u} \begin{bmatrix}1&\
u&0\\ \
u&1&0\\ 0&0&(1-\
u)/2\end{bmatrix}  \tag{2}\]  
where \(\
u\) is the Poisson's ratio. The element stiffness matrix is computed based on the material properties (such as Young's modulus) and the geometry of the element.