## 2 Materials and Method  
### Deterministic Topology Optimization  
In this study, the focus is on the minimum compliance topology optimization algorithm, as illustrated in Equation (1). The objective here is to discover the material distribution (represented by \(\mathbf{\rho}\)) throughout the design space that results in maximum stiffness while adhering to a specific volume constraint. \[ \begin{split}\min& C(\mathbf{\rho})=\mathbf{d}^{T}\mathbf{Kd }=\sum_{\mathbf{\epsilon}=1}^{N}{(\rho_{\mathbf{\epsilon}})^{p}d_{\mathbf{\epsilon}}^{T}k_ {0}d_{\mathbf{\epsilon}}}\\ \text{subject to}&\frac{V(\rho)}{V_{0}}\leq f \\ \mathbf{Kd}&=\mathbf{f}\\ 0<\rho_{min}&\leq\rho_{\mathbf{\epsilon}}\leq 1\end{split}  \tag{1}\]  
The vector \(\rho\) represents the design variables within the minimum compliance topology optimization algorithm. The stiffness matrix, displacement vector, and force vector are denoted by \(\mathbf{K}\), \(\mathbf{d}\), and \(\mathbf{f}\), respectively. \(N\) refers to the number of elements that discretize the design area while \(d_{\mathbf{\epsilon}}\) and \(k_{0}\) are the displacement of an element and elemental stiffness matrix, respectively. The material volume and the design area volume are represented by \(V(\rho)\) and \(V_{0}\), with \(f\) as a pre-determined upper limit in the volume fraction. \(\rho_{min}\) is the minimum relative density used to prevent singularity in the model. When \(\rho_{\mathbf{\epsilon}}\cong 0\), it signifies that the corresponding element within the structure is considered empty, meaning it does not contribute to the physical attributes of the design. The design domain is discretized using square-shaped finite elements, adhering to a fundamentally density-based topology optimization methodology. A revised version of the Solid Isotropic Material with Penalization (SIMP) method is employed, allowing for the implementation of filters within the optimization process. \[E_{\mathbf{\epsilon}}(\overline{\rho}_{\mathbf{\epsilon}})=E_{min}+\rho_{\mathbf{\epsilon }}^{p}(E_{0}-E_{min}) \tag{2}\]Here, \(E_{0}\) represents the stiffness of the solid elements while \(E_{min}\) is the density value attributed to the void regions to prevent the stiffness matrix from becoming singular. Additionally, \(p\) is a penalization factor, usually set to 3, used to ensure definitive black-and-white solutions. A significant advantage of the modified SIMP method is its ability to allow for the straightforward application of additional filters [40]. Certain filters must be employed to address the topology optimization problem and prevent the creation of checkerboard patterns, as seen in Figure 1. To establish areas of medium density, a smooth transition between solid and empty regions is required, facilitating a more controlled and desired material distribution. A prevalent technique for achieving this is the application of a filter radius (\(r_{min}\)) to the sensitivities or densities. The specific sensitivity filter utilized in the topology optimization (TO) problem is detailed as follows [40]:  
\[\frac{\partial\hat{c}}{d\rho_{e}}=\frac{1}{\max(\gamma,\rho_{e})\sum_{i\in N_{e }}H_{ei}}\sum_{i\in N_{e}}H_{ei}\rho_{i}\frac{\partial c}{\partial\rho_{i}}. \tag{3}\]  
Here, the term \(\gamma(\cong 10^{-3})\) is a small positive number, and \(H_{ei}\) is a weighting factor defined as follows:  
\[H_{ei}=\max(0,r_{min}-\Delta(\mathbf{e},\mathbf{i})). \tag{4}\]  
In the classical SIMP approach, the density variables cannot be zero, and the \(\gamma\) term is not required. The density filter used for TO problems is given as follows:  
\[\widetilde{\rho}_{e}=\frac{1}{\sum_{i\in N_{e}}H_{ei}}\sum_{i\in N_{e}}H_{ei} \rho_{i}. \tag{5}\]  
The filtered densities, referred to as \(\widetilde{\rho}_{e}\), are known as physical densities. Therefore, the filtered density field \(\widetilde{\rho}_{e}\) should always be presented instead of the original density field \(\rho_{e}\) as a solution to the optimization problem.