## 3 Analysis of the Influencing Factors in Topology Optimization
### Initial Density  
The initial density, \(x_{\text{initial}}\), and material properties are crucial factors assigned to each unit during the initial calculation of TO. These factors include consistent values for \(E\), \(\mu\), and density. In TO, \(x_{\text{initial}}\) is typically defaulted as \(\overline{v}\). For instance, in the case of \(\overline{v}\) = 40%, each unit is assigned a value of \(x_{\text{initial}}\) = 0.4. Although the mature MMA is employed in this study to reduce the impact of the initial parameters on the TO results and ensure a global optimum solution rather than a local optimum, further research is still required to examine whether various values of \(x_{\text{initial}}\) affect the global optimal solution. Table 4 presents the TO structural configurations obtained with various \(x_{\text{initial}}\) values under the conditions of a hexahedral mesh size of 2 mm and a tetrahedral mesh size of 4 mm. Figure 6 analyzes the influence of various \(x_{\text{initial}}\) values on the compliance and computation time. For the hexahedral mesh, the TO structural configurations for different values of \(x_{\text{initial}}\) remain consistent along the primary load path, with only slight variations in the secondary load path. Using the compliance value of \(x_{\text{initial}}\) = 0.1 as a baseline, the maximum compliance variation rate is 1.42%, whereas the maximum computation time variation rate is 22.44%. In the case of \(0.5\leq x_{\text{initial}}\leq 1\), the compliance variation rate and the computation time variation indicate two opposite trend behaviors. The structural configuration, compliance, and computation time all demonstrate superior performance at  
Figure 5: Effect of the material properties on the compliance and computation time: (**a**) hexahedral mesh and (**b**) tetrahedral mesh (\(\overline{v}\) = 41.75%, a hexahedral mesh size of 2 mm, and a tetrahedral mesh size of 4 mm). \begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Material Properties** & **Hexahedral Mesh** & **Tetrahedral Mesh** \\ \hline
3 & \(E\) = 118 GPa; \(\mu\) = 0.33 & \\
4 & \(E\) = 120 GPa; \(\mu\) = 0.342 & \\
5 & \(E\) = 209 GPa; \(\mu\) = 0.269 & \\ \hline \end{tabular}
\end{table} 
Table 3: _Cont._\(x_{\text{initial}}\) = 0.5. For the tetrahedral mesh, the TO structural configurations remain essentially unchanged for different \(x_{\text{initial}}\) values. Using the compliance value of \(x_{\text{initial}}\) = 0.1 as a baseline, the maximum compliance variation rate is obtained as 0.72%, and the maximum computation time variation rate is attained as 23.66%. In the case of \(x_{\text{initial}}\) > 0.5, compliance reaches near convergence, with the optimal compliance observed at \(x_{\text{initial}}\) = 0.5. The structural configuration exhibits no substantial features affecting performance, and the computation time is at the optimal level. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(x_{\text{initial}}\) & Hexahedral & Tetrahedral & \(x_{\text{initial}}\) & Hexahedral & Tetrahedral \\ \hline \hline \end{tabular}
\end{table} 
Table 4: TO structural configurations obtained with various initial densities \(x_{\text{initial}}\) (\(\overline{v}\) = 41.75%, hexahedral mesh size = 2 mm, tetrahedral mesh size = 4 mm, \(E\) = 110. 3 GPa, and \(\mu\) = 0.33). Figure 6: Influence of the initial density on the compliance and computation times: (**a**) hexahedral mesh and (**b**) tetrahedral mesh (\(\overline{v}\) = 41.75%, hexahedral mesh size = 2 mm, tetrahedral mesh size = 4 mm, \(E\) = 110. 3 GPa, and \(\mu\) = 0.33). In summary, the impact of \(x_{\text{initial}}\) on the structural configuration and compliance is minimal. Compared to the hexahedral mesh, the influence of \(x_{\text{initial}}\) on the tetrahedral mesh is even weaker. This indirectly validates the effectiveness of the MMA algorithm in mitigating local extrema issues in the TO procedure. Furthermore, for both hexahedral and tetrahedral meshes, a value of \(x_{\text{initial}}=0.5\) yields the ideal structural configuration.