## 3 Analysis of the Influencing Factors in Topology Optimization
### Filter Radius  
As illustrated in Equations (3) and (4), the main principle of the density filtering method is to modify the density of the central unit using the density information of all units within the filter radius (\(r\)). Specifically, the weighted average density of units within the filter radius replaces the density of the central unit. The size of \(r\) determines the number of units involved in density filtering. When \(r\) is smaller than the average edge length of the unit, only the central unit falls within the filter range, and the density filter method fails to function effectively. This can lead to issues such as checkerboard patterns and mesh dependency, which in turn negatively affect the TO structural configuration.  
Table 5 presents the TO structural configurations obtained with various filter radii (\(r\)) for hexahedral mesh sizes of 2 mm and 4 mm. Figure 7 presents the analysis results of the effect of different \(r\) values on compliance and computation time. The predicted results provided in Table 5 and Figure 7 reveal that for the hexahedral mesh, in the case of \(r=0.5\), the number of units in the filter range is small, and the efficiency of the density filtering method is limited. This leads to a large density discrepancy between adjacent units, which yields excessive branching in the structural configuration. With the increase in \(r\), the number of neighboring units used to adjust the central unit's density grows, and the density discrepancies among units reduce. The penalty factor applied to the unit densities further reduces the branching, resulting in a clearer and more stable structure. In the case of \(4.6\leq r\leq 5\), the compliance value converges, and the structural configuration remains consistent. For the case of \(0.25\leq r\leq 2\), several performance-affecting features appear in the tetrahedral mesh. With the growth of \(r\), the structural configuration becomes clearer, and the number of performance-affecting features lessens. In the case of \(4\leq r\leq 6.8\), the compliance value converges, and the structural configuration stabilizes, with no performance-affecting features remaining. However, in the case of \(r>7.2\), the number of units becomes too large, resulting in overly small density discrepancies among units after filtering, which makes the overall density distribution too uniform and introduces features that affect structural performance.  
In conclusion, \(r\) has a significant effect on the structural configuration. For hexahedral meshes, the recommended filter radius is between 4.6 and 5, i.e., 2.3 to 2.5 times the average unit edge length. For tetrahedral meshes, the recommended filter radius is between 5.2 and 6, i.e., 1.3 to 1.5 times the average unit edge length. The suggested coefficient for tetrahedral meshes is smaller than that for hexahedral meshes because, for the same \(r\), a tetrahedral mesh has more units than a hexahedral mesh, and therefore, the density filter method has a greater effect. In the subsequent analysis of other factors, the default value of \(r\) is set to 2.4 times the average unit edge length for hexahedral meshes and 1.4 times for tetrahedral meshes.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(r\) & Hexahedral Mesh & \(r\) & Hexahedral Mesh & \(r\) & Tetrahedral Mesh & \(r\) \\ (mm) & Hexahedral Mesh & \((mm)\) & Hexahedral Mesh & \((mm)\) & Tetrahedral Mesh & \((mm)\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: TO structural configurations obtained for various values of the filter radius \((r)\) (\(\overline{\varphi}=41.75\%\), hexahedral mesh size = 2 mm, tetrahedral mesh size = 4 mm, \(E=110.3\) GPa, \(\mu=0.33\), and \(x_{\text{initial}}=0.5\)).