## 3 Analysis of the Influencing Factors in Topology Optimization
### Penalty Factor  
According to Equation (1), the SIMP interpolation model of the variable density theory transforms the discrete problem into a continuous optimization problem by introducing intermediate-density units. [PERSON] and [PERSON] [6] proved the physical existence of this model in 1999; however, in engineering applications, due to the difficulty of intermediate-density units, it is typically necessary to avoid their formation or at least minimize their quantity. For this purpose, a penalty should be applied to intermediate-density units to ensure that their density approaches 0 or 1. [PERSON] and [PERSON] [44] proposed a formula to calculate the minimum value of the penalty factor as follows:  
\[ \begin{array}{l}p\geq\max\Bigl{\{}\frac{2}{1-\mu},\frac{4}{1+\mu}\Bigr{\}} \hskip 28.452756 pt(\text{in 2-D})\\ p\geq\max\Bigl{\{}15\frac{1-\mu}{7-5\mu},\frac{3}{2}\frac{1-\mu}{1-2\mu}\Bigr{\}} \hskip 28.452756 pt(\text{in 3-D})\end{array}  \tag{10}\]  
In the case of \(\mu\) = 0.33, the minimum values of \(p\) in the second and third dimensions are obtained as 3 and 2, respectively. [PERSON] and [PERSON] [44] recommended that the minimum value of \(p\) should be greater than 3 for both 2D and 3D cases, but the maximum value of \(p\) was not specified. Therefore, further analysis of \(p\) is required to verify the minimum value and also determine the maximum value. Table 6 presents the TO structural configurations obtained with different values of \(p\) for hexahedral meshes with a size of 2 mm and tetrahedral meshes with a size of 4 mm. Figure 8 demonstrates the effect of the variable \(p\) on the compliance and computation time. According to Table 6 and Figure 8, it can be seen that for hexahedral meshes, in the case of \(p\) = 1, the penalty effect disappears, which leads to an unmanufacturable structure configuration. As \(p\) increases, the penalty of unit densities is strengthened, and the units with medium density gradually converge to densities of 0 or 1, and the number of branches in the structure configuration decreases. The branches connected to the third column of the bolt connection holes are gradually eliminated; the load transfer path becomes clearer, and the compliance converges. However, in the case of \(6<p<9\), the results do not converge, which leads to calculation failure. In the case of \(p\geq 9\), certain branches in the gooseneck of the chain are removed, forming large triangular voids, and new branches appear under the voids. For the tetrahedral mesh, in the case of \(p\) = 1, the penalty effect disappears, and the structure configuration cannot be manufactured. Compared with the hexahedral mesh, the tetrahedral mesh achieves a better penalty effect at smaller values of \(p\). In the case of \(2.5\leq p\leq 3.5\), the obtained structure configuration is nearly identical to that of the  
Figure 7: Influence of different penalty factors on the compliance and computation time: (**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, \(\mu\) = 0.33, and \(x_{\text{initial}}\) = 0.5). hexahedral mesh in the range of \(3\leq p\leq 4\) with the compliance convergence. As \(p\) increases further, some branches in the structure configuration are eliminated. However, in the case of \(p>6\), the calculation results do not converge, which leads to computational failure. In conclusion, in the case of \(p\geq 6\), the penalty factor becomes too large, causing calculation failure due to non-convergence. Therefore, the maximum value of \(p\) should be set below 6. For the hexahedral mesh, it is recommended to set \(p\) in the range of [3,4], and for the tetrahedral mesh, the recommended \(p\) range is [2.5, 3.5]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(p\) & Hexahedral Mesh & Tetrahedral Mesh & \(p\) & Hexahedral Mesh & Tetrahedral Mesh \\ \hline \hline \end{tabular}
\end{table}