## 4 Results and Discussion
However, the stresses at the connections of the straight lines and arcs are increased. (5) For the optimal four-arc scheme, the radius of the basic circle \(r_{\rm fb}\) is approximately equal to the radius \(r_{\rm cr}\) of the optimal circular scheme. However, the radius of the large arc \(r_{\rm fb}\) is large enough. It significantly reduces the stresses on the bottom and top sides of the hole. Additionally, the smooth transitions between the large and small arcs reduce the stresses at the connections. Therefore, the maximum von Mises stress of the optimal four-arc scheme is significantly reduced, and the high-stress areas are fairly well-distributed around the vent hole. Figure 10 shows the von Mises stress distribution along the edges of the vent holes of the initial circular scheme and four optimal schemes, respectively. Table 5 lists the maximum, minimum, mean, and range (difference between the maximum and minimum values) values of the von Mises stresses along the edges of the vent holes of the five schemes. Combining Figures 9 and 10 and Table 5, we can see that: (1) Compared with the initial circular scheme, the maximum von Mises stresses of the optimal circular, elliptical, race-track, and four-arc schemes are significantly reduced by 13.469%, 18.906%, 18.079%, and 27.097%. (2) The minimum von Mises stresses of the optimal circular, elliptical, race-track, and four-arc schemes are increased by 4.528%, 21.319%, 22.973%, and 40.187%, respectively. (3) The mean von Mises stresses of the optimal circular, elliptical, race-track, and four-arc schemes only change slightly. (4) The range values of the optimal circular, elliptical, race-track, and four-arc schemes are significantly reduced by 27.445%, 50.147%, 49.962%, and 79.352%. (5) Obviously, the von Mises stress distribution along the edges of the vent holes of the four optimal schemes are more uniform than that of the initial circular scheme. Remarkably, the optimal four-arc scheme is the most uniform and performs the  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Schemes**} & \multirow{2}{*}{**Design Variables (mm)**} & \multirow{2}{*}{**Number of**} & \multirow{2}{*}{**Masimum Von**} & **Difference of** & **Maximum Von** \\  & & & & **Vent Holes** & **(MPa)** & **Mises Stress** \\ \hline Initial circular scheme & \(r_{\rm cr}\) & — & — & \(r_{\rm c0}\) & 225.360 & 75 & 1313.146 & 0 \\ \hline Optimal circular scheme & \(r_{\rm cr}\) & — & — & \(227.493\) & 124 & 1145.819 & \(-\)12.697\% \\ \hline Optimal elliptical scheme & \(a_{\rm e}\) & \(b_{\rm e}\) &  
best, followed by the optimal elliptical scheme, the optimal race-track scheme, and the optimal circular scheme. (6) The more uniform the von Mises stresses around the vent holes, the less the stress concentration and the smaller the maximum von Mises stress. Figure 9: Von Mises stress distribution of the initial circular and four optimal schemes. (**a**) Initial circular scheme; (**b**) Optimal circular scheme; (**c**) Optimal elliptical scheme; (**d**) Optimal race-track scheme; (**e**) Optimal four-arc scheme. Figure 11 shows the radial and circumferential stress distribution of the initial circular and four optimal schemes. Combining Figures 9 and 11, we can see that: (1) For all the five schemes, the high-radial-stress areas are distributed on the left and right sides of the hole, while the high-circumferential-stress areas are distributed on the bottom and top sides of the hole. (2) The maximum radial stress is much less than the maximum circumferential stress for the initial circular scheme. However, the maximum radial stresses are close to the maximum circumferential stresses for the four optimal schemes. (3) For all the five schemes, the maximum von Mises stresses are close to the larger values of the maximum radial stress and the maximum circumferential stress.