## 4 Results and Discussion  
To investigate the engineering practicality of SDPSO and reduce the maximum von Mises stress of the initial circular vent holes, the optimizations of vent holes with four different shapes are conducted. Considering that the computational cost of the FEM-based optimization is expensive, the particle number and the maximum generation number of SDPSO are set to 20 and 50, respectively. Figure 8 shows the convergence history of the optimization objectives (maximum von Mises stress) of the four different shapes of vent holes. It can be seen that: (1) The optimizations for the four different shapes of vent holes converge quickly, and the four corresponding optimal schemes that satisfy the predefined constraints and possess the minimum \(\sigma_{\text{eq,max}}\) are obtained. That is, the proposed SDPSO is an efficient algorithm for the stress optimization of vent holes. It is suitable and valuable for practical engineering optimization problems with constraints. (2) Compared with the initial circular scheme, the maximum von Mises stresses of the four optimal schemes corresponding to the different shapes of the vent holes are significantly reduced. That is, the developed integrated design optimization method is effective and advanced for reducing the maximum von Mises stress around the vent holes. (3) Among the four optimal schemes, the maximum von Mises stress of the four-arc scheme is the minimum. That is, the four-arc hole has more tremendous advantages in reducing the maximum von Mises stress of the vent holes, followed by the elliptical hole, the race-track hole, and the circular hole. Table 4 shows the details of the initial circular scheme, optimal circular scheme, optimal elliptical scheme, optimal race-track scheme, and optimal four-arc scheme. From it, we can see that: (1) Compared with the initial circular scheme, the total area of the vent holes corresponding to the four optimal schemes are strictly limited to a small variation range, yet the number of the vent holes change significantly. Remarkably, the number of holes corresponding to the optimal circular scheme increases from 75 to 124. (2) The maximum von Mises stresses of the optimal circular, elliptical, race-track, and four-arc schemes are reduced from 1313.146 MPa to 1145.819 MPa, 1031.590 MPa, 1042.111 MPa, and 927.395 MPa, respectively. Remarkably, the maximum von Mises stress of the optimal four  
Figure 8: Convergence history of the optimization objectives (maximum von Mises stress) of the four different shapes of vent holes. arc scheme is reduced by 29.376%. Obviously, the four-arc hole performs best, followed by the elliptical hole, the race-track hole, and the circular hole. Figure 9 shows the von Mises stress distribution of the initial circular scheme, optimal circular scheme, optimal elliptical scheme, optimal race-track scheme, and optimal four-arc scheme. Combining Table 4 and Figure 9, we can see that: (1) The high-stress area of the initial circular scheme is only distributed on the lower sides of the hole. In contrast, the high-stress areas of the four optimal schemes are distributed around the holes. That is, the von Mises stress distribution of the four optimal schemes are more uniform than that of the initial circular scheme. (2) For the optimal circular scheme, the radius \(r_{\rm cr}\) is reduced from 5.800 to 4.507, and the radial position \(r_{\rm c0}\) is increased from 225.360 to 227.493. It increases the von Mises stress on the left and right sides of the hole and decreases the stress on the bottom and top sides. (3) For the optimal elliptical scheme, the axis \(a_{\rm e}\) is increased to 8.000 and the axis \(b_{\rm e}\) is reduced to 4.733. It also increases the von Mises stress on the left and right sides of the hole and decreases the stress on the bottom and top sides. (4) For the optimal race-track scheme, the side \(a_{\rm r}\) is increased to 1.923 and the side \(b_{\rm r}\) is reduced to 4.198. It significantly decreases the von Mises stress on the bottom and top sides of the hole.