## 5 Conclusions  
To reduce the maximum von Mises stress around the vent holes of a high-pressure turbine sealing disk, this paper developed an effective integrated design optimization method, which mainly includes four different shape designs (circular, elliptical, race-track, and four-arc) for holes, an updated self-developed modelling and meshing tool, an APDL-based strength analysis, and a self-proposed efficient SDPSO algorithm. The main idea of SDPSO is: (1) by evaluating an evolutionary factor and utilizing a probability transition matrix, a non-homogeneous Markov chain is determined and auto-updated in each generation; and (2) the evolutionary factor and the Markov chain are used to  
Figure 11: Radial and circumferential stress distribution of the initial circular and four optimal schemes. (**a**) Radial stress distribution of the initial circular scheme; (**b**) Circumferential stress distribution of the initial circular scheme; (**c**) Radial stress distribution of the optimal circular scheme; (**d**) Circumferential stress distribution of the optimal circular scheme; (**e**) Radial stress distribution of the optimal elliptical scheme; (**f**) Circumferential stress distribution of the optimal elliptical scheme; (**g**) Radial stress distribution of the optimal race-track scheme; (**h**) Circumferential stress distribution of the optimal race-track scheme; (**i**) Radial stress distribution of the optimal four-arc scheme; and (**j**) Circumferential stress distribution of the optimal race-track scheme. adaptively select the inertia weight, acceleration coefficients, and delayed information to adjust the particle's velocity. Two benchmark optimization problems with constraints were selected to compare the performance of SDPSO and two popular PSO algorithms (PSO-CK and SPSO). Furthermore, SDPSO was used for the stress optimization of vent holes with four different shapes. From the results of this study, some findings could be summarized as follows:  
1. For the two benchmark optimization problems, SDPSO is superior to PSO-CK and SPSO in optimization capability, numerical robustness, and convergence speed. SDPSO has a more significant potential to find the globally optimal solution. 2. The optimizations for the vent holes with four different shapes converge quickly. SDPSO is suitable and valuable for the stress optimization of vent holes. It is an efficient algorithm for practical engineering optimization problems with constraints. 3. Compared with the initial circular scheme, 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-arc scheme is reduced by 29.376%. It could be concluded that the developed integrated design optimization method is effective and advanced for reducing the maximum von Mises stress around the vent holes. Particularly, the adopted four-arc hole has more tremendous advantages in reducing the maximum von Mises stress, followed by the elliptical hole, the race-track hole, and the circular hole. 4. 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 best, followed by the optimal elliptical scheme, the optimal race-track scheme, and the optimal circular scheme. It could be concluded that the more uniform the von Mises stresses around the vent holes, the less the stress concentration and the smaller the maximum von Mises stress. 5. By changing the shapes and corresponding design parameters of the vent holes, the maximum radial stress is adjusted to be close to the maximum circumferential stress, and, accordingly, the maximum von Mises stresses at the left and right sides of the hole are adjusted to be approximately equal to those at the bottom and top sides.