## 3 Methods  
### Integrated Design Optimization Method  
To reduce the computational cost of the FEM-based optimization, this paper selects the cyclically symmetric sector sub-model of the studied HPT sealing disk as the analytical model, as shown in Figure 2. The sub-model was proposed by the author, and its effectiveness has been validated [21]. To fully consider the pre-swirl system's aerodynamic and heat  
Figure 1: Von Mises stress distribution of the single-hole sector sub-model of the HPT sealing disk. transfer performance requirements, the vent holes' total area and radial position are strictly limited to a small variation range. In optimization, the vent hole is located in the blue area in Figure 2. To be noted, the area of the single vent hole is changed in optimization. The number of the vent holes \(n\) and the sector angle of the sub-model are changed accordingly. The change of the discrete quantity \(n\) may lead to some difficulties in geometric modelling, mesh generation, and finite element analysis. These difficulties are solved by using the developed integrated design optimization method. Figure 5 shows the flowchart of the 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 details are introduced as follows:  
1. Choose the shape of the vent hole as needed. This paper provides four different shape designs for the vent hole, namely the circular hole, the elliptical hole, the race-track hole, and the four-arc hole [21], as shown in Figure 3. The holes are described in a cylindrical coordinate system, where \(r\) represents the radial direction, \(\theta\) represents the circumferential direction, and \(z\) represents the axial direction. The shape and position of the circular hole are determined by the center \((r_{\rm e0},\theta_{\rm e0}=0,z_{\rm e0}=0)\) and the radius \(r_{\rm cr}\); while that of the elliptical hole are determined by the center \((r_{\rm e0},\theta_{\rm e0}=0,z_{\rm e0}=0)\), the axis \(a_{\rm e}\), and the axis \(b_{\rm e}\); that of the race-track hole are determined by the center \((r_{\rm e0},\theta_{\rm e0}=0,z_{\rm e0}=0)\), the side \(a_{\rm r}\), and the side \(b_{\rm r}\). The four-arc hole is biaxial symmetric and consists of four smoothly connected arcs. The large arc \(A_{\rm 1}\) is tangent to both the small arc \(A_{\rm s}\) and the basic circle \(A_{\rm b}\). The shape and position of the four-arc hole are determined by the center \((r_{\rm f0},\theta_{\rm f0}=0,z_{\rm f0}=0)\), the radius of the basic circle \(r_{\rm fb}\), the radius of the large arc \(r_{\rm fb}\), and the radius of the small arc \(r_{\rm fs}\). 2. Select the design variables according to the shape of the vent hole and determine their variation ranges. For the circular hole, the two parameters \(r_{\rm e0}\) and \(r_{\rm cr}\) are chosen as the design variables. For the elliptical hole, the three parameters \(r_{\rm e0}\), \(a_{\rm e}\), and \(b_{\rm e}\) are chosen as the design variables. For the race-track hole, the three parameters \(r_{\rm r0}\), \(a_{\rm r}\), and \(b_{\rm r}\) are chosen as the design variables. For the four-arc hole, the four parameters \(r_{\rm f0}\), \(r_{\rm fb}\), \(r_{\rm f1}\), and \(r_{\rm fs}\) are chosen as the design variables [21]. The variation ranges of these parameters are determined based on the upper and lower boundaries of the blue design region in Figure 2.
3. Calculate the area of the single vent hole \(s_{\rm s}\), the number of the vent holes \(n\), and the sector angle of the sub-model \(\frac{2\pi}{n}\). The areas of the single circular hole, the single elliptical hole, the single race-track hole, and the single four-arc hole are denoted as \(s_{\rm cs}\), \(s_{\rm ss}\), \(s_{\rm rs}\), and \(s_{\rm fs}\), respectively.