## 3 Methods
### Integrated Design Optimization Method
Their analytical expressions are derived for the convenience of calculating the number of the vent holes. \[\left\{ \begin{array}{l}s_{\rm cs}=\pi r_{\rm cr}^{2}\\ s_{\rm es}=\pi a_{\rm e}b_{\rm e}\\ s_{\rm rs}=4a_{\rm r}b_{\rm r}+\pi b_{\rm r}^{2}\\ s_{\rm fs}=2\Big{(}\arccos\frac{r_{\rm e}-r_{\rm fin}}{r_{\rm e}-r_{\rm fin}} \Big{)}\times\big{(}r_{\rm f1}^{2}-r_{\rm fs}^{2}\big{)}+\pi r_{\rm fs}^{2}-2 (r_{\rm f1}-r_{\rm fb})\times\sqrt{(r_{\rm f1}-r_{\rm fs})^{2}-(r_{\rm f1}-r_ {\rm fb})^{2}}\end{array} \right.\] (1) Considering the total area of the newly designed vent holes (namely \(n\times s_{\rm s}\)) is limited between \(s_{\rm t,low}\) and \(s_{\rm t,up}\), the number of the vent holes \(n\) is calculated. \[n=\text{ceil}\bigg{(}\frac{s_{\rm t,low}}{s_{\rm s}}\bigg{)}\] (2) where \(\text{ceil}(\bullet)\) denotes a function that rounds the variable to the nearest integer greater than or equal to it. Accordingly, the limitation to the total area could be transformed as follows: \[n\times s_{\rm s}\leq s_{\rm t,up}\] (3)Then, the sector angle \(\frac{2\pi}{n}\) is calculated and used to determine the geometry of the single-hole sector sub-model in combination with the design variables of the vent hole. 4. Automatically rebuild the geometry of the sector sub-model and generate the high-quality hexahedral mesh by using an updated self-developed modelling and meshing tool. The automatic geometry reconstruction and the high-quality mesh regeneration are two core technologies of the FEM-based optimization. To ensure high robustness, flexibility, and computational accuracy, the author develops a convenient and unified modelling and meshing tool by using Tck/Tk Scripts. Based on the original version [21], the current tool is extended to deal with four different shape designs for the vent holes. The geometry of the whole-ring sub-model with no vent hole is constructed in advance. Once the shape and parameters of the vent holes are determined, the single-hole sector sub-model will be built and partitioned into four revolving or extruding bodies. The 2D high-quality quadrilateral elements are then generated on the source faces of the regular bodies. The mapping/sweeping method is used to generate the 3D high-quality hexahedral mesh for the four regular bodies by mapping or sweeping the quadrangle mesh to the target faces. Particularly, the \"trajectory mesh\" along the sweeping direction is constructed to solve the mesh compatibility problem between neighboring bodies. Figure 4 shows the automatically generated high-quality hexahedral mesh models for the single-hole sector sub-model with the different shapes of the vent holes. We can see that the mesh quality in the areas around the vent holes of the different shapes are consistent and high. It can avoid the influences of mesh schemes on the stress distribution around the vent holes with different shapes. 5. Automatically conduct the ANSYS APDL-based strength analysis and extract the maximum von Mises stress \(\sigma_{\text{eq,max}}\). FGH96 is selected as the material, which has excellent thermal stability and endurance strength at high temperatures. The 3D 20-node SOLID 186 is selected as the computational element type of the sector sub-model. To impose the displacement boundary conditions, the coordinate systems of all the nodes of the sector sub-model are rotated into the cylindrical coordinate systems. The cyclic symmetry conditions are imposed on both sides of the sector sub-model. According to the variable dimension sub-model method developed by the author [21], the displacement boundary conditions are applied to the cut-boundary locations of the sub-model. The centrifugal, thermal, and aerodynamic loads are applied according to the actual working environment. Notably, the temperature field and the aerodynamic pressure are determined according to the analysis results of the heat transfer and aerodynamic disciplines. 6. Check whether the optimization is converged.