## 4 Example
### Structure of the Aero-Engine Turbine Casing  
In this section, reliability analysis of the aero-engine turbine casing is used as an example. The failure probability is calculated using the PSO-BPNN method to validate the engineering feasibility and practical applicability of the proposed method. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Failure Probability P\({}_{f}\)** & **Number of Samples \(N\)** & **Error** \\ \hline MCS & \(1.8619\times 10^{-2}\) & \(1\times 10^{5}\) & - \\ Proposed & \(1.7913\times 10^{-2}\) & \(1.5\times 10^{3}\) & 3.79\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Failure probability calculated by two methods. Figure 4: Cumulative distribution function of output Y obtained by two methods. The turbine casing functions as the foundation for the entire turbofan engine. Most turbine casings are thin-walled welded structures featuring localized structural characteristics, such as weld seams and transition fillets. These regions often display abrupt geometric changes, leading to stress concentrations and an increased risk of failure under loading conditions. The occurrence of a crack in these areas could critically compromise the flight safety of the aircraft. In this paper, a reliability analysis is performed on the turbine casing structure of a specific type of turbofan engine. ANSYS finite element software is employed to develop a stress-response relationship model for the turbine casing structure. (1) Simplified geometric modeling of the turbine casing. For the geometric modeling and material property definition of a certain type of turbofan engine turbine casing, the influence of the support on the mounting edge is ignored, simplifying it into an axisymmetric model. The stiffening ribs on the outer surface of the turbine casing are considered, and the mounting holes on the mounting edge of the turbine casing are appropriately simplified. The simplified turbine casing model is shown in Figure 5.  
(2) Load and constraint application. During engine operation, stages such as startup, shutdown, and working condition transitions occur. These result in changes in internal gas temperature, which induce thermal stress in the structure. The turbine casing is subjected to the combined effects of mechanical stress and thermal stress, necessitating a thermo-mechanical coupling analysis. Considering that one-way coupling analysis is more efficient and can maintain an adequate level of accuracy for relatively simple structural problems, this paper employs the one-way coupling method to perform the thermo-mechanical coupling analysis on the turbine casing. The internal-external pressure difference (\(\Delta P\)) is the primary load responsible for the mechanical stress in the turbine casing, contributing approximately 90% of the total stress level. Therefore, in the thermo-mechanical coupling analysis, the mechanical load is simplified to only account for the internal-external pressure difference. The load spectrum of the turbine casing is depicted in Figure 6. The boundary condition is defined as a fixed support applied at the rear mounting edge of the turbine casing, as illustrated in Figure 7.  
Figure 5: Schematic diagram of the turbine casing geometric model. The application of mechanical loads and boundary conditions on the turbine casing structure is illustrated in Figure 8.  
Figure 8: Schematic diagram of mechanical loads and constraints of turbine casing structure. Figure 6: Schematic diagram of the turbine casing load chart. Figure 7: Schematic diagram of the turbine casing forces and constraints. In this study, six random variables are considered, including the material yield strength \(\sigma_{P0.1}(\text{MPa})\), elastic modulus \(E(\text{GPa})\), Poisson's ratio \(\mu\), linear expansion coefficient \(\alpha(10^{-6}/^{\circ}\text{C})\), external pressure \(P_{1}(\text{MPa})\), and internal pressure \(P_{2}(\text{MPa})\). These variables are treated as random inputs to capture their inherent uncertainties, while other parameters are treated as deterministic.