## 4 Reliability Analysis of Compressor Disks
### Simulation-Based Uncertainty Analysis of Compressor Disks  
Considering the periodic symmetry of the structure, 1/36 th of the wheel sector was selected for analysis, and a 20-node hexahedral mesh was established, with the wheel material as GH4169 and a density of 8240 kg/m\({}^{3}\). The blade material was DD6, and the equivalent pressure on the contact surface of the tongue and groove was 426.91 MPa during rotation. The deterministic results of the finite-element analysis are shown in Figure 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Bayes Method Prediction** & **Simulation Prediction** & **Fusion Method Predicted** & **True Results** \\  & **Results** & **Results** & **Results** & **True Results** \\ \hline Value & 6000 & 5994 & 6042 & 6385 \\ Error & 385 & 391 & 343 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Life reliability prediction results. Figure 2: Schematic of the compressor disk (1/36 th of the wheel sector). The uncertainty of the load and geometric input variables and the uncertainty of the life model were considered in the reliability analysis. Probability distributions were used to characterize the uncertainties of the speed and temperature in the load, and \(\mathrm{r}_{1}\) (inner diameter of the wheel disk), \(\mathrm{r}_{2}\) (outer diameter of the wheel disk), \(\mathrm{w}_{1}\) (width of the wheel disk), \(\mathrm{w}_{7}\) (width of the tenon slot), and \(\mathrm{ra}_{7}\) (thickness of the tenon slot) were selected as geometric uncertainty parameters via a sensitivity analysis [29], as shown in Figure 5. To obtain the distributions, measurements of 10 actual compressor disks were performed, followed by a _K-S_ test [30]. The Ramberg-Osgood equation was used to describe the stress-strain relationship [31], and a modified Smith-Watson-Topper model based on the theory of critical distance [32] was used for life prediction. They are expressed as follows:  
\[\frac{\Delta\varepsilon_{t}}{2}=\frac{\Delta\varepsilon_{e}}{2}+\frac{\Delta \varepsilon_{p}}{2}=\frac{\Delta\sigma}{2E}+\left(\frac{\Delta\sigma}{2K^{ \prime}}\right)^{1/n^{\prime}} \tag{20}\]  
\[\left(\sigma_{max}\frac{\Delta\varepsilon_{t}}{2}\right)\bigg{|}_{r=L}=\frac{ \left(\sigma^{\prime}_{f}\right)^{2}}{E}\left(2N_{f}\right)^{2b}+\sigma^{ \prime}_{f}\varepsilon^{\prime}_{f}\left(2N_{f}\right)^{b+c} \tag{21}\]  
where \(\Delta\varepsilon_{t}\) represents the total strain range; \(\Delta\varepsilon_{e}\) and \(\Delta\varepsilon_{p}\) represent the elastic and plastic strain ranges, respectively; \(\Delta\sigma\) represents the stress range; \(\sigma_{max}\) represents the maximum stress; \(N_{f}\) represents the LCF life; and \(E\) represents elastic modulus of the material. The parameters \(\sigma^{\prime}_{f},\;\;b,\;e^{\prime}_{f}\), and \(c\) are the fatigue strength coefficient, fatigue strength index, fatigue ductility coefficient, and fatigue ductility index, respectively, which can be obtained from material fatigue tests. Figure 4: Equivalent stress distribution of the compressor disk. Figure 5: Parametric model of the compressor disk. The \(r\) represents the distance from the edge of the notch to a position on the maximum stress gradient line, and \(L\) represents the critical distance. These two variables have a power-function relationship with the life:  
\[L=AN_{f}^{B} \tag{22}\]  
To evaluate the uncertainty of the model parameters, 42 test specimens of the actual compressor disk were used for fatigue tests, as shown in Figure 6. According to [PERSON] theorem, the distribution type and distribution parameters of the model parameters were obtained using the Markov chain Monte Carlo method [29].