## 2 Methodology
### Data Set of Fatigue-Creep Lifetime  
In practical operation, strain plays a significant role in the fatigue life of components like turbine blades. [PERSON] [29] and [PERSON] [30] proposed the well-known Manson-Coffin Equation (7) to calculate the fatigue life. \[\frac{\Delta\varepsilon_{eq}}{2}=\frac{\sigma_{f}^{\prime}}{E}\Big{(}2N_{f} \Big{)}^{b}+\varepsilon_{f}^{\prime}\Big{(}2N_{f}\Big{)}^{c} \tag{7}\]  
where \(\frac{\Delta\varepsilon_{eq}}{2}\) means the amplitude of the Mises effect, \(E\) is the elastic modulus, \(\sigma_{f}^{\prime}\) and \(\varepsilon_{f}^{\prime}\) are the fatigue strength and plasticity coefficient, \(b\) and \(c\) are the fatigue strength and plasticity index, and \(N_{f}\) is the number of fatigue cycles. The Manson-Coffin equation is an empirical formula summarized on the basis of a large amount of test data. It lacks a specific mechanical interpretation and does not account for the influence of stress on fatigue life during operation. [PERSON] [31] took into account the influence of mean stress, thus making adjustments to the elastic deformation component of the Manson-Coffin equation, resulting in the formulation of Equation (8):  
\[\varepsilon_{a}=\frac{\sigma_{f}^{\prime}-\sigma_{m}}{\sigma_{f}^{\prime}} \Bigg{[}\frac{\sigma_{f}^{\prime}}{E}\Big{(}2N_{f}\Big{)}^{b}+\varepsilon_{f }^{\prime}\Big{(}2N_{f}\Big{)}^{c}\Bigg{]} \tag{8}\]  
where \(\varepsilon_{a}\) means the average strain amplitude and \(\sigma_{m}\) is the average stress. \begin{table}
\begin{tabular}{c c} \hline \hline
**No.** & **Reason for Check** \\ \hline
1 & Equivalent stress value is maximum \\
2 & Equivalent strain value is maximum \\
3 & Temperature is maximum \\
4 & Special geometric features \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Principles for the selection of checkpoints. Figure 5: Integration of fluid–thermal–structural finite element analysis. The turbine blade possesses a complex structure, experiencing multidirectional stress and strain. In this paper, the widely utilized Larson-Miller parameter method was employed for creep life prediction. Equation (9) involves calculating the stress endured by the high-temperature components and the ambient environmental temperature to estimate the creep life of the component. \[P(\sigma)=T(\text{lg}t_{r}+\text{C}) \tag{9}\]  
where \(P(\sigma)\) is a function related to the stress \(\sigma\). \(T\) represents the temperature, \(t_{r}\) is the creep failure time, and \(C\) is a material constant. A synergistic creep-fatigue interaction causes the population of creep voids to rise and the rate of fatigue fracture propagation to accelerate. Due to the interaction between the fatigue and creep, considering fatigue damage or creep damage in isolation is insufficient to describe the actual scenario. Therefore, the utilization of a time-to-failure fraction approach incorporating interaction damage factors becomes necessary for conducting damage calculations, given in Equation (10). \[D=\sum\frac{n_{i}}{N_{i}}+B\left(\sum\frac{n_{i}}{N_{i}}\times\sum\frac{t_{j}}{ T_{j}}\right)^{z}+\sum\frac{t_{j}}{T_{j}} \tag{10}\]  
where \(n_{i}\) and \(N_{i}\) are the number of cycles and the allowable number of cycles under the stress level, respectively, and \(t_{j}\) and \(T_{j}\) are the cumulative load holding time and lasting strength at the stress level, respectively. \(\frac{n_{i}}{N_{i}}\) is the total pure fatigue damage amount, \(\frac{t_{j}}{T_{j}}\) is the total net creep damage amount, and both B and \(Z\) are interaction factors of damage, ranging from 0 to 1. Various companies employ diverse metrics for lifetime prediction. GE Energy considers the maximum allowable uptime and the number of start-up operations. Siemens and the Russian Federation calculate the Equivalent Operating Hours (EOH) based on their maintenance systems. EOH refers to the total operating hours equal to a continuous load at rated capacity. EOH can provide reference information for equipment maintenance. Additionally, the actual RUL can also be obtained from the EOH results.