# Fatigue Crack and Residual Life Prediction Based on an Adaptive Dynamic Bayesian Network
## 2 Principle and Methods
### Particle Filter (PF) Equation for Crack Propagation  
2.5.1. PF State Equation  
The final predicted residual life in the crack propagation model is obtained by the cyclic time at which the crack with initial length \(a_{0}\) reaches the critical length \(a_{c}\), which can be regarded as a function of \(a\). Therefore, the goal of particle filtering is to correct the theoretical value of the accumulated crack length output by the crack propagation formula with experimental data \(a_{\text{obs}}\). The estimated crack length in the model is the theoretical length obtained by the crack propagation formula, and \(\Theta\) refers to all parameters of the crack propagation formula. The process error is denoted as \(v_{a}\), which is zero-mean Gaussian noise from model error. The formula for the state equation can be expressed as follows:  
\[a_{t}=f(\Theta)+v_{a} \tag{16}\]  
This error \(v_{a}\) is added after each theoretical crack length at output by the crack propagation formula.  
2.5.2. PF Measurement Equation  
The uncertainty of crack observation data depends on the instrument error of the crack observation and the influence of the simplified conversion of irregular crack shape. It is assumed that the measurement error is zero-mean Gaussian white noise, that is, \(\varepsilon_{a}\sim N\big{(}0,\sigma_{a}^{2}\big{)}\). The observation data can also be written as \(a_{obs}=a+N\big{(}0,\sigma_{a}^{2}\big{)}\), which can prove the following:  
\[p(a_{obs}|a)=p(a|a_{obs})=\frac{1}{\sigma_{a}\sqrt{2\pi}}\exp\left(-\frac{(a-a _{obs})^{2}}{2\sigma_{a}^{2}}\right) \tag{17}\]  
Equation (17) is equivalent to  
\[p(a_{obs}|a)p(a)\propto p(a|a_{obs}) \tag{18}\]  
Among them, \(a\) is the real length of the crack, which is an unknown value. This error is included in the crack measurement data, that is, the real length of the crack \(a=a_{obs}-\varepsilon_{a}\).  
\(\varepsilon_{a}\) is the measurement error of the crack observation data, which reflects the error between the true value and the measured value of the crack. \(v_{a}\) is the model error of the crack propagation formula, which reflects the error between the true value and the simulated value of the crack. Particle filtering selects whether to rely more on measurement data or model data based on their magnitude.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Value** & **Parameter** \\ \hline \(\Theta\) & Crack propagation parameters \(\Theta\) \\ \(\Delta a\) & Crack increment \\ \(\Delta\sigma\) & Equivalent stress amplitude \\ \(P\) & Real load \\ \(P_{\text{obs}}\) & Observed load \\ \(a\) & Real crack length \\ \(a^{0}\) & Crack length before current time step \\ \(a_{\text{obs}}\) & Observed crack length \\ F & Crack shape factor \\ \hline \hline \end{tabular}
\end{table}
Table 1: Bayesian network node variables.  
The crack length data are obtained from fiber optic data and experimental annotations, which brings two sources of uncertainty: measurement error and data sparsity. Similar to the uncertainty of the load, the measurement error in the crack length data depends on the accuracy of the annotation and the processing of the non-uniform crack equivalent length on the layered surface, usually assumed to have a zero-mean Gaussian distribution \(\varepsilon_{a}\sim N\big{(}0,\sigma_{a}^{2}\big{)}\). This method can also handle other measurement error distributions. Since crack observation requires manual collection by instruments, crack length data are rarely applicable for every time step. Even if a data point is obtained after each observation and applied to the DBN for diagnosis and prediction, data sparsity introduces data uncertainty.