# Fatigue Crack and Residual Life Prediction Based on an Adaptive Dynamic Bayesian Network
## 1 Introduction
Various types of sensors are utilized within the monitoring data collection module of ADT systems for the identification of damage, including ultrasonic sensors, X-ray devices, piezoelectric sensors, fiber Bragg gratings, and distributed fiber optics [11; 12; 13; 14; 15]. These sophisticated sensing devices enable the timely detection of crack damages, thus providing a robust guarantee for the safety assessment of structures. However, due to uncertainties associated with, e.g., material geometry, internal defects, irregular damages, and mechanical environments, the prediction of damage evolution in aerospace structures exhibits unsatisfactory accuracy. Therefore, it is necessary to establish a probabilistic model that integrates diagnosis and prediction, which can reduce the uncertainty of time-independent state variables through continuous observation (diagnosis) and probabilistically predicts the evolution of future crack damages (prognosis), thereby obtaining a more accurate assessment of the structural health state with multiple sources of uncertainty. Within the digital twin framework, the Dynamic Bayesian Network (DBN) is among the most widely applied probabilistic methods [16; 17]. Its robust system reliability framework makes it a primary method for representing and managing uncertainties in physical models (related to geometry, materials, connection stiffness, etc.) in digital twins. DBNs excel at providing direct representations of complex systems and are highly effective in identifying inter-component relationships, making them suitable for monitoring, diagnosis, and prediction tasks in uncertain environments. A unique advantage of DBNs is their ability to integrate a wide range of information sources, including experimental data, historical records, and expert opinions, which is particularly valuable in fault-tolerant systems where data are often sparse and varied. Additionally, DBNs stand out for their capability to model the temporal dynamics of cascading events, offering a more accurate description of how systems evolve over time. This enhances prediction accuracy and aids in better decision-making, resource planning, and allocation in complex and evolving systems. [PERSON] et al. [18] proposed an object-oriented modeling approach, building a DBN for the entire aviation system health in a bottom-up manner based on Bayesian theory, and validated its effectiveness and robustness through experiments on a specific aviation fuel delivery system. [PERSON] et al. [19] introduced a digital twin health monitoring method based on non-parametric Bayesian networks, combining an improved Gaussian particle filter and a Dirichlet process mixture model for real-time model updating, significantly enhancing monitoring accuracy. [PERSON] et al. [20] developed an intelligent task planning method based on digital twins, employing a DBN to predict the probability distribution of crack propagation, considering the quantification of uncertainties and encompassing damage diagnosis, prediction, and task optimization, to optimize fatigue crack growth and maintenance intervals. [PERSON] et al. [21] proposed a DBN framework for the online integration of health assessment information from empirical crack growth models, structural health monitoring, and regular inspections, updating crack size distribution and model parameters for prediction. [PERSON] et al. [22] introduced a framework utilizing a DBN with MCMC for predicting and updating crack length in fatigued structural components, including parameter probability density identification and updating. [PERSON] et al. [23] proposed a Bayesian method to estimate initial crack length distribution for PRA of repaired structures, improving parameter selection and analyzing KT-1 aircraft repair risk. However, Bayesian networks have some defects. For instance, they are highly dependent on prior probabilities, and prior probabilities often depend on assumptions, leading to poor prediction results.