Real-time inverse crack tracking in uncertain microstructures using PGD-based model reduction and extended Kalman filtering

Despite recent advances in sensing techniques and rapid growth of computational mechanics, it remains a challenge to monitor mechanical structures in real-time and evaluate their sustainability. In the present work, crack propagation is investigated by dynamically connecting experimental information and physics-based models. In order to perform on-the-fly characterization of the material state, we combine reduced order modeling strategies with data assimilation tools. First, proper generalized decomposition (PGD)-based model reduction is used to effectively solve multi-parametric problems associated with crack propagation, in which crack position and material microstructures are parametrized. The PGD framework is then integrated with an extended Kalman filter (EKF) algorithm in order to perform fast state observation, inverse analysis, and uncertainty quantification from noisy sensor data collected sequentially in time. The computation of the sensitivity matrices, which connect the output variable and model parameters, is a well-known difficulty in EKF; we show here that the developed PGD approximation provides a very convenient way of getting these matrices. The proposed method is implemented and validated on a number of numerical cases with varying degrees of complexity in terms of recovered microstructure. Additionally, the optimal sensor placement for parameter estimation is investigated.