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Estimating operational loading conditions from wave measurements using Kalman filters

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DDE - The mathematical and statistical foundation of future data-driven engineering

In leading-edge industrial applications, assessing structural integrity is an important aspect of safety requirements. Structural Health Monitoring (SHM) proposes to use sensors and signal processing units in situ. One of the most attractive SHM techniques relies on ultrasonic guided waves. Operational conditions can change wave propagation and therefore affect the interpretation of recorded signals [1]. Modeling and simulation can be helpful tools for the design or the reliability assessment of SHM solutions. In [2], we have proposed a wave propagation model to take into account effects of operational conditions such as internal stresses caused by mechanical loading. In this work, we develop a strategy for estimating these load-induced (large) deformations from ultrasonic measurements. In the context of least-squares optimization, we minimize the difference between measurements and the observed direct model. Among the available methods there exist variational methods, such as Full Waveform Inversion [3], and sequential approaches, such as Kalman Filtering. As the linearized aspect of the direct model leads to an unwieldy tangent dynamics, tangent-free methods are preferable. After reducing the dimension of the parametric space by decomposing the deformation on selected eigenmodes of a static problem, we apply iteratively the Reduced-Order Unscented Kalman Filter [4] as estimation method. We will show how this method can be understood as a gradient free alternative to an iterated Levenberg-Marquardt based minimisation of the least squares functional. Moreover it allows to launch multiple direct solvers in parallel leading to an efficient exploration of the parametric space and sensitivity analysis w.r.t. extended sets of eigenmodes. We illustrate the proposed inversion strategy by applying it to realistic cases. [1] Gorgin, R. et al. 2020. “Environmental and Operational Conditions Effects on Lamb Wave Based Structural Health Monitoring Systems: A Review.” Ultrasonics.[2] Dalmora, A. et al. 2022. “A Generic Numerical Solver for Modeling the Influence of Stress Conditions on Guided Wave Propagation for SHM Applications.” 49th Annual Review of Progress in Quantitative Nondestructive Evaluation.[3] Virieux, J. et al. 2014. “An Introduction to Full Waveform Inversion.” In Encyclopedia of Exploration Geophysics, Geophysical References Series.[4] Moireau, P. and Chapelle, D. 2011. “Reduced-Order Unscented Kalman Filtering with Application to Parameter Identification in Large-Dimensional Systems.” ESAIM : Control, Optimisation and Calculus of Variations. André Luiz Dalmora1,2,3, Alexandre Imperiale1, Sébastien Imperiale2,3, Philippe Moireau2,3 1 Université Paris-Saclay, CEA , List, F-91120, Palaiseau, France2 Project-Team M3DISIM , Inria Saclay-Ile-de-France, Inria, 91128 Palaiseau, France3 LMS , École Polytechnique, CNRS , Institut Polytechnique de Paris, 91128 Palaiseau, France    

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