Chamoin, Ludovic

Farahbakhsh, Sahar

Poncelet, M.

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Model updating with a Modified Dual Kalman Filter acting ondistributed strain measurementsS. Farahbakhsh∗, L. Chamoin∗,† and M. Poncelet∗∗ Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPSLaboratoire de Mécanique Paris-Saclay91190, Gif-sur-Yvette, France.† Institut Universitaire de France (IUF)1 rue Descartes, 75005 Paris, Francee-mail: sahar.farahbakhsh@ens-paris-saclay.fr, ludovic.chamoin@ens-paris-saclay.fr,martin.poncelet@ens-paris-saclay.frABSTRACTFollowing the advances in measurement technology and its vast availability, mechanical systems andstructures are increasingly equipped with sensors to obtain continuous information regarding thesystem state. Coupled with robust numerical models, this information can be used to build a numericaltwin of the structure that is linked to its physical twin via a feedback loop. This results in the conceptof Dynamic Data Driven Application Systems (DDDAS) that can predict and control the evolutionof the physical phenomena at stake on the structure, as well as dynamically updating the numericalmodel with the help of real-time measurements [1, 2].The physical evolution control is not addressed here, as the focus is mainly on the model updatingpart of the DDDAS process. This step requires data assimilation and sequentially solving a potentiallyill-posed inverse problem. A robust approach towards solving inverse problems regarding numericalmodels with experimental inputs is the modified Constitutive Relation Error (mCRE) [3]. One of thecritical features of this method is the distinction between reliable and unreliable information so thatonly reliable ones, such as equilibrium, known boundary conditions, and sensor positions, are stronglyimposed in the definition of the functional. In contrast, unreliable information, namely constitutiverelation, unknown boundary conditions, and sensor measurements, are dealt with in a more relaxedsense. This energy-based functional can be conceived as a least squares minimization problem onmeasurement error, regularized by a model error term, aka Constitutive Relation Error (CRE), whichaccounts for the physical sense of the minimization problem while improving the convexity of thefunctional. In other words, mCRE can be analyzed as a compromise between measurement and modelerror, and is adjusted with reference to the level of confidence one has in each of these terms. Hence,it yields impressive results even when confronting noisy and corrupted measurement data [4].Despite all its advantages, the mCRE method can become computationally expensive and might notbe suitable for real-time applications, including DDDAS. Furthermore, it does not comply with thesequential nature of DDDAS. Thus, it makes sense to use it within a sequential data assimilationframework such as Kalman Filters.Acquired from Bayesian inference, Kalman filtering is a widely known sequential data assimilationmethod for predicting evolving systems, which incrementally corrects the predictions through assimi-lated measurements [5]. Various extensions of the Kalman filter have been developed to improve themethod performance for different operations. Notably, its nonlinear extensions have been employedto solve inverse problems [6, 7]. Kalman filters are shown to be sensitive to noisy or corrupted mea-surements [8, 9]. Thereby, integrating the mCRE within this stochastic approach provides a cheap,robust and sequential model updating tool named Modified Dual Kalman Filter (MDKF) [10, 11].2In the present work, sensor data is collected from optic fiber sensors embedded inside a cement-basedmortar beam undergoing a quasi-static 4-point bending test. The optic fibers provide longitudinalstrain measures along the fiber with a spatial resolution as high as 1500 points per meter [12]. Thesemeasurements are then implemented in the MDKF approach to identify model parameters and detectpossible damage at each timestep. Furthermore, the CRE part of the functional is analyzed to detectzones with large modeling error and adapt the parametric space accordingly.This project has received funding from the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation program (grant agreement No. 101002857).REFERENCES[1] F. Darema, “Dynamic Data Driven Applications Systems: A New Paradigm for ApplicationSimulations and Measurements,” in Computational Science - ICCS 2004 (M. Bubak, G. D. vanAlbada, P. M. A. Sloot, and J. Dongarra, eds.), (Berlin, Heidelberg), pp. 662–669, Springer BerlinHeidelberg, 2004.[2] L. Chamoin, “Merging advanced sensing techniques and simulation tools for future structuralhealth monitoring technologies,” The Project Repository Journal, vol. 10, pp. 124–127, 09 2021.[3] P. Ladevèze and A. Chouaki, “Application of a posteriori error estimation for structural modelupdating,” Inverse problems, vol. 15, no. 1, p. 49, 1999. Publisher: IOP Publishing.[4] O. Allix, P. Feissel, and H. Minh Nguyen, “Identification strategy in the presence of corruptedmeasurements,” Engineering Computations, vol. 22, pp. 487–504, 2005. Publisher: Emerald.[5] R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” Journal of BasicEngineering, vol. 82, pp. 35–45, Mar. 1960.[6] M. Hoshiya and E. Saito, “Structural Identification by Extended Kalman Filter,” Journal ofEngineering Mechanics, vol. 110, pp. 1757–1770, Dec. 1984.[7] S. Mariani and A. Ghisi, “Unscented Kalman filtering for nonlinear structural dynamics,” Non-linear Dynamics, vol. 49, pp. 131–150, July 2007.[8] H.-M. Nguyen, O. Allix, and P. Feissel, “A robust identification strategy for rate-dependentmodels in dynamics,” Inverse Problems, vol. 24, p. 065006, Dec. 2008.[9] W. Li, S. Sun, Y. Jia, and J. Du, “Robust unscented Kalman filter with adaptation of processand measurement noise covariances,” Digital Signal Processing, vol. 48, pp. 93–103, Jan. 2016.[10] B. Marchand, L. Chamoin, and C. Rey, “Real-time updating of structural mechanics modelsusing Kalman filtering, modified constitutive relation error, and proper generalized decomposi-tion: Real-time updating of structural mechanics models,” International Journal for NumericalMethods in Engineering, vol. 107, pp. 786–810, Aug. 2016.[11] M. Diaz, P.-. Charbonnel, and L. Chamoin, “A new Kalman filter approach for structural param-eter tracking: Application to the monitoring of damaging structures tested on shaking-tables,”Mechanical Systems and Signal Processing, vol. 182, p. 109529, Jan. 2023.[12] L. Chamoin, S. Farahbakhsh, and M. Poncelet, “An educational review on distributed optic fibersensing based on Rayleigh backscattering for damage tracking and structural health monitoring,”Measurement Science and Technology, vol. 33, p. 124008, Dec. 2022.

Model updating with a Modified Dual Kalman Filter acting on distributed strain measurements

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Model updating with a Modified Dual Kalman Filter acting on distributed strain measurements

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