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Bayesian updating of constitutive laws for Finite Element simulation using full field measurements
(2023)
Developed finite element (FE) models have been recognized as powerful tools for predicting the mechanical behaviour of engineered systems. As a prerequisite, those models need to be improved with respect to various uncertainties; most notably, concerning underlying physics assumptions and unknown parameters. This is very often accomplished by comparing the performance of a model (e.g. the model response) against available data measured from real experiments. Another challenge emerges in doing that, however, which is accounting for uncertainties of measured data. Bayesian methods have been widely considered and utilized as a suitable approach for coping with and quantifying the aforementioned uncertainties. Phenomena like damage - in particular in quasi-brittle materials - introduce further uncertainties due to the complexity underlying the crack propagation of phenomenon. This implies that, the fitting of a numerical model and an associated constitute law that can adequately describe such effects is non-trivial. The standard approach of finite element model updating (FEMU) is therefore modified to account for tracking of the crack propagation, as recorded during an experiment under increasing loading, via full field displacement measurements. The latter are fed as Dirichlet constraints to an available finite element model, leading to the evaluation of force residuals, which quantifies the accuracy of the model. This approach - which is known as FEMU-F (force-version of the standard FEMU) [1] - is here further equipped with a Bayesian technique, which accounts for the measurement uncertainties in the full field displacement. This is achieved by penalizing the discrepancy between the measured displacements and the modeled Dirichlet constraints, where the latter are considered as further unknowns. We specifically employ the Variational Bayesian technique, proposed in [2], as an approximating tool for the estimation of posterior parameters, including displacement variables that are allowed to deviate from the measurements. A Markov chain Monte Carlo (MCMC) is also used for sampling the posterior distribution of the unknown model parameters. The model updating procedure is first demonstrated through a numerically simulated example of threepoint bending, where the parameters of a gradient-enhanced damage material model [4] are identified in accordance with synthetic noisy data (displacements and reaction forces). For the validation, experimental data from a three-point bending test are used, where full field displacements are collected through a digital image correlation (DIC) analysis (raw data taken from [3]). The data is then used for the parameter identification of a gradient damage constitutive law, which is employed as an ansatz model