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Finite element (FE) models are widely used to capture the mechanical behavior of structures. Uncertainties in the underlying physics and unknown parameters of such models can heavily impact their performance. Thus, to satisfy high precision and reliability requirements, the performance of such models is often validated using experimental data. In such model updating processes, uncertainties in the incoming measurements should be accounted for, as well. In this context, Bayesian methods have been recognized as a powerful tool for addressing different types of uncertainties. Quasi-brittle materials subjected to damage pose a further challenge due to the increased uncertainty and complexity involved in modeling crack propagation effects. In this respect, techniques such as Digital Image Correlation (DIC) can provide full-field displacement measurements that are able to reflect the crack path up to a certain accuracy. In this study, DIC-based full field measurements are incorporated into a finite element model updating approach, to calibrate unknown/uncertain parameters of an ansatz constitutive model. In contrast to the standard FEMU, where measured displacements are compared to the displacements from the FE model response, in the force-version of the standard FEMU, termed FEMU-F [1], displacements are applied as Dirichlet constraints. This enables the evaluation of the internal forces, which are then compared to measured external forces, thus quantifying the fulfillment of the momentum balance equation as a metric for the model discrepancy. In the present work, the FEMU-F approach is further equipped with a Bayesian technique that accounts for uncertainties in the measured displacements, as well. Via this modification, displacements are treated as unknown variables to be subsequently identified, while they are allowed to deviate from the measured values up to a certain measurement accuracy. To be able to identify many unknown variables; including constitutive parameters and the aforementioned displacements, the Variational Bayesian technique proposed in [2] is utilized as an approximative technique. A numerical example of a three-point bending case study is presented first to demonstrate the effectiveness of the proposed approach. The parameters of a gradient-enhanced damage material model [4] are identified using noisy synthetic data, and the effect of measurement noise is studied. The ability of the suggested approach on identifying constitutive parameters is then validated using real experimental data from a three-point bending test from [3]. The full field displacements required as input to the inference setup are extracted through a digital image correlation (DIC) analysis of the provided raw images.
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