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This is a python library for finite element (FE) modelling of static/quasi-static structural mechanics problems with legacy FEniCS, which contains the following main modules. Module structure: for defining a structural mechanics experiment, including geometry, mesh, boundary conditions (BCs), and time-varying loadings. The time is quasi-static, i.e. no dynamic (inertia) effects will be accounted for in the problem module as follows. Module material: for handling constitutive laws such as elasticity, gradient damage, plasticity, etc. Module problem: for establishing structural mechanics problems for desired structures and material laws coming from the two above modules, and solving the problems built up. These can be performed for two main cases: static (no time-evolution) that also includes homogenization, and quasi-static (QS).
A python implementation of an analytical variational Bayes algorithm of "Variational Bayesian inference for a nonlinear forward model", Chappell, Michael A., Adrian R. Groves, Brandon Whitcher, and Mark W. Woolrich, IEEE Transactions on Signal Processing 57, no. 1 (2008): 223-236, with an updated free energy equation to correctly capture the log evidence. The algorithm requires a user-defined model error allowing an arbitrary combination of custom forward models and measured data.
In this paper, we present a Bayesian framework for the identification of the parameters of nonlinear constitutive material laws using full-field displacement measurements. The concept of force-based Finite Element Model Updating (FEMU-F) is employed, which relies on the availability of measurable quantities such as displacements and external forces. The proposed approach particularly unfolds the advantage of FEMU-F, as opposed to the conventional FEMU, by directly incorporating information from full-field measured displacements into the model. This feature is well-suited for heterogeneous materials with softening, where the localization zone depends on the random microstructure. Besides, to account for uncertainties in the measured displacements, we treat displacements as additional unknown variables to be identified, alongside the constitutive parameters. A variational Bayesian scheme is then employed to identify these unknowns via approximate posteriors under the assumption of multivariate normal distributions. An optimization problem is then formulated and solved iteratively, aiming to minimize the discrepancy between true and approximate posteriors. The benefit of the proposed approach lies in the stochastic nature of the formulation, which allows to tackle uncertainties related to model parameters and measurement noise. We verify the efficacy of our proposed framework on two simulated examples using gradient damage model with a path-dependent nonlinear constitutive law. Based on a nonlocal equivalent strain norm, this constitutive model can simulate a localized damage zone representing softening and cracking. The first example illustrates an application of the FEMU-F approach to cracked structures including sensitivity studies related to measurement noise and parameters of the prior distributions. In this example, the variational Bayesian solver demonstrates a sizable advantage in terms of computational efficiency compared to a traditional least-square optimizer. The second example demonstrates a sub-domain analysis to tackle challenges associated with limited domain knowledge such as uncertain boundary conditions.
Mechanical structures are widely analysed by means of numerical simulators developed based on physical laws. In the context of structural mechanics, while the formulations of both kinetics and kinematics are transparently established in accordance with Newton’s laws and the compatibility condition of deformation, the constitutive relation often suffers from a considerable amount of uncertainties. Such uncertainties are mainly rooted in insufficient physics and imprecise assumptions and simplifications. In recent decades, Bayesian techniques have been utilized for coping with these uncertainties and for quantifying the level of confidence that may be attributed to the inferred model. Among various approaches, some of which are reviewed in [1], this work focuses on the parameter identification of a candidate parameterized physics-based constitutive model, on the basis of full-field measurements of displacements. These measurements are assumed to be obtained in the form of Digital Image Correlation (DIC) measurements, complemented with external force measurements at the structural level aggregated over certain sub-boundaries. The principal idea is to minimize a suitable objective function that measures the discrepancy between the parameterized model’s response and the measured quantities. This objective function is formulated in terms of the internal forces and the measured aggregated forces, and the associated Bayesian inverse problem is introduced. A variational Bayesian scheme, originally proposed in [2], is employed to compute the unknown posteriors considered as multivariate normal distributions. We verify our framework on two synthetic examples described by a gradient damage model [3] and an elastic law, with the former reflecting a path-dependent nonlinear constitutive law. The first example illustrates an application of the proposed procedure to cracked structures. In the second example, a regularizer, in terms of admissible stresses, is further incorporated to the identification procedure, offering robustness against contamination of measurements with noise.
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
Physics-based models of mechanical structures are widely adopted for assessing and predicting the behaviour of structures. In the context of structural mechanics, the constitutive law that describes the stress-strain relation forms an important modelling component, which suffers from a considerable amount of uncertainties. These uncertainties primarily arise due to the inherent simplifications and assumptions placed in favor of facilitating the modeling process. Bayesian techniques have been proven to be effective for tackling
uncertainties associated with the identification of material model parameters and quantifying the confidence level that can be associated with the placed modeling assumptions. We present a Bayesian framework for the identification of constitutive parameters of quasi-brittle materials suffering strain localization effects, via the use of full-field displacement measurements. The proposed framework explores the idea of force-based Finite Element Model Updating (FEMU-F), which relies on measured full-field displacements and aggregated forces. In particular, the scheme takes advantage of FEMU-F, in contrast to the conventional FEMU, where the information from full-field displacements is directly incorporated into the model. We also address the uncertainties involved in the measured displacements, by treating them as additional unknown variables to be identified, alongside the constitutive parameters. These unknown variables collectively form the inputs to a well-defined objective function, which forms the basis for the Bayesian inference problem. To efficiently solve this Bayesian problem, we employ a variational Bayesian scheme that relies on approximate posteriors represented as multivariate normal distributions. We demonstrate the proposed framework for the parameter identification of a nonlinear path-dependent gradient damage constitutive law, which exhibits strain localization and softening behaviour. The first example illustrates the effectiveness of the inference procedure, highlighting the advantage of FEMU-F in incorporating information about cracks. The second example demonstrates a sub-domain analysis suitable for inferring models with limited domain knowledge; e.g. with uncertain Dirichlet boundary conditions.
Numerical simulators, such as finite element models, have become increasingly capable of predicting the behaviour of structures and components owing to more sophisticated underlying mathematical models and advanced computing power. A common challenge lies, however, in calibrating these models in terms of their unknown/uncertain parameters. When measurements exist, this can be achieved by comparing the model response against measured data. Besides uncertain model parameters, phenomena like damage can give rise to further uncertainties; in particular, quasi-brittle materials, like concrete, experience damage in a heterogeneous manner due to various imperfections, e.g. in geometry and boundary conditions. This hardens an accurate prediction of the damaged behaviour of real structures that comprise such materials.
In this study, which draws from a data-driven approach, we use the force-version of the finite element model updating method (FEMU-F) to incorporate measured displacements into the identification of the damage parameters, in order to cope with heterogeneity. In this method, instead of conducting a forward evaluation of the model and comparing the model response (displacements) against the data, we impose displacements to the model and compare the resulting force residuals with measured reaction forces. To account for uncertainties in the measurement of displacements, we endow this approach with a penalty term, which reflects the discrepancy between measured and imposed displacements, where the latter is assumed as unknown random variables to be identified as well. A Variational Bayesian approach is used as an approximating tool for computing posterior parameters. The underlying damage model considered in this work is a gradient-enhanced damage model.
We first establish the identification procedure through two virtual examples, where synthetic data (displacements) are generated over a certain spatially-dense set of points over the domain. The procedure is then validated on an experimental case-study; namely a 3-point bending experiment with displacement measurements resulting from a digital image correlation (DIC) analysis.
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.
