Bayesian-EUCLID: discovering hyperelastic material laws with uncertainties

Abstract

Within the scope of our recent approach for Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID), we propose an unsupervised Bayesian learning framework for discovery of parsimonious and interpretable constitutive laws with quantifiable uncertainties. As in deterministic EUCLID, we do not resort to stress data, but only to realistically measurable full-field displacement and global reaction force data; as opposed to calibration of an a priori assumed model, we start with a constitutive model ansatz based on a large catalog of candidate functional features; we leverage domain knowledge by including features based on existing, both physics-based and phenomenological, constitutive models. In the new Bayesian-EUCLID approach, we use a hierarchical Bayesian model with sparsitypromoting priors and Monte Carlo sampling to efficiently solve the parsimonious model selection task and discover physically consistent constitutive equations in the form of multivariate multi-modal probabilistic distributions. We demonstrate the ability to accurately and efficiently recover isotropic and anisotropic hyperelastic models like the Neo-Hookean, Isihara, Gent-Thomas, Arruda-Boyce, Ogden, and Holzapfel models in both elastostatics and elastodynamics. The discovered constitutive models are reliable under both epistemic uncertainties -i.e. uncertainties on the true features of the constitutive catalog -and aleatoric uncertainties -which arise from the noise in the displacement field data, and are automatically estimated by the hierarchical Bayesian model.

Integrated digital image correlation for micro-mechanical parameter identification in multiscale experiments

“…Identification of micromechanical parameters within the multiscale regime has been experimentally performed by Fedele et al (2006), where differences between the experimentally observed homogenized stress field and predictions from numerical simulations carried out on a representative volume were minimized, assuming certain micromechanical parameters to be known a priori. Finally, parameter identification within the probabilistic realm, e.g., Rappel et al (2020), Hu et al (2016), Oskay and Fish (2008), and Rosić et al (2013), and machine learning realm, e.g., Flaschel et al (2021Flaschel et al ( , 2022, Joshi et al (2022), have been discussed as well.…”

Learning hyperelastic anisotropy from data via a tensor basis neural network

“…Later, De Lorentzis and co-workers [13,14] utilized sparse regression to discover interpretable constitutive laws for a wide array of material classes. An interesting extension to this work was the development of an unsupervised Bayesian framework for discovering hyperelasticity models which accounts for uncertainty [15]. Neural networks and Gaussian process models have been widely employed as replacements for human-selected, traditional model forms.…”

NN-EUCLID: deep-learning hyperelasticity without stress data

“…In the (material) model-free paradigm proposed by Kirchdoerfer and Ortiz (2016) and first formulated for elasticity, a material's state of deformation is directly mapped to a stress state closest to the stress-strain pair available in the dataset (subject to physical compatibility constraints) (Kirchdoerfer and Ortiz, 2016;Ibañez et al, 2017;Kirchdoerfer and Ortiz, 2017;Conti et al, 2018;Nguyen and Keip, 2018;Eggersmann et al, 2019;Carrara et al, 2020;Karapiperis et al, 2021). Alternative approaches keep the concept of a material model and surrogate it by learning an approximate mapping between the strains and stresses (referring again to the easiest case of elasticity) using, e.g., sparse regression with feature engineering (Flaschel et al, 2021(Flaschel et al, , 2022Joshi et al, 2022;Wang et al, 2021), manifold learning methods and polynomial approximations (Ibañez et al, 2018(Ibañez et al, , 2017González et al, 2019b), Gaussian process regression (Rocha et al, 2021;Fuhg et al, 2022), and artificial neural networks (NNs) (Ghaboussi et al, 1991;Fernández et al, 2021;Klein et al, 2022;Vlassis and Sun, 2021;Kumar et al, 2020;Bastek et al, 2022;Zheng et al, 2021;Mozaffar et al, 2019;Bonatti and Mohr, 2021;Vlassis et al, 2020;Kumar and Kochmann, 2021;As'ad et al, 2022;Liang et al, 2022), with the list being roughly in order of decreasing physical interpretability and increasing approximation power. Further hybrid approaches use data to construct automatic corrections to existing models (Ibáñez et al, 2019;González et al, 2019a).…”

“…To address the aforementioned limitations of classical approaches and supervised data-driven methods, the authors recently proposed a novel framework denoted as Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID) (Flaschel et al, 2021(Flaschel et al, , 2022Joshi et al, 2022). The method is unsupervised, i.e., it requires no stress data but only global reaction forces and full-field displacement data realistically obtainable through, e.g., DIC or DVC; it uses sparse regression over a large catalog of candidate functions to deliver interpretable constitutive laws given by parsimonious mathematical expressions.…”

“…EUCLID departs from data-driven learning to achieve physics-driven learning, i.e., the need for stress labels is circumvented by enforcing that the full-field displacement data satisfy the conservation of linear momentum -possibly along with additional physics-based requirements. We recently demonstrated the performance of EUCLID in isotropic hyperelasticity (Flaschel et al, 2021), rate-independent plasticity (Flaschel et al, 2022), anisotropic hyperelasticity and elastodynamics (Joshi et al, 2022). In the latter contribution, we developed Bayesian-EUCLID, a probabilistic framework based on Bayesian learning which automatically discovers material models along with the related uncertainties.…”