Bayesian-EUCLID: discovering hyperelastic material laws with\n uncertainties

Abstract

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