  We propose a new approach for data-driven automated discovery of isotropic
hyperelastic constitutive laws. The approach is unsupervised, i.e., it requires
no stress data but only displacement and global force data, which are
realistically available through mechanical testing and digital image
correlation techniques; it delivers interpretable models, i.e., models that are
embodied by parsimonious mathematical expressions discovered through sparse
regression of a large catalogue of candidate functions; it is one-shot, i.e.,
discovery only needs one experiment - but can use more if available. The
problem of unsupervised discovery is solved by enforcing equilibrium
constraints in the bulk and at the loaded boundary of the domain. Sparsity of
the solution is achieved by l_p regularization combined with thresholding,
which calls for a non-linear optimization scheme. The ensuing fully automated
algorithm leverages physics-based constraints for the automatic determination
of the penalty parameter in the regularization term. Using numerically
generated data including artificial noise, we demonstrate the ability of the
approach to accurately discover five hyperelastic models of different
complexity. We also show that, if a "true" feature is missing in the function
library, the proposed approach is able to surrogate it in such a way that the
actual response is still accurately predicted.

Computational Engineering, Finance, and Science