Mesh d-refinement: A data-based computational framework to account for complex material response

Model-free data-driven computational mechanics (DDCM) is a new paradigm for simulations in solid mechanics. The modeling step associated to the definition of a material constitutive law is circumvented through the introduction of an abstract phase space in which, following a pre-defined rule, physically-admissible states are matched to observed material response data (coming from either experiments or lower-scale simulations). In terms of computational resources, the search procedure that performs these matches is the most onerous step in the algorithm. One of the main advantages of DDCM is the fact that it avoids regression-based, bias-prone constitutive modeling. However, many materials do display a simple linear response in the small-strain regime while also presenting complex behavior after a certain deformation threshold. Motivated by this fact, we present a novel refinement technique that turns regular elements (equipped with a linear-elastic constitutive law) into data-driven ones if they are expected to surpass the threshold known to trigger material non-linear response. We term this technique "data refinement'', "d-refinement'' for short. It works both with data-driven elements based on either DDCM or strain-stress relations learned from data using neural networks. Starting from an initially regular FEM mesh, the proposed algorithm detects where the refinement is needed and iterates until all elements presumed to display non-linearity become data-driven ones. Insertion criteria are discussed. The scheme is well-suited for simulations that feature non-linear response in relatively small portions of the domain while the rest remains linear-elastic. The method is validated against a traditional incremental solver (i.e., Newton-Raphson method) and we show that the d-refinement framework can outperform it in terms of speed at no loss of accuracy. We provide an application that showcases the advantage of the new method: bridging scales in architected metamaterials. For this application, we also succinctly outline how d-refinement can be used in conjunction with a neural network trained on microscale data.