Finite element solver for data-driven finite strain elasticity

Abstract

A nominal finite element solver is proposed for data-driven finite strain elasticity. It bypasses the need for a constitutive model by considering a database of deformation gradient/first Piola–Kirchhoff stress tensors pairs. The boundary value problem is reformulated as the constrained minimization problem of the distance between (i) the mechanical states, i.e. strain–stress, in the body and (ii) the material states coming from the database. The corresponding constraints are of two types: kinematical, i.e. displacement–strain relation, and mechanical, i.e. conservation linear and angular momenta. The solver uses alternated minimization: the material states are determined from a local search in the database using an efficient tree-based nearest neighbor search algorithm, and the mechanical states result from a standard constrained minimization addressed with an augmented Lagrangian approach. The performance of the solver is demonstrated by means of 2D sanity check examples: the data-driven solution converges to the classical finite element solution when the material database increasingly approximates the constitutive model. In addition, we demonstrate that the balance of angular momentum, which was classically not taken into account in previous data-driven studies, must be enforced as a constraint to ensure the convergence of the method.