Manifold embedding data-driven mechanics

Abstract

This article introduces a manifold embedding data-driven paradigm to solve small- and finite-strain elasticity problems without a conventional constitutive law. This formulation follows the classical data-driven paradigm by seeking the solution that obeys the balance of linear momentum and compatibility conditions while remaining consistent with the material data by minimizing a distance measure. Our key point of departure is the introduction of a global manifold embedding as a means to learn the geometrical trend of the constitutive data mathematically represented by a smooth manifold. By training an invertible neural network to embed the data of an underlying constitutive manifold onto a Euclidean space, we reformulate the local distance-minimization problem that requires a computationally intensive combinatorial search to identify the optimal data points closest to the conservation law with a cost-efficient projection step. Meanwhile, numerical experiments performed on path-independent elastic materials of different material symmetries suggest that the geometrical inductive bias learned by the neural network is helpful to ensure more consistent predictions when dealing with data sets of limited sizes or those with missing data.