The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics

Abstract

In the absence of external loads or in the presence of symmetries (i.e., translational and rotational invariance) the nonlinear dynamics of continuum systems preserves the total linear and the total angular momentum. Furthermore, under assumption met by all classical models, the internal dissipation in the system is non-negative. The goal of this work is the systematic design of conserving algorithms that preserve exactly the conservation laws of momentum and inherit the property of positive dissipation forany step-size. In particular, within the specific context of elastodynamics, a second order accurate algorithm is presented that exhibits exact conservation of both total (linear and angular) momentum and total energy. This scheme is shown to be amenable to a completely straightforward (Galerkin) finite element implementation and ideally suited for long-term/large-scale simulations. The excellent performance of the method relative to conventional time-integrators is conclusively demonstrated in numerical simulations exhibiting large strains coupled with a large overall rigid motion.