Convergence of Asynchronous Variational Integrators in Linear Elastodynamics

Within the setting of linear elastodynamics of simple bodies, we prove that the discrete action functional obtained by following the scheme of asynchronous variational integrators converges in time. The convergence in space is assured by standard arguments when the finite element mesh is progressively refined. Our strategy exploits directly the action functional. In particular, we show that, if the asynchronicity of time steps and nodal initial data satisfy a boundedness condition, any sequence of stationary points of the discrete action functional is pre-compact in the weak−∗ W 1,∞ topology and all its cluster points are stationary points for the continuous (in time) action. In this sense our proof is new with respect to existing ones. 1. A convergence theorem A simple elastic body undergoing infinitesimal deformations occupies a regular region B of the three-dimensional ambient space. The differentiable map (x, t) 7−→ u := u (x, t) ∈ R, with x ∈ B, t ∈ [t0, tf ], n = 1, 2, 3, represents the displacement field. The map (x, t) 7−→ ε := sym∇u (x, t) associates the standard measure ε of infinitesimal deformations to each point at a given instant. In linear elastic constitutive setting and infinitesimal deformation regime, the dynamics of a simple body is governed by the action functional (1.1) A (B, [t0, tf ] ;u) := ∫ tf