Variational r‐adaption in elastodynamics

Abstract

AbstractWe develop a variational r‐adaptive finite element framework for solid dynamic applications and explore its conceptual links with the theory of dynamic configurational forces. The central idea underlying the proposed approach is to allow Hamilton's principle of stationary action to determine jointly the evolution of the displacement field and the discretization of the reference configuration of the body. This is accomplished by rendering the action stationary with respect to the material and spatial nodal coordinates simultaneously. However, we find that a naive consistent Galerkin discretization of the action leads to intrinsically unstable solutions. Remarkably, we also find that this unstable behavior is eliminated when a mixed, multifield version of Hamilton's principle is adopted. Additional features of the proposed numerical implementation include the use of uncoupled space and time discretizations; the use of independent space interpolations for velocities and deformations; the application of these interpolations over a continuously varying adaptive mesh; and the application of mixed variational integrators with independent time interpolations for velocities and nodal parameters. The accuracy, robustness and versatility of the approach are assessed and demonstrated by way of convergence tests and selected examples. Copyright © 2007 John Wiley & Sons, Ltd.