Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact

Abstract

It is now well established that discrete energy conservation/dissipation plays a key-role for the unconditional stability of time integration schemes in nonlinear elastodynamics. In this paper, from a rigorous conservation analysis of the Hilber–Hughes–Taylor time integration scheme [H. Hilber, T. Hughes, R. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Engrg. Struct. Dynam. 5 (1977) 283–292], we propose an original way of introducing a controllable energy dissipation while conserving momenta in conservative strategies like [J. Simo, N. Tarnow, The discrete energy–momentum method: conserving algorithms for nonlinear elastodynamics, Z. Angew. Math. Phys. 43 (1992) 757–792]. Moreover, we extend the technique proposed in [O. Gonzalez, Exact energy and momentum conserving algorithms for general models in nonlinear elasticity, Comput. Methods Appl. Mech. Engrg. 190 (13–14) (2000) 1763–1783] to provide energy-controlling time integration schemes for frictionless contact problems enforcing the standard Kuhn–Tucker conditions at time discretization points. We also extend this technique to viscoelastic models. Numerical tests involving the impact of incompressible elastic or viscoelastic bodies in large deformation are proposed to confirm the theoretical analysis.