Energy conservation and dissipation properties of time-integration methods for the nonsmooth elastodynamics with contact

Abstract

This research report is devoted to the study of the conservation and the dissipation properties of the mechanical energy of several time--integration methods dedicated to the elasto--dynamics with unilateral contact. Given that the direct application of the standard schemes as the Newmark schemes or the generalized--$α$ schemes leads to energy blow-up, we study two schemes dedicated to the time--integration of nonsmooth systems with contact: the Moreau--Jean scheme and the nonsmooth generalized--$α$ scheme. The energy conservation and dissipation properties of the Moreau--Jean is firstly shown. In a second step, the nonsmooth generalized--$α$ scheme is studied by adapting the previous works of Krenk and Høgsberg in the context of unilateral contact. Finally, the known properties of the Newmark and the Hilber--Hughes--Taylor (HHT) scheme in the unconstrained case are extended without any further assumptions to the case with contact.