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

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--$\alpha$ 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--$\alpha$ scheme. The energy conservation and dissipation properties of the Moreau--Jean is firstly shown. In a second step, the nonsmooth generalized--$\alpha$ scheme is studied by adapting the previous works of Krenk and H{\o}gsberg 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.