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This research report is devoted to the study of the conservation and thedissipation properties of the mechanical energy of several time--integrationmethods dedicated to the elasto--dynamics with unilateral contact. Given thatthe direct application of the standard schemes as the Newmark schemes or thegeneralized--$\alpha$ schemes leads to energy blow-up, we study two schemesdedicated to the time--integration of nonsmooth systems with contact: theMoreau--Jean scheme and the nonsmooth generalized--$\alpha$ scheme. The energyconservation and dissipation properties of the Moreau--Jean is firstly shown.In a second step, the nonsmooth generalized--$\alpha$ scheme is studied byadapting the previous works of Krenk and H{\o}gsberg in the context ofunilateral contact. Finally, the known properties of the Newmark and theHilber--Hughes--Taylor (HHT) scheme in the unconstrained case are extendedwithout any further assumptions to the case with contact.

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