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

TL;DR

This paper investigates energy conservation and dissipation in time-integration methods for nonsmooth elastodynamics with contact, focusing on schemes like Moreau–Jean and nonsmooth generalized-α, and extending properties of classical methods to contact problems.

Contribution

It analyzes and extends energy properties of specific time-integration schemes for nonsmooth contact problems, including the Moreau–Jean and nonsmooth generalized-α schemes.

Findings

Moreau–Jean scheme conserves energy under certain conditions

Nonsmooth generalized–α scheme's properties are adapted for contact systems

Classical schemes like Newmark and HHT are extended to nonsmooth contact cases

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--$\alpha$ 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.