Moreau-type integrators based on the time finite element discretization of the virtual action

In this paper, we derive and compare three integrators for nonsmooth mechanical systems by discretizing the principle of virtual action with finite elements in time. The weak as well as the strong variational form of the principle are discretized using a piecewise linear shape function and different quadrature rules. After introducing a suitable constitutive law for the contact forces arising in the discretized system, this approach leads to the well established time-stepping scheme of Moreau [1], the variational Moreau-type scheme derived in [3] and another related scheme, which we call the symmetric Moreau-type scheme. It is shown using a benchmark system that the symmetric and the variatonal Moreau-type schemes, in contrast to Moreau's scheme, show an excellent longterm energy behavior.