Projected event-capturing time-stepping schemes for nonsmooth mechanical systems with unilateral contact and Coulomb’s friction

This work addresses the problem of the numerical time-integration of nonsmooth mechanical systems subjected to unilateral contacts, impacts and Coulomb’s friction. The considered systems are the space-discretized continuous systems obtained by using a Finite Element Method (FEM) approach or the multi-body systems, or a mix of them as in flexible multibody dynamics. Up to now, two main numerical schemes are available for this purpose: the Moreau–Jean scheme which solves the constraints at the velocity level together with a Newton impact law and the Schatzman–Paoli scheme which directly considers the constraints at the position level. In both schemes, the position and velocity constraints are not both satisfied in discrete time. A first attempt to improve the time simulation is made by directly using the Gear–Gupta–Leimkuhler (GGL) approach for Differential Algebraic Equations (DAE), that solves, in discrete time, the constraints on both position and velocity levels. This obtained direct projection scheme succeeds in solving in discrete time both position and velocity constraints, but introduces some chattering at contact after a finite accumulation of impacts. A second new scheme is proposed that improves the direct projected scheme by combining several steps of activation and projection to avoid the chattering effect. The stability and the local order of the scheme will be discussed. The usefulness of the scheme is demonstrated on several academic examples and is illustrated on an industrial application: the modeling and simulation of an electrical circuit breaker.