Discontinuous variational time integrators for complex multibody collisions

The objective of the present work is to formulate a new class of discontinuous variational time integrators that allow the system to adopt two possibly different configurations at each sampling time tk, representing predictor and corrector configurations of the system. The resulting sequence of configuration pairs then represents a discontinuous—or non-classical—trajectory. Continuous or classical trajectories are recovered simply by enforcing a continuity constraint at all times. In particular, in systems subject to one-sided contact constraints simulated via discontinuous variational time integrators, the predictor configuration is not required to satisfy the one-sided constraints, whereas the corrector configuration is obtained by a closest-point projection (CPP) onto the admissible set. The resulting trajectories are generally discontinuous, or non-classical, but are expected to converge to classical or continuous solutions for decreasing time steps. We account for dissipation, including friction, by means of a discrete Lagrange–d'Alembert principle, and make extensive use of the spacetime formalism in order to ensure exact energy conservation in conservative systems, and the right rate of energy decay in dissipative systems. The structure, range and scope of the discontinuous variational time integrators, and their accuracy characteristics are illustrated by means of examples of application concerned with rigid multibody dynamics