Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems

The purpose of this work is twofold. First, we demonstrate analytically tha t the classical Newmark family as well as related integration algorithms ar e variational in the sense of the Veselov formulation of discrete mechanics . Such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behaviour. This analyt ical result is verified through numerical examples and is believed to be on e of the primary reasons that this class of algorithms performs so well. Second, we develop algorithms for mechanical systems with forcing, and in p articular, for dissipative systems. In this case, we develop integrators th at are based on a discretization of the Lagrange d'Alembert principle as we ll as on a variational formulation of dissipation. It is demonstrated that these types of structured integrators have good numerical behaviour in term s of obtaining the correct amounts by which the energy changes over the int egration run. Copyright (C) 2000 John Wiley & Sons, Ltd.