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Published December 10, 2000 | Updated
Journal Article Open

Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems


The purpose of this work is twofold. First, we demonstrate analytically that the classical Newmark family as well as related integration algorithms are 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 behavior. This analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well. Second, we develop algorithms for mechanical systems with forcing, and in particular, for dissipative systems. In this case, we develop integrators that are based on a discretization of the Lagrange d'Alembert principle as well as on a variational formulation of dissipation. It is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run.

Additional Information

© 2000 John Wiley & Sons, Ltd. Received 2 August 1999. Revised 24 November 1999. March, 1999, This version: November 25, 1999. We thank Petr Krysl and Sanjay Lall for helpful comments and inspiration, and Yuri Suris for suggesting how Newmark might be directly expressed as a variational algorithm. We are also grateful to the referees for their constructive and helpful comments.

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