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Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems
Author(s) -
Kane C.,
Marsden J. E.,
Ortiz M.,
West M.
Publication year - 2000
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/1097-0207(20001210)49:10<1295::aid-nme993>3.0.co;2-w
Subject(s) - variational integrator , discretization , dissipative system , mathematics , dissipation , integrator , symplectic geometry , newmark beta method , mechanical system , numerical integration , algorithm , computer science , mathematical analysis , finite element method , physics , computer network , bandwidth (computing) , quantum mechanics , artificial intelligence , thermodynamics
Abstract 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 behaviour. This analytical result is verified 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 behaviour in terms of obtaining the correct amounts by which the energy changes over the integration run. Copyright © 2000 John Wiley & Sons, Ltd.