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

*(English)*Zbl 0969.70004Summary: 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.

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.

##### MSC:

70-08 | Computational methods for problems pertaining to mechanics of particles and systems |

##### Keywords:

discretization of Lagrange-d’Alembert principle; Newmark algorithm; variational integrator; conservative systems; Veselov formulation of discrete mechanics; mechanical systems with forcing; dissipative systems; variational formulation
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\textit{C. Kane} et al., Int. J. Numer. Methods Eng. 49, No. 10, 1295--1325 (2000; Zbl 0969.70004)

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