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

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 behavior. 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 behavior in terms of obtaining the correct amounts by which the energy changes over the integration run.

Structure-preserving model reduction for mechanical systems

“…This approach gives rise to structure-preserving integration algorithms. In fact, the ideas of structure preservation used in this paper are also tied to the numerical techniques we employ; we make use of the Newmark integrator, which has recently been shown to be a variational integrator by Kane et al [13]; see also [20] for an asynchronous generalization to the PDE context and with applications to nonlinear elastodynamic simulations. As such, the model reduction gives rise to a time-discretized evolution equation which exactly preserves momentum and the symplectic form, and approximately preserves energy.…”

“…The Newmark algorithm is second-order accurate if and only if γ = 1/2, otherwise the algorithm is only consistent, and so for the remainder of this paper we choose γ = 1/2. The Newmark integrator has recently been shown to be variational, for all values of β, by Kane et al [13]. Because of the variational structure, this numerical integration method is given by the solution to a variational principle in discrete-time, and it has the property that it exactly conserves momentum and the symplectic form.…”

Simulation of floating structure dynamics in waves by implicit coupling of a fully non-linear potential flow model and a rigid body motion approach

“…The predictorcorrector scheme is initialized with a cubic extrapolation of the force exerted on the body at next time step t i+1 . The Newmark scheme has the advantage to better preserve the energy of conservative mechanical systems (see Kane 1999) as compared to the classical fourh-order Runge-Kutta method. Besides, no sub-step is required for this scheme.…”

Symplectic-energy-momentum preserving variational integrators

“…In fact, the whole Newmark family of algorithms is variational. 18 Our methods can be used to make these integrators also preserve energy by using time-adaptive stepping. We also mention that the popular Verlet methods and shake algorithms are variational integrators ͑see Refs.…”