Variational and linearly-implicit integrators, with applications

TL;DR

This paper establishes a connection between symplectic linearly-implicit integrators and variational principles, enabling efficient simulation of constrained mechanical systems and systems on Lie groups with coarse time-stepping.

Contribution

It reveals that certain integrators are variational linearizations of Newmark methods and links penalty methods to Lagrange multiplier approaches, enhancing simulation efficiency.

Findings

Integrators are variational linearizations of Newmark methods.

Penalty methods can be used for coarse time-stepping of constrained systems.

Efficient simulation of mechanical systems on Lie groups is enabled.

Abstract

We show that symplectic and linearly-implicit integrators proposed by [Zhang and Skeel, 1997] are variational linearizations of Newmark methods. When used in conjunction with penalty methods (i.e., methods that replace constraints by stiff potentials), these integrators permit coarse time-stepping of holonomically constrained mechanical systems and bypass the resolution of nonlinear systems. Although penalty methods are widely employed, an explicit link to Lagrange multiplier approaches appears to be lacking; such a link is now provided (in the context of two-scale flow convergence [Tao, Owhadi and Marsden, 2010]). The variational formulation also allows efficient simulations of mechanical systems on Lie groups.