Variational and linearly-implicit integrators, with applications
This work offers a theoretical unification and practical improvement for simulating holonomically constrained mechanical systems, benefiting computational mechanics and robotics.
The paper establishes that symplectic and linearly-implicit integrators from Zhang and Skeel (1997) are variational linearizations of Newmark methods, enabling coarse time-stepping for constrained mechanical systems without solving nonlinear systems. It also provides a link between penalty methods and Lagrange multiplier approaches via two-scale flow convergence.
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.