Variational integrators for constrained dynamical systems

Leyendecker, S., J. E. Marsden and M. Ortiz

Z. Angew. Math. Mech., 88, (2008), 677-708

Abstract:

A variational formulation of constrained dynamics is presented in the continuous and in the discrete setting. The existing theory on variational integration of constrained problems is extended by aspects on the initialization of simulations, the discrete Legendre transform and certain postprocessing steps. Furthermore, the discrete null space method which has been introduced in the framework of energy-momentum conserving integration of constrained systems is adapted to the framework of variational integrators. It eliminates the constraint forces (including the Lagrange multipliers) from the time-stepping scheme and subsequently reduces its dimension to the minimal possible number. While retaining the structure preserving properties of the specific integrator, the solution of the smaller dimensional system saves computational costs and does not suffer from conditioning problems. The performance of the variational discrete null space method is illustrated by numerical examples dealing with mass point systems, a closed kinematic chain of rigid bodies and flexible multibody dynamics and the solutions are compared to those obtained by an energy-momentum scheme.