Variational integrators for constrained dynamical systems
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 timestepping 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.
© 2008 Wiley-VCH. Received 28 December 2007; accepted 27 May 2008; published online 10 September 2008. The authors would like to thank P. Betsch and P. Steinmann whose works with the first author on energy-momentum conserving integration of multibody dynamics are cited frequently here. Thanks also to S. Ober-Blöbaum for helpful discussions. Furthermore, the support of the Alexander von Humboldt-Foundation is gratefully acknowledged. Research partially supported by AFOSR grant FA9550-08-1-0173.