Search

Multistep methods of BDF type are the method-of-choice in several industrial multibody system simulation packages. In the present paper, BDF is applied to constrained systems in nonlinear configuration spaces with Lie group structure that allows, e.g., a representation of multibody systems with large rotations without singularities. The k -step Lie group integrator BLieDF avoids order reduction by a slightly perturbed argument of the exponential map for representing the nonlinearity of the numerical flow in the configuration space without any time-consuming re-parametrization. This integrator is compared with multistep methods on Lie groups suggested by Faltinsen, Marthinsen and Munthe-Kaas (2001) and the advantages of the novel BLieDF integrator are shown.

In this paper we consider numerical methods for solving nonlinear equations on matrix Lie groups. Recently Owren and Welfert (Technical Report Numerics, No 3/1996, Norwegian University of Science and Technology, Trondheim, Norway, 1996) have proposed a method where the original nonlinear equation F(Y)=0 is transformed into a nonlinear equation on the Lie algebra of the group, thus Newton-type methods may be applied which require the evaluation of exponentials of matrices. Here the previous transformation will be performed by the Cayley approximant of the exponential map. This approach has the advantage that no exponentials of matrices are needed. The numerical tests reported in the last section seem to show that our approach is less expensive and provides a larger convergence region than the method of Owren and Welfert.