A Lie algebra approach to Lie group time integration of constrained systems

[en] Lie group integrators preserve by construction the Lie group structure of a nonlinear configuration space. In multibody dynamics, they support a representation of (large) rotations in a Lie group setting that is free of singularities. The resulting equations of motion are differential equations on a manifold with tangent spaces being parametrized by the corresponding Lie algebra. In the present paper, we discuss the time discretization of these equations of motion by a generalized-alpha Lie group integrator for constrained systems and show how to exploit in this context the linear structure of the Lie algebra. This linear structure allows a very natural definition of the generalized-alpha Lie group integrator, an efficient practical implementation and a very detailed error analysis. Furthermore, the Lie algebra approach may be combined with analytical transformations that help to avoid an undesired order reduction phenomenon in generalized-alpha time integration. After a tutorial-like step-by-step introduction to the generalized-alpha Lie group integrator, we investigate its convergence behaviour and develop a novel initialization scheme to achieve second-order accuracy in the application to constrained systems. The theoretical results are illustrated by a comprehensive set of numerical tests for two Lie group formulations of a rotating heavy top.

Disciplines :

Mechanical engineering

Author, co-author :

Arnold, Martin

Cardona, Alberto; Universidad Nacional del Litoral - CONICET > CIMEC

Bruls, Olivier ; Université de Liège > Département d'aérospatiale et mécanique > Laboratoire des Systèmes Multicorps et Mécatroniques

Language :

English

Title :

A Lie algebra approach to Lie group time integration of constrained systems

Publication date :

2016

Main work title :

Structure-preserving integrators in nonlinear structural dynamics and flexible multibody dynamics

Author, co-author :

Betsch, Peter

Publisher :

Springer International Publishing

Collection name :

CISM Courses and Lectures, Volume 565

Pages :

91-158

Peer reviewed :

Peer reviewed

Available on ORBi :

since 08 July 2016

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