Conference abstracts

Session A5 - Geometric Integration and Computational Mechanics

July 11, 16:00 ~ 16:30 - Room 111

On unit-quaternion based Galerkin Lie group variational integrators

Sigrid Leyendecker

University of Erlangen-Nuremberg, Germany   -

Lie-group variational integrators are often used to simulate rigidy body dynamics~\cite{Leitz2014,Terze2016}. Using unit quaternions is an efficient way to represent the rotational degrees of freedom of a rigid body~\cite{Betsch2009}. This is in part due to the fact, that they facilitate a simple interpolation method presented in this talk. This interpolation method, toghether with an appropriate quadrature rule, is then used to approximate the action of the dynamical system with rotational degrees of freedom and a Lie-group variational integrator of arbitrary order is derived.

We present a numerical convergence analysis, both on the main grid, as well as for the whole Galerkin curve. Our convergence analysis for the whole Galerkin curve is in agreement with the results of Hall et al.~\cite{Hall2015,Hall2015a}, but we also investigate the convergence on the main grid, where the convergence rate is considerably higher. Furthermore, we use unit quternions instead of rotation matrices, which might be computationally more efficient. Our interpolation method does not require a change of coordinates in the momentum matching part of the discrete Euler-Lagrange equations as the one used in~\cite{Hall2015}. The computational efficiency is investigated by showing the relationship between the error and the CPU-time.

The same method can be extended to derive multisymplectic Galerkin Lie-group variational integrators, e.g.~for the simulation of geometrically exact beam dynamics, by applying the interpolation method to the two-dimensional space time domain.


\bibitem{Betsch2009} Peter Betsch and Ralf Siebert. \newblock Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration. \newblock {\em International Journal for Numerical Methods in Engineering}, 79(4):444--473, 2009.

\bibitem{Hall2015} James Hall and Melvin Leok. \newblock Lie group spectral variational integrators. \newblock {\em Foundations of Computational Mathematics}, pages 1--59, 2015.

\bibitem{Hall2015a} James Hall and Melvin Leok. \newblock Spectral variational integrators. \newblock {\em Numerische Mathematik}, 130(4):681--740, 2015.

\bibitem{Leitz2014} Thomas Leitz, Sina Ober-Bl{\"{o}}baum, and Sigrid Leyendecker. \newblock Variational integrators for dynamical systems with rotational degrees of freedom. \newblock In {\em Proceedings of WCCM XI - ECCM V - ECFD VI}, pages 3148--3159, 2014.

\bibitem{Terze2016} Zdravko Terze, Andreas M{\"{u}}ller, and Dario Zlatar. \newblock Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations. \newblock {\em Multibody System Dynamics}, pages 1--25, 2016.


Joint work with Thomas Leitz (University of Erlangen-Nuremberg).

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