LieGroupIntegration_with_constraints

Lie group time integration for constrained systems

The numerical solution of initial value problems for differential equations on manifolds has been considered for about four decades.¹ Since the 1990's Lie group methods that exploit the specific properties of differentiable manifolds with Lie group structure have found great interest.²

Their practical application in flexible multibody dynamics benefits much from the seminal work of Brüls and Cardona³ who defined and implemented a Lie group generalized-α method for constrained mechanical systems on Lie groups in terms of solution increments in the corresponding Lie algebra.⁴ This Lie algebra approach⁵ proved to be useful as well in the convergence analysis of these methods⁶ and paved the path for the construction of higher order Lie group BDF methods⁷ and a RATTLE type variational Lie group integrator⁸.

In his recently completed PhD thesis⁹, Hante considered solution increments in the Lie algebra as derivative vectors that may be used for a consistent discretization of a geometrically exact beam model in space and time. Part of this work is a comprehensive open source software repository including efficient implementations of

• the Lie group generalized-α method³,
• a second order Lie group BDF method⁷ and
• the variational Lie group integrator RATTLie⁸,⁹

for the stabilized index-2 formulation of constrained mechanical systems on Lie groups.⁵,⁶ The methods, their implementation and numerical test results are discussed in detail in the PhD thesis⁹. Furthermore, the GitHub repositories include online documentations.

In view of THREAD's Deliverable D3.2, the relevant repositories are

• gena: Lie group generalized-α method,
• BLieDF: Lie group BDF methods of order $k\leq 2$,
• RATTLie: RATTLE type variational Lie group integrator,
• liegroup: basic components of Lie group integrators like exponential map, tangent operator, ... ,
• heavy_top: benchmark problem Heavy Top³.

Footnotes

1. W.C. Rheinboldt: Differential-algebraic systems as differential equations on manifolds. Math. Comp. 43(1984)473-482. https://doi.org/10.1090/S0025-5718-1984-0758195-5 ↩

2. A. Iserles, H.Z. Munthe-Kaas, S. Nørsett, and A. Zanna: Lie-group methods. Acta Numerica 9(2000)215-365. https://doi.org/10.1017/S0962492900002154 ↩

3. O. Brüls and A. Cardona: On the use of Lie group time integrators in multibody dynamics. J. Comput. Nonlinear Dynam. 5(2010)031002 (13 pages). https://doi.org/10.1115/1.4001370 ↩ ↩² ↩³

4. O. Brüls, A. Cardona, and M. Arnold: Lie group generalized-α time integration of constrained flexible multibody systems. Mechanism and Machine Theory 48(2012)121-137. https://doi.org/10.1016/j.mechmachtheory.2011.07.017 ↩

5. M. Arnold, A. Cardona, and O. Brüls: A Lie algebra approach to Lie group time integration of constrained systems. In P. Betsch, editor, Structure-Preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics, volume 565 of CISM Courses and Lectures, pages 91-158. Springer International Publishing, Cham, 2016. https://doi.org/10.1007/978-3-319-31879-0_3 ↩ ↩²

6. M. Arnold, O. Brüls, and A. Cardona: Error analysis of generalized-α Lie group time integration methods for constrained mechanical systems. Numerische Mathematik 129(2015)149-179. https://doi.org/10.1007/s00211-014-0633-1 ↩ ↩²

7. V. Wieloch and M. Arnold: BDF integrators for constrained mechanical systems on Lie groups. Journal of Computational and Applied Mathematics 387(2021)112517 (17 pages). https://doi.org/10.1016/j.cam.2019.112517 ↩ ↩²

8. S. Hante and M. Arnold: RATTLie: A variational Lie group integration scheme for constrained mechanical systems. Journal of Computational and Applied Mathematics 387(2021):112492 (14 pages). https://doi.org/10.1016/j.cam.2019.112492 ↩ ↩²

9. S. Hante: Geometric Integration of a Constrained Cosserat Beam Model. PhD thesis, Martin Luther University Halle-Wittenberg, Institute of Mathematics, June 2022. https://doi.org/10.25673/91397 ↩ ↩² ↩³