A Lie Algebra Approach to Lie Group Time Integration of Constrained Systems
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.
The work of this author was supported by the German Minister of Education and Research, BMBF grant 05M13NHB: “Modelling and structure preserving discretization of inelastic components in system simulation”.
- Arnold, M., Brüls, O.,& Cardona, A. (2011a). Convergence analysis of generalized-\(\alpha \) Lie group integrators for constrained systems. In J. C. Samin & P. Fisette (Eds.), Proceedings of Multibody Dynamics 2011 (ECCOMAS Thematic Conference), Brussels, Belgium.Google Scholar
- Arnold, M., Brüls, O.,& Cardona, A. (2011b). Improved stability and transient behaviour of generalized-\(\alpha \) time integrators for constrained flexible systems. In Fifth International Conference on Advanced COmputational Methods in ENgineering (ACOMEN 2011), Liège, Belgium, 14–17 November 2011.Google Scholar
- Arnold, M., Burgermeister, B., Führer, C., Hippmann, G., & Rill, G. (2011c). Numerical methods in vehicle system dynamics: State of the art and current developments. Vehicle System Dynamics, 49, 1159–1207. doi: 10.1080/00423114.2011.582953.
- Arnold, M., Cardona, A., & Brüls, O. (2014). Order reduction in time integration caused by velocity projection. In Proceedings of the 3rd Joint International Conference on Multibody System Dynamics and the 7th Asian Conference on Multibody Dynamics, 30 June–3 July 2014. BEXCO. Korea: Busan.Google Scholar
- Betsch, P., & Leyendecker, S. (2006). The discrete null space method for the energy consistent integration of constrained mechanical systems. Part II: Multibody dynamics. International Journal for Numerical Methods in Engineering, 67, 499–552. doi: 10.1002/nme.1639.MathSciNetCrossRefMATHGoogle Scholar
- Bottasso, C. L., Bauchau, O. A., & Cardona, A. (2007). Time-step-size-independent conditioning and sensitivity to perturbations in the numerical solution of index three differential algebraic equations. SIAM Journal on Scientific Computing, 29, 397–414. doi: 10.1137/050638503.MathSciNetCrossRefMATHGoogle Scholar
- Brüls, O., & Arnold, M. (2008). The generalized-\(\alpha \) scheme as a linear multistep integrator: Towards a general mechatronic simulator. Journal of Computational and Nonlinear Dynamics, 3(4), 041007. doi: 10.1115/1.2960475.
- Brüls, O., Arnold, M.,& Cardona, A. (2011). Two Lie group formulations for dynamic multibody systems with large rotations. In Proceedings of IDETC/MSNDC 2011, ASME 2011 International Design Engineering Technical Conferences, Washington, USA.Google Scholar
- Cardona, A.,& Géradin, M. (1994). Numerical integration of second order differential-algebraic systems in flexible mechanism dynamics. In M. F. O. Seabra Pereira & J. A. C. Ambrósio (Eds.), Computer-Aided Analysis of Rigid and Flexible Mechanical Systems, NATO ASI Series (Vol. E–268). Dordrecht: Kluwer Academic Publishers. doi: 10.1007/978-94-011-1166-9_16.
- Crouch, P. E., & Grossman, R. (1993). Numerical integration of ordinary differential equations on manifolds. Journal of Nonlinear Science, 3, 1–33. doi: 10.1007/BF02429858.
- García de Jalón, J., & Gutiérrez-López, M. D. (2013). Multibody dynamics with redundant constraints and singular mass matrix: Existence, uniqueness, and determination of solutions for accelerations and constraint forces. Multibody System Dynamics, 30, 311–341. doi: 10.1007/s11044-013-9358-7.MathSciNetCrossRefMATHGoogle Scholar
- Géradin, M., & Cardona, A. (2001). Flexible multibody dynamics: A finite element approach. Chichester: Wiley.Google Scholar
- Golub, G. H., & van Loan, Ch. F. (1996). Matrix computations (3rd ed.). Baltimore London: The Johns Hopkins University Press.Google Scholar
- Jay, L. O.,& Negrut, D. (2008). A second order extension of the generalized-\(\alpha \) method for constrained systems in mechanics. In C. Bottasso (Ed.), Multibody Dynamics. Computational Methods and Applications, Computational Methods in Applied Sciences (Vol. 12, pp. 143–158). Dordrecht: Springer. doi: 10.1007/978-1-4020-8829-2_8.
- Kobilarov, M., Crane, K.,& Desbrun, M. (2009). Lie group integrators for animation and control of vehicles. ACM Transactions on Graphics, 28(2, Article 16), 1–14. doi: 10.1145/1516522.1516527.
- Müller, A., & Terze, Z. (2014b). The significance of the configuration space Lie group for the constraint satisfaction in numerical time integration of multibody systems. Mechanism and Machine Theory, 82, 173–202. doi: 10.1016/j.mechmachtheory.2014.06.014.
- Negrut, D., Rampalli, R., Ottarsson, G.,& Sajdak, A. (2005). On the use of the HHT method in the context of index 3 differential algebraic equations of multi-body dynamics. In J. M. Goicolea, J. Cuadrado & J. C. García Orden (Eds.), Proceedings of Multibody Dynamics 2005 (ECCOMAS Thematic Conference), Madrid, Spain.Google Scholar
- Sanborn, G. G., Choi, J., & Choi, J. H. (2014). Review of RecurDyn integration methods. In Proceedings of the 3rd Joint International Conference on Multibody System Dynamics and the 7th Asian Conference on Multibody Dynamics, 30 June–3 July (2014). BEXCO. Korea: Busan.Google Scholar
- Walter, W. (1998). Ordinary Differential Equations. Number 182 in Graduate Texts in Mathematics. Berlin: Springer.Google Scholar