Order reduction in time integration caused by velocity projection
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Holonomic constraints restrict the configuration of a multibody system to a subset of the configuration space. They imply so called hidden constraints at the level of velocity coordinates that may formally be obtained from time derivatives of the original holonomic constraints. A numerical solution that satisfies hidden constraints as well as the original constraint equations may be obtained considering both types of constraints simultaneously in each time step (Stabilized index-2 formulation) or using projection techniques. Both approaches are well established in the time integration of differential-algebraic equations. Recently, we have introduced a generalized-α Lie group time integration method for the stabilized index-2 formulation that achieves second order convergence for all solution components. In the present paper, we show that a separate velocity projection would be less favourable since it may result in an order reduction and in large transient errors after each projection step. This undesired numerical behaviour is analysed by a one-step error recursion that considers the coupled error propagation in differential and algebraic solution components. This one-step error recursion has been used before to prove second order convergence for the application of generalized-α methods to constrained systems.
KeywordsGeneralized-α method Lie group time integration Velocity projection
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- M. Géradin and A. Cardona, Flexible multibody dynamics: A finite element approach, John Wiley & Sons, Ltd., Chichester (2001).Google Scholar
- K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical solution of initial-value problems in differential-algebraic equations, 2nd ed., SIAM, Philadelphia (1996).Google Scholar
- D. Negrut, R. Rampalli, G. Ottarsson and A. Sajdak, 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, and J.C. García Orden, editors, Proc. of Multibody Dynamics 2005 (ECCOMAS), Madrid, Spain (2005).Google Scholar
- C. Führer, Differential-algebraische Gleichungssysteme in mechanischen Mehrkörpersystemen, Theorie, numerische Ansätze und Anwendungen, Technical report, TU München (1988).Google Scholar
- L. O. Jay and D. Negrut, A second order extension of the generalized-a method for constrained systems in mechanics, C. Bottasso, editor, Multibody Dynamics. Computational Methods and Applications, Springer, Dordrecht (2008) 143–158.Google Scholar
- E. Hairer and G. Wanner, Solving Ordinary Differential Equations, II. Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin Heidelberg New York (1996).Google Scholar
- O. Brüls, M. Arnold and A. Cardona, Two Lie group formulations for dynamic multibody systems with large rotations, Proceedings of IDETC/MSNDC 2011, ASME 2011 International Design Engineering Technical Conferences, Washington, USA (2011).Google Scholar
- M. Arnold, O. Brüls and A. Cardona, Improved stability and transient behaviour of generalized-a time integrators for constrained flexible systems, Fifth International Conference on Advanced Computational Methods in ENgineering (ACOMEN 2011), Liège, Belgium, 14-17 November (2011).Google Scholar
- Z. Terze, A. Müller and D. Zlatar, DAE index 1 formulation for multibody system dynamics in Lie-group setting, In P. Eberhard and P. Ziegler, eds., Proc. of The 2nd Joint International Conference on Multibody System Dynamics, Stuttgart, Germany, May 29- June 1 (2012).Google Scholar