Abstract
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.
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This paper was presented at the Joint Conference of the 3rd IMSD and the 7th ACMD, Busan, Korea, June, 2014. Recommended by Guest Editor Sung-Soo Kim and Jin Hwan Choi
Martin Arnold is Professor of Numerical mathematics at Martin Luther University Halle-Wittenberg. In 1990, he received his Ph.D. in mathematics from Martin Luther University. From 1990 to 1997 he was assistant professor at the University of Rostock and obtained a habilitation degree. He worked as research scientist and head of a Computational methods team at DLR German Aerospace Center and as Privatdozent at Munich University of Technology. Since 2003 he is full professor at Martin Luther University. His research interests include tim-dependent coupled and constrained systems of differential equations and numerical methods and software for modelbased simulation in science and engineering.
Alberto Cardona is head of the Research Center of Computational Methods at Universidad Nacional del Litoral and Conicet, Santa Fe, Argentina. The center performs research in several fields in computational mechanics, ranging from computational fluid mechanics to structural dynamics, parallel computations and materials modeling. Alberto Cardona specialized in multibody dynamics and is co-author of the book “Flexible Multibody Dynamics: A Finite Element Approach”. He has co-authored more than seventy papers in international journals. He co-founded a company in computational mechanics software, which develops the software Oofelie. His research interests include modeling in flexible multibody dynamics, contact mechanics, nonlinear heat transfer and computational welding.
Olivier Brüls is Professor in the Department of Aerospace and Mechanical Engineering (LTAS) at the University of Liège in Belgium. After a Master degree in Mechanical Engineering obtained in 2001, he achieves a Ph.D. thesis at the University of Liège in 2005. In 2005, he spends a postdoctoral year within the Institute of Engineering and Computational Mechanics at the University of Stuttgart, Germany. Since 2008, he is the head of the Multibody and Mechatronic Systems Lab and his research interests include flexible multibody dynamics, mechatronics, numerical simulation, control and optimization methods.
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Arnold, M., Cardona, A. & Brüls, O. Order reduction in time integration caused by velocity projection. J Mech Sci Technol 29, 2579–2585 (2015). https://doi.org/10.1007/s12206-015-0501-7
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DOI: https://doi.org/10.1007/s12206-015-0501-7