×

Regularized numerical integration of multibody dynamics with the generalized \(\alpha \) method. (English) Zbl 1220.70009

Summary: This paper discusses the consistent regularization property of the generalized \(\alpha \) method when applied as an integrator to an initial value high index and singular differential-algebraic equation model of a multibody system. The regularization comes from within the discretization itself and the discretization remains consistent over the range of values the regularization parameter may take. The regularization involves increase of the smallest singular values of the ill-conditioned Jacobian of the discretization and is different from Baumgarte and similar techniques which tend to be inconsistent for poor choice of regularization parameter. This regularization also helps where pre-conditioning the Jacobian by scaling is of limited effect, for example, when the scleronomic constraints contain multiple closed loops or singular configuration or when high index path constraints are present. The feed-forward control in Kane’s equation models is additionally considered in the numerical examples to illustrate the effect of regularization. The discretization presented in this work is adopted to the first order DAE system (unlike the original method which is intended for second order systems) for its \(A\)-stability and same order of accuracy for positions and velocities.

MSC:

70E55 Dynamics of multibody systems
65L80 Numerical methods for differential-algebraic equations

Software:

RODAS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arnold, M.; Brüls, O., Convergence of the generalized-\( \alpha\) scheme for constrained mechanical systems, Multibody System Dynamics, 18, 7, 185-202 (2007) · Zbl 1121.70003
[2] U. Ascher, P. Lin, Sequential regularization methods for simulating mechanical systems with many closed loops, SIAM Journal on Scientific Computing 21 (4) (1999) 1244-1262. URL <http://link.aip.org/link/?SCE/21/1244/1; U. Ascher, P. Lin, Sequential regularization methods for simulating mechanical systems with many closed loops, SIAM Journal on Scientific Computing 21 (4) (1999) 1244-1262. URL <http://link.aip.org/link/?SCE/21/1244/1
[3] Betts, J.; Biehn, N.; Campbell, S., Convergence of nonconvergent IRK discretizations of optimal control problems with state inequality constraints, SIAM Journal on Scientific Computing, 23, 6, 1981-2007 (2002) · Zbl 1009.49025
[4] Bottasso, C.; Bauchau, O.; Cardona, A., A 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 (2007) · Zbl 1133.65057
[5] Bottasso, C. L.; Dopico, D.; Trainelli, L., On the optimal scaling of index three DAEs in multibody dynamics, Multibody System Dynamics, 19, 1-2, 3-20 (2008) · Zbl 1136.70001
[6] K. Brenan, S. Campbell, L. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, second revised ed., Society for Industrial and Applied Mathematics (SIAM), 1996.; K. Brenan, S. Campbell, L. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, second revised ed., Society for Industrial and Applied Mathematics (SIAM), 1996. · Zbl 0844.65058
[7] Campbell, S., High-index differential-algebraic equations, Mechanics Based Design of Structures and Machines, 23, 2, 199-222 (1995)
[8] Chung, J.; Hulbert, G., Time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\( \alpha\) method, Journal of Applied Mechanics, Transactions ASME, 60, 2, 371-375 (1993) · Zbl 0775.73337
[9] D. Cruz-Uribe, C. Neugebauer, Sharp error bounds for the trapezoidal rule and Simpson’s rule, Journal of Inequalities in Pure and Applied Mathematics 3:4.49 (4:49) (2002) 1-49.; D. Cruz-Uribe, C. Neugebauer, Sharp error bounds for the trapezoidal rule and Simpson’s rule, Journal of Inequalities in Pure and Applied Mathematics 3:4.49 (4:49) (2002) 1-49. · Zbl 1030.41016
[10] Erlicher, S.; Bonaventura, L.; Bursi, O., The analysis of the generalized-method for non-linear dynamic problems, Computational Mechanics, 28, 83-104 (2002) · Zbl 1146.74327
[11] M. Fliess, J. Uvine, P. Rouchon, A simplified approach of crane control via a generalized state-space model, in: Proceedings 30th lEEE Conference on Decision and Control, 1991, pp. 736-741.; M. Fliess, J. Uvine, P. Rouchon, A simplified approach of crane control via a generalized state-space model, in: Proceedings 30th lEEE Conference on Decision and Control, 1991, pp. 736-741.
[12] Gear, C.; Leimkuhler, B.; Gupta, G., Automatic integration of Euler-Lagrange equations with constraints, Journal of Computational and Applied Mathematics, 12-13, 77-90 (1985) · Zbl 0576.65072
[13] E. Hairer, G. Wanner, Solving Ordinary Differential Equations 2 Stiff and Differential-Algebraic Problems, third printing of second revised ed., Springer-Verlag, 2004.; E. Hairer, G. Wanner, Solving Ordinary Differential Equations 2 Stiff and Differential-Algebraic Problems, third printing of second revised ed., Springer-Verlag, 2004.
[14] Jansen, K.; Whiting, C.; Hulbert, G., A generalized-alpha method for integrating the filtered Navier-Stokes equations with a stabilized finite element method, Computer Methods in Applied Mechanics and Engineering, 190, 305-319 (2000) · Zbl 0973.76048
[15] L.O. Jay, D. Negrut, A second order extension of the generalized-alpha method for constrained systems in mechanics, in: E. Onate (Ed.), Computational Methods in Applied Sciences Series.; L.O. Jay, D. Negrut, A second order extension of the generalized-alpha method for constrained systems in mechanics, in: E. Onate (Ed.), Computational Methods in Applied Sciences Series. · Zbl 1303.70004
[16] Lunk, C.; Simeon, B., Solving constrained mechanical systems by the family of Newmark and alpha-methods, Journal of Applied Mathematics and Mechanics (ZAMM), 86, 772-784 (2006) · Zbl 1116.70008
[17] Mattsson, S.; Söderlind, G., Index reduction in differential-algebraic equations using dummy derivatives, SIAM Journal on Scientific Computing, 14, 677-692 (1993) · Zbl 0785.65080
[18] S. Moberg, S. Hanssen, A DAE approach to feed-forward control of flexible manipulators, in: Robotics and Automation, 2007 IEEE International Conference, 2007, pp. 3439-3444.; S. Moberg, S. Hanssen, A DAE approach to feed-forward control of flexible manipulators, in: Robotics and Automation, 2007 IEEE International Conference, 2007, pp. 3439-3444.
[19] N. C. Parida, S. Raha, The \(\operatorname{Α;}\) http://link.aip.org/link/?SCE/31/2386/1; N. C. Parida, S. Raha, The \(\operatorname{Α;}\) http://link.aip.org/link/?SCE/31/2386/1 · Zbl 1196.65106
[20] Petzold, L. R.; Ren, Y.; Maly, T., Regularization of higher-index differential-algebraic equations with rank-deficient constraints, SIAM J. Scientific Computing, 18, 753-774 (1993) · Zbl 0872.65063
[21] Raha, S.; Petzold, L. R., Constraint partitioning for structure in path-constrained dynamic optimization problems, Applied Numerical Mathematics, 39, 105-126 (2001) · Zbl 0989.65083
[22] Rajan, N.; Raha, S., The Stochastic \(\alpha\) method: a numerical method for simulation of noisy second order dynamical systems, Computer Modeling in Engineering and Sciences, 23, 2, 91-116 (2008) · Zbl 1232.65014
[23] T. Söderström, Perturbation results for singular values, Tech. Rep. 1999-001, Technical Report of Information Technology Department, Uppsala University, Uppsala, Sweden, 1999.; T. Söderström, Perturbation results for singular values, Tech. Rep. 1999-001, Technical Report of Information Technology Department, Uppsala University, Uppsala, Sweden, 1999.
[24] Yen, J.; Petzold, L.; Raha, S., A time integration algorithm for flexible mechanism dynamics: the DAE-\( \alpha\) method, Journal of Computational Methods and Applied Mechanical Engineering (CMAME), 158, 341-355 (1998) · Zbl 0949.70006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.