zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Simulation of unilateral constrained systems with many bodies. (English) Zbl 1146.70317
Summary: Nowadays the theory of multi-body systems including unilateral constraints is quite well established. However, the tendency towards more and more detailed and complex models may not be compensated with increasing computer power. In fact the growing computational effort demands for improved numerical methods in order to solve large systems. In this paper a time-stepping method is proposed for the computation of multi-body systems with many unilateral constraints. Stability and accuracy are discussed with respect to the given discretisation. In order to handle many contacts an iterative algorithm is applied based on a Gauss-Seidel relaxation scheme. A numerical example shows the efficiency of the relaxation scheme in comparison with Lemke’s method and an Augmented Lagrangian approach.

70E55Dynamics of multibody systems
70-08Computational methods (mechanics of particles and systems)
70F40Problems with friction (particle dynamics)
ATLAS; RODAS; Meschach
Full Text: DOI
[1] Pfeiffer, F. and Glocker, Ch., Multibody Dynamics with Unilateral Contacts, Wiley Series in Nonlinear Science, Wiley, New York, 1996. · Zbl 0922.70001
[2] Jean, M., ’The non-smooth contact dynamics method’, Computational Methods in Applied and Mechanical Engineering 177, 1999, 235--257. · Zbl 0959.74046 · doi:10.1016/S0045-7825(98)00383-1
[3] Stewart, D. E. and Trinkle, J. C., ’An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and coulomb friction’, International Journal of Numerical Methods in Engineering 39, 1996, 2673--2691. · Zbl 0882.70003 · doi:10.1002/(SICI)1097-0207(19960815)39:15<2673::AID-NME972>3.0.CO;2-I
[4] Stiegelmeyr, A., ’A time stepping algorithm for mechanical systems with unilateral contacts’, in Proceedings of DETC’99, ASME, Las Vegas, 1999.
[5] Johansson, L., ’A linear complementarity algorithm for rigid body impact with friction’, European Journal of Mechanics A/Solids 18, 1999, 703--717. · Zbl 0952.70012 · doi:10.1016/S0997-7538(99)00119-9
[6] Rockafellar, R. T., ’Augmented Lagrangians and applications of the proximal point algorithm in convex programming’, Mathematics of Operations Research 1(2), 1976, 97--116. · Zbl 0402.90076 · doi:10.1287/moor.1.2.97
[7] Alart, P. and Curnier, A., ’A mixed formulation for frictional contact problems prone to Newton like solution methods,’ Computer Methods in Applied Mechanics and Engineering 92, 1991, 353--375. · Zbl 0825.76353 · doi:10.1016/0045-7825(91)90022-X
[8] Leine, R. I. and Glocker, Ch., ’A set-valued force law for spatial Coulomb--Contensou friction’, in Proceedings of DETC’03, ASME, Chicago, 2003. · Zbl 1038.74513
[9] Glocker, Ch., Dynamik von Starrkörpersystemen mit Reibung und Stöen, Fortschritt-berichte VDI, Reihe 18, Nr. 182, VDI Verlag Düsseldorf, 1995.
[10] Glocker, Ch., Set-Valued Force Laws, Dynamics of Non-Smooth Systems, Lecture Notes in Applied Mechanics, Vol. 1, Springer-Verlag, Berlin, 2001. · Zbl 0979.70001
[11] Moreau, J. J., Unilateral Contact and Dry Friction in Finite Freedom Dynamics, Volume 302 of International Centre for Mechanical Sciences, Courses and Lectures. J.J. Moreau P.D. Panagiotopoulos, Springer, Vienna, 1988. · Zbl 0703.73070
[12] Stiegelmeyr, A., Zur numerischen Berechnung strukturvarianter Mehrkörpersysteme, VDI Verlag, Reihe 18, No. 271, Düsseldorf, 2001.
[13] Cottle, R. W., Pang, J.-S. and Stone, R. E., The Linear Complementarity Problem, Computer Science and Scientific Computing, Academic Press, San Diego, 1992. · Zbl 0757.90078
[14] Moreau, J. J., ’Some numerical methods in multibody dynamics: Application to granular materials’, European Journal of Mechanics A/Solids 13(4), Suppl., Special Issue; 2nd European Solid Mechanics Conference Euromech, Genoa, Italy, 1994, pp. 93--114. · Zbl 0815.73009
[15] Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer Verlag, Berlin, 1991. · Zbl 0729.65051
[16] Funk, K., Simulation eindimensionaler Kontinua mit Unstetigkeiten, Fortschritt-berichte VDI, Reihe 18, Nr. 294, VDI Verlag Düsseldorf, 2003.
[17] Funk, K. and Pfeiffer, F., ’A time-stepping algorithm for stiff mechanical systems with unilateral constraints’, in Proceedings of the International Conference on Nonsmooth/Nonconvex Mechanics with Applications in Engineering, pp. 307--314, Thessaloniki, 2002.
[18] Arnold, M., Zur Theorie und zur numerischen Lösung von Anfangswertproblemen für differential-algebraische Systeme von höherem Index, Fortschritt-berichte VDI, Reihe 20, Nr. 264, VDI Verlag Düsseldorf, 1998.
[19] Rockafellar, R. T., Convex Analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, 1970. · Zbl 0193.18401
[20] Hestenes, M., ’Multiplier and gradient methods’, Journal of Optical Theory Applications 4, 303--320, 1969. · Zbl 0174.20705 · doi:10.1007/BF00927673
[21] Powell, B. T., ’A method for nonlinear constraints in minimization problems’, in Optimization, Ed. R. Fletcher, Academic Press, London, 1969. · Zbl 0194.47701
[22] Dennis, J. E. and Schable, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1983.
[23] ATLAS, Automatically Tuned Linear Algebra Software package, http://math-atlas.source-forge.net.