By using SIAM Journals Online you agree to abide by the
Terms and Conditions of Use.

©  SIAM

 

SIAM Journal on Applied Dynamical Systems

Previous Article
Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs
We investigate the following family of evolutionary 1+1 PDEs that describes the balance between convection and stretching for small viscosity in the dynamics of one-dimensional nonlinear waves in flu...
Next Article
Vortex Dynamics on a Cylinder
Point vortices on a cylinder (periodic strip) are studied geometrically. The Hamiltonian formalism is developed, a nonexistence theorem for relative equilibria is proved, equilibria are classified wh...

You are logged in to this journal.

Nonsmooth Lagrangian Mechanics and Variational Collision Integrators

SIAM J. Appl. Dyn. Syst. Volume 2, Issue 3, pp. 381-416 (2003)

Issue Date: 2003
FULL TEXT OPTIONS   (FREE)
Download PDF (401 kB) View Cart

Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated.

©2003 Society for Industrial and Applied Mathematics
Permalink: http://dx.doi.org/10.1137/S1111111102406038

KEYWORDS and AMS

Keywords
AMS Subject Classifications
37M15, 70F35, 58E30

PUBLICATION DATA

ISSN:
1536-0040 (online)
Publisher:
AIP is a member of CrossRef SIAM

REFERENCES (61)
View in Separate Window/Tab
  1. R. Abraham, J. Marsden, T. Ratiu, Manifolds, tensor analysis, and applications, Applied Mathematical Sciences, Vol. 75, Springer-Verlag, 1988x+654 doi:10.1063/1.532892 [ISI] [MathRev]
  2. Mihai Anitescu, Florian Potra, David Stewart, Time-stepping for three-dimensional rigid body dynamics, Comput. Methods Appl. Mech. Engrg., 177 (1999), 183–197, Computational modeling of contact and friction [MathRev]
  3. F. Armero, E. Petöcz, A new dissipative time-stepping algorithm for frictional contact problems: formulation and analysis, Comput. Methods Appl. Mech. Engrg., 179 (1999), 151–178
  4. Eric Barth, Benedict Leimkuhler, Sebastian Reich, A time-reversible variable-stepsize integrator for constrained dynamics, SIAM J. Sci. Comput., 21 (1999), 1027–1044
  5. Bernard Brogliato, Silviu-Iulian Niculescu, Pascal Orhant, On the control of finite-dimensional mechanical systems with unilateral constraints, IEEE Trans. Automat. Control, 42 (1997), 200–215
  6. Bernard Brogliato, Nonsmooth impact mechanics, Lecture Notes in Control and Information Sciences, Vol. 220, Springer-Verlag London Ltd., 1996xvi+400, Models, dynamics and control [MathRev]
  7. Bernard Brogliato, On the control of non-smooth complementarity dynamical systems, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 2369–2383, Non-smooth mechanics
  8. N. J. Carpenter, R. L. Taylor, and M. G. Katona, Lagrange constraints for transient finite-element surface-contact, Internat. J. Numer. Methods Engrg., 32 (1991), pp. 103–128. [ISI]
  9. F. Cirak and M. West, Scalable Explicit Collision Response Using Momentum Decomposition, manuscript.
  10. Frank Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., 1983xiii+308, A Wiley-Interscience Publication [ZentralblattMath] [MathRev]
  11. Asen Dontchev, Frank Lempio, Difference methods for differential inclusions: a survey, SIAM Rev., 34 (1992), 263–294 [MathRev]
  12. A. Filippov, Differential equations with discontinuous right-hand side, Mat. Sb. (N.S.), 51 (93) (1960), 99–128
  13. A. Filippov, Classical solutions of differential equations with multi-valued right-hand side, SIAM J. Control, 5 (1967), 609–621 [MathRev]
  14. A. Filippov, Differential equations with discontinuous righthand sides, Mathematics and its Applications (Soviet Series), Vol. 18, Kluwer Academic Publishers Group, 1988x+304, Translated from the Russian [ZentralblattMath] [MathRev]
  15. M. Gotay, J. Isenberg, and J. E. Marsden, Momentum Maps and the Hamiltonian Structure of Classical Relativistic Field Theories I, http://www.cds.caltech.edu/marsden/ (1997).
  16. Ernst Hairer, Christian Lubich, Long-time energy conservation of numerical methods for oscillatory differential equations, SIAM J. Numer. Anal., 38 (2000), 414–441 [MathRev]
  17. E. Hairer, S. Nørsett, G. Wanner, Solving ordinary differential equations. I, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, 1993xvi+528, Nonstiff problems [ZentralblattMath] [MathRev]
  18. E. Hairer, G. Wanner, Solving ordinary differential equations. II, Springer Series in Computational Mathematics, Vol. 14, Springer-Verlag, 1996xvi+614, Stiff and differential-algebraic problems [MathRev]
  19. Magnus Hestenes, Calculus of variations and optimal control theory, John Wiley & Sons Inc., 1966xii+405 [ZentralblattMath] [MathRev]
  20. Y. A. Houndonougbo and B. B. Laird, Constant-temperature molecular dynamics algorithms for mixed hard-core/continuous potentials, J. Chem. Phys., 117 (2002), pp. 1001–1009. [ISI]
  21. Y. A. Houndonougbo, B. B. Laird, and B. J. Leimkuhler, A molecular dynamics algorithm for mixed hard-core/continuous potentials, Mol. Phys., 98 (2000), pp. 309–316. [Inspec] [ISI]
  22. Arieh Iserles, Hans Munthe-Kaas, Syvert Nørsett, Antonella Zanna, Lie-group methods, Acta Numer., Vol. 9, Cambridge Univ. Press, Cambridge, 2000, 215–365 [MathRev]
  23. M. Jean, Unilateral contact with dry friction: time and space discrete variables formulation, Arch. Mech. (Arch. Mech. Stos.), 40 (1988), 677–691
  24. C. Kane, J. Marsden, M. Ortiz, Symplectic-energy-momentum preserving variational integrators, J. Math. Phys., 40 (1999), 3353–3371 [MathRev]
  25. C. Kane, E. Repetto, M. Ortiz, J. Marsden, Finite element analysis of nonsmooth contact, Comput. Methods Appl. Mech. Engrg., 180 (1999), 1–26 [MathRev]
  26. C. Kane, J. Marsden, M. Ortiz, M. West, Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, Internat. J. Numer. Methods Engrg., 49 (2000), 1295–1325
  27. Markus Kunze, Manuel Monteiro Marques, An introduction to Moreau's sweeping process, Lecture Notes in Phys., Vol. 551, Springer, Berlin, 2000, 1–60 [MathRev]
  28. C.-Y. Lee, J. Oden, Theory and approximation of quasistatic frictional contact problems, Comput. Methods Appl. Mech. Engrg., 106 (1993), 407–429
  29. Ben Leimkuhler, Sebastian Reich, A reversible averaging integrator for multiple time-scale dynamics, J. Comput. Phys., 171 (2001), 95–114 [ISI] [MathRev]
  30. Frank Lempio, Vladimir Veliov, Discrete approximations of differential inclusions, Bayreuth. Math. Schr., (1998), 149–232 [MathRev]
  31. A. Lew, J. Marsden, M. Ortiz, M. West, Asynchronous variational integrators, Arch. Ration. Mech. Anal., 167 (2003), 85–146 [MathRev]
  32. Mongi Mabrouk, A unified variational model for the dynamics of perfect unilateral constraints, Eur. J. Mech. A Solids, 17 (1998), 819–842 [MathRev]
  33. Manuel Monteiro Marques, Chocs inélastiques standards: un résultat d'existence, Sém. Anal. Convexe, 15 (1985), 0–0Exp. No. 4, 32
  34. Manuel Monteiro Marques, Differential inclusions in nonsmooth mechanical problems, Progress in Nonlinear Differential Equations and their Applications, 9, Birkhäuser Verlag, 1993x+179, Shocks and dry friction [MathRev]
  35. Jerrold Marsden, George Patrick, Steve Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351–395
  36. Jerrold Marsden, Tudor Ratiu, Introduction to mechanics and symmetry, Texts in Applied Mathematics, Vol. 17, Springer-Verlag, 1994xvi+500, A basic exposition of classical mechanical systems [MathRev]
  37. J. E. Marsden and M. West, Discrete mechanics and variational integrators, in Acta Numerica, Acta Numer. 10, Cambridge University Press, Cambridge, UK, 2001, pp. 357–514.
  38. Jean-Jacques Moreau, Une formulation du contact à frottement sec; application au calcul numérique, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 302 (1986), 799–801 [MathRev]
  39. J.-J. Moreau, Unilateral contact and dry friction in finite freedom dynamics, in Nonsmooth Mechanics and Applications, J.-J. Moreau and P. D. Panagiotopoulos, eds., CISM Courses and Lectures 302, Springer-Verlag, Vienna, 1988, pp. 1–82.
  40. J. Moreau, Numerical aspects of the sweeping process, Comput. Methods Appl. Mech. Engrg., 177 (1999), 329–349, Computational modeling of contact and friction
  41. A. Pandolfi, C. Kane, J. Marsden, M. Ortiz, Time-discretized variational formulation of non-smooth frictional contact, Internat. J. Numer. Methods Engrg., 53 (2002), 1801–1829
  42. Laetitia Paoli, Michelle Schatzman, Schéma numérique pour un modèle de vibrations avec contraintes unilatérales et perte d'énergie aux impacts, en dimension finie, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 211–215
  43. L. Paoli, M. Schatzman, Mouvement à un nombre fini de degrés de liberté avec contraintes unilatérales: cas avec perte d'énergie, RAIRO Modél. Math. Anal. Numér., 27 (1993), 673–717 [MathRev]
  44. Laetitia Paoli, Michelle Schatzman, Approximation et existence en vibro-impact, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 1103–1107
  45. D. Peric and D. R. J. Owen, Computational model for 3-d contact problems with friction based on the penalty method, Internat. J. Numer. Methods Engrg., 35 (1992), pp. 1289–1309.
  46. F. Pfeiffer, Unilateral problems of dynamics, Archive of Applied Mechanics, 69 (1999), pp. 503–527. [ISI]
  47. Friedrich Pfeiffer, Christoph Glocker, Multibody dynamics with unilateral contacts, Wiley Series in Nonlinear Science, John Wiley & Sons Inc., 1996xiv+317, A Wiley-Interscience Publication [MathRev]
  48. E. Pires, J. Oden, Analysis of contact problems with friction under oscillating loads, Comput. Methods Appl. Mech. Engrg., 39 (1983), 337–362
  49. R. Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, 1970xviii+451 [ZentralblattMath] [MathRev]
  50. J. Sanz-Serna, M. Calvo, Numerical Hamiltonian problems, Applied Mathematics and Mathematical Computation, Vol. 7, Chapman & Hall, 1994xii+207 [MathRev]
  51. Michelle Schatzman, Sur une classe de problèmes hyperboliques non linéaires, C. R. Acad. Sci. Paris Sér. A-B, 277 (1973), 0–0A671–A674
  52. Michelle Schatzman, A class of nonlinear differential equations of second order in time, Nonlinear Anal., 2 (1978), 355–373 [MathRev]
  53. Michelle Schatzman, Claude-Henri Lamarque, Jérôme Bastien, An ill-posed mechanical problem with friction, Eur. J. Mech. A Solids, 18 (1999), 415–420
  54. Juan Simo, Peter Wriggers, Robert Taylor, A perturbed Lagrangian formulation for the finite element solution of contact problems, Comput. Methods Appl. Mech. Engrg., 50 (1985), 163–180
  55. David Stewart, Convergence of a time-stepping scheme for rigid-body dynamics and resolution of Painlevé's problem, Arch. Ration. Mech. Anal., 145 (1998), 215–260 [Inspec] [ISI] [MathRev]
  56. David Stewart, Rigid-body dynamics with friction and impact, SIAM Rev., 42 (2000), 3–39 [MathRev]
  57. R. L. Taylor and P. Papadopoulos, On a finite-element method for dynamic contact impact problems, Internat. J. Numer. Methods Engrg., 36 (1993), pp. 2123–2140. [Inspec] [ISI]
  58. Luther White, J. Oden, Dynamics and control of viscoelastic solids with contact and friction effects, Nonlinear Anal., 13 (1989), 459–474
  59. P. Wriggers, T. V. Van, and E. Stein, Finite-element formulation of large deformation impact-contact problems with friction, Comput. & Structures, 37 (1990), pp. 319–331.
  60. P. Wriggers and G. Zavarise, Application of augmented Lagrangian techniques for nonlinear constitutive laws in contact interfaces, Comm. Numer. Methods Engrg., 9 (1993), pp. 815–824. [Inspec]
  61. L. Young, Lectures on the calculus of variations and optimal control theory, Foreword by Wendell H. Fleming, W. B. Saunders Co., 1969xi+331 [ZentralblattMath] [MathRev]