Conference article

Regularized; Stabilized; Variational Methods for Multibodies

Claude Lacoursière
HPC2N/Vrlab, Sweden \ Department of Computing Science, Umeå University, Sweden

Download article

Published in: The 48th Scandinavian Conference on Simulation and Modeling (SIMS 2007); 30-31 October; 2007; Göteborg (Särö)

Linköping Electronic Conference Proceedings 27:5, p. 40-48

Show more +

Published: 2007-12-21


ISSN: 1650-3686 (print), 1650-3740 (online)


A time-discrete formulation of the variational principle of mechanics is used to construct a novel first order; fixed time step integration method for multibody systems subject to mixed constraints. The new stepper; coined Spook; includes physics motivated constraint regularization and stabilization terms. The stepper is proved to be stable for the case of linear constraints; for non-zero regularization and stabilization parameters. For fixed stabilization value; the regularization can be made arbitrarily small; corresponding to arbitrarily stiff penalty forces. The “relaxed” constraint formulation permits a separation of time scales so that stiff forces are treated as relaxed constraints. Constraint stabilization makes the stiff forces modeled this way strictly dissipative; and thus; the stepper essentially filters out the high oscillations; but is rigorously symplectic for the rest of the motion. Spook solves a single linear system per time step and is insensitive to constraint degeneracies for non-zero regularization. In addition; it keeps the constraint violations within bounds of O(h2); where h is the time step. Because it is derived from the discrete variational principle; the stepping scheme globally preserves the symmetries of the physical system. The combination of these features make Spook a very good choice for interactive simulations. Numerical experiments on simple multibody systems are presented to demonstrate the performance and stability properties.


No keywords available


[1] H. C. Andersen; RATTLE: A velocity versionof the SHAKE algorithm for molecular dynamics calculations; J. Comp. Phys.; 52 (1983); pp. 24–34.

[2] U. Ascher; H. Chin; L. Petzold; and S. Reich; Stabilization of constrained mechanical sys-tems with DAEs and invariant manifolds; Mech. Struct. & Mach.; 23 (1995); pp. 135–158.

[3] U. Ascher and P. Lin; Sequential regularization methods for nonlinear higher index DAEs; SIAM J. Scient. Comput; 18 (1997); pp. 160–181.

[4] ; Sequential regularization methods for simu- lating mechanical systems with many closed loops.; SIAM J. Scient. Comput.; 21 (1999); pp. 1244– 1262.

[5] J. Baumgarte; Stabilization of constraints and integrals of motion in dynamical systems; Computer Methods in Applied Mechanics and Engineering; 1 (1972); pp. 1–16.

[6] F. A. Bornemann; Homogenization in Time of Singularly Perturbed Mechanical Systems; vol. 1687 of Lecture notes in mathematics; 0075- 8434; Springer; Berlin; 1998.

[7] V. N. Brendelev; On the realization of con- straints in nonholonomic mechanics; J. Appl. Math. Mech.; 45 (1981); pp. 481–487.

[8] J. Cort´es and S. Mart´inez; Non-holonomic in- tegrators; Nonlinearity; 14 (2001); pp. 1365–1392.

[9] T. A. Davis; Algorithm 832: UMFPACK— an unsymmetric-pattern multifrontal method; ACM Transactions on Mathematical Software; 30 (2004); pp. 196–199.

[10] J. W. Eaton; GNU Octave Manual; 2.9.12 ed.; 2007.

[11] R. E. Gillilan and K. R. Wilson; Shadowing; rare events; and rubber bands - a variational Ver- let algorithm for molecular-dynamics; J. Chem. Phys.; 97 (1992); pp. 1757–1772.

[12] R. Goldenthal; D. Harmon; R. Fattal; M. Bercovier; and E. Grinspun; Efficient simulation of inextensible cloth; in SIGGRAPH ’07: ACM SIGGRAPH 2007 papers; New York; NY; USA; 2007; ACM Press; p. 49.

[13] E. Hairer; C. Lubich; and G. Wanner; Ge- ometric Numerical Integration; vol. 31 of Spring Series in Computational Mathematics; Springer- Verlag; Berlin; Heidelberg; New York; London; Paris; Tokyo; Hong Kong; 2001.

[14] E. Hairer and G. Wanner; Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems; vol. 14 of Springer Series in Computational Mathematics; Springer-Verlag; Berlin; Heidelberg; New York; London; Paris; Tokyo; Hong Kong; second revised edition ed.; 1996.

[15] C. Kane; J. E. Marsden; M. Ortiz; and M. West; Variational integrators and the New- mark algorithm for conservative and dissipative mechanical systems; International Journal forNumerical Methods in Engineering; 49 (2000); p p. 1295–1325.

[16] A. V. Karapetian; On realizing nonholonomic constraints by viscous friction forces and Celtic stones stability; J. Appl. Math. Mech.; 45 (1981); pp. 42–51.

[17] A. J. Kurdila and F. J. Narcowich; Sufficient conditions for penalty formulation methods in an- alytical dynamics; Computational Mechanics; 12 (1993); pp. 81–96.

[18] C. Lacoursi`ere; Ghosts and Machines: Regu- larized Variational Methods for Interactive Sim- ulations of Multibodies with Dry Frictional Con-tacts; PhD thesis; Department of Computing Science; Ume°a University; Sweden; SE-901 87; Ume°a; Sweden; June 2007.

[19] C. Lanczos; The Variational Pinciples of Mechanics; Dover Publications; New York; fourth ed.; 1986.

[20] B. Leimkuhler and S. Reich; Simulating Hamiltonian Dynamics; vol. 14 of Cambridge Monographs on Applied and Computational Mathematics; Cambridge University Press; Cambridge; 2004.

[21] C. Lubich; C. Engstler; U. Nowak; and U. P¨ohle; Numerical integration of constrained mechanical systems using MEXX; Mech. Struct. & Mach.; 23 (1995); pp. 473–497.

[22] J. E. Marsden and M. West; Discrete mechan- ics and variational integrators; Acta Numer.; 10 (2001); pp. 357–514.

[23] H. Rubin and P. Ungar; Motion under a strong constraining force; Communications on Pure and Applied Mathematics; X (1957); pp. 65–87.

[24] J.-P. Ryckaert; G. Ciccotti; and H. J. C. Bendersen; Numerical integration of the Carte- sian equations of motion of a system with con- straints: Molecular dynamics of n-alkanes; J. Comp. Phys.; 23 (1977); pp. 327–341. shake.

[25] L. Verlet; Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard- Jones molecules.; Phys. Rev.; 159 (1967); pp. 98– 103.

Citations in Crossref