Explicit Newmark/Verlet algorithm for time integration of the rotational dynamics of rigid bodies

International Journal for Numerical Methods in Engineering

Volumn 62, Issue 15, 2005, Pages 2154-2177

  Krysl, P ^a   Endres, L ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

3D rotation; Explicit integrator; Newmark; Rigid body dynamics; SO(3) group

Indexed keywords

ALGORITHMS; VECTORS;

EXPLICIT NEWMARK/VERLET ALGORITHMS; RIGID BODIES; ROTATIONAL DYNAMICS; VECTOR SPACE ALGORITHM;

RIGIDITY;

ALGORITHM; RIGIDITY; ROTATION;

EID: 17344369847     PISSN: 00295981     EISSN: None     Source Type: Journal    
DOI: 10.1002/nme.1272     Document Type: Article

References (45)

1. Kane C, Marsden JE, Ortiz M, West M. Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. International Journal for Numerical Methods in Engineering 2000; 49(10):1295-1325.
2. Cardona A, Geradin M. A beam of the finite element nonlinear theory with finite rotations. International Journal for Numerical Methods in Engineering 1988; 26:2403-2438.
3. Moser J, Veselov AP. Discrete versions of some classical integrable systems and factorization of matrix polynomials. Communications in Mathematical Physics 1991; 139:217-243.
4. Austin M, Krishnaprasad PS, Wang LS. Almost Lie-Poisson integrators for the rigid body. Journal of Computational Physics 1993; 107 105-117.
5. McLachlan RI. Explicit Lie-Poisson integration and the Euler equations. Physical Review Letters 1993; 71:3043-3046.
6. Leimkuhler B, Reich S. Symplectic integration of constrained Hamiltonian systems. Mathematics of Computation 1994; 65:589-605.
7. Reich S. Symplectic integrators for systems of rigid bodies. Physica D 1994; 76:375-383.
8. Lewis D, Simo JC. Conserving algorithms for the dynamics of Hamiltonian systems of Lie groups. Journal of Nonlinear Science 1995; 4 253-299.
9. Munthe-Kaas H. Lie-Butcher theory for Runge-Kutta methods. BIT 1995; 35:572-587.
10. McLachlan RI, Scovel C. Equivariant constrained symplectic integration. Journal of Nonlinear Science 1995; 16:233-256.
11. Zanna A. The method of iterated commutators for ordinary differential equations on Lie groups. Technical Report 1996/NA12, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England, 1996.
12. Marthinsen A, Munthe-Kaas H, Owren B. Simulation of ordinary differential equations on manifolds: some numerical experiments and verifications. Modelling Identification and Control 1997; 18 l):75-88.
13. Dullweber A, Leimkuhler B, McLachlan R. Symplectic splitting methods for rigid body molecular dynamics. Journal of Chemical Physics 1997; 107(15):5840-5851.
14. Wendlandt JM, Marsden JE. Mechanical integrators derived from a discrete variational principle. Physica D 1997; 106:223-246.
15. Bobenko AI, Lorbeer B, Suris YuB. Integrable discretization's of the Euler top. Journal of Mathematical Physics 1998; 39 12):6668-6683.
16. Munthe-Kaas H. Runge-Kutta methods on Lie groups. BIT 1998; 38 1):92-111.
17. Engo K, Faltinsen S. Numerical integration of Lie-Poisson systems while preserving coadjoint orbits and energy. Reports in Informatics 179 University of Bergen, 1999.
18. Zanna A. Collocation and relaxed collocation for the Fer and Magnus expansions. SIAM Journal of Numerical Analysis 1999; 36 4):1145-1182.
19. Pekarsky S, Marsden JE, Shkoller S. Discrete Euler-Poincaré and Lie-Poisson equations. Nonlinearity 1999; 12:1647-1662.
20. Iserles A, Munthe-Kaas HZ, Norsett SP, Zanna A. Lie-group methods. Acta Numerica 2000; 9:215-365.
21. Rochinha FA, Sampaio R. Nonlinear rigid body dynamics: energy and momentum conserving algorithm. Computational Methods in Engineering Sciences 2000; 1(2):7-18.
22. Holder T, Leimkuhler B, Reich S. Explicit variable step size and time reversible integration. Applied Numerical Mathematics 2001; 39 367-377.
23. Zanna A, Eng K, Munthe-Kaas H. Adjoint and selfadjoint Lie-group methods. BIT 2001; 41(2):395-421.
24. Puso MA. An energy and momentum conserving method for rigid-flexible body dynamics. International Journal for Numerical Methods in Engineering 2002; 53:1393-1414.
25. McLachlan RI, Zanna A. The discrete Moser-Veselov algorithm for the free rigid body (revisited). Reports in Informatics 255, University of Bergen, 2003.
26. Fasso F. Comparison of splitting algorithms for the rigid body. Journal of Computational Physics 2003; 189:527-538.
27. Celledoni E, Owren B. Lie methods for rigid body dynamics and time integration on manifolds. Computer Methods in Applied Mechanics and Engineering 2003; 192:421-438.
28. Budd CJ, Iserles A. Geometric integration: numerical solution of differential equations on manifolds. Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences 1999; 357(1754):945-956.
29. Newmark NM. A method of computation for structural dynamics. Journal of Engineering Mechanics (ASCE) 1959; 67-94.
30. Verlet L. Computer 'experiments' on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Physical Review D 1967; 98-103.
31. Kane C, Marsden JE, Ortiz M. Symplectic-energy-momentum preserving variational integrators. Journal of Mathematical Physics 1999; 40(7):3353-3371.
32. Lubich C, Hairer E, Wanner G. Geometric numerical integration illustrated by the Störmer-Verlet method. Acta Numerica 2003; 399-450.
33. Simo JC, Wong KK. Unconditionally stable algorithms for the orthogonal group that exactly preserve energy and momentum. International Journal for Numerical Methods in Engineering 1991; 31:19-52.
34. Simo JC, Vu-Quoc L. Three-dimensional finite-strain rod nidek: Part II. Computational aspects. Computer Methods in Applied Mechanics and Engineering 1988; 58:79-116.
35. Borri M, Trainelli L, Bottasso CL. On representations and parametrizations of motion. Multibody System Dynamics 2000; 4 129-193.
36. Simo JC, Vu-Quoc L. On the dynamics in space of rods undergoing large motions-a geometrically exact approach. Computer Methods in Applied Mechanics and Engineering 1988; 66:125-161.
37. Svanberg M. An improved leap-frog rotational algorithm. Molecular Physics 1997; 92(6):1085-1088.
38. Vu-Quoc L, Zhang X, Walton OR. A 3-D discrete-element method for dry granular flows of ellipsoidal particles. Computer Methods in Applied Mechanics and Engineering 2000; 187:483-528.
39. Hulbert G. Explicit momentum conserving algorithms for rigid body dynamics. Computers and Structures 1992; 44(6):1291-1303.
40. Thompson WT. Introduction to Space Dynamics. Dover Publications: New York, 1986.
41. Engo K, Marthinsen A. Application of geometric integration to some mechanical problems. Multibody System Dynamics 1998; 2:71-88.
42. Engo K, Marthinsen A. A note on the numerical solution of the heavy top equations. Technical Report, Department of Informatics, University of Bergen, 1999.
43. Crouch PE, Grossman R. Numerical integration of ordinary differential equations on manifolds. Journal of Nonlinear Science 1993; 3 1-33.
44. Simo JC, Tarnow N, Wong KK. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering 1992; 100:63-116.
45. Hairer E, Lubich C, Wanner G. Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, vol. 31. Springer: Berlin, 2002.