Energy-dissipative momentum-conserving time-stepping algorithms for the dynamics of nonlinear Cosserat rods

Computational Mechanics

Volumn 31, Issue 1-2 SPEC., 2003, Pages 3-26

  Armero, F ^a   Romero, I ^a,b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSIDAD POLITÉCNICA DE MADRID   (Spain)

Author keywords

Cosserat rods; Finite elements; High frequency dissipative time stepping algorithms; Nonlinear dynamics

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; COMPUTER SIMULATION; DEGREES OF FREEDOM (MECHANICS); FINITE ELEMENT METHOD; INTEGRATION; STRAIN MEASUREMENT; STRUCTURAL ANALYSIS;

COSSERAT RODS; ENERGY-DISSIPATIVE MOMENTUM; TIME-STEPPING ALGORITHMS;

DYNAMIC RESPONSE;

EID: 0038495539     PISSN: 01787675     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00466-002-0389-9     Document Type: Article

References (39)

1. Antman SS (1992) Nonlinear problems of elasticity. Springer-Verlag
2. Armero F, Petocz E (1996) Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems. Comput. Meth. Appl. Mech. Eng. 158: 269-300
3. Armero F, Romero I (2001a) On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part I: low order methods for two model problems and nonlinear elastodynamics. Comput. Meth. Appl. Mech. Eng. 190: 2603-2649
4. Armero F, Romero I (2001b) On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics, Part II: high order methods. Comput. Meth. Appl. Mech. Eng. 190: 6783-6824.
5. Armero F, Romero I (2001c) On the objective and conserving integration of geometrically exact rod models. In: Wall WA, Bletzinger KU, Schweizerhof K (Eds) Trends in Computational Structural Mechanics. CIMNE, Barcelona, Spain.
6. Bauchau OA, Theron NJ (1996) Energy decaying scheme for non-linear beam models. Comput. Meth. Appl. Mech. Eng. 134: 37-56
7. Bauchau OA, Damilano G, Theron NJ (1995) Numerical integration of non-linear elastic multibody systems. Int. J. Numer. Meth. Eng. 38(16): 2727-2751
8. Betsch P, Steinman P (2002) Frame-indifferent beam finite elements based upon the geometrically exact beam theory. Int. J. Numer. Meth. Eng. 54: 1775-1788
9. Botasso CL, Borri M (1997) Energy preserving/decaying schemes for non-linear beam dynamics using the helicoidal approximation. Comput. Meth. Appl. Mech. Eng. 143: 393-415
10. Botasso CL, Borri M, Trainelli L (2001) Integration of elastic multibody systems by invariant conserving/dissipating algorithms. II. Numerical schemes and applications. Comput. Meth. Appl. Mech. Eng. 190: 3701-3733
11. Cardona A, Geradin M (1989) Time integration of the equations of motion in mechanism analysis. Comput. Struct. 33(3): 801-820
12. Cosserat E, Cosserat F (1909) Théorie des corps déformables, Traité de Physique. Paris, second edition
13. Crisfield MA (1997) Non-linear finite element analysis of solids and structures. vol. 2: Advanced Topics, John-Wiley & Sons, Chichester
14. Crisfield MA, Galvanetto U, Jelenić G (1997) Dynamics of 3D co-rotational beams. Comput. Mech. 20: 507-519
15. Crisfield MA, Jelenić G (1998) Objectivity of strain measures in geometrically exact 3D beam theory and its finite element implementation. Proc. Roy. Soc. London 455: 1125-1147
16. Crisfield M, Shi J (1994) A Co-rotational element/time-integration strategy for non-linear dynamics. Int. J. Numer. Meth. Eng. 37, 1897-1913
17. Galvanetto U, Crisfield MA (1996) An energy-conserving co-rotational procedure for the dynamics of planar beam structures. Int. J. Numer. Meth. Eng. 39: 2265-2282
18. González O (2000) Exact energy-momentum conserving algorithms for general models in nonlinear elasticity. Comput. Meth. Appl. Mech. Eng. 190: 1763-1783
19. Gruttman F, Sauer R, Wagner W (1998) A geometrically nonlinear eccentric 3D-beam element with arbitrary cross-sections. Comput. Meth. Appl. Mech. Eng. 160: 383-400
20. Hairer E, Wanner G (1991) Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin
21. Hilber HM, Hughes TJR, Taylor RL (1977) Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Eng. Struct. Dynamics 5: 283-292
22. Hughes TJR (1987) The finite element method. Prentice-Hall
23. Ibrahimbegovic A (1995) On finite element implementation of geometrically nonlinear Reissner's beam theory: three-dimensional curved beam elements. Comput. Meth. Appl. Mech. Eng. 122: 11-26
24. Jelenić G, Crisfield MA (1998) Interpolation of rotational variables in nonlinear dynamics of 3D beams. Int. J. Numer. Meth. Eng. 43: 1193-1222
25. Jelenić G, Crisfield MA (1999) Geometrically exact 3d beam theory: implementation of a strain-invariant finite element for statics and dynamics. Comput. Meth. Appl. Mech. Eng. 171: 141-171
26. Jelenić G, Crisfield MA (2000) Dynamic analysis of 3D beams with joints in presence of large rotations. Comput. Meth. Appl. Mech. Eng. (in press)
27. Kane C, Marsden JE, Ortiz M, West M (2000) Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Meth. Eng. 49, 1295-1325
28. Kuhl D, Crisfield MA (1997) Energy conserving and decaying algorithms in non-linear structural dynamics. Int. J. Numer. Meth. Eng. 45: 569-599
29. Kuhl D, Ramm E (1996) Constraint energy momentum algorithm and its application to non-linear dynamics of shells. Comput. Meth. App. Mech. Eng. 136: 293-315
30. Kuhl D, Ramm E (1999) Generalized energy-momentum method for nonlinear adaptive shell analysis. Comput. Meth. App. Mech. Eng. 178: 343-366
31. Prothero A, Robinson A (1974) On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput. 28: 145-162
32. Reissner E (1972) A one-dimensional finite strain beam theory: the plane problem. J. Appl. Math. Phys. (ZAMP) 23: 795-804
33. Romero I, Armero F (2002a) An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy-momentum conserving scheme. Int. J. Numer. Meth. Eng. 54: 1683-1716
34. Romero I, Armero F (2002b) Numerical integration of the stiff dynamics of geometrically exact shells. Int. J. Numer. Meth. Eng. 54: 1043-1086
35. Simo JC (1985) A three-dimensional finite-strain rod model. Part I. The three-dimensional dynamic problem. Comput. Meth. Appl. Mech. Eng. 72(3): 267-304
36. Simo JC, Tarnow N (1992) The discrete energy-momentum method. conserving algorithms for nonlinear elastodynamics. ZAMP 43: 757-793
37. Simo JC, Tarnow N, Doblaré M (1995) Non-linear dynamics of three-dimensional rods: exact energy and momentum conserving algorithms. Int. J. Numer. Meth. Eng. 38(9): 1431-1473
38. Simo JC, Vu-Quoc L (1988) On the dynamics in space of rods undergoing large motions - a geometrically exact approach. Comput. Meth. Appl. Mech. Eng. 66(2): 125-161
39. Wilson EL (1986) A computer program for the dynamic stress analysis of underground structures. SEL Report 68-1, University of California at Berkeley