On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. II: Second-order methods. (English) Zbl 1068.74022

[For part I see the authors, ibid. 190, No. 20–21, 2603–2649 (2001; Zbl 1008.74035).]
From the summary: We present the formulation of a new high-frequency dissipative time-stepping algorithm for nonlinear elastodynamics that is second-order accurate in time. The new scheme exhibits unconditional energy dissipation and momentum conservation (and thus the given name of energy-dissipative, momentum-conserving second-order scheme), leading also to the conservation of the relative equilibria of the underlying physical system. The unconditional character of these properties applies not only with respect to the time step size but, equally important, with respect to the considered elastic potential. Moreover, the dissipation properties are fully controlled through an algorithmic parameter, reducing to existing fully conserving schemes, is desired. The design of the new algorithm if described in detail, including a complete analysis of its dissipation/conservation properties in the fully nonlinear range of finite elasticity.


74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74B20 Nonlinear elasticity


Zbl 1008.74035


Full Text: DOI


[1] Armero, F.; Petocz, E., Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems, Comput. Methods Appl. Mech. Eng., 158, 269-300 (1996) · Zbl 0954.74055
[2] Arrmero, F.; Romero, I., 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. Methods Appl. Mech. Eng., 190, 2603-2649 (2001) · Zbl 1008.74035
[3] Bauchau, O. A.; Damilano, G.; Theron, N. J., Numerical integration of nonlinear elastic multi-body systems, Int. J. Numer. Meth. Eng., 38, 2727-2751 (1995) · Zbl 0840.73057
[4] Bauchau, O. A.; Theron, N. J., Energy decaying scheme for non-linear beam models, Comput. Methods Appl. Mech. Eng., 134, 37-56 (1996) · Zbl 0918.73311
[5] Bauchau, O. A.; Joo, T., Computational schemes for non-linear elasto-dynamics, Int. J. Numer. Meth. Eng., 45, 693-719 (1999) · Zbl 0941.74078
[6] Botasso, C.; Borri, M., Integrating rotations, Comput. Methods Appl. Mech. Eng., 164, 307-331 (1998) · Zbl 0961.74029
[7] Chung, J.; Hulbert, G. M., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method, J. Appl. Mech., 60, 371-375 (1993) · Zbl 0775.73337
[8] Crisfield, M. A.; Galvanetto, U.; Jelenic, G., Dynamics of 3-D co-rotational beams, Comput. Mech., 20, 507-519 (1997) · Zbl 0904.73076
[9] Crisfield, M.; Shi, J., A co-rotational element/time-integration strategy for non-linear dynamics, Int. J. Numer. Meth. Eng., 37, 1897-1913 (1994) · Zbl 0804.70002
[10] Dacorogna, B., Direct Methods in the Calculus of Variations (1989), Springer: Springer New York · Zbl 0703.49001
[11] Gonzalez, O.; Simo, J. C., On the stability of symplectic and energy-momentum algorithms for nonlinear hamiltonian systems with symmetry, Comput. Methods Appl. Mech. Eng., 134, 197-222 (1996) · Zbl 0900.70013
[12] Gonzalez, O., Exact energy-momentum conserving algorithms for general models in nonlinear elasticity, Comput. Methods Appl. Mech. Eng., 190, 1763-1783 (2000) · Zbl 1005.74075
[13] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (1991), Springer: Springer Berlin · Zbl 0729.65051
[14] Hilber, H. M.; Hughes, T. J.R.; Taylor, R. L., Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Eng. Struct. Dyn., 5, 283-292 (1977)
[15] Hughes, T. J.R., The Finite Element Method (1987), Prentice-Hall: Prentice-Hall New York
[17] Hughes, T. J.R.; Hulbert, M., Space-time finite element methods for elastodynamics: Formulation and error estimates, Comput. Methods Appl. Mech. Eng., 66, 339-363 (1988) · Zbl 0616.73063
[18] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Eng., 45, 285-312 (1984) · Zbl 0526.76087
[19] Kuhl, D.; Crisfield, M. A., Energy conserving and decaying algorithms in non-linear structural dynamics, Int. J. Numer. Meth. Eng., 45, 569-599 (1997) · Zbl 0946.74078
[20] Kuhl, D.; Ramm, E., Constraint energy momentum algorithm and its application to non-linear dynamics of shells, Comput. Methods Appl. Mech. Eng., 136, 293-315 (1996) · Zbl 0918.73327
[21] Kuhl, D.; Ramm, E., Generalized energy-momentum method for non-linear adaptive shell dynamics, Comput. Methods Appl. Mech. Eng., 178, 343-366 (1999) · Zbl 0968.74030
[23] Newmark, N. M., A method of computation for structural dynamics, J. Eng. Mech. Div., ASCE, 85, 67-94 (1959)
[24] Prothero, A.; Robinson, A., On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comp., 28, 145-162 (1974) · Zbl 0309.65034
[26] Simo, J. C.; Tarnow, N., The discrete energy-momentum method conserving algorithms for nonlinear elastodynamics, ZAMP, 43, 757-793 (1992) · Zbl 0758.73001
[28] Wood, W. L., Practical Time-Stepping Schemes (1990), Clarendon Press: Clarendon Press Oxford · Zbl 0694.65043
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