×

Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. (English) Zbl 0764.73096

It is shown that widely used implicit schemes, in particular the classical Newmark family of algorithms and its variants, generally fail to conserve total angular momentum for nonlinear Hamiltonian systems including classical rigid body dynamics, nonlinear elastodynamics, nonlinear rods and nonlinear shells. A general class of implicit time- stepping algorithms is presented which preserves exactly the conservation laws present in a general Hamiltonian system with symmetry, in particular the total angular momentum and the total energy. A complete analysis of these algorithms and a related class of schemes referred to as symplectic integrators is given.

MSC:

74S20 Finite difference methods applied to problems in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
70-08 Computational methods for problems pertaining to mechanics of particles and systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arnold, V. I., Mathematical Methods of Classical Mechanics (1988), Springer: Springer New York
[2] Richtmyer, R. D.; Morton, K. W., Difference Methods for Initial Value Problems (1967), Interscience: Interscience New York · Zbl 0155.47502
[3] Morton, K. W., Initial value problems by finite and other methods, (Jacobs, D. A.H., The State of the Art in Numerical Analysis (1977), Academic Press: Academic Press London), 699-756
[4] Labudde, R. A.; Greenspan, D., Numer. Math., 26, 1-16 (1976), Part II · Zbl 0382.65031
[5] Marciniak, A., Energy conserving, arbitrary order numerical solutions of the \(N\)-body problem, Numer. Math., 45, 207-218 (1984) · Zbl 0527.65057
[6] Simo, J. C.; Marsden, J. E.; Krishnaprasad, P. S., The Hamiltonian structure of nonlinear elasticity: The convected representation of solids, rods and plates, Arch. Rational Mech. Anal., 104, 125-183 (1988) · Zbl 0668.73014
[7] Dahlquist, G., A special stability problem for linear multistep methods, BIT, 3, 27-43 (1963) · Zbl 0123.11703
[8] Kan, Feng, Differences schemes for Hamiltonian formalism and symplectic geometry, J. Comput. Math., 4, 279-289 (1986) · Zbl 0596.65090
[9] Hughes, T. J.R.; Liu, W. K.; Caughy, P., Transient finite element formulations that preserve energy, J. Appl. Mech., 45, 366-370 (1978) · Zbl 0392.73075
[10] Newmark, N. M., A method of computation for structural dynamics, ASCE J. Engrg. Mech. Division, 85, 67-94 (1959)
[11] Goldstein, H., Classical Mechanics (1982), Addison-Wesley: Addison-Wesley Reading, MA
[12] De Vogelaere, R., Methods of integration which preserve the contact transformation property of Hamiltonian equations, Department of Mathematics, University of Notre Dame, Report 4 (1956)
[13] Lasagni, F. M., Canonical Runge-Kutta methods, Z. Angew. Math. Phys., 39, 952-953 (1988) · Zbl 0675.34010
[14] Sanchez-Serna, J. M., Runge-Kutta schemes for Hamiltonian systems, BIT, 28, 877-883 (1973) · Zbl 0655.70013
[15] Chanell, P. J.; Scovel, J. C., Sympletic integration of Hamiltonian systems (1989), Preprint
[16] Zhong, G.; Marsden, J. E., Lie Poisson Hamilton Jacobi theory and Lie Poisson integrators, Physica A (1990), in press · Zbl 1369.70038
[17] Bathe, K. J.; Wilson, E. L., Stability and accuracy analysis of direct integration methods, Earthquake Engrg. Structural Dynamics, 1, 283-291 (1973) · Zbl 0273.65027
[18] Hilber, H. M.; Hughes, T. J.R.; Taylor, R. L., Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Engrg. Structural Dynamics, 5, 283-292 (1977)
[19] Wood, W. L.; Bossak, M.; Zienkiewicz, O. C., An alpha modification of Newmark’s method, Internat. J. Numer. Methods Engrg., 15, 1562-1566 (1981) · Zbl 0441.73106
[20] Hoff, C.; Pahl, P. J., Development of an implicit method with numerical dissipation from a generalized single-step algorithm for structural dynamics, Comput. Methods Appl. Mech. Engrg., 67, 367-385 (1988) · Zbl 0619.73002
[21] Love, A. E.H., The Mathematical Theory of Elasticity (1944), Dover: Dover New York · Zbl 0063.03651
[22] Reissner, E., On a one-dimensional large-displacement finite-strain beam theory, Stud. Appl. Math., 52, 87-95 (1973) · Zbl 0267.73032
[23] Atman, S. S., Kirchhoff problem for nonlinearly elastic rods, Quart. J. Appl. Math., XXXII, 3, 221-240 (1974) · Zbl 0302.73031
[24] Simo, J. C., A finite strain beam formulation. The three-dimensional dynamic problem, Comput. Methods Appl. Mech. Engrg., 49, 55-70 (1985), Part I · Zbl 0583.73037
[25] Simo, J. C.; Wong, K. K., Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum, Internat. J. Numer. Methods Engrg., 31, 19-52 (1991) · Zbl 0825.73960
[26] Simo, J. C.; Fox, D. D., On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization, Comput. Methods Appl. Mech. Engrg., 72, 267-304 (1989) · Zbl 0692.73062
[27] Cardona, A.; Geradin, M., Time integration of the equations of motion in mechanism analysis, Comput. & Structures, 33, 801-820 (1989) · Zbl 0697.73065
[28] Simo, J. C.; Rifai, M. S.; Fox, D. D., On a stress resultant geometrically shell model. Part VI: Conserving algorithms for nonlinear dynamics, Comput. Methods Appl. Mech. Engrg. (1992), to appear · Zbl 0760.73045
[29] Wanner, G., A short proof of nonlinear A-stability, BIT, 16, 226-227 (1976) · Zbl 0329.65048
[30] Simo, J. C., Nonlinear stability of the time-discrete variational problem in nonlinear heat conduction, plasticity and elastoplasticity, Comput. Methods Appl. Mech. Engrg., 88, 111-131 (1991) · Zbl 0751.73066
[31] Simo, J. C.; Doblare, M., Momentum conserving algorithms for nonlinear (geometrically exact) rods and beams (1991), Preprint
[32] Greenspan, D., Conservative numerical methods for “\(x = f(x)\)”, J. Comput. Phys., 56, 28-41 (1984) · Zbl 0561.65056
[33] Marsden, J. E., Elementary Classical Analysis (1974), Freeman: Freeman San Francisco, CA · Zbl 0285.26005
[34] Dennis, J. E.; Schnable, R. L., Numerical Methods for Unconstrained Optimization and Nonlinear Equations (1983), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0579.65058
[35] Golub, G. H.; Van Loan, C. F., Matrix Computations (1989), Johns Hopkins Univ. Press: Johns Hopkins Univ. Press Baltimore, MD · Zbl 0733.65016
[36] Arnold, V. I.; Kozlov, V. V.; Neihstadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics (1987), Springer: Springer Berlin · Zbl 0674.70003
[37] Argyris, J. H., An excursion into large rotations, Comput. Methods Appl. Mech. Engrg., 32, 85-155 (1982) · Zbl 0505.73064
[38] Pars, L. A., A Treatise on Analytical Dynamics (1965), Wiley: Wiley New York · Zbl 0125.12004
[39] Simo, J. C.; Lewis, D.; Marsden, J. E., Stability of relative equilibria. Part I: The reduced energy momentum method, Arch. Rational Mech. Anal. (1991), in press · Zbl 0738.70010
[40] Simo, J. C.; Posbergh, T.; Marsden, J. E., Stability of relative equilibria. Part II: Application to nonlinear elasticity, Arch. Rational Mech. Anal. (1991), in press · Zbl 0738.70011
[41] Argyris, J. H.; Dunne, P. C.; Angelopolous, T., Dynamic response by large step integration, Earthquake Engrg. Structural Dynamics, 2, 185-203 (1973)
[42] Simo, J. C.; Vu-Quoc, L., On the dynamics in space of rods undergoing large motions - A geometrically exact approach, Comput. Methods Appl. Mech. Engrg., 66, 125-161 (1988) · Zbl 0618.73100
[43] Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (1976), Dover: Dover New York · Zbl 0061.41806
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.