Abstract
We describe a class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are characterized by the following distinguishing attributes: i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; ii) the algorithms derive from a spacetime form of a discrete version of Hamilton’s principle. As a consequence of this variational structure, the algorithms conserve local energy and momenta exactly, subject to solvability of the local time steps. Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorithm.
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References
Belytschko, T. [1981], Partitioned and adaptive algorithms for explicit time integration. In W. Wunderlich, E. Stein, and K.-J. Bathe, editors, Nonlinear Finite Element Analysis in Structural Mechanics, 572–584. Springer-Verlag.
Belytschko, T. and R. Mullen [1976], Mesh partitions of explicit-implicit time integrators. In K.-J. Bathe, J.T. Oden, and W. Wunderlich, editors, Formulations and Computational Algorithms in Finite Element Analysis, 673–690. MIT Press, 1976.
Bridges, T.J. and S. Reich [1999], Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. (preprint).
Ge, Z. and J. E. Marsden [1988], Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory, Phys. Lett. A, 133, 134–139.
Gonzalez, O. [1996], Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6, 449–468.
Gonzalez, O. and J.C. Simó [1996], On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry, Comp. Meth. Appl. Mech. Eng., 134, 197–222.
Gotay, M.J., J. Isenberg, J.E. Marsden and R. Montgomery [1997], Momentum maps and classical relativistic fields, Part I: Covariant field theory. (Unpublished). Available from http://www.cds.caltech.edu/~marsden.
Kane, C., J. E. Marsden, and M. Ortiz [1999], Symplectic energy-momentum integrators, J. Math. Phys., 40, 3353–3371.
Kane, C., J. E. Marsden, M. Ortiz, and M. West [2000], Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, Int. J. Num. Math. Eng., 49, 1295–1325.
Knuth, D. [1998], The art of computer programming, Addison-Wesley.
Hughes, T. J. R. [1987] The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs, N.J..
Hairer, E. and C. Lubich [2000], Long-time energy conservation of numerical methods for oscillatory differential equations, SIAM Journal on Numerical Analysis, 38, 414–441.
Hairer, E. and C. Lubich [2000], The life-span of backward error analysis for numerical integrators, Numerische Mathematik, 76, 441–462.
Lew, A., M. West, J. E. Marsden, and M. Ortiz, Asynchronous Variational Integrators, in preparation.
Marsden, J. E. and T. J. R. Hughes [1994], Mathematical Foundations of Elasticity. Prentice Hall, 1983. Reprinted by Dover Publications, NY, 1994.
Marsden, J. E., G. W. Patrick, and S. Shkoller [1998], Multisymplectic geometry, variational integrators and nonlinear PDEs, Comm. Math. Phys. 199, 351–395.
Marsden, J. E., S. Pekarsky, S. Shkoller, and M. West [2001], Variational methods, multisymplectic geometry and continuum mechanics, J. Geometry and Physics, 38, 253–284.
Marsden, J. E. and S. Shkoller [1999], Multisymplectic geometry, covariant Hamiltonians and water waves, Math. Proc. Camb. Phil. Soc., 125, 553–575.
Marsden, J. E. and M. West [2001], Discrete variational mechanics and variational integrators, Acta Numerica, 10, 357–514.
Neal, M. O. and T. Belytschko [1989], Explicit-explicit subcycling with noninteger time step ratios for structural dynamic systems, Computers & Structures, 6, 871–880.
Ortiz, M. [1986], A note on energy conservation and stability of nonlinear time stepping algorithms, Computers and Structures, 24, 167–168.
Reich, S. [1999], Backward error analysis for numerical integrators, SIAM Journal on Numerical Analysis, 36, 1549–1570.
Simó, J. C., N. Tarnow, and K. K. Wong [1992], Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Comp. Meth. Appl. Mech. Eng., 100, 63–116.
Smolinski, P. and Y.-S. Wu [1998], An implicit multi-times tep integration method for structural dynamics problems, Computational Mechanics, 22, 337–343.
West, M. [2001], Variational Runge-Kutta methods for ODEs and PDEs. (preprint).
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To Jerry Marsden on the occasion of his 60th birthday
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Lew, A., Ortiz, M. (2002). Asynchronous Variational Integrators. In: Newton, P., Holmes, P., Weinstein, A. (eds) Geometry, Mechanics, and Dynamics. Springer, New York, NY. https://doi.org/10.1007/0-387-21791-6_3
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DOI: https://doi.org/10.1007/0-387-21791-6_3
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