On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part II: second-order methods

We present in this paper 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 (EDMC-2)), 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, if desired, The design of the new algorithm is described in detail, including a complete analysis of its dissipation/conservation properties in the fully nonlinear range of finite elasticity. To motivate the different constructions that lead to the dissipative properties of the final scheme, the same arguments are used first in the construction of new linear time-stepping algorithms for the system of linear elastodynamics, including first- and second-order schemes. The new schemes exhibit a rigorous decay of the physical energy, with the second-order schemes formulated in a general two-stage formula accommodating the aforementioned control parameter in the dissipation of the energy. A complete spectral analysis of the new schemes is presented in this linear range to evaluate their different numerical properties. In particular, the dissipative character of the proposed schemes in the high-frequency range is fully demonstrated. In fact, it is shown that the new second-order scheme is L-stable. Most remarkably, the extension of these ideas to the nonlinear range is accomplished avoiding the use of multi-stage formulae, given the freedom gained in using general nonlinear relations, while preserving the conservation laws of the momenta and the corresponding relative equilibria. Several representative numerical simulations are presented in the context of nonlinear elastodynamics to evaluate the performance of the newly proposed schemes. (C) 2001 Elsevier Science B.V. All rights reserved.