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

We present in this paper the development of a class of time-stepping algorithms for nonlinear elastodynamics that exhibits the controllable numerical dissipation in the high-frequency range required for the robust solution of the resulting numerically stiff systems. To motivate and illustrate better the developments in this general case, we present first the formulation and analysis of these methods for two simple model problems. Namely, we consider a nonlinear elastic spring/mass system and a simplified model of thin elastic beams. As it is discussed in detail in this paper, the conservation by the numerical algorithm of the momenta and corresponding relative equilibria of these characteristic Hamiltonian systems with symmetry is of the main importance. These conservation properties lead for a fixed and finite time step to a correct qualitative picture of the phase space where the discrete dynamics takes place, even in the presence of the desired and controlled numerical dissipation of the energy. This situation is contrasted with traditional “dissipative” numerical schemes, which are shown through rigorous analyses to not only lose their dissipative character in the general nonlinear range, but also the aforementioned conservation properties, thus leading to a qualitatively distorted approximation of the phase dynamics. The key for a successful algorithm in this context is the incorporation of the numerical dissipation in the internal modes of the motion while not affecting the group motions of the systems. The algorithms presented in this work accomplish these goals. The focus in this first part is given to first-order methods. Representative numerical simulations, ranging from applications in nonlinear structural dynamics to nonlinear continuum three-dimensional elastodynamics, are presented in the context of the finite element method to illustrate these ideas and results.