Effective implementation of a time discontinuous Galerkin method for non-linear structural dynamics

In the last years much research has been devoted to developing accurate and robust time integration methods for non-linear structural dynamics. To this end, filtering out the unwanted high-frequency response has been recognized as a desirable feature and various attempts have been made to introduce such a dissipative feature (see [1] and the references therein, amongst other). An attractive approach is offered by the time discontinuous Galerkin (TDG) method, which has been successfully applied to linear and non-linear structural dynamics, owing to its good stability and accuracy properties (see for example [2,3]). Although it properly combines higher order accuracy and high-frequency dissipation, standard implementations turn out to be much more expensive than classical algorithms, since a larger system of equations should be solved at each time step. Thus, developing effective implementations plays a key role in the practical applicability of the method. The purpose of this work is to establish an efficient iterative procedure to implement the TDG method based on piecewise linear time interpolation for non-linear dynamics. This is achieved by generalizing the algorithm proposed by the authors in [3] for linear problems. The solution strategy is rather different from previous efforts, which are essentially based on conventional iterative schemes (such as Gauss-Jacobi or Gauss-Seidel scheme). In particular, the present procedure is based on an implicit predictor-two-corrector scheme, designed to search for the TDG solution while preserving the TDG stability and dissipative properties at each iteration. Improving accuracy is demanded to iterative corrections. The proposed procedure is simple and offers several advantages. Remarkable computational savings are achieved, since iterative corrections are performed using an effective stiffness matrix, as it occurs in standard algorithms. In linear regime, no more than two iterations are needed to obtain the same overall accuracy of the TDG method. Moreover, the algorithm can be easily implemented into existing finite element codes. Numerical tests confirm the attractiveness of the present procedure, which appears to be competitive with commonly used algorithms and available TDG implementations. REFERENCES [1] D. Kuhl, M. A. Crisfield, Energy conserving and decaying algorithms in non-linear structural dynamics, Int. J. Numer. Meth. Engng., 45, (1999), 569-600. [2] N. E. Wiberg, X. D. Li, Adaptive finite element procedures for linear and non-linear dynamics, Int. J. Numer. Meth. Engng., 46, (1999), 1781-1802. [3] M. Mancuso, F. Ubertini, An efficient integration procedure for linear dynamics based on a time discontinuous Galerkin formulation, Comput. Mech., 32, (2003), 154-168.