Structural Dynamic Analysis by a Time-Discontinuous Galerkin Finite Element Method

Abstract

SUMMARYThis paper studies a time-discontinuous Galerkin finite element method for structural dynamic problems, by which both displacements and velocities are approximated as piecewise linear functions in the time domain and may be discontinuous at the discrete time levels. A new iterative solution algorithm which involves only one factorization for each fixed time step size and a few iterations at each step is presented for solving the resulted system of coupled equations. By using the jumps of the displacements and the velocities in the total energy norm as error indicators, an adaptive time-stepping procedure for selecting the proper time step size is described. Numerical examples including both single-DOF and multi-DOF problems are used to illustrate the performance of these algorithms. Comparisons with the exact results and/or the results by the Newmark integration scheme are given. It is shown that the time-discontinuous Galerkin finite element method discussed in this study possesses good accuracy (third order) and stability properties, its numerical implementation is not difficult, and the higher computational cost needed in each time step is compensated by use of a larger time step size.

De3: Yet Another High-Order Integration Scheme for Linear Structural Dynamics

“…Related to the time discontinuous Galerkin method [6] that is built on the assumption of possible discontinuities of the underlying integrated field, the method proposed by Bottasso and Trainelli [3] is similarly formulated in terms of the left and right limits of the field variables at the time nodes. Assuming equation (1) to hold with constant symmetric stiffness, damping and mass matrices under the typical positiveness assumptions (K, C ≥ 0 and M > 0), the integration scheme they proposed reads…”

Iterative solutions for implicit Time Discontinuous Galerkin methods applied to non-linear elastodynamics

“…Li and Wiberg (1996) proposed the use of a Jacobi iterative method on block matrices implemented in the unknown velocities for the solution of (9) in the linear case. That implementation involved the factorization of a matrix of the same dimension of that required in a standard finite difference-based time stepping scheme.…”

“…In this paper, initially we analyse the convergence properties of the iterative solutions proposed by Li and Wiberg (1996) and Wiberg and Li (1999) in the non-linear case; and we prove that the iterative solution relying on the…”

“…Therefore, there is an interest in reducing the computational expense without degrading the accuracy and stability properties. Predictor-multicorrector solution procedures are an attractive approach towards this objective and some competitive algorithms of this type have been developed for linear systems (Hulbert 1989;Li and Wiberg 1996;Bonelli et al 2001) as well as for non-linear systems (Wiberg and Li 1997;Wiberg and Li 1999;Bonelli et al 2002;Bursi and Mancuso 2002). In view of very largescale problems, Li and Wiberg (1996) and Wiberg and Li (1999) proposed a reduced matrix for each fixed time step resolving the system of coupled equations in the unknown velocities applying the iterative methods of Jacobi and of Gauss-Seidel, respectively.…”

“…Based on a similar strategy, Chien and Wu (2000) investigated the accuracy and convergence properties as well as the computational effort of Jacobi and Gauss-Seidel iterative solutions applied to linear multiple-degrees-of-freedom (M.D.o.F.) systems.In this paper, initially we analyse the convergence properties of the iterative solutions proposed by Li and Wiberg (1996) and Wiberg and Li (1999) in the non-linear case; and we prove that the iterative solution relying on the …”