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

Abstract

Time Discontinuous Galerkin methods require the factorization of a matrix larger than that exploited in standard implicit schemes. Therefore, they lend themselves to implementations based on predictor-multicorrector solution algorithms. In this paper, various convergent and computationally efficient iterative methods implemented in the unknown displacements for determining the solution of non linear systems are proposed. The iterative solutions presented here differ from those implemented in the unknown velocities in that they are computationally superior. The results of numerical simulations relevant to Duffing oscillators and to a stiff spring pendulum discretized with finite elements which are designed to evaluate the efficacy of these iterative methods with non-linear systems, show a low-computational expense when compared to earlier iterative schemes.Keywords Time Discontinuous Galerkin methods, Iterative methods, Low-computational expense, Non-linear dynamics IntroductionThe numerical solution of problems in structural dynamics requires the time integration of a system of ordinary differential equations. Finite difference time-stepping algorithms are widely used tools (Hughes 1987, p. 490), but particular attention is required for application to nonlinear problems as they were mainly developed in the context of linear problems. For instance, the trapezoidal rule which is unconditionally stable (A-stable) and nondissipative in the linear regime, does not guarantee a stable time integration in non-linear dynamics and is not able to dissipate high frequencies. These drawbacks have been reported in several studies, see, among others, Simo and Tarnow (1992) and Bauchau et al. (1995). Hence, much research has been devoted to the development of more robust algorithms for non-linear dynamics, in which numerical dissipation in the resolution of the high-frequency range has been introduced (see Kuhl and Ramm 1998; Kuhl and Crisfield 1999).A recent alternative approach to numerical time integration is based on Galerkin formulations in the time domain (Borri and Bottasso 1993). The time interval of interest is partitioned in a number of subintervals, where the response is approximated by means of trial functions in the time variable. The use of discontinuous displacement and velocity fields leads to the class of Time Discontinuous Galerkin (TDG) methods. TDG schemes have been analysed extensively in the context of linear problems (Borri and Bottasso 1993;Cannarozzi and Mancuso 1995;Fung and Leung 1996; Fan et al. 1997a, b) and their superiority in comparison with standard finite difference methods has been demonstrated. These methods are L-stable and, therefore, the higher modes are eliminated almost in a single time step. Recent applications to nonlinear dynamics have confirmed these features (Ruge 1996; Bar-Yoseph et al. 1996a, b;Wiberg and Li 1997), thus indicating that TDG methods can compete successfully with standard schemes when high-quality numerical simulations are required.Nonetheless, these methods require...

Energy–momentum consistent finite element discretization of dynamic finite viscoelasticity

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A new high-order accurate continuous Galerkin method for linear elastodynamics problems

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“…The predictor/multi-corrector solver for the TDG method suggested in [17] is only conditionally stable for thirdorder time approximations and is studied for problems without damping. Many different iterative solvers were developed for the TDG method but only for the linear time approximations (see [1,2,4,5,[18][19][20]25] and others). Simple 1-D numerical tests show that even with a direct solver the new TCG method reduces the computation time by 5-25 times in comparison to that of the second-order methods and is much faster than the TDG method.…”