Time Adaptive Computations Using Discontinuous and Continuous Galerkin Integration Schemes

Abstract eng:
The present paper is concerned with the time adaptive numerical integration of linear and non-linear structural dynamics using p-Galerkin time integration schemes. Different versions of single and two field formulations of discontinuous as well as continuous Galerkin schemes [1,2] with arbitrary polynomial degree are compared. The order of accuracy, the numerical dissipation and the numerical effort are investigated for this generalized family of Galerkin time integration schemes. The error analyses, defining the order of accuracy, are performed by the comparison with analytically calculated solutions and numerically calculated improved solutions in linear and non-linear dynamics, respectively. It will be shown that the order of accuracy can be arbitrarily chosen by the polynomial degree of the temporal approximations, energy conservation is obtained for continuous Galerkin schemes, numerical dissipation occurs only by using discontinuous Galerkin schemes and, finally, dissipation of artificial higher frequency responses significantly reduces the time integration error in real life structural simulations. Numerically efficient Babuska-Rheinboldt type [3] error estimates as main ingredients of adaptive time integrations are proposed. By numerical simulations it can be seen that this residual based error estimate leads to results in accordance with the error compared to the analytical solution in linear dynamics and the expensive error estimate using a sup-stepping procedure in non-linear dynamics. Finally, the adaptive time stepping procedure by [4] is adapted for higher order accurate Galerkin time integration schemes. Selected examples demonstrate the properties and the performance of the proposed adaptive integration schemes. References [1] G.M. Hulbert, Time Finite Element Methods for Structural Dynamics. International Journal for Numerical Methods in Engineering, 33, 307-331, 1992. [2] M. Borri, C. Bottasso, A General Framework for Interpreting Time Finite Element Formulations. Computational Mechanics, 13, 133-142, 1993. [3] I. Babuska, C. Rheinboldt, A-Posteriori Error Estimates for the Finite Element Method. International Journal for Numerical Methods in Engineering, 12, 1597-1615, 1978. [4] O.C.Zienkiewicz, Y.M. Xie, A Simple Error and Adaptive Time Stepping Procedure for Dynamic Analysis. Earthquake Engineering and Structural Dynamics, 20, 871-887, 1991.

• Export as: BibTeX | MARC | MARCXML | DC | EndNote | NLM | RefWorks
• Add to your basket