Accurate time integration of linear elastodynamics problems

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Abstract

The paper deals with the following issues of existing time-integration methods for a semi-discrete system of elastodynamics equations: a) the quantification and the suppression of spurious high frequencies; b) the selection of the amount of numerical dissipation for a time-integration method; and c) accurate time integration of low modes. The finite element method used in the paper or other methods can be applied for the space discretization. A new two-stage time-integration procedure consisting of basic computations and the filtering stage is developed. For accurate integration of all frequencies, a time-integration method with zero (or small) numerical dissipation is applied for basic computations which allow spurious high-frequency oscillations. To filter these oscillations, pre- or/and post-processing is applied using a time-integration method with large numerical dissipation. New implicit first-order and explicit second-order time-continuous Galerkin (TCG) methods with large numerical dissipation are developed for the filtering stage of the two-stage time-integration procedure. A new general expression related to the selection of the minimum necessary amount of numerical dissipation (in terms of a time increment) for pre- or post-processing is suggested. The application of the two-stage time-integration procedure to 1-D and 2-D elastodynamics problems shows its effectiveness. Using the two-stage time-integration procedure, wave propagation and structural dynamics problems are uniformly solved. In contrast to existing approaches that use a method with the same dissipation (or artificial viscosity) for all calculations, the new technique requires no interaction between user and computer code for wave propagation and impact problems (the selection of the size of time increments for existing time-integration methods with numerical dissipation is an issue and is user-defined) and yields accurate and non-oscillatory results.

Original languageEnglish
Pages (from-to)111-148
Number of pages38
JournalCMES - Computer Modeling in Engineering and Sciences
Volume71
Issue number2
StatePublished - 2011

Keywords

  • Elastic waves
  • Elastodynamics
  • Finite elements
  • Numerical dissipation
  • Spurious oscillations
  • Time integration

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