Accurate Numerical Solutions of Structural Dynamics and Wave Propagation Problems Based on New Dispersion-Reduction Technique and New Two-Stage Time-Integration Technique

Abstract eng:
There are the following issues with existing numerical methods for elastodynamics problems (including wave propagation and structural dynamics problems): a) a large dispersion error of space-discretization methods may lead to a great error in space, especially in the 2-D and 3-D cases; b) due to spurious high-frequency oscillations, the lack of reliable numerical techniques that yield an accurate solution of wave propagation in solids; c) the treatment of the error accumulation for long-term integration; d) the selection of an effective time-integration method among known ones; e) the selection of the size of a time increment for a time-integration method with numerical dissipation; f) the increase in accuracy and the reduction of computation time for real-world dynamic problems. A new numerical approach for computer simulation of the dynamic response of linear elastic structures is suggested, resolves the issues listed and includes two main components: a) a new dispersion reduction technique for linear finite elements based on the extension of the modified integration rule method to elastodynamics problems, and b) a new two-stage time-integration technique with the filtering stage. The suggested two-stage time-integration technique includes the stage of basic computations and the filtering stage, new first-, second- and high-order accurate time-integration methods for elastodynamics, and a new calibration procedure for the selection of the minimum necessary amount of numerical dissipation for time-integration methods, new criteria for the selection of time-integration methods for elastodynamics. In contrast to existing approaches, the new technique does not require guesswork for the selection of numerical dissipation and does not require interaction between users and computer codes for the suppression of spurious high-frequency oscillations. Different discretization methods in space such as the finite element method, the spectral element method, the boundary element method, and others can be used with the suggested two-stage time-integration approach. 1-D and 3-D numerical examples show that the new approach used with the finite element method yields an accurate non-oscillatory solution for impact and wave propagation problems and considerably reduces the number of degrees of freedom and the computation time in comparison with existing methods.

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