Accurate finite element modeling of linear elastodynamics problems with the reduced dispersion error

摘要

It is known that the reduction in the finite element space discretization error for elastodynamics problems is related to the reduction in numerical dispersion of finite elements. In the paper, we extend the modified integration rule technique for the mass and stiffness matrices to the dispersion reduction of linear finite elements for linear elastodynamics. The analytical study of numerical dispersion for the modified integration rule technique and for the averaged mass matrix technique is carried out in the 1-D, 2-D and 3-D cases for harmonic plane waves. In the general case of loading, the numerical study of the effectiveness of the dispersion reduction techniques includes the filtering technique (developed in our previous papers) that identifies and removes spurious high-frequency oscillations. 1-D, 2-D and 3-D impact problems for which all frequencies of the semi-discrete system are excited are solved with the standard approach and with the new dispersion reduction technique. Numerical results show that compared with the standard mass and stiffness matrices, the simple dispersion reduction techniques lead to a considerable decrease in the number of degrees of freedom and computation time at the same accuracy, especially for multi-dimensional problems. A simple quantitative estimation of the effectiveness of the finite element formulations with reduced numerical dispersion compared with the formulation based on the standard mass and stiffness matrices is suggested.