Finite Element Modeling of Linear Elastodynamics Problems With Explicit Time-Integration Methods and Linear Elements With Reduced Dispersion. Comparative Study of Different Finite Element Techniques Used for Elastodynamics.

Abstract eng:
We have developed two finite element techniques with reduced dispersion for linear elastodynamics that are used with explicit time-integration methods. These techniques are based on the modified integration rule for the mass and stiffness matrices and on the averaged mass matrix approaches that lead to the numerical dispersion reduction for linear finite elements. The analytical study of numerical dispersion for the new techniques is carried out in the 1-D, 2-D and 3-D cases. The numerical study of the effectiveness of the dispersion reduction techniques includes two-stage time-integration approach with the filtering stage (developed in our previous papers) that quantifies and removes spurious high-frequency oscillations from numerical results. We have found that in contrast to the standard linear elements with explicit time-integration methods and the lumped mass matrix, the finite element techniques with reduced dispersion yield more accurate results at small time increments (smaller than the stability limit) in the 2D and 3-D cases. The advantages of the new technique are illustrated by the solution of the 1-D and 2-D impact problems. The new approaches with reduced dispersion can be easily implemented into existing finite element codes and lead to significant reduction in computation time at the same accuracy compared with the standard finite element formulations. Finally, we compare the accuracy of the linear elements with reduced dispersion, the spectral low- and high-order elements as well as the isogeometric elements by the solution of the 1-D impact problem. For all these solutions we use two-stage time integration technique with the filtering stage that removes spurious oscillations and allows an accurate comparison of different space discretization techniques used for elastodynamics. It is also interesting to mention that the amount of numerical dissipation at the filtering stage can be used as a quantitative measure for the comparison of accuracy of the different numerical formulations used for elastodynamics.

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