Accurate solutions of elastic and acoustic waves propagation problems by the linear elements with reduced dispersion

Finite element techniques with reduced dispersion (RD) are suggested for linear acoustic and elastodynamics wave propagation problems using explicit time-integration methods. The linear elements with RD are based on the modified integration rule approach for the mass and stiffness matrices and on averaged mass matrix approach. First, the analytical study of numerical dispersion of the new techniques is applied to 1-D, 2-D and 3-D cases for elastodynamics and 2-D case acoustic wave propagation problems. The 1-D case of elastodynamics coincides to the 1-D case of acoustic wave propagation problems. In contrast to standard linear elements with explicit time-integration methods, the analytical study of the numerical dispersion shows that linear elements with RD yield more accurate results at small time increments; i.e., smaller than the stability limit, in the 2-D and 3-D cases for elastodynamics problems. For the 2-D acoustic wave propagation problems, the most accurate results can be obtained with time increments close to the stability limit.

Numerical solutions of wave propagation problems with high frequency and impact loading may lead to divergent numerical results at mesh refinement. These are explained by the appearance of large, spurious high-frequency oscillations. In order to quantify and suppress these oscillations, a two-stage time-integration technique including a stage of basic computations and a filtering stage, is applied to obtain accurate convergent results as well as significantly reduces the numerical anisotropy of solutions. Numerical results show that for elastodynamics and acoustic problems, compared with standard linear elements at the same accuracy, linear elements with RD significantly reduce the number of degrees of freedom (dof) by factors of $2-3$ or more, $4-9$ or more, and $8-27$ or more in the 1-D, 2-D, and 3-D cases, respectively.

Second, we compare the accuracy of numerical solutions obtained with high-order, standard finite elements, isogeometric elements, spectral elements, and linear elements with RD based on implicit and explicit time-integration methods. We use the 1-D impact problem with simple analytical solutions as a benchmark. The two-stage time-integration technique is applied to obtain accurate solutions for elastodynamics problems using different space-discretization methods. The comparison shows that, at the same number of dof, high-order isogeometric and spectral elements yield more accurate results than the high-order standard elements and linear elements with RD. However, linear elements with RD require less computation time for the same accuracy than other high-order elements.

We also analyze the effect of the size of time increments in the stage of basic computations on the accuracy of numerical results. With the exception of linear elements with lumped mass matrices, accurate solutions with other low- and high-orders elements require small time increments. Increments smaller than the stability limit should be used for long observation times. The time increments should also be inversely proportional to the square root of the observation times for the 2nd-order time-integration methods.