Spatial Dispersion of Elastic Waves in Transversely Isotropic Media Using Lagrange Spectral Element Method

Abstract

Dispersion is studied for the two-dimensional propagation of elastic waves in transversely isotropic media using the Lagrange spectral element method. Spectral element matrices are derived as the tensor product of elementary second-order tensors. Gauss-Lobatto-Legendre points are used for the interpolation of Lagrange-type shape functions as well as for the numerical integration to obtain elementary matrices. The Rayleigh quotient approximation technique is employed to find the solution of the eigenvalue problem, which is obtained from the semidiscretized form of the elastic wave equation for propagation of plane harmonic waves. Variations of errors in the phase/group velocities of bulk waves are depicted graphically with the order of interpolation polynomial, angle with the symmetry axis, and the time discretization. Error analysis clearly demonstrated the effectiveness of the Lagrange spectral element method for wave simulation in a transversely isotropic medium.