Spurious reflection of elastic waves in nonuniform finite element grids

Elastic wave propagation in a one-dimensional grid of finite elements whose size if uniform from element to element except at one node is analyzed, using complex variables. It is found that spurious wave reflection, along with an increase of amplitude of the diffracted wave, takes place when a wave passes between two finite elements of different sizes. Spurious reflection is significant only for relatively small wavelengths (less than ten times the size of the larger elements) and is important even for very small differences in element size (10%). If the wave arrives from finite elements of smaller size, the transmitted wave has a larger amplitude than the incident wave although the mean energy flux is less. The consistent mass matrix is found to give much smaller spurious reflections than the lumped mass matrix and to enable resolution of smaller wavelengths. This contrasts with the fact that for numerical stability (and suppression of spurious grid oscillations) the lumped mass matrix is better, and for suppression of wave dispersion a combination of lumped and consistent mass matrices is best. The study is restricted to explicit time-step algorithm, second-order (central) difference formulas, and finite elements with linear spatial expansions. In this case it is found that the time step has negligible effects.