A review of temporal and spatial dispersions of linear and quadratic finite elements in linear elastic wave propagation problems

Abstract

The dispersion behaviour of the finite element method, applied to the treatment of stress wave propagation tasks in an elastic solid continuum, is reviewed and complemented with the authors’ contributions in the field, along with substantial details of finite element technology. It is shown how finite element dispersion disqualifies to a certain extent the stress wave propagation modelling and, as such, cannot be completely eradicated. The paper, however, reveals the ways how the dispersion effect (actually, modelling errors) could be minimized. The effects of spatial and temporal dispersions of the finite element method are treated. 1D and 2D linear and quadratic finite elements and their suitability are analysed for use with implicit and explicit integration methods. Historical as well as new, up-to-date approaches are also reviewed. The paper closes with recommendations for values of mesh size and timestep size, mass matrices and direct time integrations with respect to dispersion errors in finite element modelling of elastic wave propagation problems in solids.

Keywords

• dispersion
• elastic wave propagation
• finite element method
• direct time integration
• spurious oscillations