Keywords: wave propagation, dispersion error, numerical stability, implicit integration, explicit integration, serendipity elements.

The finite element method (FEM) is often employed for the numerical solution of elastic wave propagation. It is known that the spatial and temporal discretization introduces dispersion errors [1]. These parasitic effects do not exist in an ideal elastic continuum. The dispersion behaviour of the FEM results in the change of wave speeds and the distortion of wave fronts. Furthermore, a FE mesh behaves like a frequency filter. This text overviews the fundamentals, computational methods and results accomplished by the dispersion analysis. The emphasis is laid on higher order elements, namely the quadratic serendipity elements and the numerical stability issues.

Belytschko and Mullen published in the late seventies the pioneering paper [1] on dispersion of quadratic one-dimensional elements. They showed that spurious branches in the spectrum existed, called the optical modes. The existence of such modes incurs the presence of the noise associated with the propagation of discontinuities and spurious oscillations. In terms of numerical computation, stability deteriorates. The dispersion analysis also revealed the existence of band gaps known as the stop bands in the frequency spectrum of higher order elements. Within these bands the solution decays exponentially.

In principle, the dispersion errors and spurious oscillations cannot be removed from a numerical solution of the fast transient dynamic problems. Such numerical effects can nonetheless be partially reduced by a suitable choice of the mesh size, direct time integration method, time step and also by the modification of the mass matrix or the quadrature rule. One of the essential pieces of knowledge is that the formulation with the consistent mass matrix overestimates the value of the numerical wave speed, causing the wave front to smear, whereas the lumped mass matrix slows down the wave to lag behind, thereby yielding sharp wave fronts.

[1]

T. Belytschko, R. Mullen, "On dispersive properties of finite element solutions", in J. Miklowitz et al., (Editors), "Modern Problems in Elastic Wave Propagation", John Wiley, New York, 67-82, 1978.

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