P‐wave and S‐wave decomposition in boundary integral equation for plane elastodynamic problems

Abstract

SUMMARYThe method of plane wave basis functions, a subset of the method of Partition of Unity, has previously been applied successfully to ÿnite element and boundary element models for the Helmholtz equation. In this paper we describe the extension of the method to problems of scattering of elastic waves. This problem is more complicated for two reasons. First, the governing equation is now a vector equation and second multiple wave speeds are present, for any given frequency. The formulation has therefore a number of novel features. A full development of the necessary theory is given. Results are presented for some classical problems in the scattering of elastic waves. They demonstrate the same features as those previously obtained for the Helmholtz equation, namely that for a given level of error far fewer degrees of freedom are required in the system matrix. The use of the plane wave basis promises to yield a considerable increase in e ciency over conventional boundary element formulations in elastodynamics.

A comparative study of three BEM for transient dynamic crack analysis of 2-D anisotropic solids

“…To overcome this difficulty, several advanced methods have been proposed in recent years. Here, we just mention the fast multipole method (FMM) [15,18,19], the wave boundary element method (WBEM) or the wave basis functions method [4,16,17] based on the partition of unity method (PUM) [13]. A review of some advanced computational methods for wave simulation in high frequency range has been presented by Bettess [4].…”

Numerical modelling of elastic wave scattering in frequency domain by the partition of unity finite element method

“…For simplicity of the notations, the dependency of the numerical solution u h on the numbers of approximating P and S plane waves, m P and m S , is not indicated. The above approximation can be derived from the Helmholtz decomposition theorem, as discussed in [26]. The unknowns are no longer the nodal values of the displacement u h , but are now the amplitudes A P z,l , and A S z,l , attached to a node z and corresponding to P and S plane waves travelling in the directions d l P and d l S , respectively.…”

“…For time-harmonic elastic wave problems, the plane wave basis decomposition by pressure and shear waves was successfully implemented by Perrey-Debain et al [26] for the two-dimensional boundary integral equation. It has also been successfully applied via the UWVF method by Huttunen et al [27] to elastic wave propagation in two space dimension.…”

Numerical analysis of two plane wave finite element schemes based on the partition of unity method for elastic wave scattering

“…For elastic wave problems, the plane wave basis decomposition was successfully implemented for the two-dimensional boundary integral equation [8]. It has also been successfully implemented via the Ultra Weak variational formulation (UWVF) [9] and via the discontinuous enrichment method (DEM) [10].…”