Guided elastic waves and perfectly matched layers

On the use of perfectly matched layers in the presence of long or backward propagating guided elastic waves

“…It makes essential use of the orthogonality (or biorthogonality) properties enjoyed by the guided modes and the a priori knowledge that some of these modes are associated with backward and/or long waves. It therefore bears some strong similarities with the method proposed by Skelton et al in [8] to overcome the very same issue, which uses the biorthogonality relations satisfied by the Rayleigh-Lamb modes to separate the forward propagating waves from the backward ones in order to treat them appropriately within the PML. It also shares a bond with the approach proposed by Barnett and Greengard in [11] for an integral representation for quasi-periodic scattering problems, in the sense that it involves the computation of a finite number of "corrections", which measure in some way the failure of the approximate solution to satisfy a radiation condition.…”

“…For classical scattering problems, one of the simplest ways to accomplish this is to employ perfectly matched layers, as explained in the next section. However, this technique is sometimes discarded in elastic waveguides, because it fails when backward propagating modes are present (see [8] for instance). Alternative solutions, which use the modal representation, have been developed [20,14,21], but they may require specific implementation efforts.…”

“…References pertaining to PML for time-harmonic elastic wave propagation problems are, for instance, [22,23,8,24]. Here, only the most straighforward kind of PML, that is, with constant coefficients, is considered, but the whole reasoning still applies if these coefficients are complex-valued functions of the space variables satisfying the conditions stated below.…”

“…We should also add that, compared to the approach of Skelton et al in [8], which involves at least two different PML coefficients (one for the forward propagating modes and one for the backward (or long) ones), our procedure is only a corrective post-processing of the solution obtained using the PML technique in a classical way.…”

“…Investigations connected this behavior to the existence of so-called backward waves, whose group and phase velocities have opposite signs. 1 In their presence, exponentional growth occurs in the layers, rendering the method completely unusable [8,9]. While the generated instabilities are largely discernible in transient simulations, it should be emphasized that it is not the case in time-harmonic ones, the solution effectively computed, usually in a numerically stable manner, simply being not an approximation to the outgoing solution of the problem.…”