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

• We analyze Perfectly Matched Layers in elastic waveguides and time-harmonic regime.
• The boundary conditions at the end of the layer are designed to avoid the coupling of modes.
• PMLs do not select the outgoing solution in the presence of backward propagating modes.
• PMLs can be used however to compute a kind of reduced basis of solutions of the equations.
• The outgoing solution is recovered a posteriori as a linear combination of these solutions.

An efficient method to compute the scattering of a guided wave by a localized defect, in an elastic waveguide of infinite extent and bounded cross section, is considered. It relies on the use of perfectly matched layers (PML) to reduce the problem to a bounded portion of the guide, allowing for a classical finite element discretization. The difficulty here comes from the existence of backward propagating modes, which are not correctly handled by the PML. We propose a simple strategy, based on finite-dimensional linear algebra arguments and using the knowledge of the modes, to recover a correct approximation to the solution with a low additional cost compared to the standard PML approach. Numerical experiments are presented in the two-dimensional case involving Rayleigh–Lamb modes.