Hardy space infinite elements for time-harmonic two-dimensional elastic waveguide problems

TL;DR

This paper introduces a Hardy space infinite element method for time-harmonic elastic waveguide problems that effectively handles modes with different velocity signs, avoiding modal separation and working across frequency intervals.

Contribution

It develops a novel Hardy space infinite element approach based on Laplace transforms that accurately discretizes waveguide problems with mixed mode velocities without modal separation.

Findings

Method successfully computes resonances and converges in numerical tests.

Operates on frequency intervals with frequency-independent operators.

Avoids modal separation, simplifying the analysis of waveguide problems.

Abstract

We consider time-harmonic linear elasticity equations in domains containing two-dimensional semi-infinite strips. Since for such problems there exist modes with different signs of group and phase velocity, standard perfectly matched layer (PML) as well as standard Hardy space infinite element methods fail. We apply a recently developed infinite element method for a physically correct discretization of such waveguide problems which is based on a Laplace transform in propagation direction. In the Laplace domain the space of transformed solutions can be separated into a sum of a space of incoming and a space of outgoing functions where both function spaces are certain Hardy spaces. The Hardy space is chosen such that the construction of a simple infinite element is possible. The method does not use a modal separation and works on intervals of frequencies. On those intervals the involved operators are frequency independent and hence lead to linear eigenvalue problems when computing resonances. Numerical experiments containing convergence tests and resonance problems are included.