Perfectly matched layers for two-dimensional open waveguides: Semi-infinite Laguerre-based and unbounded-function finite-thickness formulations

This paper presents an alternative finite element method for the computation of trapped and leaky waves in two-dimensional elastic open waveguides, relying on a Laguerre spectral representation of perfectly matched layers (PMLs). Using a Gauss-Radau-Laguerre quadrature rule, the formulation leads to a semi-infinite approximation of the absorbing layer that naturally captures the outgoing-field decay of leaky waves in complex stretching coordinates, while preserving a linear eigenvalue problem for the axial wavenumber. With a proper choice for the scaling factor of Laguerre functions, the multiple user-defined damping parameters of PMLs can be reduced to a single parameter, namely the PML angle, which governs the amount of leaky modes revealed by the rotation of continua of radiation modes. The formulation is implemented for two-dimensional waveguides within a semi-analytical finite element method and validated on representative examples involving both trapped and leaky modes. For comparison purposes, a finite-thickness PML based on an unbounded hyperbolic absorbing function, originally introduced for acoustic scattering problems, is also adapted here to elastic open waveguide eigenproblems. Compared with conventional finite-thickness PMLs, both formulations achieve a substantial reduction in the number of degrees of freedom required to discretize the exterior domain, while maintaining accurate dispersion characteristics. Both formulations also lead to frequency-explicit PML finite element matrices, thereby avoiding matrix reassembly at each frequency step. The proposed semi-infinite Laguerre formulation further reduces the number of PML degrees of freedom, requiring only three to five PML points to achieve good accuracy, while eliminating the need for tuning a characteristic scaling velocity, leaving the PML angle as the only user-defined parameter.