Spectral Element Method for the Elastic/Acoustic Waveguide Problem in Anisotropic Metamaterials

Waveguide problems are fundamental to elastic and acoustic wave propagation, where we are interested in finding the propagation constants and modal patterns of waveguide modes. As the waveguide is assumed uniform in one direction, the original 3-D problem can be converted into a so-called 2.5-D problem by using the Fourier transform in that direction. However, the introduction of elastic metamaterials (EMM) broadens the horizon of this subject, and new features are required in EMM waveguides that cannot be obtained by most traditional waveguide solvers. In this work, a spectral element method (SEM) is developed to simulate the elastic/acoustic waveguide problem in anisotropic media with anisotropic mass density and/or negative index parameters. We design an anisotropic density EMM waveguide with our SEM solver to demonstrate several intriguing phenomena. Comparisons with the traditional finite element method (FEM) for several examples show the significant advantages of the SEM in term of accuracy and efficiency.