An extended semi-analytical finite element method for modeling guided waves in plates with pillared metasurfaces

This work presents a semi-analytical finite element (SAFE) scheme for modeling guided waves in plates equipped with arrays of resonators, known as metasurfaces. The method requires only the finite element discretization of the waveguide's cross-section, with the resonators’ dynamic effects incorporated as a traction condition on the plate surface. Through this approach, the general SAFE framework is extended to account for a metasurface of linear resonators atop the plate, thereby broadening its applicability to metamaterial design. Dispersion properties, including wave propagation and attenuation, band gap information, and wave shapes, are obtained by solving a linearized eigenvalue problem. Furthermore, the method enables the computation of both frequency and time responses for a generic forcing function at an arbitrary source–receiver distance. A viscoelastic steel plate, either in vacuum or in water, coupled to an array of mass–spring–dashpot oscillators, is considered, and Lamb, quasi-Lamb, and quasi-Scholte modes, existing in the fluid-coupled scenario, are computed and discussed. The algorithm offers enormous computational advantages over other techniques established in the field of metamaterials. Consequently, it can greatly aid the development of pillared metaplates and potentially other engineered structures, as the method can be further generalized to address waveguides of arbitrary cross-sections and anisotropic media.