Dynamic response of large finite doubly periodic elastic cylinders by the wave finite element method

Periodic media have been extensively investigated due to their industrial relevance and their remarkable ability to inhibit wave propagation within specific frequency bands. For one-dimensional periodic systems, the Wave Finite Element (WFE) method provides an efficient means to compute dispersion relations and dynamic responses. This approach has been successfully extended to two-dimensional periodic structures, incorporating periodicity in both directions and employing finite element modeling along with reduction techniques such as Craig–Bampton and Bloch modal projections, including configurations with damping and anisotropy. Beyond planar systems, curved and helical periodic structures have been modeled using the WFE and semi-analytical finite element methods to capture complex cyclic or helical symmetries for dispersion analysis. Cylindrical configurations have also received attention, ranging from basic vibration studies to wave propagation in multilayered, ribbed, or metamaterial shells, mainly for analyzing dispersion characteristics or representing infinite structures. Recognizing that real structures are finite, the present work focuses on bounded elastic cylinders exhibiting double periodicity. The Wave Finite Element (WFE) method is applied to compute their dynamic response. Starting from the finite element matrices of a substructure, circumferential wavenumbers are imposed to extract axial modes, and the total response is obtained through a linear combination of these modes. Numerical examples demonstrate the efficiency of the method in modeling finite, doubly periodic cylindrical systems, including homogeneous, perforated, and resonant configurations. The short computation time of this approach makes it suitable for structures composed of a large number of substructures.