Wave Propagation in Polar Periodic Structures Using Floquet Theory and Finite Element Analysis

Abstract

In this paper, a generalised approximated approach to study wave propagation in structures that exhibit radial and/or circumferential periodicity is presented. Only a circular sector of the structure is studied, which could be a circumferential period or an arbitrary slice according to the kind of periodicity of the structure (radial, circumferential, both radial and circumferential). The slice is then approximated using piecewise Cartesian waveguides, whose wave characteristics are obtained by the theory of wave propagation in periodic Cartesian structures and Finite Element analysis. Wave amplitudes change due to the changes in the geometry of the slice are accommodated in the model assuming that the energy flow through the interfaces of each Cartesian waveguide is the same. Results are validated considering the response of an infinite isotropic thin plate excited by a point harmonic force, showing the accuracy of the method and the computational advantage compared to a standard FE harmonic analysis for infinite structures. A numerical example of a polar periodic structure, mimicking a spider web, is also presented to investigate the potential applications of the method.

Free and Forced Wave Motion in a Two-Dimensional Plate with Radial Periodicity

“…Following a previous study by the same authors [33], this paper presents a simplified adaptation of this WFE technique to structures in polar coordinates exhibiting periodicity in the radial directions. Cylindrical wave propagation is thus estimated exploiting the Floquet theory formulation for an infinite periodic structure in one dimension.…”

A scaled wave finite element method for computing scalar wave radiation and scattering in exterior domains

“…Compared to the SAFE method, the WFE can deal with more general structures and do not need the development of special matrices but can use matrices produced by commercial FE software. This Floquet's theory was applied to exterior problems by [75,76] by dividing the exterior domain into layers and trying to apply the WFE on them with possible corrections on the solution to obtain constant energy flux through the layers. However, as the matrices in the different layers of the exterior domain are not constant, only approximate solutions have been obtained.…”

“…In this paper, we propose to improve these last attempts of [75,76] by coupling the equilibrium equations between adjacent layers as in the WFE approach and which only needs classical finite element matrices, to the expansion of the solution suggested by the scaled boundary finite element. The solution in the exterior domain will be decomposed as a sum of elementary functions made of harmonic waves multiplied by polynomials and finite element approximation vectors.…”