A Mathieu function boundary spectral method for scattering by multiple variable poro-elastic plates, with applications to metamaterials and acoustics

Abstract

Many problems in fluid mechanics and acoustics can be modelled by Helmholtz scattering off poro-elastic plates. We develop a boundary spectral method, based on collocation of local Mathieu function expansions, for Helmholtz scattering off multiple variable poro-elastic plates in two dimensions. Such boundary conditions, namely the varying physical parameters and coupled thin-plate equation, present a considerable challenge to current methods. The new method is fast, accurate and flexible, with the ability to compute expansions in thousands (and even tens of thousands) of Mathieu functions, thus making it a favourable method for the considered geometries. Comparisons are made with elastic boundary element methods, where the new method is found to be faster and more accurate. Our solution representation directly provides a sine series approximation of the far-field directivity and can be evaluated near or on the scatterers, meaning that the near field can be computed stably and efficiently. The new method also allows us to examine the effects of varying stiffness along a plate, which is poorly studied due to limitations of other available techniques. We show that a power-law decrease to zero in stiffness parameters gives rise to unexpected scattering and aeroacoustic effects similar to an acoustic black hole metamaterial.

Do we need non-linear corrections? On the boundary Forchheimer equation in acoustic scattering

“…We first consider the case of a single plate, where we compute the solution using separation of variables following the method outlined in [16,17,11]. This induces the well-known expansion of the pressure field in terms of Mathieu functions, initially with unknown coefficients that must be determined by the boundary conditions (see the discussions and references in the above papers for the history of earlier methods that use Mathieu functions, such as the early work of Morse and Rubenstein [19] or the recent boundary integral method of Nigro [20], discussed also in [21,22], for a single rigid impermeable plate).…”

“…Additionally, we expand η a in Chebyshev polynomials (of the first kind). To cope with the resulting non-linear system of equations that determine the coefficients in these expansions (as opposed to the linear systems of equations appearing in [16,17]), we collocate and use Newton's method. Through a local separation of variables, we can also handle multiple plates, as outlined in §3.3 and demonstrated in §4.2, though this is not the focus of the current study.…”

“…where se m (τ ) = se m (Q; τ ) denote sine-elliptic functions and Hse m (Q; ν) = Hse m (ν) denote Mathieu-Hankel functions. Full details of this process can be found in, for example, [21,16,17].…”

“…To implement the non-linear condition, we extend a previous linear boundary collocation method [16,17,11], which represents the solution in terms of local Mathieu function expansions. In the context of the current paper, this method, along with a partitioning of the system according to the different (kinematic and non-linear Forchheimer) boundary conditions, gives rise to a non-linear system of equations (see (3.10)) for the unknown coefficients, which we solve via Newton's method.…”

Acoustic emission of a vortex ring near a porous edge. Part 1: theory

“…Therefore, poroelasticity may enable broadband noise reduction for finite edge sections or aerofoils. More recent investigations and extensions involving finite geometries include a finite one-dimensional rigid plate with a poroelastic extension (Ayton 2016), multiple finite plates with various material properties (Colbrook & Ayton 2019; Colbrook & Kisil 2020), and generalised two-dimensional poroelastic plates with straight, swept or serrated edges (Pimenta, Wolf & Cavalieri 2018).…”