Fast and spectrally accurate numerical methods for perforated screens (with applications to Robin boundary conditions)

Abstract

This paper considers the use of compliant boundary conditions to provide a homogenized model of a finite array of collinear plates, modelling a perforated screen or grating. While the perforated screen formally has a mix of Dirichlet and Neumann boundary conditions, the homogenized model has Robin boundary conditions. Perforated screens form a canonical model in scattering theory, with applications ranging from electromagnetism to aeroacoustics. Interest in perforated media incorporated within larger structures motivates interrogating the appropriateness of homogenized boundary conditions in this case, especially as the homogenized model changes the junction behaviour considered at the extreme edges of the screen. To facilitate effective investigation we consider three numerical methods solving the Helmholtz equation: the unified transform and an iterative Wiener–Hopf approach for the exact problem of a set of collinear rigid plates (the difficult geometry of the problem means that such methods, which converge exponentially, are crucial) and a novel Mathieu function collocation approach to consider a variable compliance applied along the length of a single plate. We detail the relative performance and practical considerations for each method. By comparing solutions obtained using homogenized boundary conditions to the problem of collinear plates, we verify that the constant compliance given in previous theoretical research is appropriate to gain a good estimate of the solution even for a modest number of plates, provided we are sufficiently far into the asymptotic regime. We further investigate tapering the compliance near the extreme endpoints of the screen and find that tapering with $\tanh $ functions reduces the error in the approximation of the far field (if we are sufficiently far into the asymptotic regime). We also find that the number of plates and wavenumber has significant effects, even far into the asymptotic regime. These last two points indicate the importance of modelling end effects to achieve highly accurate results.

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).…”

“…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.…”

Reducing aerofoil–turbulence interaction noise through chordwise-varying porosity

“…Here, we now solve the problem using the Mathieu function collocation method of Colbrook & Priddin (2020) which provides an expansion of in Mathieu functions using separation of variables in elliptic coordinates. A full discussion of this method can be found in Colbrook & Priddin (2020), and user-friendly code for the method can be found at .…”

“…The acoustic response will be achieved through a Mathieu collocation method (Colbrook & Priddin 2020). This method restricts us to solving for the acoustics in zero-lift configurations (zero angle of attack in uniform mean flow).…”

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

“…The convergence to the eigenvalues and eigenfunctions depends on the parameter Q , in general being slower for larger Q . However, the convergence is exponential, yielding machine precision for small truncation parameter n , even for very large Q [42].…”