A high-order integral solver for scalar problems of diffraction by screens and apertures in three dimensional space

TL;DR

This paper introduces a high-order integral solver for 3D diffraction problems involving screens and apertures, achieving high accuracy and fast convergence through novel formulations and quadrature rules.

Contribution

The authors develop new integral formulations and quadrature rules that effectively handle singularities, enabling super-algebraic convergence and efficient high-frequency diffraction simulations.

Findings

High-order quadrature rules resolve singularities with super-algebraic convergence.

The solver achieves high accuracy with few iterations for low and high frequencies.

Numerical results demonstrate effectiveness on classical and complex diffraction problems.

Abstract

We present a novel methodology for the numerical solution of problems of diffraction by infinitely thin screens in three dimensional space. Our approach relies on new integral formulations as well as associated high-order quadrature rules. The new integral formulations involve weighted versions of the classical integral operators associated with the thin-screen Dirichlet and Neumann problems as well as a generalization to the open surface problem of the classical Calderon formulae. The high-order quadrature rules we introduce for these operators, in turn, resolve the multiple Green function and edge singularities (which occur at arbitrarily close distances from each other, and which include weakly singular as well as hypersingular kernels) and thus give rise to super-algebraically fast convergence as the discretization sizes are increased. When used in conjunction with Krylov-subspace linear algebra solvers such as GMRES, the resulting solvers produce results of high accuracy in small numbers of iterations for low and high frequencies alike. We demonstrate our methodology with a variety of numerical results for screen and aperture problems at high frequencies---including simulation of classical experiments such as the diffraction by a circular disc (including observation of the famous Poisson spot), interference fringes resulting from diffraction across two nearby circular apertures, as well as more complex geometries consisting of multiple scatterers and cavities.