Superalgebraically Convergent Smoothly-Windowed Lattice Sums for Doubly Periodic Green Functions in Three-Dimensional Space

TL;DR

This paper introduces a highly efficient, superalgebraically convergent algorithm for evaluating quasi-periodic Green functions in 3D, significantly improving computational speed for wave scattering problems with doubly periodic structures.

Contribution

It presents a novel smooth windowing technique that ensures superalgebraic convergence of lattice sums, enhancing the numerical evaluation of Green functions in periodic scattering problems.

Findings

Lattice sums converge faster than any power of the number of terms used.

Numerical results demonstrate high efficiency and accuracy of the proposed method.

The approach is effective except at certain Wood frequencies, which are addressed in Part II.

Abstract

This paper, Part I in a two-part series, presents (i) A simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) An associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain "Wood frequencies" at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on use of smooth windowing functions, gives rise to lattice sums which converge superalgebraically fast--that is, faster than any power of the number of terms used--in sharp contrast with the extremely slow convergence exhibited by the corresponding sums in absence of smooth windowing. (The Wood-frequency problem is treated in Part II.) A proof presented in this paper establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety of numerical results demonstrate the practical efficiency of the proposed approach.