Efficient numerical solution of acoustic scattering from doubly-periodic arrays of axisymmetric objects

Abstract

We present a high-order accurate boundary-based solver for three-dimensional (3D) frequency-domain scattering from a doubly-periodic grating of smooth axisymmetric sound-hard or transmission obstacles. We build the one-obstacle solution operator using separation into P azimuthal modes via the FFT, the method of fundamental solutions (with N proxy points lying on a curve), and dense direct least-squares solves; the effort is O(N 3 P) with a small constant. Periodizing then combines fast multipole summation of nearest neighbors with an auxiliary global Helmholtz basis expansion to represent the distant contributions, and enforcing quasiperiodicity and radiation conditions on the unit cell walls. Eliminating the auxiliary coefficients, and preconditioning with the one-obstacle solution operator, leaves a well-conditioned square linear system that is solved iteratively. The solution time per incident wave is then O(NP) at fixed frequency. Our scheme avoids singular quadratures, periodic Green's functions, and lattice sums, and its convergence rate is unaffected by resonances within obstacles. We include numerical examples such as scattering from a grating of period 13λ × 13λ comprising highly-resonant sound-hard "cups" each needing NP = 64800 surface unknowns, to 10-digit accuracy, in half an hour on a desktop.

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

“…By further augmenting the linear system to include decay/radiation conditions, the scheme has already proven useful for cases when the periodicity is less than the space dimension, such as singly periodic in 2D [13,22,58] or doubly periodic in 3D [56]. The latter case has applications in electrostatics [54] and Stokes flow [16,52].…”

“…However, with the exception of layered media that vary in only one dimension, finding this tensor requires the numerical solution of "cell problems" [62,66], namely BVPs in which the solution is periodic up to some additive constant expressing the macroscopic driving. In this work we focus on the first two (nonoscillatory) PDEs above, noting that the methods that we present also apply with minor changes to the oscillatory Helmholtz and Maxwell cases, at least up to moderate frequencies [5,13,56]. In this work we focus on the first two (nonoscillatory) PDEs above, noting that the methods that we present also apply with minor changes to the oscillatory Helmholtz and Maxwell cases, at least up to moderate frequencies [5,13,56].…”

“…Application areas span all of the major elliptic PDEs, including the Laplace equation (thermal/electrical conductivity, electrostatics and magnetostatics of composites [12,27,35,38]); the Stokes equations (porous flow in periodic solids [18,26,50,75], sedimentation [1], mobility [69], transport by cilia carpets [16], vesicle dynamics in microfluidic flows [58]); elastostatics (microstructured periodic or random composites [29,36,61,64]); and the Helmholtz and Maxwell equations (phononic and photonic crystals, bandgap materials [42,65]). In this work we focus on the first two (nonoscillatory) PDEs above, noting that the methods that we present also apply with minor changes to the oscillatory Helmholtz and Maxwell cases, at least up to moderate frequencies [5,13,56].…”

An FFT-accelerated direct solver for electromagnetic scattering from penetrable axisymmetric objects

“…We made use of OpenMP for parallelism across decoupled Fourier modes; linear systems were solved via LU -factorization using a standard LAPACK library and the code was compiled using the GCC Fortran compiler. Various fast direct solvers such as [26,37,42,44] could be applied if larger problems were involved, but our examples did not warrant such methods.…”

High-NA achromatic metalenses by inverse design

“…60λ size range. With recent developments in fast electromagnetic solvers [50,51], one can anticipate applying this approach to devices orders of magnitude larger, with small and controllable errors, in the near future. The full-device optimizations enable us to overcome the efficiency losses and single-functionality limitations associated with assuming small, periodic unit cells, which are especially prominent at high NA.…”