A High Order Numerical Method for Scattering from Locally Perturbed Periodic Surfaces

Abstract

In this paper, we will introduce a high order numerical method to solve the scattering problems with non-periodic incident fields and (locally perturbed) periodic surfaces. For the problems we are considering, the classical methods to treat quasi-periodic scattering problems no longer work, while a Bloch transform based numerical method was proposed in [LZ17b]. This numerical method, on one hand, is able to solve this kind of problems convergently; on the other hand, it takes up a lot of time and memory during the computation. The motivation of this paper is to improve this numerical method, from the regularity results of the Bloch transform of the total field, which have been studied in [Zha17]. As the set of the singularities of the total field is discrete in R, and finite in one periodic cell, we are able to improve the numerical method by designing a proper integration contour with special conditions at the singularities. With a good choice of the transformation, we can prove that the new numerical method could possess a super algebraic convergence rate. This new method improves the efficient significantly. At the end of this paper, several numerical results will be provided to show the fast convergence of the new method. The method also provides a possibility to solve more complicated problems efficiently, e.g., three dimensional problems, or electromagnetic scattering problems.

Reconstruction of a local perturbation in inhomogeneous periodic layers from partial near field measurements

“…We show equivalence of the two problems and consider the latter to prove existence of the solution to the scattering problem by applying Fredholm theory for the reduced problem. Moreover, we stay in the framework of the equivalent formulation to introduce a numerical method to approximate the solution to the original problem, which is based on [LZ17a] and [Zha18], where the algorithm for the sound-soft scattering layer is developed. Considering the regularity of the transformed solution w.r.t.…”

“…The Bloch-Floquet transform is a well-known approach in electrical engineering, which is called the array scanning method, see, e.g., [MB79], [Val+08]. Nevertheless, the consideration of applying the transform to scattering problems was given just recently by constructing a numerical scheme and analyzing error bounds for the acoustic and electromagnetic scattering problem in the case of sound-soft boundary conditions (see [LZ17a], [Zha18], [LZ17b]). Moreover, in [HN17] the acoustic scattering problem for an inhomogeneous layer was studied by applying the Bloch-Floquet transform and considering integral equations.…”

Numerical methods for scattering problems in periodic waveguides

“…We apply the finite element method to solve the periodic problem (14). Suppose that Ω 0 is covered by a family of regular and quasi-uniform meshes (see [32,33]) M h with the largest mesh size h 0 > 0.…”

“…Recently, the Floquet-Bloch transform has been applied to both theoretical analysis and numerical simulation for scattering problems in periodic structures. We refer to [9,10,19] for scattering problems with (perturbed) periodic media, to [11][12][13][14] for problems with periodic surfaces, and to [2][3][4] for periodic waveguides. From the Floquet-Bloch theory, the unique solution of the periodic waveguide problem with absorption is written as a contour integral on the unit circle, where the integrand is a family of quasi-periodic solutions depending analytically on the quasi-periodicities.…”

Near-field imaging of locally perturbed periodic surfaces

“…Moreover, if the incident field u i ∈ H 2 r (Ω p H ) and the surfaces are C 2,1 , then the solution belongs to the space H r 0 (W * ; H 2 α (D 2π )). In [LZ17b], a convergent numerical method based on (8) has been proposed for the numerical solution, and a high order method has been proposed in [Zha18].…”