A Floquet--Bloch Transform Based Numerical Method for Scattering from Locally Perturbed Periodic Surfaces

Abstract

Scattering problems for periodic structures have been studied a lot in the past few years. A main idea for numerical solution methods is to reduce such problems to one periodicity cell. In contrast to periodic settings, scattering from locally perturbed periodic surfaces is way more challenging. In this paper, we introduce and analyze a new numerical method to simulate scattering from locally perturbed periodic structures based on the Bloch transform. As this transform is applied only in periodic domains, we firstly rewrite the scattering problem artificially in a periodic domain. With the help of the Bloch transform, we secondly transform this problem into a coupled family of quasiperiodic problems posed in the periodicity cell. A numerical scheme then approximates the family of quasiperiodic solutions (we rely on the finite element method) and backtransformation provides the solution to the original scattering problem. In this paper, we give convergence analysis and error bounds for a Galerkin discretization in the spatial and the quasiperiodicity's unit cells. We also provide a simple and efficient way of implementation that does not require numerical integration in the quasiperiodicity, together with numerical examples for scattering from locally perturbed periodic surfaces computed by this scheme.

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.…”

Near-field imaging of locally perturbed periodic surfaces

“…(1) 0 (k|x − y|) belongs to the space H 1 r (Ω p H ) only if r < 0. From [LZ17b], the direct solver introduced in Section 2.2 does not converge.…”

“…Following the arguments in [Lec17,LZ17b], it is easy to prove that when the functions ζ and ζ p are Lipschitz continuous, the variational problem (8) is equivalent to (5). When (5) has a unique solution of H 1 r (Ω p H ) for some |r| < 1, the problem (8) has a unique solution in H r 0 (W * ; H 1 α (D 2π )).…”

“…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].…”