78 research outputs found
Near-field imaging of locally perturbed periodic surfaces
This paper concerns the inverse scattering problem to reconstruct a locally
perturbed periodic surface. Different from scattering problems with
quasi-periodic incident fields and periodic surfaces, the scattered fields are
no longer quasi-periodic. Thus the classical method for quasi-periodic
scattering problems no longer works. In this paper, we apply a Floquet-Bloch
transform based numerical method to reconstruct both the unknown periodic part
and the unknown local perturbation from the near-field data.
By transforming the original scattering problem into one defined in an
infinite rectangle, the information of the surface is included in the
coefficients. The numerical scheme contains two steps. The first step is to
obtain an initial guess, i.e., the locations of both the periodic surfaces and
the local perturbations, from a sampling method. The second step is to
reconstruct the surface. As is proved in this paper, for some incident fields,
the corresponding scattered fields carry little information of the
perturbation. In this case, we use this scattered field to reconstruct the
periodic surface. Then we could apply the data that carries more information of
the perturbation to reconstruct the local perturbation. The Newton-CG method is
applied to solve the associated optimization problems. Numerical examples are
given at the end of this paper to show the efficiency of the numerical method
Numerical methods for scattering problems from multi‐layers with different periodicities
In this paper, we consider a numerical method to solve scattering problems with multi-periodic layers with different periodicities. The main tool applied in this paper is the Bloch transform. With this method, the problem is written into an equivalent coupled family of quasi-periodic problems. As the Bloch transform is only defined for one fixed period, the inhomogeneous layer with another period is simply treated as a non-periodic one. First, we approximate the refractive index by a periodic one where its period is an integer multiple of the fixed period, and it is decomposed by finite number of quasi-periodic functions. Then the coupled system is reduced into a simplified formulation. A convergent finite element method is proposed for the numerical solution, and the numerical method has been applied to several numerical experiments. At the end of this paper, relative errors of the numerical solutions will be shown to illustrate the convergence of the numerical algorithm
Higher order convergence of perfectly matched layers in 3D bi-periodic surface scattering problems
The perfectly matched layer (PML) is a very popular tool in the truncation of
wave scattering in unbounded domains. In Chandler-Wilde & Monk et al. 2009, the
author proposed a conjecture that for scattering problems with rough surfaces,
the PML converges exponentially with respect to the PML parameter in any
compact subset. In the author's previous paper (Zhang et al. 2022), this result
has been proved for periodic surfaces in two dimensional spaces, when the wave
number is not a half integer. In this paper, we prove that the method has a
high order convergence rate in the 3D bi-periodic surface scattering problems.
We extend the 2D results and prove that the exponential convergence still holds
when the wavenumber is smaller than . For lareger wavenumbers, although
exponential convergence is no longer proved, we are able to prove that a higher
order convergence for the PML method
Fast convergence for of perfectly matched layers for scattering with periodic surfaces: the exceptional case
In the author's previous paper (Zhang et al. 2022), exponential convergence
was proved for the perfectly matched layers (PML) approximation of scattering
problems with periodic surfaces in 2D. However, due to the overlapping of
singularities, an exceptional case, i.e., when the wave number is a half
integer, has to be excluded in the proof. However, numerical results for these
cases still have fast convergence rate and this motivates us to go deeper into
these cases. In this paper, we focus on these cases and prove that the fast
convergence result for the discretized form. Numerical examples are also
presented to support our theoretical results
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