This manuscript presents a fast direct solution technique for solving two dimensional wave scattering problems from quasi-periodic multilayered structures. The fast solver is built from the linear system that results from the discretization of a boundary integral formulation that is robust at Wood's anomalies. When the interface geometries are complex, the linear system is too large to be handled via dense linear algebra. The key building block of the proposed solver is a fast direct direct solver for the large sparse block system that corresponds to the discretization of boundary integral equations. The solver makes use of hierarchical matrix inversion techniques, has a cost that scales linearly with respect to the number of unknowns on the interfaces and the precomputation can be used for all choices of boundary data. By partitioning the remainder of the precomputation into parts based on their dependence on incident angle, the proposed direct solver is efficient for problems involving many incident angles like those that arise in applications. For example for a problem on an eleven layer geometry where the solution is desired for 287 incident angles, the proposed solution technique is 87 times faster than building a new fast direct solver for each new incident angle. An additional feature of the proposed solution technique is that solving a problem where an interface or layer property is changed require an update in the precomputation that cost linearly with respect to the number points on the affected interfaces with a small constant prefactor. The efficiency for modified geometries and multiple solves make the solution technique well suited for optimal design and inverse scattering applications.
Direct discretization of ) is possible via finite difference or finite element methods @cite_13 but it faces two challenges, (i) meshing to the interfaces to maintain accuracy and (ii) enforcing the radiation condition. Meshing can be effectively be handled using mesh generation software such as GMESH @cite_40 . Techniques such as perfectly matched layers @cite_14 can artificially enforce radiation conditions but lose accuracy if the points per wavelength remains fixed (the so-called effect) @cite_20 . Another alternative method for directly discretizing ) is the rigorous-coupled wave analysis (RCWA) or Fourier Modal Method. It is designed for multilayer gratings @cite_33 which is dependent on an iterative solve. While a Fourier factorization method @cite_29 @cite_36 can be used to accelerate convergence of an iterative solver, this solution approach is not ideal for problems with many right hand sides that arise in applications. An additional challenge of RCWA is that it is difficult to apply to arbitrary shaped interfaces.
@cite_14 @cite_33 @cite_36 @cite_29 @cite_40 @cite_13 @cite_20
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SUMMARY The perfectly matched layer absorbing boundary condition has proven to be very efficient for the elastic wave equation written as a first-order system in velocity and stress. We demonstrate how to use this condition for the same equation written as a second-order system in displacement. This facilitates use in the context of numerical schemes based upon such a system, e.g. the finite-element method, the spectral-element method and some finite-difference methods. We illustrate the efficiency of this second-order perfectly matched layer based upon 2-D benchmarks with body and surface waves. A rigorous coupled-wave approach is used to analyze diffraction by general planar gratings bounded by two different media. The grating fringes may have any orientation (slanted or unslanted) with respect to the grating surfaces. The analysis is based on a state-variables representation and results in a unifying, easily computer-implementable matrix formulation of the general planar-grating diffraction problem. Accurate diffraction characteristics are presented for the first time to the authors’ knowledge for general slanted gratings. This present rigorous formulation is compared with rigorous modal theory, approximate two-wave modal theory, approximate multiwave coupled-wave theory, and approximate two-wave coupled-wave theory. Typical errors in the diffraction characteristics introduced by these various approximate theories are evaluated for transmission, slanted, and reflection gratings. Inclusion of higher-order waves in a theory is important for obtaining accurate predictions when forward-diffracted orders are dominant (transmission-grating behavior). Conversely, when backward-diffracted orders dominate (reflection-grating behavior), second derivatives of the field amplitudes and boundary diffraction need to be included to produce accurate results. Two recursive and numerically stable matrix algorithms for modeling layered diffraction gratings, the S-matrix algorithm and the R-matrix algorithm, are systematically presented in a form that is independent of the underlying grating models, geometries, and mountings. Many implementation variants of the algorithms are also presented. Their physical interpretations are given, and their numerical stabilities and efficiencies are discussed in detail. The single most important criterion for achieving unconditional numerical stability with both algorithms is to avoid the exponentially growing functions in every step of the matrix recursion. From the viewpoint of numerical efficiency, the S-matrix algorithm is generally preferred to the R-matrix algorithm, but exceptional cases are noted. The recent reformulation of the coupled-wave method by Lalanne and Morris [ J. Opt. Soc. Am. A13, 779 ( 1996)] and by Granet and Guizal [ J. Opt. Soc. Am. A13, 1019 ( 1996)], which dramatically improves the convergence of the method for met allic gratings in TM polarization, is given a firm mathematical foundation in this paper. The new formulation converges faster because it uniformly satisfies the boundary conditions in the grating region, whereas the old formulations do so only nonuniformly. Mathematical theorems that govern the factorization of the Fourier coefficients of products of functions having jump discontinuities are given. The results of this paper are applicable to any numerical work that requires the Fourier analysis of products of discontinuous periodic functions. Gmsh is an open-source 3-D finite element grid generator with a build-in CAD engine and post-processor. Its design goal is to provide a fast, light and user-friendly meshing tool with parametric input and advanced visualization capabilities. This paper presents the overall philosophy, the main design choices and some of the original algorithms implemented in Gmsh. Copyright (C) 2009 John Wiley & Sons, Ltd. The development of numerical methods for solving the Helmholtz equation, which behaves robustly with respect to the wave number, is a topic of vivid research. It was observed that the solution of the Galerkin finite element method (FEM) differs significantly from the best approximation with increasing wave number. Many attempts have been presented in the literature to eliminate this lack of robustness by various modifications of the classical Galerkin FEM. However, we will prove that, in two and more space dimensions, it is impossible to eliminate this so-called pollution effect. Furthermore, we will present a generalized FEM in one dimension which behaves robustly with respect to the wave number.