Preconditioning iterative algorithm for the electromagnetic scattering from a large cavity

A preconditioning iterative algorithm is proposed for solving electromagnetic scattering from an open cavity embedded in an infinite ground plane. In this iterative algorithm, a physical model with a vertically layered medium is employed as a preconditioner of the model of general media. A fast algorithm developed in (SIAM J. Sci. Comput. 2005; 27 :553–574) is applied for solving the model of layered media and classical Krylov subspace methods, restarted GMRES, COCG, and BiCGstab are employed for solving the preconditioned system. Our numerical experiments on cavity models with large numbers of mesh points and large wave numbers show that the algorithm is efficient and the number of iterations is independent of the number of mesh points and dependent upon the wave number. Copyright © 2008 John Wiley & Sons, Ltd.     

The method of integral equation for scattering problem with a mixed crack

We consider the scattering of an electromagnetic time‐harmonic plane wave by an infinite cylinder having a mixed open crack (or arc) in R² as the cross section. The crack is made up of two parts, and one of the two parts is (possibly) coated by a material with surface impedance λ. We transform the scattering problem into a system of boundary integral equations by adopting a potential approach, and establish the existence and uniqueness of a weak solution to the system by the Fredholm theory. Copyright © 2011 John Wiley & Sons, Ltd.     

Boundary integral method for scattering problems with cracks buried in a piecewise homogeneous medium

We consider the scattering of time‐harmonic acoustic plane waves by a crack buried in a piecewise homogeneous medium. The integral representation for a solution is obtained in the form of potentials by using Green's formula. The density in potentials satisfies the uniquely solvable Fredholm integral equation. Then we obtain the existence and uniqueness of the solution. Copyright © 2011 John Wiley & Sons, Ltd.     

Computation of waveguide scattering matrix near thresholds

A waveguide occupies infinite strip with one or several narrows on a two-dimensional (2D) plane and is governed by the Helmholtz equation with Dirichlet boundary condition. On the waveguide continuous spectrum, which coincides with a half-axis, a scattering matrix is defined. At each point of the continuous spectrum this matrix has finite size, which changes at thresholds. The thresholds form a sequence of positive numbers increasing to infinity. Approximate calculation of the scattering matrix in a threshold vicinity requires special treatment. We discuss and compare two methods of numerical approximation to the scattering matrix near a threshold.     

A remark on the single scattering preconditioner applied to boundary integral equations

This article deals with boundary integral equation preconditioning for the multiple scattering problem. The focus is put on the single scattering preconditioner, corresponding to the diagonal part of the integral operator, for which two results are proved. Indeed, after applying this geometric preconditioner, it appears that, firstly, every direct integral equations become identical to each other, and secondly, that the indirect integral equation of Brakhage–Werner becomes equal to the direct integral equations, up to a change of basis. These properties imply in particular that the convergence rate of a Krylov subspaces solver will be exactly the same for every preconditioned integral equations. To illustrate this, some numerical simulations are provided at the end of the paper.     

Water wave scattering by finite arrays of circular structures

The scattering of small amplitude water waves by a finite arrayof locally axisymmetric structures is considered. Regions ofvarying quiescent depth are included and their axisymmetricnature, together with a mild-slope approximation, permits anadaptation of well-known interaction theory which ultimatelyreduces the problem to a simple numerical calculation. Numericalresults are given and effects due to regions of varying depthon wave loading and free-surface elevation are presented.     

On radiation conditions for rough surface scattering problems

^** Email: arens{at}numathics.com^*** Email: hohage{at}math.uni-goettingen.de It is well known that Sommerfeld's radiation condition is nota valid characterization of outgoing waves for scattering problemsat rough surfaces. Instead, a radiation condition called upwardpropagating radiation condition (UPRC) is commonly used. Recently,a different radiation condition called the pole condition hasbeen investigated for scattering problems at bounded obstacles.In this paper we show the equivalence between the UPRC and thepole condition. In doing so, we give a rigorous interpretationof a formula called the angular spectrum representation forDirichlet data in the space of bounded continuous functions.     

Variational formulations for scattering in a three‐dimensional acoustic waveguide

Variational formulations for direct time‐harmonic scattering problems in a three‐dimensional waveguide are formulated and analyzed. We prove that the operators defined by the corresponding forms satisfy a Gårding inequality in adequately chosen spaces of test and trial functions and depend analytically on the wavenumber except at the modal numbers of the waveguide. It is also shown that these operators are strictly coercive if the wavenumber is small enough. It follows that these scattering problems are uniquely solvable except possibly for an infinite series of exceptional values of the wavenumber with no finite accumulation point. Furthermore, two geometric conditions for an obstacle are given, under which uniqueness of solution always holds in the case of a Dirichlet problem. Copyright © 2007 John Wiley & Sons, Ltd.     

求解二维多区域声波散射问题的边界元方法

对于多散射区域的声波散射问题的外Neumann边值问题,用单层位势来逼近每个散射域上的散射波,再利用位势理论的跳跃关系将问题转换为第二类边界积分方程组的求解问题,然后用Nystrom方法进行了求解.对多个随机散射区域的声波散射问题,数值例子体现了该求解方法的可行性和准确性.     

Boundary layer techniques for solving the Helmholtz equation in the presence of small inhomogeneities

We consider solutions to the Helmholtz equation in two and three dimensions. Based on layer potential techniques we provide for such solutions a rigorous systematic derivation of complete asymptotic expansions of perturbations resulting from the presence of diametrically small inhomogeneities with constitutive parameters different from those of the background medium. It is expected that our results will find important applications for developing effective algorithms for reconstructing small dielectric inhomogeneities from boundary measurements.     

Reconstructing small perturbations of an obstacle for acoustic waves from boundary measurements on the perturbed shape itself

We derive relationships between the shape deformation of an impenetrable obstacle and boundary measurements of scattering fields on the perturbed shape itself. Our derivation is rigorous by using a systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation). We extend these techniques to derive asymptotic expansions of the Dirichlet-to-Neumann (DNO) and Neumann-to-Dirichlet (NDO) operators in terms of the small perturbations of the obstacle as well as relationships between the shape deformation of an obstacle and boundary measurements of DNO or NDO on the perturbed shape itself. All relationships lead us to very effective algorithms for determining lower order Fourier coefficients of the shape perturbation of the obstacle.     

Convergence in competition models with small diffusion coefficients

It is well known that for reaction-diffusion 2-species Lotka-Volterra competition models with spatially independent reaction terms, global stability of an equilibrium for the reaction system implies global stability for the reaction-diffusion system. This is not in general true for spatially inhomogeneous models. We show here that for an important range of such models, for small enough diffusion coefficients, global convergence to an equilibrium holds for the reaction-diffusion system, if for each point in space the reaction system has a globally attracting hyperbolic equilibrium. This work is planned as an initial step towards understanding the connection between the asymptotics of reaction-diffusion systems with small diffusion coefficients and that of the corresponding reaction systems.     

Decay and scattering of small solutions for Rosenau equations

We study the long-time behavior of small solutions of the Cauchy problem for a Rosenau equation. For a class of nonlinearity of the perturbation, the global small solution was obtained, and the decay and scattering for small amplitude solution are established.     

Sharp norm estimates for a class of integral operators

ABSTRACT

Norm comparison inequalities for two integral operators with radial kernels are established. Sharp norm estimates for operators with monotone and convex/concave kernels are obtained. Integral analogues of Bennett's estimates for summability matrices are given. The exact operator norms with power weights are also obtained for a class of integral operators with radial quasimonotone kernels.     

Analysis of the scattering by an unbounded rough surface

This paper is concerned with the mathematical analysis of the solution for the wave propagation from the scattering by an unbounded penetrable rough surface. Throughout, the wavenumber is assumed to have a nonzero imaginary part that accounts for the energy absorption. The scattering problem is modeled as a boundary value problem governed by the Helmholtz equation with transparent boundary conditions proposed on plane surfaces confining the scattering surface. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. Furthermore, the scattering problem is investigated for the case when the scattering profile is a sufficiently small and smooth deformation of a plane surface. Under this assumption, the problem is equivalently formulated into a set of two‐point boundary value problems in the frequency domain, and the analytical solution, in the form of an infinite series, is deduced by using a boundary perturbation technique combined with the transformed field expansion approach. Copyright © 2012 John Wiley & Sons, Ltd.     

The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces

This paper is concerned with the inverse problem of reconstructing an infinite, locally rough interface from the scattered field measured on line segments above and below the interface in two dimensions. We extend the Kirsch-Kress method originally developed for inverse obstacle scattering problems to the above inverse transmission problem with unbounded interfaces. To this end, we reformulate our inverse problem as a nonlinear optimization problem with a Tikhonov regularization term. We prove the convergence of the optimization problem when the regularization parameter tends to zero. Finally, numerical experiments are carried out to show the validity of the inversion algorithm.     

Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems

We consider the approximation of the frequency domain three-dimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the time-harmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition. A truncated (computational) domain is then defined, which covers the transition region. The truncated domain need only have a minimally smooth outer boundary (e.g., Lipschitz continuous). We consider the truncated PML problem which results when a perfectly conducting boundary condition is imposed on the outer boundary of the truncated domain. The existence and uniqueness of solutions to the truncated PML problem will be shown provided that the truncated domain is sufficiently large, e.g., contains a sphere of radius . We also show exponential (in the parameter ) convergence of the truncated PML solution to the solution of the original scattering problem inside the transition layer.

Our results are important in that they are the first to show that the truncated PML problem can be posed on a domain with nonsmooth outer boundary. This allows the use of approximation based on polygonal meshes. In addition, even though the transition coefficients depend on spherical geometry, they can be made arbitrarily smooth and hence the resulting problems are amenable to numerical quadrature. Approximation schemes based on our analysis are the focus of future research.

     

High frequency scattering by convex curvilinear polygons

We consider the scattering of a time-harmonic acoustic incident plane wave by a sound soft convex curvilinear polygon with Lipschitz boundary. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency of the incident wave. Here we propose a novel Galerkin boundary element method with a hybrid approximation space, consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes; a uniform mesh on illuminated sides, and graded meshes refined towards the corners of the polygon on illuminated and shadow sides. Numerical experiments suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy need only grow logarithmically as the frequency of the incident wave increases.     

A stable well-conditioned integral equation for electromagnetism scattering

Finding a formulation for electromagnetic scattering of surfaces which is both well-posed and produces a well-conditioned linear system is still a challenging problem. We here propose one such formulation valid in the high-frequency regime. The mathematical analysis is provided and numerical results on rather complex geometries show the performance of the method.