A Second-Order Method for the Electromagnetic Scattering from a Large Cavity

In this paper,we study the electromagnetic scattering from a two dimen- sional large rectangular open cavity embedded in an infinite ground plane,which is modelled by Helmholtz equations.By introducing nonlocal transparent boundary con- ditions,the problem in the open cavity is reduced to a bounded domain problem.A hypersingular integral operator and a weakly singular integral operator are involved in the TM and TE cases,respectively.A new second-order Toeplitz type approximation and a second-order finite difference scheme are proposed for approximating the hyper- singular integral operator on the aperture and the Helmholtz in the cavity,respectively. The existence and uniqueness of the numerical solution in the TE case are established for arbitrary wavenumbers.A fast algorithm for the second-order approximation is pro- posed for solving the cavity model with layered media.Numerical results show the second-order accuracy and efficiency of the fast algorithm.More important is that the algorithm is easy to implement as a preconditioner for cavity models with more general media.     

A solution method for hypersingular integral equations

The Neumann problem for the Helmholtz equation is considered. The double-layer potential is used to reduce the problem to a hypersingular integral equation. The properties of the hypersingular operator in a neighborhood lead to a method for approximate solution of the hypersingular equation with an arbitrary contour. Some numerical results are reported.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 130–136, 1993.     

Effective semi-analytic integration for hypersingular Galerkin boundary integral equations for the Helmholtz equation in 3D

We deal with the Galerkin discretization of the boundary integral equations corresponding to problems with the Helmholtz equation in 3D. Our main result is the semi-analytic integration for the bilinear form induced by the hypersingular operator. Such computations have already been proposed for the bilinear forms induced by the single-layer and the double-layer potential operators in the monograph The Fast Solution of Boundary Integral Equations by O. Steinbach and S. Rjasanow and we base our computations on these results.     

On efficient preconditioners for iterative solution of a Galerkin boundary element equation for the three-dimensional exterior Helmholtz problem

The paper presents a Galerkin numerical method for solving the hyper-singular boundary integral equations for the exterior Helmholtz problem in three dimensions with a Neumann's boundary condition. Previous work in the topic has often dealt with the collocation method with a piecewise constant approximation because high order collocation and Galerkin methods are not available due to the presence of a hypersingular integral operator. This paper proposes a high order Galerkin method by using singularity subtraction technique to reduce the hyper-singular operator to a weakly singular one. Moreover, we show here how to extend the previous work (J. Appl. Numer. Math. 36 (4) (2001) 475–489) on sparse preconditioners to the Galerkin case leading to fast convergence of two iterative solvers: the conjugate gradient normal method and the generalised minimal residual method. A comparison with the collocation method is also presented for the Helmholtz problem with several wavenumbers.     

On the solvability of a complete two-dimensional hypersingular integral equation

We study the solvability of a complete two-dimensional linear hypersingular integral equation that contains a hypersingular integral operator in which the integral is understood in the sense of Hadamard finite value as well as an integral operator in which the integral is understood in the sense of principal value, an integral operator with a weakly singular kernel, and an integral-free term. We consider smooth solutions in the class of functions that have Hölder continuous derivatives outside a neighborhood of the boundary. We prove the Fredholm alternative and estimate the norm of the solution in a special metric.     

On the numerical solution of a complete two-dimensional hypersingular integral equation by the method of discrete singularities

We study the solvability of a complete two-dimensional linear integral equation with a hypersingular integral understood in the sense of the Hadamard principal value. We justify the convergence of a quadrature-type numerical method for the case in which the equation in question is uniquely solvable. We present an application of the results to the numerical solution of the Neumann boundary value problem on a plane screen for the Helmholtz equation by the surface potential method.     

A Burton-Miller boundary element-free method for Helmholtz problems

A Burton-Miller boundary element-free method is developed by using the Burton-Miller formulation for meshless and boundary-only analysis of Helmholtz problems. The method can produce a unique solution at all wavenumbers and is valid for Dirichlet, Neumann and mixed problems simultaneously. An efficient numerical integration procedure is presented to handle both strongly singular and hypersingular boundary integrals directly and uniformly. Numerical results reveal that this direct meshless method only involves boundary nodes and can deal with Helmholtz problems at extremely large wavenumbers.     

Numerical solution to a two-dimensional hypersingular integral equation and sound propagation in urban areas

A mathematical model of sound propagation from a noise source in urban areas is constructed. The exterior Neumann problem for the scalar Helmholtz equation is reduced to a system of hypersingular integral equations. A numerical method for solving the system of integral equations is described. The convergence of the quadrature formulas underlying the numerical method is estimated. Numerical results are presented for particular applications.     

Analysis of a Diagonal Form of the Fast Multipole Algorithm for Scattering Theory

Discretisation of the integral equations of acoustic scattering yields a system of linear equations with full coefficient matrices. In recent years a number of fast algorithms for the solution of this system have been proposed. In this paper we present a complete analysis for a fast multipole method for the Helmholtz equation. A one-level diagonal form of the multipole method is applied to a hypersingular integral equation arising from 2d scattering theory. The error of the approximation is analysed and the results used to establish the complexity of the method.     

Singular value expansion for the Green function of Helmholtz operator

We consider the integral operator defined on a circular disk, and with kernel the Green function of the Helmholtz operator. We present an analytic framework for the explicit computation of the singular system of this kernel. In particular, the main formulas of this framework are given by a characteristic equation for the singular values and explicit expressions for the corresponding singular functions. We provide also a property of the singular values, that gives an important information for the numerical evaluation of the singular system. Finally, we present a simple numerical experiment, where the singular system computed by a simple implementation of these analytic formulas is compared with the singular system obtained by a discretization of the Green function of the Helmholtz operator.     

Comparison between the shifted-Laplacian preconditioning and the controllability methods for computational acoustics

Processes that can be modelled with numerical calculations of acoustic pressure fields include medical and industrial ultrasound, echo sounding, and environmental noise. We present two methods for making these calculations based on Helmholtz equation. The first method is based directly on the complex-valued Helmholtz equation and an algebraic multigrid approximation of the discretized shifted-Laplacian operator; i.e. the damped Helmholtz operator as a preconditioner. The second approach returns to a transient wave equation, and finds the time-periodic solution using a controllability technique. We concentrate on acoustic problems, but our methods can be used for other types of Helmholtz problems as well. Numerical experiments show that the control method takes more CPU time, whereas the shifted-Laplacian method has larger memory requirement.     

Use of Shifted Laplacian Operators for Solving Indefinite Helmholtz Equations

A shifted Laplacian operator is obtained from the Helmholtz operator by adding a complex damping. It serves as a basic tool in the most successful multigrid approach for solving highly indefinite Helmholtz equations — a Shifted Laplacian preconditioner for Krylov-type methods. Such preconditioning significantly accelerates Krylov iterations, much more so than the multigrid based on original Helmholtz equations. In this paper, we compare approximation and relaxation properties of the Helmholtz operator with and without the complex shift, and, based on our observations, propose a new hybrid approach that combines the two. Our analytical conclusions are supported by two-dimensional numerical results.     

Partial radiation conditions and hypersingular integral operators

The structure of the partial radiation conditions is analyzed. It is found that the principal part of the operator defining these conditions defines a hypersingular operator. The series corresponding to the operator associated with the boundary conditions is transformed to a rapidly converging series.     

Spectrum created by line defects in periodic structures

We study a Helmholtz‐type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a three‐dimensional periodic medium; the defect is infinitely extended in one direction, but compactly supported in the remaining two. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We will show that even small perturbations lead to additional spectrum in the spectral gaps of the unperturbed operator and investigate some properties of the spectrum that is created.     

The numerical solution of an evolution problem of second order in time on a closed smooth boundary

We consider an initial value problem for the second-order differential equation with a Dirichlet-to-Neumann operator coefficient. For the numerical solution we carry out semi-discretization by the Laguerre transformation with respect to the time variable. Then an infinite system of the stationary operator equations is obtained. By potential theory, the operator equations are reduced to boundary integral equations of the second kind with logarithmic or hypersingular kernels. The full discretization is realized by Nyström's method which is based on the trigonometric quadrature rules. Numerical tests confirm the ability of the method to solve these types of nonstationary problems.     

An integral equation technique for scattering problems with mixed boundary conditions

This paper presents an integral formulation for Helmholtz problems with mixed boundary conditions. Unlike most integral equation techniques for mixed boundary value problems, the proposed method uses a global boundary charge density. As a result, Calderón identities can be utilized to avoid the use of hypersingular integral operators. Numerical results illustrate the performance of the proposed solution technique.     

Singular boundary method for 3D time-harmonic electromagnetic scattering problems

A frequency domain singular boundary method is presented for solving 3D time-harmonic electromagnetic scattering problem from perfect electric conductors. To avoid solving the coupled partial differential equations with fundamental solutions involving hypersingular terms, we decompose the governing equation into a system of independent Helmholtz equations with mutually coupled boundary conditions. Then the singular boundary method employs the fundamental solutions of the Helmholtz equations to approximate the scattered electric field variables. To desingularize the source singularity in the fundamental solutions, the origin intensity factors are introduced. In the novel formulation, only the origin intensity factors for fundamental solutions of 3D Helmholtz equations and its derivatives need to be considered which have been derived in the paper. Several numerical examples involving various perfectly conducting obstacles are carried out to demonstrate the validity and accuracy of the present method.     

On the convergence of a numerical method for a hypersingular integral equation on a closed surface

We consider a linear integral equation with a hypersingular integral treated in the sense of the Hadamard finite value. This equation arises in the solution of the Neumann boundary value problem for the Laplace equation with a representation of a solution in the form of a double-layer potential. We consider the case in which the interior or exterior boundary value problem is solved in a domain; whose boundary is a smooth closed surface, and an integral equation is written out on that surface. For the integral operator in that equation, we suggest quadrature formulas like the method of vortical frames with a regularization, which provides its approximation on the entire surface for the use of a nonstructured partition. We construct a numerical scheme for the integral equation on the basis of suggested quadrature formulas, prove an estimate for the norm of the inverse matrix of the related system of linear equations and the uniform convergence of numerical solutions to the exact solution of the hypersingular integral equation on the grid.     

Integral representations in quaternionic analysis related to the helmholtz operator

As the Laplacian also the Helmholtz operator can be factorized in Clifford analysis. Using the fundamental solution of the Helmholtz equation, a fundamental solution can be constructed for the factors of the Helmholtz operator. These are used in the case of quaternions to prove Cauchy–Pompeiu type representation formulas in terms of powers of the factors as well as in terms of powers of the Helmholtz operator.     

Multi‐periodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n‐torus

In this paper, we study the solutions to the generalized Helmholtz equation with complex parameter on some conformally flat cylinders and on the n‐torus. Using the Clifford algebra calculus, the solutions can be expressed as multi‐periodic eigensolutions to the Dirac operator associated with a complex parameter λ∈?. Physically, these can be interpreted as the solutions to the time‐harmonic Maxwell equations on these manifolds. We study their fundamental properties and give an explicit representation theorem of all these solutions and develop some integral representation formulas. In particular, we set up Green‐type formulas for the cylindrical and toroidal Helmholtz operator. As a concrete application, we explicitly solve the Dirichlet problem for the cylindrical Helmholtz operator on the half cylinder. Finally, we introduce hypercomplex integral operators on these manifolds, which allow us to represent the solutions to the inhomogeneous Helmholtz equation with given boundary data on cylinders and on the n‐torus. Copyright © 2009 John Wiley & Sons, Ltd.