Point source excitation in direct and inverse scattering: The soft and the hard small sphere

A spherical wave emanating from a point source is scatteredby either a soft or a hard body. The incident spherical wavehas a wavelength which is much larger than the characteristicdimension of the scatterer and it is modified in such a wayas to recover the plane wave incidence when the source pointrecedes to infinity. Using low frequency expansions the scatteringproblem is transformed to a sequence of exterior potential problemsin the presence of a monopole singularity located at the sourceof the incident wave field. Complete expansions for the scatteringamplitude are provided. The method is applied to the cases ofa soft and a hard sphere and the first three approximationsfor the near, as well as the far, field are evaluated. It isobserved that every one, after the first, low frequency approximationof the far field, involves one spherical multipole more thanthe corresponding approximation for the case of an incidentplane wave. As the point singularity tends to infinity, therelative results recover all the known expressions for planeincidence. It is shown that for point excitation the Rayleighapproximation of the scattering amplitude for a hard sphereis of the second order, in contrast to the case of plane excitationwhich is of the third order. Simple algorithms that specifythe radius and the position of a soft and a hard sphere areproposed, which are based on the additional dependence of thescattering amplitude represented by the distance from the pointsource to the centre of the scatterer. The inversion algorithmis shown to be stable whenever the source point is not too faraway from the target sphere. A simple way to decide whetherthe sphere is a soft or a hard body is also provided.     

The resistive coated sphere in the presence of a point generated wave field

We consider the low‐frequency scattering problem of a point source generated incident field by a small penetrable sphere. The sphere, which is also lossy, contains in its interior a co‐ecentric spherical core on the boundary of which an impedance boundary condition is satisfied. An appropriate modification of the incident wave field allows for the reduction of the solution to the corresponding scattering problem of plane wave incidence, by moving the point source to infinity. For the near field, we obtain the low‐frequency coefficients of the zeroth and the first order. This was done with the help of the corresponding solution for the hard core problem and an appropriate use of linearity with respect to the Robin parameter. In the far field, we derive the leading non‐vanishing terms for the normalized scattering amplitude and the scattering cross‐section, which are both of the second order, as well as for the absorption cross‐section, which is of the zeroth order. The special cases of a lossy or a lossless penetrable sphere, of a resistive sphere, and of a hard sphere are recovered by an appropriate choice of the physical or the geometrical parameters. Copyright © 1999 John Wiley & Sons, Ltd.     

关于Helmholtz外问题的边界积分方程解的唯一性问题

本文用能量分析的观点探讨了用边界积分方程描述Helmholtz外问题时,解的唯一性不能保持的原因.文中证明了,当利用积分方程来描述问题时,实际上将无穷远处的Sommerfeld条件改成了既适合于外向波(辐射波),又适合于内向波(吸收波),即整个系统的能量保持守恒.并根据此观点解释了保持唯一性的算法.     

A New Non-Overlapping Domain Decomposition Method for a 3D Laplace Exterior Problem

We propose a method for solving three-dimensional boundary value problems for Laplace’s equation in an unbounded domain. It is based on non-overlapping decomposition of the exterior domain into two subdomains so that the initial problem is reduced to two subproblems, namely, exterior and interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To match the solutions on the interface between the subdomains (the sphere), we introduce a special operator equation approximated by a system of linear algebraic equations. This system is solved by iterative methods in Krylov subspaces. The performance of the method is illustrated by solving model problems.     

Simple Asymptotics for a Generalized Wave Equation with Degenerating Velocity and Their Applications in the Linear Long Wave Run-Up Problem

Asymptotic solutions of the wave equation degenerating on the boundary of the domain (where the wave propagation velocity vanishes as the square root of the distance from the boundary) can be represented with the use of a modified canonical operator on a Lagrangian submanifold, invariant with respect to theHamiltonian vector field, of the nonstandard phase space constructed by the authors in earlier papers. The present paper provides simple expressions in a neighborhood of the boundary for functions represented by such a canonical operator and, in particular, for the solution of the Cauchy problem for the degenerate wave equation with initial data localized in a neighborhood of an interior point of the domain.     

Diffraction d'ondes acoustiques par un guide semi-infini

We establish an existence and uniqueness result for an acoustic waves scattering problem by a bounded obstacle located in a homogeneous medium which contains a semi-infinite wave guide. We use a well-suited radiation condition. A Green function is calculated by using the Wiener-Hopf method and a priori estimates are proved to obtain the uniqueness property.     

The exterior Robin problem for the Helmholtz equation

A new method is given for solving the exterior Robin problem for the Helmholtz equation. The problem is reformulated as a new integral equation which is continuous as the field point approaches the boundary. It is shown that its solution can be represented as a convergent Neumann series for convex surfaces, for small values of the wave number. Examples are included which illustrate the method.     

A System of Plane Elasticity Canonical Integral Equations and Its Application

In this paper, we obtain a new system of canonical integral equations for the plane elasticity problem over an exterior circular domain, and give its numerical solution. Coupling with the classical finite element method, it can be used for solving general plane elasticity exterior boundary value problems. This system of highly singular equations is also an exact boundary condition on the artificial boundary. It can be approximated by a series of nonsingular integral boundary conditions.     

The Atkinson-Wilcox expansion theorem for electromagnetic chiral waves

Consider the problem of scattering of a time-harmonic electromagnetic wave by a three-dimensional bounded and smooth obstacle. The infinite space outside the obstacle is filled by a homogeneous isotropic chiral medium. In the region exterior to a sphere that includes the scatterer, any solution of the generalized Helmholtz's equation that satisfies the Silver-Müller radiation condition has a uniformly and absolutely convergent expansion in inverse powers of the radial distance from the center of the sphere. The coefficients of the expansion can be determined from the leading coefficient, “the radiation pattern”, by a recurrence relation.     

Potential flow past a porous body with a core of different permeability

It is well known that a uniform flow past a non-permeable rigid body does not exert a total force upon the surface of the body, however this is not the case when the body is permeable. Power et. al. (1984, 1986) first solved the problem of uniform potential flow past a two-dimensional permeable circular cylinder, with constant permeability, and found that the exterior flow exerts a drag force upon the surface of the cylinder independent of its size and secondly the problem when the uniform potential flow past a porous sphere, with constant permeability, in this case the exterior flow exerts a drag force on the sphere which is linearly dependent on the radius of the sphere. Here we will present the solution of two problems, a uniform potential flow past a porous circular cylinder and past a porous sphere, for each case the porous body is composed of two materials with different permeabilities. In both cases the total force exerted by the exterior flow upon the body is dependent on the thickness of the porous materials, and in the limit when the two permeabilities are equal, the previous results, circular cylinder and sphere, with constant permeability, are recovered. Atlhough, the mathematics involved in the solution of the present problem is simple, due to the nice boundary geometry of the bodies, the final expression for the total force found in each case is quite interesting on the way it depends on the permeability relation, in particular, in the limiting cases of a porous body with solid or hollow core.     

Determining the parameters of a stratified piecewise constant medium for the unknown shape of an impulse source

For a hyperbolic wave equation with some parameter λ, we consider the problem of finding the piecewise constant wave propagation speed and a series of parameters in the conjugation condition. Moreover, the shape is assumed unknown of the impulse point source that excites the oscillation process. We prove that, under certain assumptions on the structure of the medium, its sought parameters are determined uniquely from the displacements of points of the boundary given for two different values of λ. We give an algorithm for solving the problem.     

A boundary parametric approximation to the linearized scalar potential magnetostatic field problem

We consider the linearized scalar potential formulation of the magnetostatic field problem in this paper. Our approach involves a reformulation of the continuous problem as a parametric boundary problem. By the introduction of a spherical interface and the use of spherical harmonics, the infinite boundary conditions can also be satisfied in the parametric framework. That is the field in the exterior of a sphere is expanded in a ‘harmonic series’ of eigenfunctions for the exterior harmonic problem. The approach is essentially a finite element method coupled with a spectral method via a boundary parametric procedure. The reformulated problem is discretized by finite element techniques which leads to a discrete parametric problem which can be solved by well conditioned iteration involving only the solution of decoupled Neumann type elliptic finite element systems and L² projection onto subspaces of spherical harmonics. Error and stability estimates given show exponential convergence in the degree of the spherical harmonics and optimal order convergence with respect to the finite element approximation for the resulting fields in L².     

海洋波导中三维可穿透目标的快速声学成像

研究了用声传播远场分布信息来成像海洋波导环境中三维可穿透目标的反问题.建立了求解这类反问题的远场方程,基于内透射边界值问题的分析,讨论了远场方程解的唯一性和可解性,证明了总能找到远场方程的一个在最小平方意义下的近似解,其模在可穿透目标内部的取值是小的,而在外部的取值是大的,进而发展了一种快速成像可穿透目标的一种指示器样本方法.数值试验表明了这种方法是有效的,即使在有限孔径测量方式的情况,也能够得到未知目标的一个理想成像,而且不需要先验知道可穿透目标的任何几何与物理信息.     

A modification of the factorization method for scatterers with different physical properties

We study an inverse acoustic scattering problem by the factorization method when the unknown scatterer consists of two objects with different physical properties. Especially, we consider the following two cases: One is the case when each object has the different boundary condition, and the other one is when different penetrability. Our idea here is to modify the far‐field operator depending on the cases to avoid unnecessary a priori assumptions.     

Classification of Harmonic Functions in the Exterior of the Unit Ball

We solve the Laplace equation in an exterior infinite spherical domain with nonlinear (quadratic) boundary conditions on the spherical boundary. We linearize the problem and, under the additional assumption that the distinguishing function is spherically symmetric, write the solution by using the formal power series method with recursion of the series coefficients. Applying the Poincaré--Perron theorem, we describe the space of convergent formal power series and calculate its dimension. Estimating the roots of the fourth-degree characteristic polynomial corresponding to the given problem, we also calculate the dimension of the space of functions whose gradient at each point of the sphere is orthogonal to the linear combination of an axially symmetric dipole and a quadrupole. In conclusion, we state several unsolved problems arising in geophysical applications.     

Regularized Combined Field Integral Equations

Summary. Many boundary integral equations for exterior boundary value problems for the Helmholtz equation suffer from a notorious instability for wave numbers related to interior resonances. The so-called combined field integral equations are not affected. However, if the boundary is not smooth, the traditional combined field integral equations for the exterior Dirichlet problem do not give rise to an L²()-coercive variational formulation. This foils attempts to establish asymptotic quasi-optimality of discrete solutions obtained through conforming Galerkin boundary element schemes.This article presents new combined field integral equations on two-dimensional closed surfaces that possess coercivity in canonical trace spaces. The main idea is to use suitable regularizing operators in the framework of both direct and indirect methods. This permits us to apply the classical convergence theory of conforming Galerkin methods.     

A domain decomposition method for linear exterior boundary value problems

In this paper, we present a domain decomposition method, based on the general theory of Steklov-Poincaré operators, for a class of linear exterior boundary value problems arising in potential theory and heat conductivity. We first use a Dirichlet-to-Neumann mapping, derived from boundary integral equation methods, to transform the exterior problem into an equivalent mixed boundary value problem on a bounded domain. This domain is decomposed into a finite number of annular subregions, and the Dirichlet data on the interfaces is introduced as the unknown of the associated Steklov-Poincaré problem. This problem is solved with the Richardson method by introducing a Dirichlet-Robin-type preconditioner, which yields an iteration-by-subdomains algorithm well suited for parallel computations. The corresponding analysis for the finite element approximations and some numerical experiments are also provided.     

Numerical Experiments with an Interior-Exterior Point Method for Nonlinear Programming

The paper presents an algorithm for solving nonlinear programming problems. The algorithm is based on the combination of interior and exterior point methods. The latter is also known as the primal-dual nonlinear rescaling method. The paper shows that in certain cases when the interior point method (IPM) fails to achieve the solution with the high level of accuracy, the use of the exterior point method (EPM) can remedy this situation. The result is demonstrated by solving problems from COPS and CUTE problem sets using nonlinear programming solver LOQO that is modified to include the exterior point method subroutine.     

A linear sampling method for inverse problems of diffraction gratings of mixed type

This paper is concerned with the direct and inverse problem of scattering of a time‐harmonic wave by a Lipschitz diffraction grating of mixed type. The scattering problem is modeled by the mixed boundary value problem for the Helmholtz equation in the unbounded half‐plane domain above a periodic Lipschitz surface on which a mixed Dirichlet and impedance boundary condition is imposed. We first establish the well‐posedness of the direct problem, employing the variational method, and then extend Isakov's method to prove uniqueness in determining the Lipschitz diffraction grating profile by using point sources lying above the structure. Finally, we develop a periodic version of the linear sampling method to reconstruct the diffraction grating. In this case, the far field equation defined on the unit circle is replaced by a near field equation defined on a line above the surface, which is a linear integral equation of the first kind. Numerical results are also presented to illustrate the efficiency of the method in the case when the height of the unknown grating profile is not very large and the noise level of the near field measurements is not very high. Copyright © 2012 John Wiley & Sons, Ltd.     

Bifurcation of Solutions for an Inverse Problem in Potential Theory

Local existence and global numerical continuation of some solution branches are studied in an inverse potential problem in the exterior of a sphere. It is shown how multiple solutions arise where the solution field is tangent to the sphere. This work provides a tractable example of bifurcation arising from the edge of a continuum of eigenvalues.