Estimates for the low‐frequency electromagnetic fields scattered by two adjacent metal spheres in a lossless medium

Inductive electromagnetic means, currently employed in real physical applications and dealing with voluminous bodies embedded in lossless media, often call for analytically demanding tools of field calculation at modeling stage and later on at numerical stage. Here, one is considering two closely adjacent perfect conductors, possibly almost touching one another, for which the 3D bispherical geometry provides a good approximation. The particular scattering problem is modeled with respect to the two solid impenetrable metallic spheres, which are excited by a time‐harmonic magnetic dipole, arbitrarily orientated in the 3D space. The incident, the scattered, and the total non‐axisymmetric electromagnetic fields yield rigorous low‐frequency expansions in terms of positive integral powers of the real‐valued wave number in the exterior medium. We keep the most significant terms of the low‐frequency regime, that is, the static Rayleigh approximation and the first three dynamic terms, while the additional terms are small contributors and they are neglected. The typical Maxwell‐type problem is transformed into intertwined either Laplace's or Poisson's potential‐type boundary value problem with impenetrable boundary conditions. In particular, the fields are represented via 3D infinite series expansions in terms of bispherical eigenfunctions, obtaining analytical closed‐form solutions in a compact fashion. This procedure leads to infinite linear systems, which can be solved approximately within any order of accuracy through a cutoff technique.     

Low‐frequency electromagnetic scattering by a metal torus in a lossless medium with magnetic dipolar illumination

The present contribution is concerned with an analytical presentation of the low‐frequency electromagnetic fields, which are scattered off a highly conductive ring torus that is embedded within an otherwise lossless ambient and interacting with a time‐harmonic magnetic dipole of arbitrary orientation, located nearby in the three‐dimensional space. Therein, the particular 3‐D scattering boundary value problem is modeled with respect to the solid impenetrable torus‐shaped body, where the toroidal geometry fits perfectly. The incident, the scattered, and the total non‐axisymmetric magnetic and electric fields are expanded in terms of positive integral powers of the real‐valued wave number of the exterior medium at the low‐frequency regime, whereas the static Rayleigh approximation and the first three dynamic terms provide the most significant part of the solution, because all the additional terms are small contributors and, hence, they are neglected. Consequently, the typical Maxwell‐type physical problem is transformed into intertwined either Laplace's or Poisson's potential‐type boundary value problems with the proper conditions, attached to the metallic surface of the torus. The fields of interest assume representations via infinite series expansions in terms of standard toroidal eigenfunctions, obtaining in that way analytical closed‐form solutions in a compact fashion. Although this mathematical procedure leads to infinite linear systems for every single case, these can be readily and approximately solved at a certain level of desired accuracy through standard cut‐off techniques. Copyright © 2016 John Wiley & Sons, Ltd.     

Stabilized boundary element methods for low‐frequency electromagnetic scattering

The electromagnetic scattering at a perfectly conducting object is usually initiated by an incoming electromagnetic field. It is well known that the classical boundary element implementations solving for the scattered electric field are not uniformly stable with respect to the frequency of the incoming signal. The subject of this article is to develop a stabilized boundary element formulation that does not suffer from the so‐called low‐frequency breakdown. The mathematical theory is verified by numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.     

Electromagnetic scattering by a perfectly conducting obstacle in a homogeneous chiral environment: solvability and low‐frequency theory

The scattering of plane time‐harmonic electromagnetic waves propagating in a homogeneous isotropic chiral environment by a bounded perfectly conducting obstacle is studied. The unique solvability of the arising exterior boundary value problem is established by a boundary integral method. Integral representations of the total exterior field, as well as of the left and right electric far‐field patterns are derived. A low‐frequency theory for the approximation of the solution to the above problem, and the derivation of the far‐field patterns is also presented. Copyright © 2002 John Wiley & Sons, Ltd.     

Inverse scattering by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is first established by using the integral equation method. We then proceed to establish two tools that play important roles for the inverse problem: one is a mixed reciprocity relation and the other is a priori estimates of the solution on some part of the interfaces between the layered media. For the inverse problem, we prove in this paper that both the penetrable interfaces and the possible inside inhomogeneity can be uniquely determined from a knowledge of the far field pattern for incident plane waves.     

The factorization method for inverse acoustic scattering by a penetrable anisotropic obstacle

This paper is devoted to studying the factorization method applied to the inverse problem of reconstructing a penetrable anisotropic obstacle from far field patterns. We proved the validity of the factorization method. Copyright © 2013 John Wiley & Sons, Ltd.     

The inverse electromagnetic scattering problem by a penetrable cylinder at oblique incidence

In this work, we consider the method of non-linear boundary integral equation for solving numerically the inverse scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder in three dimensions. We consider the indirect method and simple representations for the electric and the magnetic fields in order to derive a system of five integral equations, four on the boundary of the cylinder and one on the unit circle where we measure the far-field pattern of the scattered wave. We solve the system iteratively by linearizing only the far-field equation. Numerical results illustrate the feasibility of the proposed scheme.     

The inverse scattering problem for a partially coated penetrable cavity with interior measurements

In this paper we consider the inverse scattering problem for a cavity that is bounded by a partially coated penetrable inhomogeneous medium of compact support and recover the shape of the cavity and the surface conductivity from a knowledge of measured scattered waves due to point sources located on a curve or surface inside the cavity. First, we prove that both the shape of the cavity and the surface conductivity on the coated part can be uniquely determined from a knowledge of the measured data. Next, we establish a linear sampling method for determining both the shape of the cavity and the surface conductivity. A central role in our justification is played by an eigenvalue problem which we call the exterior transmission eigenvalue problem. Finally, we present some numerical examples to illustrate the validity of our method.     

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.     

Low-frequency dipolar electromagnetic scattering by a solid ellipsoid in lossless environment

Electromagnetic wave scattering phenomena for target identification are important in many applications related to fundamental science and engineering. Here, we present an analytical formulation for the calculation of the magnetic and electric fields that scatter off a highly conductive ellipsoidal body, located within an otherwise homogeneous and isotropic lossless medium. The primary excitation source assumes a time-harmonic magnetic dipole, precisely fixed and arbitrarily orientated that operates at low frequencies and produces the incident fields. The scattering problem itself is modeled with respect to rigorous expansions of the electromagnetic fields at the low-frequency regime in terms of positive integral powers of the real wave number of the ambient. Obviously, the Rayleigh static term and a few dynamic terms are sufficient for the purpose of the present work, as the additional terms are neglected due to their minor contribution. Therein, the classical Maxwell's theory is suitably modified, leading to intertwined either Laplace's or Poisson's equations, accompanied by the impenetrable boundary conditions for the total fields and the limiting behavior at infinity. On the other hand, the complete spatial anisotropy of the three-dimensional space is secured via the introduction of the genuine ellipsoidal coordinate system, being appropriate for tackling incrementally such scattering boundary value problems. The nonaxisymmetric fields are obtained via infinite series expansions in terms of ellipsoidal harmonic eigenfunctions, providing handy closed-form solutions in a compact fashion, whose validity is verified by a straightforward reduction to simpler geometries of the metal object. The main idea is to demonstrate an efficient methodology, according to which the constructed analytical formulae can offer the appropriate environment for a fast numerical estimation of the scattered electromagnetic fields that could be useful for real data inversion.     

Regularized Newton iteration method for a penetrable cavity with internal measurements in inverse scattering problem

In this paper, we consider the inverse scattering problem of determining the shape of a cavity with a penetrable inhomogeneous medium of compact support from one source and a knowledge of measurements placed on a curve inside the cavity. First, the boundary value problem of the partial differential equations can be transformed into an equivalent system of nonlinear and ill-posed integral equations for the unknown boundary. Then, we apply the regularized Newton iterative method to reconstruct the boundary and prove the injectivity for the linearized system. Finally, we present some numerical examples to show the feasibility of our method.     

Low‐gain adaptive stabilization of semilinear second‐order hyperbolic systems

In this paper low‐gain adaptive stabilization of undamped semilinear second‐order hyperbolic systems is considered in the case where the input and output operators are collocated. The linearized systems have an infinite number of poles and zeros on the imaginary axis. The adaptive stabilizer is constructed by a low‐gain adaptive velocity feedback. The closed‐loop system is governed by a non‐linear evolution equation. First, the well‐posedness of the closed‐loop system is shown. Next, an energy‐like function and a multiplier function are introduced and the exponential stability of the closed‐loop system is analysed. Some examples are given to illustrate the theory. Copyright © 2004 John Wiley & Sons, Ltd.     

Low‐rank approximation of tensors via sparse optimization

The goal of this paper is to find a low‐rank approximation for a given nth tensor. Specifically, we give a computable strategy on calculating the rank of a given tensor, based on approximating the solution to an NP‐hard problem. In this paper, we formulate a sparse optimization problem via an l₁‐regularization to find a low‐rank approximation of tensors. To solve this sparse optimization problem, we propose a rescaling algorithm of the proximal alternating minimization and study the theoretical convergence of this algorithm. Furthermore, we discuss the probabilistic consistency of the sparsity result and suggest a way to choose the regularization parameter for practical computation. In the simulation experiments, the performance of our algorithm supports that our method provides an efficient estimate on the number of rank‐one tensor components in a given tensor. Moreover, this algorithm is also applied to surveillance videos for low‐rank approximation.     

Adaptive Fourier decomposition‐based Dirac type time‐frequency distribution

The Dirac‐type time‐frequency distribution (TFD), regarded as ideal TFD, has long been desired. It, until the present time, cannot be implemented, due to the fact that there has been no appropriate representation of signals leading to such TFD. Instead, people have been developing other types of TFD, including the Wigner and the windowed Fourier transform types. This paper promotes a practical passage leading to a Dirac‐type TFD. Based on the proposed function decomposition method, viz., adaptive Fourier decomposition, we establish a rigorous and practical Dirac‐type TFD theory. We do follow the route of analytic signal representation of signals founded and developed by Garbo, Ville, Cohen, Boashash, Picinbono, and others. The difference, however, is that our treatment is theoretically throughout and rigorous. To well illustrate the new theory and the related TFD, we include several examples and experiments of which some are in comparison with the most commonly used TFDs. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximation by an iterative method of a low‐Mach model with temperature dependent viscosity

In this work, we prove the existence and the uniqueness of the strong solution of a low‐Mach model, for which the dynamic viscosity of the fluid is a given function of its temperature. The method is based on the convergence study of a sequence towards the solution, for which the rates are also given. The originality of the approach is to consider the system in terms of the temperature and the velocity, leading to a nonlinear temperature equation and the development of some specific tools and results.     

Well‐conditioned boundary integral formulations for high‐frequency elastic scattering problems in three dimensions

We construct and analyze a family of well‐conditioned boundary integral equations for the Krylov iterative solution of three‐dimensional elastic scattering problems by a bounded rigid obstacle. We develop a new potential theory using a rewriting of the Somigliana integral representation formula. From these results, we generalize to linear elasticity the well‐known Brakhage–Werner and combined field integral equation formulations. We use a suitable approximation of the Dirichlet‐to‐Neumann map as a regularizing operator in the proposed boundary integral equations. The construction of the approximate Dirichlet‐to‐Neumann map is inspired by the on‐surface radiation conditions method. We prove that the associated integral equations are uniquely solvable and possess very interesting spectral properties. Promising analytical and numerical investigations, in terms of spherical harmonics, with the elastic sphere are provided. Copyright © 2014 John Wiley & Sons, Ltd.     

Matrix solution to electromagnetic scattering by a conducting cylinder with an eccentric metamaterial coating

Closed series solution to scattering by an eccentric coated cylinder is realized in matrix form. Diffracted radiation characteristics are investigated for N incident plane transverse electric (TE) waves. The solution is obtained by the boundary value analysis and the addition theorem of the Bessel's functions. Wave transformation and orthogonality of the complex exponentials are also used to find an infinite series in the solution. Numerical results are shown by reducing the infinite series to a limited number of terms and compared to previously published works.     

Directional ‐matrix compression for high‐frequency problems

Standard numerical algorithms, such as the fast multipole method or ‐matrix schemes, rely on low‐rank approximations of the underlying kernel function. For high‐frequency problems, the ranks grow rapidly as the mesh is refined, and standard techniques are no longer attractive. Directional compression techniques solve this problem by using decompositions based on plane waves. Taking advantage of hierarchical relations between these waves' directions, an efficient approximation is obtained. This paper is dedicated to directional ‐matrices that employ local low‐rank approximations to handle directional representations efficiently. The key result is an algorithm that takes an arbitrary matrix and finds a quasi‐optimal approximation of this matrix as a directional ‐matrix using a prescribed block tree. The algorithm can reach any given accuracy, and the approximation requires only units of storage, where n is the matrix dimension, κ is the wave number, and k is the local rank. In particular, we have a complexity of if κ is constant and for high‐frequency problems characterized by κ²～n. Because the algorithm can be applied to arbitrary matrices, it can serve as the foundation of fast techniques for constructing preconditioners.     

Low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions

In this paper, we consider low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions. The time discretization is based on the Euler implicit scheme, and the spatial discretization is based on the low‐order element (P₁−P₁ or P₁−P₀) for the approximation of the velocity and pressure. Moreover, some error estimates for the numerical solution of fully discrete stabilized finite element scheme are obtained. Finally, numerical experiments are performed to confirm our theoretical results.     

Time‐dependent scattering of generalized plane waves by a wedge

We obtain explicit formulas for the scattering of plane waves with arbitrary profile by a wedge under Dirichlet, Neumann and Dirichlet‐Neumann boundary conditions. The diffracted wave is given by a convolution of the profile function with a suitable kernel corresponding to the boundary conditions. We prove the existence and uniqueness of solutions in appropriate classes of distributions and establish the Sommerfeld type representation for the diffracted wave. As an application, we establish (i) stability of long‐time asymptotic local perturbations of the profile functions and (ii) the limiting amplitude principle in the case of a harmonic incident wave. Copyright © 2015 John Wiley & Sons, Ltd.