Volume and surface integral equations for electromagnetic scattering by a dielectric body

We derive and analyze two equivalent integral formulations for the time-harmonic electromagnetic scattering by a dielectric object. One is a volume integral equation (VIE) with a strongly singular kernel and the other one is a coupled surface-volume system of integral equations with weakly singular kernels. The analysis of the coupled system is based on standard Fredholm integral equations, and it is used to derive properties of the volume integral equation.     

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 electromagnetic scattering problem for a partially coated dielectric

We use the linear sampling method to determine the shape and surface conductivity of a partially coated dielectric infinite cylinder from knowledge of the far field pattern of the scattered TM polarized electromagnetic wave at fixed frequency. A mathematical justification of the method is provided based on the use of a complete family of solutions. Numerical examples are given showing the efficiency of our method.     

Numerical solution of some classes of integral equations using Bernstein polynomials

This paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using Bernstein polynomials as basis. The integral equations considered are Fredholm integral equations of second kind, a simple hypersingular integral equation and a hypersingular integral equation of second kind. The method is explained with illustrative examples. Also, the convergence of the method is established rigorously for each class of integral equations considered here.     

Temporal discretization choices for stable boundary element methods in electromagnetic scattering problems

Diverse alternative temporal discretization schemes are analyzed for stable numerical solution of the surface integral equations in obtaining the transient scattering response of arbitrarily shaped conducting bodies. Streamlined formulations for three main categories including using either the conventional time integrators or the subdomain temporal basis functions, or the entire-domain time bases are presented in conceptually similar frameworks for solving types of the electric, magnetic, and combined field integral equations. To this end, first compatible temporal interpolations with conveniently usable time integrators are introduced based on stability analysis of the delay differential equations (DDE). Detailed guidelines for effective implementation of appropriate subdomain time basis functions are then studied. It is demonstrated that since in the latter approach the time derivatives are handled analytically, the extension of the stable region tremendously enhances while approaching small time step sizes. Eventually, the orthogonal weighted Laguerre polynomials are set forth to provide unconditionally stable schemes. Besides, adaptive partitioning of triangular patches is proposed to efficiently control the precision of numerical quadratures over the surface of source distribution. Numerical results are verified through comparison with the results obtained using the finite integration technique (FIT). Convergence behaviour of the widely used schemes is also investigated.     

Time domain scattering and inverse scattering problems in a locally perturbed half-plane

This paper is concerned with efficient numerical methods for solving the time-dependent scattering and inverse scattering problems of acoustic waves in a locally perturbed half-plane. By symmetric continuation, the scattering problem is reformulated as an equivalent symmetric problem defined in the whole plane. The retarded potential boundary integral equation method is modified to solve the forward problem. Then we consider the inverse scattering problem of determinating the local perturbation from the measured scattered data. The time domain linear sampling method is employed to deal with the inverse problem. The computation schemes proposed in this paper are relatively simple and easy to implement. Several numerical examples are presented to show the effectiveness of the proposed methods.     

Three dimensional electromagnetic scattering T-matrix computations

The infinite T-matrix method is a powerful tool for electromagnetic scattering simulations, particularly when one is interested in changes in orientation of the scatterer with respect to the incident wave or changes of configuration of multiple scatterers and random particles, because it avoids the need to solve the fully reconfigured systems. The truncated T-matrix (for each scatterer in an ensemble) is often computed using the null-field method. The main disadvantage of the null-field based T-matrix computation is its numerical instability for particles that deviate from a sphere. For large and/or highly non-spherical particles, the null-field method based truncated T-matrix computations can become slowly convergent or even divergent. In this work, we describe an electromagnetic scattering surface integral formulation for T-matrix computations that avoids the numerical instability. The new method is based on a recently developed high-order surface integral equation algorithm for far field computations using basis functions that are tangential on a chosen non-spherical obstacle. The main focus of this work is on the mathematical details required to apply the high-order algorithm to compute a truncated T-matrix that describes the scattering properties of a chosen perfect conductor in a homogeneous medium. We numerically demonstrate the stability and convergence of the T-matrix computations for various perfect conductors using plane wave incident radiation at several low to medium frequencies and simulation of the associated radar cross of the obstacles.     

On approximation and numerical solution of Fredholm integral equations of second kind using quasi-interpolation

In this article a method is presented, which can be used for the numerical treatment of integral equations. Considered is the Fredholm integral equation of second kind with continuous kernel, since this type of integral equation appears in many applications, for example when treating potential problems with integral equation methods.The method is based on the approximation of the integral operator by quasi-interpolating the density function using Gaussian kernels. We show that the approximation of the integral equation, gained with this method, for an appropriate choice of a certain parameter leads to the same numerical results as Nyström’s method with the trapezoidal rule. For this, a convergence analysis is carried out.     

Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Gauss product quadrature rules

The Gauss product quadrature rules and collocation method are applied to reduce the second-kind nonlinear two-dimensional Fredholm integral equations (FIE) to a nonlinear system of equations. The convergence of the proposed numerical method is proved under certain conditions on the kernel of the integral equation. An iterative method for approximating the solution of the obtained nonlinear system is provided and its convergence is proved. Also, some numerical examples are presented to show the efficiency and accuracy of the proposed method.     

用Backus-Gilbert方法求解声波散射问题

利用位势理论将散射问题的外边界问题转化为第一类边界积分方程求解,再利用Backus-Gilbert方法给出了二维空间的数值结果,与Tikhonov正则化方法比较,虽然精度稍差一些,但是计算方法和计算机实现比较简单.     

On Taylor-series expansion methods for the second kind integral equations

In this paper, we comment on the recent papers by Yuhe Ren et al. (1999) [1] and Maleknejad et al. (2006) [7] concerning the use of the Taylor series to approximate a solution of the Fredholm integral equation of the second kind as well as a solution of a system of Fredholm equations. The technique presented in Yuhe Ren et al. (1999) [1] takes advantage of a rapidly decaying convolution kernel k(|s−t|) as |s−t| increases. However, it does not apply to equations having other types of kernels. We present in this paper a more general Taylor expansion method which can be applied to approximate a solution of the Fredholm equation having a smooth kernel. Also, it is shown that when the new method is applied to the Fredholm equation with a rapidly decaying kernel, it provides more accurate results than the method in Yuhe Ren et al. (1999) [1]. We also discuss an application of the new Taylor-series method to a system of Fredholm integral equations of the second kind.     

A preconditioned GMRES for complex dense linear systems from electromagnetic wave scattering problems

We consider the numerical solution of linear systems arising from the discretization of the electric field integral equation (EFIE). For some geometries the associated matrix can be poorly conditioned making the use of a preconditioner mandatory to obtain convergence. The electromagnetic scattering problem is here solved by means of a preconditioned GMRES in the context of the multilevel fast multipole method (MLFMM). The novelty of this work is the construction of an approximate hierarchically semiseparable (HSS) representation of the near-field matrix, the part of the matrix capturing interactions among nearby groups in the MLFMM, as preconditioner for the GMRES iterations. As experience shows, the efficiency of an ILU preconditioning for such systems essentially depends on a sufficient fill-in, which apparently sacrifices the sparsity of the near-field matrix. In the light of this experience we propose a multilevel near-field matrix and its corresponding HSS representation as a hierarchical preconditioner in order to substantially reduce the number of iterations in the solution of the resulting system of equations.     

Utilizing feed-back neural network approach for solving linear Fredholm integral equations system

This paper intended to offer an architecture of artificial neural networks (NNs) for finding approximate solution of a second kind linear Fredholm integral equations system. For this purpose, first we substitute the N-th truncation of the Taylor expansion for unknown functions in the origin system. By applying the suggested neural network for adjusting the real coefficients of given expansions in resulting system. The proposed NN is a two-layer feed-back neural network such that it can get a initial vector and then calculates it’s corresponding output vector. In continuance, a cost function is defined by using output vector and the target outputs. Consequently, the reported NN using a learning algorithm that based on the gradient descent method, will adjust the coefficients in given Taylor series. Eventually, we have showed this method in comparison with existing numerical methods such as trapezoidal quadrature rule provides solutions with good generalization and high accuracy. The proposed method is illustrated by several examples with computer simulations.     

An all-frequency weakly-singular surface integral equation for electromagnetism in dielectric media: Reformulation and well-posedness analysis

We develop and analyze a surface integral equation (SIE) whose solution pertains to numerical simulations of propagating time-harmonic electromagnetic waves in three-dimensional dielectric media. The formulae to evaluate the far-field pattern and propagation of the electric and magnetic fields in the interior and exterior of a dielectric body, through surface integrals, require the solution of a 2×2$2\times 2$ system of weakly-singular SIEs for the two unknown electric and magnetic fields at the interface surface of the dielectric body. The SIE is governed by an operator that is of the classical identity plus compact form. The tangential surface currents and normal surface charges of the dielectric model can be easily computed from the surface electric and magnetic fields.     

An efficient boundary element method for two-dimensional transient wave propagation problems

The boundary element method (BEM) has been recognized by its unique feature of requiring neither internal cells nor their associated domain integrals in the computation. The method preserves its elegance for transient problems by means of a certain time-stepping scheme that initiates the time integration always from the initial time. Unfortunately, this time-marching scheme becomes rather difficult to apply, because the computation time and storage requirement grow dramatically with the increasing number of time steps. This paper shows that a reduction of one half of the computation time as well as the storage requirement can be achieved by an efficient truncation scheme for two-dimensional transient wave propagation problems. In particular, a guiding parameter for the determination of the truncation limit is proposed, and the overall measure of the error with respect to the truncation guide parameter is established.     

基于自适应小波神经网络的第二类Fredholm积分方程数值解法

该文构造了一类三层前馈自适应小波神经网络,将小波分析中平移因子和伸缩因子的拟合设置为输入层到隐层的权值与阈值,采用小波基函数作为隐层激活函数,并根据梯度下降算法自适应地调整参数.应用自适应小波神经网络数值求解第二类Fredholm积分方程,通过数值算例验证了该方法的可行性和有效性.     

An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate

This article is concerned with uniqueness for reconstructing a periodic inhomogeneous medium sitting on a perfectly conducting plate. We deal with the problem in the framework of time-harmonic Maxwell systems without TE or TM polarization. An orthogonal relation is obtained for two refractive indices and then used to prove that the refractive index can be uniquely identified from a knowledge of the incident fields and the total tangential electric field on a plane above the inhomogeneous medium, utilizing the eigenvalues and eigenfunctions of a quasi-periodic Sturm–Liouville eigenvalue problem.     

An L refined projection approximate solution of the radiation transfer equation in stellar atmospheres

This paper deals with the numerical approximation of the solution of a weakly singular integral equation of the second kind which appears in Astrophysics. The reference space is the complex Banach space of Lebesgue integrable functions on a bounded interval whose amplitude represents the optical thickness of the atmosphere. The kernel of the integral operator is defined through the first exponential-integral function and depends on the albedo of the media. The numerical approximation is based on a sequence of piecewise constant projections along the common annihilator of the corresponding local means. In order to produce high precision solutions without solving large scale linear systems, we develop an iterative refinement technique of a low order approximation. For this scheme, parallelization of matrix computations is suitable.