Combining an extrapolation technique with the method of moments forsolving large scattering problems involving bodies of revolution

The purpose of this work is to combine an extrapolation technique with the method of moments (MoM) to solve scattering problems involving large bodies. It has been shown in a previous work that the current induced on the smooth parts of large scatterers may be represented as a series of complex exponential functions with a few terms. Based on this concept, a hybrid set of basis functions is constructed using entire domain functions of complex exponential type on the smooth portion of the scatterer, complemented by subdomain basis functions near edges and discontinuities. An extrapolation procedure is developed in which the scattering problem is first solved for a portion of the scatterer using the conventional MoM. Next, a set of entire-domain basis functions, whose behaviour could be extrapolated with an increase in the size of the scatterer, is extracted from this original solution. The procedure outlined has the very desirable feature that the total number of basis functions remains unchanged even as the scatterer size is increased, allowing for large scatterers to be handled with a relatively small number of unknowns. The extrapolation technique is applied to scattering problems from bodies of revolution (BORs), and numerical results for an open cylinder and a barrel-shaped BOR are presented     

A new numerical approach to the calculation of electromagnetic scattering properties of two-dimensional bodies of arbitrary cross section

A new method is introduced for formulating the scattering problem in which the scattered fields (and the interior fields in the case of a dielectric scatterer) are represented in an expansion in terms of free-space modal wave functions in cylindrical coordinates, the coefficients of which are the unknowns. The boundary conditions are satisfied using either an analytic continuation procedure, in which the far-field pattern (in Fourier series form) is continued into the near field and the boundary conditions are applied at the surface of the scatterer; or the completeness of the modal wave functions, to approximately represent the fields in the interior and exterior regions of the scatterer directly. The methods were applied to the scattering of two-dimensional cylindrical scatterers of arbitrary cross section and only the TM polarization of the excitation is considered. The solution for the coefficients of the modal wave functions are obtained by inversion of a matrix which depends only on the shape and material of the scatterer. The methods are illustrated using perfectly conducting square and elliptic cylinders and elliptic dielectric cylinders. A solution to the problem of multiple scattering by two conducting scatterers is also obtained using only the matrices characterizing each of the single scatterers. As an example, the method is illustrated by application to a two-body configuration.     

Radar scattering from bodies of revolution using an efficientpartial differential equation algorithm

A technique is presented for solving the problem of scattering by a three-dimensional body of revolution using a partial differential equation (PDE) technique in conjunction with a radiation boundary condition applied in the Fresnel region of the scatterer. The radiation boundary condition, which is used to truncate the PDE mesh, is based on an asymptotic expansion derived by Wilcox (1956). Numerical results illustrating the procedure and verifying the accuracy of the results are included     

Reduction of the number of unknowns in large stacked planarstructures by a fringe aperture formulation

For scattering problems comprising a combination of planar structures, the total number of unknowns may be significantly reduced if an aperture formulation is employed rather than a patch formulation. The rationale behind using the aperture formulation is based on the recognition that the decay rate of the scattered aperture field is independent of the size of the scatterer. Therefore, any scatterer may be surrounded by an aperture of fixed width over which an integral equation is formulated. The area of this aperture is proportional to the perimeter of the scatterer rather than its area, and it becomes much smaller compared with the entire scatterer area as the size of the scatterer increases, hence the reduction in the number of independent unknowns. A truncation criterion for the finite aperture is determined via a numerical study of the aperture field behavior for various angles of incidence. In addition, the a priori knowledge of the physical optics component is also taken into account, reducing the unknown function to an aperture field component that is the outcome of the remaining fringe current only. The total current distribution can be subsequently derived from this field by adding the known physical optics field and invoking the inverse of the Green's function in the spectral domain. This analysis of isolated planar scatterers results in a spectral scattering matrix representation that is subsequently used for cascading of stacked structures     

A novel efficient algorithm for scattering from a complex BOR usingmixed finite elements and cylindrical PML

An efficient finite-element method (FEM) is developed to compute scattering from a complex body of revolution (BOR). The BOR is composed of a perfect conductor and impedance surfaces and arbitrary inhomogeneous materials. The method uses edge-based vector basis functions to expand the transverse field components and node-based scalar basis functions to expand the angular component. The use of vector basis functions eliminates the problem of spurious solutions suffered by other three component FEM formulations. The FEM mesh is truncated with a perfectly matched layer (PML) in cylindrical coordinates. The use of PML in cylindrical coordinates avoids the wasted computation which results from a spherical mesh boundary with an elongated scatterer. The FEM equations are solved by ordering the unknowns with a reverse Cuthill-McKee algorithm and applying a banded-matrix solution algorithm. The method is capable of handling large, realistic radar targets, and good agreement with measured results is achieved for benchmark targets     

Numerical Solution of Steady-State Electromagnetic Scattering Problems Using the Time-Dependent Maxwell's Equations

A numerical method is described for the solution of the electromagnetic fields within an arbitrary dielectric scatterer of the order of one wavelength in diameter. The method treats the irradiation of the scatterer as an initial value problem. At t = 0, a plane-wave source of frequency f is assumed to be turned on. The diffraction of waves from this source is modeled by repeatedly solving a finite-difference analog of the time-dependent Maxwell's equations. Time stepping is continued until sinusoidual steady-state field values are observed at all points within the scatterer. The envelope of the standing wave is taken as the steady-state scattered field. As an example of this method, the computed results for a dielectric cylinder scatterer are presented. An error of less than /spl plusmn/10 percent in locating and evaluating the standing-wave peaks within the cylinder is achieved for a program execution time of 1 min. The extension of this method to the solution of the fields within three-dimensional dielectric scatterers is outlined.     

On the analysis of scattering and antenna problems using the singularity expansion technique

In this paper, the possibility of analyzing scattering and antenna problems from a singularity expansion point of view is discussed. As an example of the method, a thin-wire scatterer is considered by first determining the locations of the exterior natural resonant frequencies and then constructing the time response of the current on the body, much in the same manner as in classical circuit theory. The numerical techniques used will be presented, and some advantages of the singularity expansion method over the other conventional ways of treating this problem will be mentioned.     

The three-dimensional algorithm of solving the electric fieldintegral equation using face-centered node points, conjugate gradientmethod, and FFT

It has been known for a long time that the accuracy of solving the scattering by a dielectric body using the electric field integral equation (EFIE) is poor when the permittivity of the scatterer becomes large. Recently, this problem has been settled by using a procedure involving face-centered node points. Such a procedure is efficient, since it preserves the convolution property in the EFIE and, hence, the applicability of the fast Fourier transform (FFT). This procedure is generalized to the three-dimensional and anisotropic case. The generalization is quite straightforward in both the formulation and the programming. A calculation for a scatterer with a relative permittivity as high as 100 indicates that the proposed procedure converges quite rapidly, whereas the conventional using the conjugate gradient method approach fails to converge     

Average dielectric properties of discrete random media using multiple scattering theory

Using rigorous multiple scattering theory in determining the average or bulk dielectric properties of discrete random media is the objective of this communication. The random medium is modeled as a random distribution of identical, spherical scatterers imbedded in an homogeneous unbounded background medium. At high scatterer concentration, the form of the radial distribution function becomes important; two forms are considered here, viz., virial series and the self-consistent form. The average loss tangent of the bulk medium is computed as a function of frequency and scatterer concentration, and compared with a frequently used mixture formula, e.g., Maxwell-Garnett. The results show that multiple scattering losses are significant at the higher concentrations and must be accounted for whenka gtrsim 0.05. The theory and the computational procedure can thus he used as a mixture formula forkain the range0 < ka lesssim 5.0and concentrations in the range0.01 < c lesssim 0.40.     

The null field approach to electromagnetic scattering fromcomposite objects

An improved null-field approach to scattering from composite objects is reported. Several alternative expressions for the transition matrix of a composite object are derived that partly or completely avoid the geometrical constraints inherent in previous null-field results for a composite scatterer. Some of the new versions make use of Q-matrices for the open surfaces which are defined by the interfaces between the different parts of the composite scatterer. The numerical performance of the various alternatives is investigated for a number of test cases. Comparisons between alternative versions for the same scattering problem are made to provide a measure of the absolute accuracy of the computed null-field results. Whenever possible, the results are compared with other computed or measured results     

Scattering from a large body with cracks and cavities by the fast and accurate finite-element boundary-integral method

A large body with cracks and cavities is a typical structure widely existing in realistic targets. In this paper, a newly developed fast and accurate finite-element boundary-integral (FA-FE-BI) method is applied to compute scattering by this kind of scatterer. A thorough analysis on this FA-FE-BI numerical technique is presented, clearly demonstrating that this technique has computational complexity O(N log N) and memory requirement O(N) (N is the total number of surface unknowns). An inward-looking approach is employed as a preconditioner to speed up the rate of convergence of iterative solvers for this structure. Under these techniques, a powerful code is developed for this kind of scatterer whose accuracy, efficiency, and capability is well confirmed by various numerical results.     

An iterative method for solving scattering problems

An iterative method is developed for computing the current induced by plane wave excitation on conducting bodies of arbitrary shape. In this method, the scattering body is divided into lit- and shadow-side regions separated by the geometric optics boundary. The induced current at any point on the surface of the scatterer is expressed as the sum of an approximate optics current and a correction current. Both of these currents are computed by iteration for the lit and shadow regions separately. The general theory is presented and applied to the problems of scattering from a two-dimensional cylinder of circular and square cross sections. The results are compared with the method of moments and good agreement is obtained. This method does not give erroneous results at internal resonances of the scatterer, does not suffer from computer storage problems and can be extended to nonperfect conductors as well as to three-dimensional bodies.     

Radiation and scattering from electrically small conducting bodies of arbitrary shape above an infinite ground plane

A simple moment solution is given for the low-frequency electromagnetic scattering or radiation problem involving a small perfectly conducting body of arbitrary shape placed close to an infinite ground plane. The method of images is used to account for the presence of the ground plane. The dynamic problem is approximated by two uncoupled problems, an electrostatic one and a magnetostatic one. Each static problem is then solved using the method of moments. The surface of the perfectly conducting scatterer is modeled by a set of planar triangular patches. Pulse expansion and point-matching testing are used in the electrostatic problem. For the magnetostatic problem, a set of solenoidal vector expansion functions is used. The induced dipole moments are computed from the induced electrostatic charge and the magnetostatic current densities. The scattered field is the field of these induced dipoles oscillating with the frequency of the incident field. Scatterers of various shapes are studied. Special attention is given to a conducting box on the ground plane.     

Electromagnetic scattering by a dielectric body with arbitraryinhomogeneity and anisotropy

The scattering problem for a dielectric body is formulated in terms of the electric field integral equation where the scatterer is of general shape, inhomogeneity, and anisotropy. On applying the pulse-function expansion and the point-matching technique, the integral equation is solved using an efficient procedure involving the conjugate-gradient method and the fast Fourier transform (FFT). The solution procedure runs parallel to that of the two-dimensional case previously presented by the author (see ibid., vol.AP-35, p.1418-25, Dec. 1987). Most of the work presented involves generalizing two-dimensional Green's function and operations into corresponding three-dimensional ones     

A boundary-element solution of the Leontovitch problem

A boundary-element method is introduced for solving electromagnetic scattering problems in the frequency domain relative to an impedance boundary condition (IBC) on an obstacle of arbitrary shape. The formulation is based on the field approach; namely, it is obtained by enforcing the total electromagnetic field, expressed by means of the incident field and the equivalent electric and magnetic currents and charges on the scatterer surface, to satisfy the boundary condition. As a result, this formulation is well-posed at any frequency for an absorbing scatterer. Both of the equivalent currents are discretized by a boundary-element method over a triangular mesh of the surface scatterer. The magnetic currents are then eliminated at the element level during the assembly process. The final linear system to be solved keeps all of the desirable properties provided by the application of this method to the usual perfectly conducting scatterer; that is, its unknowns are the fluxes of the electric currents across the edges of the mesh and its coefficient matrix is symmetric     

Microwave imaging-Location and shape reconstruction frommultifrequency scattering data

The problem of determining the shape and location of an object embedded in a homogeneous dissipative medium from measurements of the field scattered by the object is considered in this paper. The object is assumed to be an infinite cylinder of known cross section illuminated by a TM plane wave and the scattered field is measured on a line segment perpendicular to the direction of incidence. Measurement data are carried out at three different frequencies for a homogeneous cylinder of known dielectric constant. The location and contour shape are determined using two different reconstruction algorithms, a Newton-Kantorovich (NK) method and the modified gradient (MG) method whose effectiveness and robustness are compared. Both methods are based on domain integral representations of the field in the body. They involve an iterative minimization of the defect between an integral representation of the field measured on the line and the actual measured data. The NK method involves a linearization of the nonlinear relation between the field and the contrast, as well as the solution of a direct scattering problem at each iteration. The MG method seeks the simultaneous reconstruction of the field and the characteristic function of the support of the scatterer without solving a direct problem at each step. Both methods employed the same initial guess and the a priori information that the characteristic function is nonnegative     

Electromagnetic scattering from anisotropic materials, part II: Computer code and numerical results in two dimensions

The two-dimensional problem of oblique scattering by penetrable cylinders of arbitrary cross section made of materials which are linear, lossy, anisotropic and possibly inhomogeneous is considered. The materials are characterized by arbitrary tensor susceptibilitiesbar{x}_{ec}andbar{x}_{m}. The frequency-domain volume integrodifferential equations satisfied by the electric and magnetic fields and obtained in a previous paper (Part 1) are analyzed numerically. Optimal ordering of the unknowns and transverse electric-transverse magnetic (TE-TM) decomposition in the matrix formulation of the problem are discussed. The cross section of the scatterer is broken down into a triangular mesh. The field components at the vertices of the triangles are the unknowns; within each triangle, each field component is a linear combination of its values at the vertices. Computed field distributions inside the scatterer are found to be in excellent agreement with results obtained by other methods.     

Enhancing HF received fields with large planar and cylindrical ground screens

Users of extended range HF (3 to 30 MHz) radar and communication systems employing the ionosphere desire signal reception at incidence angles near-grazing to the local earth tangent. For vertical polarization, the vanishing received fields at low incidence angles over dielectric earth may be increased by using large ground screens. In this paper a ground-screen formulation based on scattering techniques is developed. The ground screen is viewed as a scatterer in free space, excited by a plane wave. A Fresnel image wave is added to establish the air-earth interface. Formulations are developed for the semi-infinite screen and for the circular-cylinder surface segment screen. The semi-infinite screen is representative in performance to a round screen of radius equal to the distance from the edge of the semi-infinite screen at which the field is computed or measured. For a ground screen on a hill the cylindrical segment is appropriate. Computations for a simulated earth were made and corroborated by experimental simulation with scale models. Improvements in field strength of 7 to 14 dB or more can be achieved with large screens over no screen. "Relatively small" tilted or raised flat screens, and cylindrical segment screens, can give improvements equal to very large flat screens on the earth.     

区域分裂法及其在三维散射中的应用

基于区域分裂（ＤＤＭ）和频域有限差分（ＦＤＦＤ）提出了一种分析三维散射问题的精确高效算法。用区域分裂法可以把原问题分解成若干子问题,可以大大缩小稀疏矩阵的规模,使得求解大尺寸三维散射问题成为可能。文中首先在小尺寸下计算了三维立体柱的散射特性,并和没有进行区域分裂时的ＦＤＦＤ法进行了对比,验证了本算法的正确性;然后又计算了尺寸比较大的三维矩形柱的散射特性,验证了它的高效性。     

Absorbing boundary conditions on arbitrary boundaries for thescalar and vector wave equations

The solution of open region scattering problems involving inhomogeneous arbitrarily shaped objects may be performed through the use of partial differential equation techniques, which require enclosing the scatterer by an outer boundary on which an absorbing boundary condition (ABC) is applied. In order to minimize the size of the domain to be meshed and, consequently, the number of unknowns, if may be advisable to implement ABC's devised for outer boundaries of arbitrary shapes. Such ABC's are obtained for the 3D scalar and vector wave equations; they incorporate most of existing boundary conditions. When used in conjunction with a finite element technique, the numerical results derived by using a simplified form of these ABC's compare favourably to those obtained by using a rigorous hybrid finite element-integral equation formulation. These boundary conditions have been obtained in the frequency-domain framework; they may, however, be used in time-domain calculations