We discuss the foundations and state-of-the-art of the method of analytical regularization (MAR) (also called the semi-inversion method). This is a collective name for a family of methods based on conversion of a first-kind or strongly-singular second-kind integral equation to a second-kind integral equation with a smoother kernel, to ensure point-wise convergence of the usual discretization schemes. This is done using analytical inversion of a singular part of the original equation; discretization and semi-inversion can be combined in one operation. Numerous problems being solved today with this approach are reviewed, although in some of them, MAR comes in disguise.

The method of analytical regularization in wave-scattering and eigenvalue problems: foundations and review of solutions

This paper presents a fast method for electromagnetic scattering and radiation problems pertinent to three-dimensional (3D) buried objects. In this approach, a new symmetrical form of the Green's function is derived, which can reduce the number of Sommerfeld integrals involved in the buried objects problem. The integration along steepest descent paths and leading-order approximations are introduced to evaluate these Sommerfeld integrals, which can greatly accelerate the computation. Based on the fast evaluation of Sommerfeld integrals, the radiation of an arbitrarily oriented electric dipole buried in a half space is first analyzed and computed. Then, the scattering by buried dielectric objects and conducting objects is considered using the method of moments (MOM). Numerical results show that the fast method can save tremendous CPU time in radiation and scattering problems involving buried objects.

Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects

A fast algorithm for electromagnetic scattering by buried three dimensional (3-D) dielectric objects of large size is presented by using the conjugate gradient (CG) method and fast Fourier transform (FFT). In this algorithm, the Galerkin method is utilized to discretize the electric field integral equations, where rooftop functions are chosen as both basis and testing functions. Different from the 3-D objects in homogeneous space, the resulting matrix equation for the buried objects contains both cyclic convolution and correlation terms, either of which can be solved rapidly by the CG-FFT method. The near-scattered field on the observation plane in the upper space has been expressed by two-dimensional (2-D) discrete Fourier transforms (DFTs), which also can be rapidly computed. Because of the use of FFTs to handle the Toeplitz matrix, the Sommerfeld integrals' evaluation which is time consuming yet essential for the buried object problem, has been reduced to a minimum. The memory required in this algorithm is of order N (the number of unknowns), and the computational complexity is of order N/sub iter/N log N, in which N/sub iter/ is the iteration number, and N/sub iter//spl Lt/N is usually true for a large problem.

Fast algorithm for electromagnetic scattering by buried 3-D dielectric objects of large size

An analytical solution is presented for the electromagnetic scattering from a dielectric circular cylinder embedded in a dielectric half-space with a slightly rough interface. The solution utilizes the spectral (plane-wave) representation of the fields and accounts for all the multiple interactions between the rough interface and the buried cylinder. First-order coefficients from the small perturbation method are used for computation of the scattered fields from the rough surface. The derivation includes both TM and TE polarizations and can be easily extended for other cylindrical buried objects (e.g., cylindrical shell, metallic cylinder). Several scattering scenarios are examined utilizing the new solution for a dielectric cylinder beneath a flat, sinusoidal, and arbitrary rough surface profile. Results indicate that the scattering pattern of a buried object below a slightly rough surface differs from the flat surface case only when the surface roughness spectrum contains a limited range of spatial frequencies. Furthermore, the illuminated area of the incident wave is seen to be a critical factor in the visibility of a buried object below a rough surface.

Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface

An analysis is described for determining the current induced by a known excitation on a conducting cylinder located near the planar interface between two semi-infinite, homogeneous half-spaces of different electromagnetic properties. The perfectly conducting cylinder of general cross section is of infinite extent and the excitation is transverse magnetic to the cylinder axis. An integral equations for the induced current is derived and a numerical method for solving it is developed. The kernel of the integral equation contains a term corresponding to the usual open-space Green's function plus a term proportional to a Sommerfeld-type integral in two dimensions. Various forms of the Sommerfeld-type integral are given and the choices of form amenable to efficient evaluation are discussed. For a flat strip, a circular cylinder, and a rectangular cylinder, data are presented and discussed for selected parameters. Data are presented for cylinders above and below the interface as well as for a cylinder resting on the interface.

Current induced on a conducting cylinder located near the planar interface between two semi-infinite half-spaces

An analysis is described for determining the current induced by a known excitation on a conducting strip which resides on the planar interface between two semi-infinite homogeneous half-spaces of different electromagnetic properties. The perfectly conducting flat strip is of infinite extent and the excitation is transverse magnetic to the strip axis. An integral equation for the induced current is formulated and it is shown that its kernel which is in general a Sommerfeld-type integral can be expressed in closed form when the permeabilities of the two media are the same. Under this practical condition the integral equation is solved numerically and data are presented for cases of interest. For the electrically narrow strip, the integral equation is approximated and this approximate equation is solved analytically.

Current induced on a conducting strip which resides on the planar interface between two semi-infinite half-spaces

The electromagnetic scattering characteristics of an array of narrow, conducting strips can he developed readily by extending the work of Butler and Wilton who show that Chebyshev polynomials augmented with the edge condition can be used to solve the narrow-strip/narrow-slot integral equations. The strips reside in a homogeneous medium of infinite extent and are considered narrow relative to wavelength in the medium at the frequency of excitation. The unknown current distributions on the strips are represented as linear combinations of certain basis functions that are exact solutions to the approximate equation for an isolated narrow strip subject to a special excitation. The resulting power-series treatment allows easy calculation of the coupling terms among the strips in the array in a simple matrix equation by which the unknown coefficients in the current distribution expansions may be readily computed. With these coefficients, one can obtain the distribution of current on each strip and the total scattered field. The method is particularly well suited for handling large arrays with more strips than could be accommodated by the usual moment method. Numerical data-currents and scattered fields-are presented for various cases of interest.

A method for computing scattering by large arrays of narrow strips

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Current induced on a conducting cylinder located near the planar interface between two semi-infinite half-spaces

Current induced on a conducting strip which resides on the planar interface between two semi-infinite half-spaces

A method for computing scattering by large arrays of narrow strips