We consider two approaches of evaluating matrix elements of electromagnetic volume integral equations.

We report new developments in the analytical evaluation of the near-field contribution to the matrix elements of the electric and magnetic field operators for planar conducting structures embedded in a layered medium. The method is applicable to Rao-Wilton-Glisson (RWG) basis functions supported on parallel interfaces in the medium. Our method is an extension of the approach described in [1] of representing a Green function as a two-dimensional Laplacian of an auxiliary function. Such Laplacian representations can be obtained for the asymptotic forms of the Green functions, which are being subtracted in order to regularize the behavior of the Sommerfeld-type integrals. Matrix elements resulting from these asymptotic forms, given originally as quadruple surface integrals with singular integrands, are then reduced to double contour integrals over the perimeters of the surface elements, involving simple closed-form non-singular auxiliary functions.

We consider new developments in the analytical evaluation of the near-field contribution to the matrix elements of the electric and magnetic field operators for planar conducting structures embedded in a layered medium. The method is applicable to Rao-Wilton-Glisson (RWG) basis functions supported on parallel interfaces in the layered medium. The method uses suitably constructed representations of the mixed potential formulation integral kernels in terms of two-dimensional Laplacian of auxiliary functions. Such Laplacian representations can be obtained for the asymptotic forms of the Green functions, which are being subtracted in order to regularize the behavior of the Sommerfeld-type integrals. Matrix elements resulting from these asymptotic forms, given originally as quadruple surface integrals with singular integrands, are then reduced to double contour integrals over the perimeters of the surface elements, involving simple closed-form non-singular auxiliary functions M. The new developments include: · Derivation of relations between elements of the asymptotic dyadic Green functions and the kernels of the mixed-potential representation of the fields. · Inclusion of additional terms introduced in [1], which improve convergence of the Sommerfeld integrals. These additional kernel components, related to half-line source potentials, were not included in our previous paper [1]; they constitute non-leading asymptotic contributions to the mixed-potential kernels K Φ and K Ψ . · Construction of two additional auxiliary functions needed to represent the above-mentioned additional terms. The resultant auxiliary functions are expressed as integrals of the previously obtained [1] functions for the leading asymptotic kernel terms. · Construction of simplified analytical expressions for the matrix elements of the asymptotic parts of the pertinent dyadic Green functions. The asymptotic matrix elements, given in terms of quadruple surface integrals with singular integrands, are subsequently converted, by using suitably constructed Laplacian representations of the Green function, to double contour integrals over the perimeters of the surface elements, with simple, non-singular, smoothly varying integrands. The line integrals can be either evaluated analytically or by means of low order numerical quadratures. We discuss the relative merits of the direct numerical and analytic evaluation if these line integrals.

We present an approach allowing conversion of surface integrals (over planar surface elements) to line integrals, in evaluating matrix elements of the derivatives of the Helmholtz-equation Green function, in particular the vector Green function in electromagnetics. A general procedure is outlined and explicit expressions are provided in the static limit. In the latter case, the accuracy and the computational cost of the proposed technique are compared to those of a more conventional approach based on evaluating surface integrals of a known closed-form potential.

Summary form only given. We consider an approach to imaging involving range measurement with short signals propagating through dilute random discrete-scatterer media with scatterer sizes larger than the wavelength of the incident pulse, such as typical atmospheric clouds in the optical region or rain in the millimeter-wave domain. We demonstrate, through numerical simulations employing the radiative-transfer equation (as well as in terms of simple analytic model describing small angle diffractive scattering component of the propagating pulse), that intensity of a pulse, in addition to a “ballistic” (coherent) contribution and to a long late-time diffusive tail, exhibits also a characteristic sharply rising early-time signal. This early-time component can be attributed to the small-angle diffractive part of the scattering cross-section on medium particles. Correspondingly, it decays with the distance significantly more slowly than the coherent (ballistic) contribution: its attenuation rate is proportional to the large-angle, non-diffractive cross-section, rather than the total cross-section. The sharply rising leading edge of the signal is due to the same physical mechanism of diffractive scattering. We propose to take advantage of the small rise time of the early-time part of the pulse, extract it by means of high-pass filtering, and use as a short signal, providing a high range resolution - typically of the order of several centimeters. We also discuss a possible utilization of longer chirped signals, instead of short pulses. In this case it should be possible to attain by measuring and processing the mutual coherence function for two different observation times (or frequencies), rather than only the field intensity.

Summary form only given. We describe further results in the development of an approach to imaging through dilute obscuring particulate media, in which energy absorption in the scatterers is small and the attenuation is caused primarily by scattering.

We describe an approach to autofocusing for large apertures on curved SAR trajectories. It is a phase-gradient type method in which phase corrections compensating trajectory perturbations are estimated not directly from the image itself, but rather on the basis of partial" SAR data { functions of the slow and fast times { recon- structed (by an appropriate forward-projection procedure) from windowed scene patches, of sizes comparable to distances between distinct targets or localized features of the scene. The resulting partial data" can be shown to contain the same information on the phase perturbations as that in the original data, provided the frequencies of the perturbations do not exceed a quantity proportional to the patch size. The algorithm uses as input a sequence of conventional scene images based on moderate-size subapertures constituting the full aperture for which the phase corrections are to be determined. The subaperture images are formed with pixel sizes comparable to the range resolution which, for the optimal subaperture size, should be also approximately equal the cross-range resolution. The method does not restrict the size or shape of the synthetic aperture and can be incorporated in the data collection process in persistent sensing scenarios. The algorithm has been tested on the publicly available set of GOTCHA data, intentionally corrupted by random-walk-type trajectory uctuations (a possible model of errors caused by imprecise inertial navigation system readings) of maximum frequencies compatible with the selected patch size. It was able to eciently remove image corruption for apertures of sizes up to 360 degrees.

An approach for solving volumetric integral equations in acoustics, applicable to problems involving large density contrasts, is described. While the conventional Lippmann-Schwinger integral equations become under such circumstances ill conditioned, the proposed approach reformulates them and casts them into an equivalent system of well-conditioned surface and volume integral equations. The corresponding fast solver [utilizing stiffness matrix compression based on fast Fourier transforms and characterized by O(N log N) solution complexity and storage requirements, where N is the number of unknowns] was enhanced to incorporate the proposed formulation. Features of the solution method and of the solver are illustrated on representative examples of numerically large problems.

We analyze the structure of the (3→3)-nucleon amplitude appearing in proton-deuteron elastic scattering amplitude. We calculate that part of the amplitude, which can be unambiguously expressed, through analyticity and unitarity, in terms of physical (2→2)-nucleon amplitudes. By comparing the result with the experimental data on spin observables we find that it is necessary to add extra terms, which must be interpreted as 3-body effects. These 3-body forces, of the type of contact interactions, are also required by the high-energy asymptotics of the amplitudes involved, and can be, to some extent, determined from soft-pion theorems. Much more information on them should become available through systematic analysis of the spin structure of p-d scattering amplitudes.

Elements are described of a volumetric integral-equation-based algorithm applicable to accurate large-scale simulations of scattering and propagation of sound waves through inhomogeneous media. The considered algorithm makes possible simulations involving realistic geometries characterized by highly subwavelength details, large density contrasts, and described in terms of several million unknowns. The algorithm achieves its competitive performance, characterized by O(NlogN) solution complexity and O(N) memory requirements, where N is the number of unknowns, through a fast and nonlossy fast Fourier transform based matrix compression technique, the adaptive integral method, previously developed for solving large-scale electromagnetic problems. Because of its ability of handling large problems with complex geometries, the developed solver may constitute an efficient and high fidelity numerical simulation tool for calculating acoustic field distributions in anatomically realistic models, e.g., in investigating acoustic energy transfer to the inner ear via nonairborne pathways in the human head. Examples of calculations of acoustic field distribution in a human head, which require solving linear systems of equations involving several million unknowns, are presented.

We describe elements and representative application s of a new time domain integral equation solver applicable, in particular, to problems involving interaction of wide-band pulses with dispersive media . We discuss our new analytical formulation of integral equations specially tailored to problems involving dispersive media . The formulation is both general and significantly simpler than the conventional approaches: instead of using the customary integral equation operators involving the Green function and its derivatives, we construct effective integral equation operators equal (i) to the Fourier transform of the dispersive medium Green function, (ii) to the Fourier transform of the product of the dispersive medium Green function with the frequency dependent dielectric permittivity, and, (iii) to the Fourier transform of the product of the dispersive medium Green function with the inverse of the dielectric permittivity. An important benefit of such an approach is that the resulting integrals involve only single (and not double) time convolutions. The formulation is applicable to systems involving bulk dispersive regions and thin disper sive sheets represented as interfaces . We present results of complete analytical calculations and of corresponding numerical procedures for the evaluation of matrix elements of the integral operator s, executed in the framework of the full Galerkin scheme in space and time variables , for the "conductive Debye medium" (i.e., for a medium with the electric permittivity given by the Debye formula supplemented with a term responsible for the medium conductivity). The procedure employs a suitable contour integration around singularities of the effective Green function operators in the complex frequency plane .

We describe elements and representative applications of a fast time domain integral equation solver based on the AIM type spatial matrix compression, applicable to problems involving interaction of wide-band pulses with dispersive media. The formulation is both general and significantly simpler than the conventional approaches: instead of using the customary integral equation operators involving the Green function and its derivatives, we construct effective integral equation operators equal to (i) the Fourier transform of the dispersive medium Green function, (ii) the Fourier transform of the product of the dispersive medium Green function with the frequency dependent dielectric permittivity, and, (iii) the Fourier transform of the product of the dispersive medium Green function with the inverse of the dielectric permittivity. An important benefit of such an approach is that the resulting integrals involve only single (and not double) time convolutions. The formulation is applicable to systems involving bulk dispersive regions and thin dispersive sheets represented as interfaces. We discuss details of complete analytical calculations and of corresponding numerical procedures for the evaluation of matrix elements of the integral operators, executed in the framework of the full Galerkin scheme in space and time variables, for the "conductive Debye medium" (i.e., for a medium with the electric permittivity given by the Debye formula supplemented with a term responsible for the medium conductivity). The procedure employs a suitable contour integration around singularities of the effective Green function operators in the complex frequency plane. The full Galerkin discretization is used with RGW spatial basis functions and band limited temporal basis functions. Our formulation constitutes the basis for accurate numerical simulation framework for a variety of problems involving wideband pulse propagation, including wideband antenna pattern simulation or propagation of narrow pulses through dispersive media.

Abstract : The objective of our effort was to develop computational methods for constructing high-frequency asymptotic solutions in scattering on perfectly conducting objects. The emphasis of the first stage of our work was to describe high frequency phenomena in terms of numerically implemented evolution of wave-fronts associated with the propagating waves. The wave-front evolution algorithm is implemented for the leading high frequency mechanisms including: free-space propagation, reflection on smooth surfaces, wave-front splitting at the shadow boundary, generation and propagation of edge diffracted wave-fronts, and surface wave propagation. The second stage of our work was directed towards developing novel fast rigorous (direct or iterative) solution methods based on construction of economical parameterization of high frequency solutions in terms of basis functions defined on large supports. The wave-front evolution technique developed in the first stage provides a numerical prescription for the selection and determination the parameters of the postulated analytical representation of such basis functions. The numerical prescription together with numerical tools constructed during this effort will constitute an important element of the planned future, high frequency solution technique employing basis functions defined on large supports.

This paper describes a simple physically-motivated "near-field" preconditioning scheme that is effective in accelerating convergence of surface, volume, and combined surface/volume integral equations for a broad variety of electromagnetic scattering problems. It can be easily implemented numerically in method of moment (MoM) solvers (both conventional and those employing matrix-compression techniques), irrespective of the analytical form of the integral-equation kernel. It has low memory and CPU requirements, both of which scale linearly with the number of unknowns, and is easily amenable to efficient parallelization. We demonstrate the preconditioner's performance (in conjunction with the BiCGstab(ell) iterative solver) on two representative geometries, and observe a significant reduction in the number of iterations required for convergence.

The availability of accurate and fast numerical tools capable of simulating electromagnetic wave propagation through dispersive media constitutes an important element of analysis of many aspects of wide-band pulse propagation (e.g., in the context of wide frequency band antennas or foliage penetration). Although there has been recently much interest in developing fast solution methods for time domain integral formulations of Helmholtz and Maxwell’s equations (Refs. 1, 2, 3, 4), relatively little attention has been given to the development of such tools for dispersive media, which pose a challenge from both analytic and numerical point of view.

We propose a novel time-domain impedance matrix compression and solution method applicable to dispersive and/or layered media. The method, which we call the FFT time domain (FFTTD) method, is based on Toeplitz properties of the impedance matrix, in both temporal and spatial indices, allowing application of fast Fourier transforms (FFTs). It departs from the conventional marching-on-in-time (MOT) solution scheme, and utilizes instead an algorithm belonging to the category of superfast direct solutions for block-Toeplitz matrices. The computational cost of the proposed method scales as O(N/sub t/ N/sub s/ log/sup 2/ N/sub t/ log N/sub s/) and O(N/sub t/ N/sub s//sup 4/3/ log/sup 2/ N/sub t/ log N/sub s/) for volume and surface problems respectively, where by N/sub t/ and N/sub s/ we denote the number of temporal and spatial samples.

The adaptive integral method (AIM) is a fast method associated with O(N1.5) or less complexity. It has been extensively used for the analysis of metallic scatterers on the basis of the electric field integral equation (EFIE), and the AIM implementation is extended to include more general surface types such as impedance, resistive, dielectric and others. The associated multipole expansions of the basis functions are presented for all integral operators, and examples of perfect electrically conducting (PEC) and dielectric surfaces are given for validation

We demonstrate that truncation of biological models combined with application of proper transmission boundary conditions on interfaces created by the truncation allow to simulate field distributions and specific absorption rates at significant memory and computation time savings without compromising the accuracy of integral equation formulations. The proposed technique can provide a reliable and efficient approach for hyperthermia treatment planning or design of wireless personal devices.

We consider an extension of the fast Fourier transforms based impedance matrix compression method to problems involving complex conducting structures embedded in layered media of infinite extent in the transverse directions. The method requires generalization of the compression technique to the multilayered medium Green's function. The method is applicable to structures which may be electromagnetically large and, at the same time, discretized with highly sub-wavelength resolution. We analyze two approaches of compressing the far field part of the impedance matrix through an approximation to the Fourier transforms of the basis functions /spl phi//spl tilde/(q): (A) based on the Taylor expansion about q=0, and (B) based on the least-squares approximation at the q values giving dominant contribution to the impedance matrix element. While compression (A) is applicable already to distances much smaller than the wavelength, compression (B) is based on the field behavior in the asymptotic wave region, and thus applies only to distances comparable to and larger than the wavelength. Therefore, approach (A) is better suited to structures discretized with highly sub-wavelength resolution. Its other advantages are simplicity and independence of the structure of the Green's function. For densely packed structures, the method is characterized by O(NlogN) computational complexity and O(N) memory requirements with a small, when compared to other approaches, proportionality coefficient in front of the estimates. Error estimated for the proposed algorithm are discussed.