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In this paper, we present a method for solving electromagnetic scattering from a composite object at low frequencies using the method of moments (MoM) and loop-tree basis. Many applications involve composite objects which consist of several homogeneous regions. The loop-tree basis used for analyzing scattering from a homogeneous body at low frequencies could not be directly applied to analyze the low-frequency scattering from a composite object. In general, it is very difficult, if not impossible, to find a set of loop-tree basis functions that is valid for the structures on both sides of the interfaces. In this paper, we treat a composite object as a limiting case of multibody problem so that we could setup the MoM equation using the loop-tree basis found on each single body. A process is then developed to eliminate the redundant unknowns. The proposed method makes it possible to analyze low-frequency scattering from an arbitrary composite object. The validity and applications are illustrated with representative numerical examples. Index Terms--Composite object, loop-tree basis, low frequency.

Aperture coupled dielectric resonator antennas (DRAs) that reside above a layered background are analyzed using the method of moments (MoM). Dyadic Green's functions for the layered medium are derived in a simple and effective manner and Rao-Wilton-Glisson (RWG)function is used to expand both electric and magnetic currents. A transmission-line technique is used to compute the antenna-input impedance. Numerical results obtained for both hemispheric and rectangular DRAs agree well with published measurement results. Index Terms--Dielectric resonator antenna (DRA).

A diffraction tomographic (DT) algorithm has been proposed for detecting three-dimensional (3-D) dielectric objects buried in a lossy ground, using electric dipoles or magnetic dipoles as transmitter and receiver, where the air-earth interface has been taken into account and the background is lossy. To derive closed-form reconstruction formulas, an approximate generalized Fourier transform is introduced. Using this algorithm, the locations, shapes, and dielectric properties of buried objects can be well reconstructed under the low-contrast condition, and the objects can be well detected even when the contrast is high. Due to the use of fast Fourier transforms to implement the problem, the proposed algorithm is fast and quite tolerant to the error of measurement data, making it possible to solve realistic problems. Reconstruction examples are given to show the validity of the algorithm. Index Terms--Detection of objects, diffraction tomography (DT), generalized fourier transform, Green's functions, inverse scattering, lossy earth, Sommerfeld integrals.

In this paper, numerical simulations of a new enhanced very early time electromagnetic (VETEM) prototype system are presented, where a horizontal transmitting loop and two horizontal receiving loops are used to detect buried targets, in which three loops share the same axis and the transmitter is located at the center of receivers. In the new VETEM system, the difference of signals from two receivers is taken to eliminate strong direct-signals from the transmitter and background clutter and furthermore to obtain a better SNR for buried targets. Because strong coupling exists between the transmitter and receivers, accurate analysis of the three-loop antenna system is required, for which a loop-tree basis function method has been utilized to overcome the low-frequency breakdown problem. In the analysis of scattering problem from buried targets, a conjugate gradient (CG) method with fast Fourier transform (FFT) is applied to solve the electric field integral equation. However, the convergence of such CG-FFT algorithm is extremely slow at very low frequencies. In order to increase the convergence rate, a frequency-hopping approach has been used. Finally, the primary, coupling, reflected, and scattered magnetic fields are evaluated at receiving loops to calculate the output electric current. Numerous simulation results are given to interpret the new VETEM system. Comparing with other single-transmitter-receiver systems, the new VETEM has better SNR and ability to reduce the clutter. Index Terms--Buried object detection, conjugate gradient fast Fourier transform (CG-FFT), electromagnetic scattering, frequency hopping, loop antennas, loop-tree basis, low frequencies, numerical analysis.

The full-wave analysis of the generalized microstrip line on an inhomogeneous anisotropic substrate is carried out by using the finite-difference method. The resulting sparse matrix equation is solved efficiently using the bi-Lanczos algorithm. The use of the inhomogeneous wave equation to formulate the problem makes it easy to analyze structures with multilayered substrates. The algorithm can analyze complicated structures with multiple conductors at arbitrary locations. A spatial interpolation scheme is used to evaluate the contribution from the off-diagonal terms in [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. The use of the bi-Lanczos algorithm allows us to solve the problem at O([N.sup.1.5]) complexity. Storage requirements can be made to scale as O(N). This makes it possible to analyze large problems on a small computer. Very good agreement is seen between published results and results obtained using this technique. Index Terms - Anisotropic substrate, finite difference, Lanczos algorithm, microstrip lines.

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 (DFT's), which also can be rapidly computed. Because of the use of FFT's 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)< Index Terms: Buried objects, CG-FFT, fast algorithm, Sommerfeld integrals.

A very efficient three-dimensional (3-D) solver for the diffusion of the electromagnetic fields in an inhomogeneous medium is described. The proposed method employs either the node-based or the edge-based finite-element method (FEM) to discretize Maxwell's equations. The resultant matrix equation is solved by the spectral Lanczos decomposition method (SLDM), which is based on the Krylov subspace (Lanczos) approximation of the solution in frequency domain. By analyzing some practical geophysical problems, it is shown that the SLDM is extremely fast and, furthermore, the electromagnetic fields at many frequencies can be evaluated by performing the SLDM iteration only at the lowest frequency. Index Terms - Electromagnetic diffusion, fast solvers, finite-element methods.

In this paper, we present an accurate method of moments (MoM) solution of the combined field integral equation (CFIE) using the multilevel fast multipole algorithm (MLFMA) for scattering by large, three-dimensional (3-D), arbitrarily shaped, homogeneous objects. We first investigate several different MoM formulations of CFIE and propose a new formulation, which is both accurate and free of interior resonances. We then employ MLFMA to significantly reduce the memory requirement and computational complexity of the MoM solution. Numerical results are presented to demonstrate the accuracy and capability of the proposed method. The method can be extended in a straightforward manner to scatterers composed of different homogeneous dielectric and conducting objects. Index Terms - Electromagnetic scattering, moment methods.

A new technique, the steepest descent-fast multipole method (SDFMM), is developed to efficiently analyze scattering from perfectly conducting random rough surfaces. Unlike other prevailing methods, this algorithm has linear computational complexity and memory requirements, making it a suitable candidate for analyzing scattering from large rough surfaces as well as for carrying out Monte Carlo simulations. The method exploits the quasiplanar nature of rough surfaces to efficiently evaluate the dyadic Green's function for multiple source and observation points. This is achieved through a combination of a Sommerfeld steepest descent integral and a multilevel fast multipole-like algorithm based on inhomogeneous plane wave expansions. The fast evaluation of the dyadic Green's function dramatically speeds up the iterative solution of the integral equation for rough surface scattering. Several numerical examples are presented to demonstrate the efficacy and accuracy of the method in analyzing scattering from extremely large finite rough surfaces. Index Terms - Fast-multipole methods, integral equations, multilevel algorithms, rough surface scattering.

This paper studies, in detail, a variety of formulations for the hybrid finite-element and boundary-integral (FE-BI) method for three-dimensional (3-D) electromagnetic scattering by inhomogeneous objects. It is shown that the efficiency and accuracy of the FE-BI method depends highly on the formulation and discretization of the boundary-integral equation (BIE) used. A simple analysis of the matrix condition number identifies the efficiency of the different FE-BI formulations and an analysis of weighting functions shows that the traditional FE-BI formulations cannot produce accurate solutions. A new formulation is then proposed and numerical results show that the resulting solution has a good efficiency and accuracy and is completely immune to the problem of interior resonance. Finally, the multilevel fast multipole algorithm (MLFMA) is employed to significantly reduce the memory requirement and computational complexity of the proposed FE-BI method. Index Terms - Boundary-integral equations, electromagnetic scattering, finite-element methods, nonhomogeneous media.

The fast multipole method (FMM) and multilevel fast multipole algorithm (MLFMA) are reviewed. The number of modes required, block-diagonal preconditioner, near singularity extraction, and the choice of initial guesses are discussed to apply the MLFMA to calculating electromagnetic scattering by large complex objects. Using these techniques, we can solve the problem of electromagnetic scattering by large complex three-dimensional (3-D) objects such as an aircraft (VFY218) on a small computer. Index Terms - Electromagnetic scattering, numerical analysis.

Various methods for efficiently solving electromagnetic problems are presented. Electromagnetic scattering problems can be roughly classified into surface and volume problems, while fast methods are either differential or integral equation based. The resultant systems of linear equations are either solved directly or iteratively. A review of various differential equation solvers, their complexities, and memory requirements is given. The issues of grid dispersion and hybridization with integral equation solvers are discussed. Several fast integral equation solvers for surface and volume scatterers are presented. These solvers have reduced computational complexities and memory requirements. Index Terms - Numerical methods.

A local shape function (LSF) -based numerical reconstruction algorithm for biological bodies efficiently reconstructs the density and compressibility of soft tissues from ultrasonic scattering data using conjugate gradients and fast Fourier transforms. An iterative minimization technique accounts for the nonlinearity effects of the multiple scattering. Simulations and real experiments demonstrate the capability of the algorithm. The LSF overcomes singularity problems by matching the boundary conditions on the surface of the subscatterer.

The mode matching method is applied to different waveguide geometries, and a recursive scheme is defined to use the appropriate number of modes in each section of the waveguide. Numerical simulations for different types of applications are given and show very good results.

The use of wavelet basis functions for the efficient solution of electromagnetic integral equations is studied. It has previously been demonstrated that the use of wavelets for expansion and testing functions produces a sparse moment-method matrix. Here, this effect is examined and analyzed in terms of the radiation/receiving characteristics of the wavelet basis functions. The limitations of wavelets as an efficient solution technique are discussed, and a comparison is made to other fast algorithms.

A three-dimensional (3-D) version of the nested equivalent principle algorithm (NEPAL) is presented. In 3-D, a scatterer is first decomposed into N subscatterers. Then, spherical wave functions are used to represent the scattered field of the subscatterers. Subscatterers are divided into different levels of groups in a nested manner. For example, each group consists of eight subgroups, and each subgroup contains eight sub-subgroups, and so on. For each subgroup, the scattering solution is first solved and the number of subscatterers of the subgroup is then reduced by replacing the interior subscatterers with boundary subscatterers using Huygens' equivalence principle. As a result, when the subgroups are combined to form a higher level group, the group will have a smaller number of subscatterers. This process is repeated for each level, and in the last level, the number of subscatterers is proportional to that of boundary size of the scatterers. This algorithm has a computational complexity of O([N.sup.2]) in three dimensions for all excitations and has the advantage of solving large scattering problems for multiple excitations. This is in contrast to Gaussian elimination which has a computational complexity of O([N.sup.3]).

The inversion of induction tool measurements using the distorted Born iterative method (DBIM) and the conjugate gradient-fast Fourier-Hankel transform (CG-FFHT) is described. The inverse problem is formulated in terms of an integral equation of scattering where the unknown to be sought is the conductivity in the rock formation, when the measurements along a borehole axis are performed. The nonlinear problem is linearized at each stage using the distorted Born approximation. The inhomogeneous medium Green's function in the distorted Born approximation is found by solving a volume integral equation using the CG-FFHT method, which allows a rapid solution to a large problem with reduced computational complexity and memory requirement. In this manner, the inverse problem is solved with a computational complexity proportional to [N.sub.tl]N log N where [N.sub.tl] is the number of transmitter locations used in the data collection and N is the total number of pixels used to model the unknown formation. The memory requirement is of order N[N.sub.tl].

In this paper, we formulate and implement a numerical mode-matching (NMM) method to model electrode-type resistivity tools in invaded thin beds. We derive the low-frequency approximation of the Maxwell's equations to obtain the partial differential equation for the potential field. The new NMM program is validated by comparing the numerical results with those obtained from other dc programs. It is found that this new program is much faster than the program using the finite-element method (FEM), and hence is useful for routine interpretation of resistivity logs and for inversion.

Local shape function imaging uses far-field microwave scattering data to reconstruct the presence or absence of small metal cylinders throughout space, in order to model arbitrary metallic objects. The reconstructed images represent the scattering amplitude at discrete locations in space with multiple scattering effects incorporated. Super-resolution is demonstrated for monochromatic image reconstructions. Even better reconstructions are obtained with multiple frequency data. The speed of computation is increased with a fast forward solver algorithm. Also, measured data is used in the local shape function imaging algorithm and the resolution is improved over diffraction tomography.

The performance of the Born iterative method of nonlinear two-dimensional profile inversion is examined for the reconstruction of large objects and in the presence of measurement noise. Time-domain data are used. It is shown that objects at least as large as about nine wavelengths can be inverted without any convergence problems. The algorithm is also shown to perform well in the presence 10% noise, or 20 dB signal-to-noise ratio. The simultaneous inversion of permittivity and conductivity profiles is also formulated and solved using the Born iterative method. Objects with various loss tangents are reconstructed, and the limits of applicability of the algorithm are investigated

A fast recursive algorithm is used to compute the scattering properties of finite array of strip gratings on a dielectric slab. This algorithm has a computational complexity of O(N log squared N) for one incident angle and O(N squared log N) for N incident angles. It uses plane wave basis for expanding the incident wave and the scattered wave. The scattered wave is expanded in terms of a Sommerfeld-type integral with spectral distribution along a vertical branch cut, rendering its expansion very efficient. To validate the scattering solution obtained using the recursive algorithm, comparisons with the method of moments are illustrated. The current distributions on the strips and scattering patterns are both presented. Since this algorithm has reduced computational complexity and is fast compared to other conventional methods, it can be used to analyze very large strip arrays. Scattering solution of a 50-wavelength wide strip is illustrated.

Two recursive T-matrix algorithms (RTMA's) are presented and their reduced computational complexities and reduced memory requirements are demonstrated. These algorithms are applied to the problem of electromagnetic scattering from conducting strip and patch geometries. For a systematic development, canonical geometries of strips and patches are chosen. These geometries are reminiscent of finite-sized frequency selective surfaces (FSS's). Computational complexities of O(N(super 2)) and O(N(super 7/3)) and memory requirements of O(N) and O(N(super 4/3)) are shown to be feasible for two-dimensional and three-dimensional geometries, respectively. The formulation uses only two components of the electric field. Therefore, the vector electromagnetic problem of scattering from three-dimensional patch geometries can be solved using scalar - rather than vector - addition theorems for spherical harmonic wave functions. For a two-dimensional strip problem, both TM and TE polarizations can be solved simultaneously using this formulation. Numerical scattering results are presented in the form of radar cross sections (RCS's) and validated by comparison with the method of moments (MoM).

The first three-dimensional analysis of a short backfire antenna (SBA) is described using the vector mode matching method. The computation is efficient for a model that uses only the first harmonic, cos phi, and that deforms the path of numerical integration. This model is useful for SBA computer-aided designs. A larger directivity than previously reported is obtained with the aid of this model. The theoretical predictions of the radiation pattern yield better agreement than previous methods when compared with measurements performed on a prototype.