Tapan K. Sarkar

Survey of numerical methods for solution of large systems of linear equations for electromagnetic field problems

Many of the popular methods for the solution of large matrix equations are surveyed with the hope of finding an efficient method suitable for both electromagnetic scattering and radiation problems and system identification problems.

The Electrostatic Field of Conducting Bodies in Multiple Dielectric Media

A method for computing the electrostatic fields and the capacitance matrix for a multiconductor system in a multiple dielectric region is presented. The number of conductors and the number of dielectrics in this analysis are arbitrary. Some of the conductors maybe of finite volume and others may be infinitesimally thin. The conductors can be either above a single ground plane or between two parallel ground planes. The formulation is obtained by using a free-space Greens function in conjunction with total charge on the conductor-to-dielectric interfaces and polarization charge on the dielectric-to-dielectric interfaces. The solution is effected by the method of moments using triangular subdomains with piecewise constant expansion functions and point matching for testing. Computed results are given for some finite-length conducting lines, compared to previous results obtained by two-dimensional analysis.

On a class of finite step iterative methods (Conjugate directions) for the solution of an operator equation arising in electromagnetics

A class of finite step iterative methods for the solution of linear operator equations is presented. Specifically, the basic principles of the method of conjugate directions are developed. Gaussian elimination and the method of conjugate gradients are then presented as two special cases. With an arbitrary initial guess, the method of conjugate gradient always converges to the solution in at most N iterations, where N is the number of independent eigenvalues for the operator in the finite dimensional space in which the problem is being solved. The conjugate gradient method requires much less storage ( sim 5N ) than the conventional matrix methods ( sim N^{2} ) in the solution of problems of higher complexity. Also, after each iteration the quality of the solution is known in the conjugate gradient method. The conjugate gradient method is also superior to the spectral iterative method as the latter does not always converge and it doubles the complexity of a given problem, unnecessarily. Four versions of the conjugate gradient method are presented in detail, and numerical results for a thin wire scatterer are given to illustrate various properties of each version.

Some mathematical considerations in dealing with the inverse problem

Many problems of mathematical physics can be formulated in terms of the operator equation Ax = y , where A is an integro-differential operator. Given A and x , the solution for y is usually straightforward. However, the inverse problem which consists of the solution for x when given A and y is much more difficult. The following questions relative to the inverse problem are explored. 1) Does specification of the operator A determine the set {y} for which a solution x is possible? 2) Does the inverse problem always have a unique solution? 3) Do small perturbations of the forcing function y always result in small perturbations of the solution? 4) What are some of the considerations that enter into the choice of a solution technique for a specific problem? The concept of an ill-posed problem versus that of a well-posed problem is discussed. Specifically, the manner by which an ill-posed problem may be regularized to a well-posed problem is presented. The concepts are illustrated by several examples.

A note on the choice weighting functions in the method of moments

The objective is to show from a mathematical standpoint that there are certain rules that must be followed in the choice of weighting functions used in the method of moments (MM). It is shown that for a particular problem it is the operator that dictates the method (Galerkins method or another method such as the method of least squares) to be applied, and it is not computational considerations only. For example, it is shown that in solving Hallens and Pocklingtons equation by the method of moments, it is unnatural to choose the weighting functions which are zero at the ends of the domain of the solution. The deficiency of certain weighting functions is presented based on mathematical reasoning, and a numerical example is given to illustrate the effect of the choice of the weighting functions on the rate of the convergence of the solution.

An iterative method for solving electrostatic problems

The method of steepest descent is applied to the solution of electrostatic problems. The relation between this method and the Rayleigh-Ritz, Galerkins, and the method of least squares is outlined. Also, explicit error formulas are given for the rate of convergence for this method. It is shown that this method is also suitable for solving singular operator equations. In that case this method monotonically converges to the solution with minimum norm. Finally, it is shown that the technique yields as a by-product the smallest eigenvalue of the operator in the finite dimensional space in which the problem is solved. Numerical results are presented only for the electrostatic case to illustrate the validity of this procedure which show excellent agreement with other available data.

A novel window for harmonic analysis

This paper presents a new window design technique and discusses its effect on the detection of harmonic signals in the presence of nearby strong harmonic interference. Four design parameters are introduced to independently control the pattern falloff rate, the overall sidelobe level, the near-sidelobe level and the depth of a steerable wide dip. Since the deep dip can be steered to any frequency the new window has effectively improved the detectability of a small tone without degrading resolvability. Contrary to the conventional approach of initially specifying the continuous weighting, the new design technique starts with constructing the spectral window which meets the specifications and then employs the fast Fourier transform to compute the discrete weighting. No iterative sampling or perturbation procedure is required. Numerous examples are given to demonstrate the flexibility of the new window. Several examples of detecting a three-tone signal have demonstrated the superiority of the new window in its detectability, resolvability, and accuracy of measuring the tone frequencies and amplitudes.

A simple technique for solving E -field integral equations for conducting bodies at internal resonances

A simple but effective method is presented to analyze electromagnetic radiation and scattering from condueting bodies at frequencies corresponding to internal resonances of a cavity of the same shape. The advantage of this technique is that it requires only the E -field integral equation and hot both E -field and H -field as required by the combined fields formulation. It is shown theoretically that this method produces a solution with minimum norm and converges monotonically as the order of the approximation is increased. The minimum norm solution for the current density given by the E -field integral equation is not the correct current density as there is a portion of the resonant current that exists on the body. However, the minimum norm solution indeed provides the true scattering fields. This technique may also be utilized for obtaining a minimum norm solution for nearly singular and singular matrix equations. Examples are presented to illustrate the application of this technique.

Suboptimal approximation/identification of transient waveforms from electromagnetic systems by pencil-of-function method

A noniterative method for approximating signals by a linear combination of exponentials is presented. Although the technique results in a suboptimal approximation, the continuous dependence of the suboptimal exponents sim{s}_{i} on the integral square error epsilon is such that lim (epsilon = 0) sim{s}_{i} rightarrow {s}_{i} , the best least squares exponents. The method is also useful for system identification, where the system is modeled by a black box and one has access only to the input and output terminals. A technique is demonstrated for finding the multiple poles of a system along with the residues at the poles when the system output to a known input is given. Advantages of the method are natural insensitivity to noise in the data and a capability for approximately determining signal order. Representative computations are made of the poles from the transient response of a conducting pipe tested at the ATHAMAS-I EMP simulator.

Computation of Inductance of Simple Vias Between Two Striplines Above a Ground Plane (Short Papers)

In this paper, an analysis is developed for calculating the lumped inductance of a simple via connecting two infinitely thin striplines, located above a perfectly conducting ground plane. The striplines are oriented in the same direction, and the via is assumed to be in the form of an infinitely thin vertical plate, connecting the two lines. This system is analyzed by a hybrid partial-element and circuit-theory approach. Numerical results are presented to illustrate the application of this technique.