Malcolm M. Bibby

High Order Representations for Singular Currents at Corners

In connection with a high order method of moments solution of integral equations for electromagnetic scattering, several approaches are investigated for representing current and charge densities in the vicinity of corners. One approach uses the first N terms from the asymptotic solution for an infinite wedge to represent the current density. An alternative approach uses a traditional polynomial type of expansion, but augments it in the vicinity of corners with additional terms from the wedge result. The residual error obtained via the solution of an over-determined system of equations is used to judge the relative accuracy and efficiency of various approaches.

High-order numerical solutions of the MFIE for the linear dipole

The magnetic field integral equation (MFIE) was applied to a dipole using three different discretization methods and high-order basis functions. For moderate-order, and higher, basis functions it was found that the different discretization methods produced essentially the same results. Continuity of current and its first derivative was observed at cell boundaries even though continuity of current was not explicitly enforced there. The MFIE provided lower condition numbers than the Hallen equation over the range of dipole radii examined. In close proximity to surface discontinuities, including hidden ones, residual errors could not be significantly reduced by increasing the order of the basis functions, implying the need for better modeling at discontinuities and calling into question the use of faceting to represent curved surfaces.

On the use of overdetermined systems in the adaptive numerical solution of integral equations

The residual error incurred when numerically solving integral equations for a number of electromagnetic radiation and scattering problems is calculated with the aid of an overdetermined system. This error is systematically reduced by adaptively refining the model for the surface current. Error reduction is achieved by selectively shrinking cell dimensions (h-refinement), increasing the order of the basis functions representing the current (p-refinement), or a combination of both (hp-refinement). The correlation between residual error and surface current error is investigated and found to be high. The impact of edge singularities and curvature discontinuities is discussed.

An Introduction to the Locally-Corrected Nystrom Method

This lecture provides a tutorial introduction to the Nystrom and locally-corrected Nystrom methods when used for the numerical solutions of the common integral equations of two-dimensional electromagnetic fields. These equations exhibit kernel singularities that complicate their numerical solution. Classical and generalized Gaussian quadrature rules are reviewed. The traditional Nystrom method is summarized, and applied to the magnetic field equation for illustration. To obtain high order accuracy in the numerical results, the locally-corrected Nystrom method is developed and applied to both the electric field and magnetic field equations. In the presence of target edges, where current or charge density singularities occur, the method must be extended through the use of appropriate singular basis functions and special quadrature rules. This extension is also described. Table of Contents: Introduction / Classical Quadrature Rules / The Classical Nystrom Me hod / The Locally-Corrected Nystrom Method / Generalized Gaussian Quadrature / LCN Treatment of Edge Singularities

Error trends in higher-order discretizations of the EFIE and MFIE

We concern ourselves with the EFIE and MFIE applied to 2D or cylindrical perfectly conducting scatterers. The MoM procedure is used with subsectional Legendre polynomial basis functions, without the imposition of cell-to-cell continuity. The equations are enforced by point testing, or equivalently Dirac delta testing functions, located at the nodes of open Newton-Coates quadrature rules. The discretization process may yield a square system matrix, or an overdetermined system matrix, depending on the number of testing points that we employ. If used with an overdetermined matrix, the approach is also known as the boundary residual method. We investigate error trends for the square matrix case and the case where the matrix is overdetermined by a factor of 2. A benefit to the overdetermined matrix is that the residual error in the matrix equation can be obtained as part of the matrix solution process, and used to estimate the residual error in the continuous equation. For geometries with corners or edges, the polynomial basis is modified to incorporate the expected behavior of the current density at those locations.

High-order basis functions for singular currents at corners

The use of a full Motz-like expansion for the current density in the cells adjacent to the corners of a conducting scatterer provides a substantial improvement in accuracy, compared to the use of a representation that only employs the leading-order singularity. Furthermore, as the order of the expansion is increased in conjunction with that in the non-corner cells, the slope of the resulting error curve increases, confirming that higher-order behavior is obtained.

A High Order Numerical Investigation of Electromagnetic Scattering From a Torus and a Circular Loop

The normally integrated magnetic field integral equation (NIMFIE) formulation is used to generate accurate numerical results for toroidal scatterers. Numerical data for the surface currents and scattering cross section induced by a plane wave on a perfectly conducting torus are presented. The special case of a thin circular loop is also examined. These results are expected to find use in numerical validation.

Use of extrapolation to improve accuracy and enhance confidence in numerical results

Two-point and three-point are used to improve the accuracy of numerical results for electromagnetic applications. When applied to a sequence of results obtained with conventional h-refinement, extrapolation procedures permit an estimate of the quantity of interest in the limit as h tends to zero. Three-point procedures also provide an estimate of the rate at which the error decreases, and can consequently improve confidence in the numerical results.

Normally-Integrated Magnetic Field Integral Equations for Electromagnetic Scattering

Modified magnetic field integral equations (MFIEs) are proposed incorporating a testing integration in the direction normal to the boundary of the target surface. The result involves a simpler kernel than the conventional MFIE, facilitating accurate computations. By employing a complex exponential testing function, the equations can be made immune to the interior resonance problem. The influence of the limits of integration, and the overall accuracy, is investigated using high order basis functions.

Progress in developing custom cubature rules for computational electromagnetics

Computational electromagnetics often requires the creation of quadrature and cubature rules beyond those found in the literature. In many situations rules are required for the accurate evaluation of integrals containing singularities; examples include 1/R singularities and singularities in the basis functions used to represent current or charge density at sharp edges and corners. Techniques for synthesizing quadrature rules involve the solution of nonlinear systems of equations. This presentation will describe recent progress as well as some of the challenges encountered by the authors in multidimensional quadrature rule synthesis.