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 Method / The Locally-Corrected Nystrom Method / Generalized Gaussian Quadrature / LCN Treatment of Edge Singularities

An Introduction to the Locally Corrected Nyström Method

To solve problems of bodies of revolution (BoR) using the method of moments, the modal Green's function is used, which is defined as the radiation field of a circular loop antenna with sinusoidal current distribution. It is a group of oscillating integrals that forms the bottleneck for further accelerating the computation of BoR problems. In this paper, we propose closed-form expressions for near-axis far-distance modal Green's functions, which can be used to accelerate the computation of BoR problems especially when the object is slim. A criterion is presented based on a rigorous error analysis to guarantee its range of applications. Both the scattering and radiation examples are presented to demonstrate the validity and efficiency of the proposed expressions.

Closed Form Modal Green's Functions for Accelerated Computation of Bodies of Revolution

An FFT-accelerated direct solver for electromagnetic scattering from penetrable axisymmetric objects

The focus of this chapter is to provide a broad overview of the high-order Locally Corrected Nystrom (LCN) method. The LCN method is applied herein to the solution of a general hybrid surface/volume integral equation defining the electromagnetic scattering and/or radiation from arbitrary three-dimensional geometries of arbitrary material composition. The chapter starts out by providing a basic understanding of the LCN algorithm and its analogy to the more popular method of moments. The LCN method is then specialized to the solution of hybrid integral equations by providing nuts-and-bolts details regarding the application of the method. LCN discretizations of surface integral equations are presented for curvilinear quadrilateral or triangular meshes. Furthermore, LCN discretizations of volume integral equations are derived for curvilinear hexahedral, tetrahedral, or prism meshes. Quadrature rules suitable for a Nystrom discretization and basis functions suitable for local corrections are defined for each topological type. Performing the local corrections for the different kernels encountered is also discussed. Finally a number of numerical examples are presented that span the different integral operators and topological types. For all cases, data is provided that demonstrates the high-order convergence of a LCN solution method. It is hoped that the power, yet simplicity, of the LCN method is conveyed in this chapter, along with enough details for an interested reader to develop their own LCN software.

The Locally Corrected Nyström Method for Electromagnetics

In this paper we consider the classic problem of scattering of waves from perfectly conducting cylinders with piecewise smooth boundaries. The scattering problems are formulated as integral equations and solved using a Nystrom scheme, where the corners of the cylinders are efficiently handled by a method referred to as Recursively Compressed Inverse Preconditioning (RCIP). This method has been very successful in treating static problems in non-smooth domains and the present paper shows that it works equally well for the Helmholtz equation. In the numerical examples we focus on scattering of E- and H-waves from a cylinder with one corner. Even at a size kd=1000, where k is the wavenumber and $d$ the diameter, the scheme produces at least 13 digits of accuracy in the electric and magnetic fields everywhere outside the cylinder. (Less)

/pdf/an-accurate-boundary-value-problem-solver-applied-to-4lzk3r17sm.pdf

An accurate boundary value problem solver applied to scattering from cylinders with corners

Locally corrected Nyström method for EM scattering by bodies of revolution

A locally corrected Nyström method for EM scattering with a Fourier mode expansion

The locally corrected Nystro/spl uml/m method is applied to the magnetic field integral equation for a conducting body of revolution. A construction method is presented for the locally corrected weights for the resulting one-dimensional coupled scalar magnetic field integral equations. Special attention is paid to minimizing the cost for multi-frequency computations. Numerical results are presented for the sphere, oblate spheroid, and right circular cylinder. Good agreement with results from mature moment method codes is observed.

EM scattering from bodies of revolution using the locally corrected Nystro/spl uml/m method

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Locally corrected Nyström method for EM scattering by bodies of revolution

A locally corrected Nyström method for EM scattering with a Fourier mode expansion

EM scattering from bodies of revolution using the locally corrected Nystro/spl uml/m method