The Method of Moments in Electromagnetics

Walton C. Gibson

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000412505
• Edition: 3
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

The Method of Moments in Electromagnetics, Third Edition details the numerical solution of electromagnetic integral equations via the Method of Moments (MoM). Previous editions focused on the solution of radiation and scattering problems involving conducting, dielectric, and composite objects. This new edition adds a significant amount of material on new, state-of-the art compressive techniques. Included are new chapters on the Adaptive Cross Approximation (ACA) and Multi-Level Adaptive Cross Approximation (MLACA), advanced algorithms that permit a direct solution of the MoM linear system via LU decomposition in compressed form. Significant attention is paid to parallel software implementation of these methods on traditional central processing units (CPUs) as well as new, high performance graphics processing units (GPUs). Existing material on the Fast Multipole Method (FMM) and Multi-Level Fast Multipole Algorithm (MLFMA) is also updated, blending in elements of the ACA algorithm to further reduce their memory demands.

The Method of Moments in Electromagnetics is intended for students, researchers, and industry experts working in the area of computational electromagnetics (CEM) and the MoM. Providing a bridge between theory and software implementation, the book incorporates significant background material, while presenting practical, nuts-and-bolts implementation details. It first derives a generalized set of surface integral equations used to treat electromagnetic radiation and scattering problems, for objects comprising conducting and dielectric regions. Subsequent chapters apply these integral equations for progressively more difficult problems such as thin wires, bodies of revolution, and two- and three-dimensional bodies. Radiation and scattering problems of many different types are considered, with numerical results compared against analytical theory as well as measurements.

Chapter 1

Computational Electromagnetics

[1] In the beginning, the design and analysis of electromagnetic devices and structures was largely experimental. However, once the computer and numerical programming languages were developed, people immediately began using them to solve electromagnetic field problems of ever-increasing complexity. This pursuit of computational electromagnetics (CEM) has yielded many innovative, powerful analysis algorithms, and it now drives the development of electromagnetic devices people use every day. As the power of the computer continues to grow, so do the number of available algorithms as well as the size and complexity of the problems that can be solved. While the data gleaned from experimental measurements is invaluable, the entire process can be costly in terms of money and the manpower required to do the machine work and assembly, and to collect data at the measurement range. One of the fundamental drives behind reliable computational electromagnetics algorithms is the ability to simulate the behavior of devices and systems before they are actually built. This allows engineers to engage in levels of optimization that would be painstaking or even impossible if done experimentally. CEM also helps to provide fundamental insights into electromagnetic problems through the power of computation and computer visualization, making it one of the most important areas of engineering today.

1.1 CEM Algorithms

The range of electromagnetic problems is extensive, and this has led to the development of different classes of CEM algorithms, each with its own benefits and limitations. In the “early days” of CEM, many problems of practical size could not be solved unless some assumptions were made about the underlying physics and approximations made, usually under the asymptotic or high-frequency limit. These approximate algorithms are now commonly known as “high-frequency” methods. Algorithms that do not make these sorts of approximations are more demanding in terms of CPU and system memory, and historically have been limited to problems of small electrical size. These are usually referred to as “exact” or “low-frequency” algorithms. Both classes of algorithms can be further subdivided into time or frequency-domain algorithms. We will now summarize some of the most commonly used methods to provide some context in how the Moment Method fits in the CEM environment.

1.1.1 Low-Frequency Methods

Low-frequency (LF) methods are so-named because they solve Maxwell's Equations with no implicit approximations, and are typically limited to problems of small electrical size due to limitations of computation time and system memory. Though computers continue to grow more powerful and can solve problems of ever-increasing size, this nomenclature will likely remain common in the literature for the foreseeable future.

1.1.1.1 Finite Difference Time Domain Method

The Finite Difference Time Domain (FDTD) method [2, 3] uses finite differences to solve Maxwell's Equations in the time domain. The application of FDTD is usually very straightforward: the solution domain is discretized into small rectangular or curvilinear elements, and a “leap frog” in time is used to compute electric and magnetic fields from one another at discrete time steps. FDTD excels at analysis of inhomogeneous and nonlinear media, though its demands for system memory are high due to discretization of the entire solution domain, and it suffers from dispersion issues and the need to artificially truncate of the solution boundary. FDTD is typically applied in EM packaging and waveguide problems, as well as the study of wave propagation in complex (often composite) materials.

1.1.1.2 Finite Element Method

The Finite Element Method (FEM) [4, 5] is a method used to solve frequency-domain boundary-valued electromagnetic problems via variational techniques. It can be used with two- and three-dimensional canonical elements of differing shape, allowing for a highly accurate discretization of the solution domain. FEM is often used in the frequency domain for computing the field distribution in complex, closed regions such as cavities and waveguides. As in the FDTD, the solution domain must be discretized and truncated, making the FEM approach often unsuitable for radiation or scattering problems unless combined with a boundary integral equation approach such as the Method of Moments [4].

1.1.1.3 Method of Moments

The Method of Moments (MOM) is a technique used to solve electromagnetic surface¹ or volume integral equations in the frequency domain. MOM differs from FDTD and FEM as the electromagnetic sources (surface or volume currents) are the quantities of interest, and so only the surface or volume of the antenna or scatterer must be discretized. As a result, the MOM is widely used in solving radiation and scattering problems. In this book, we focus on the practical solution of surface integral equations of radiation and scattering using MOM.

1.1.2 High-Frequency Methods

Electromagnetic problems of large size have existed long before the computers that could solve them. Common examples of larger problems are those involving radar cross section prediction, or the calculation of an antenna's radiation pattern when in close proximity to a large structure. Many approximations have been made to the equations of radiation and scattering to make these problems tractable. Most of these treat the fields in their asymptotic limit, and employ ray-optics and edge diffraction. When the problem is electrically large, many asymptotic methods produce results that are accurate enough on their own, or can be used as a “first pass” before a more accurate though computationally demanding method is applied.

1.1.2.1 Geometrical Theory of Diffraction

The Geometrical Theory of Diffraction (GTD) [6, 7] uses ray-optics to determine electromagnetic wave propagation. The spreading and amplitude intensity and decay in a ray bundle are computed using Fermat's principle and the radius of curvature at the bounce points. GTD attempts to account for the fields diffracted by edges, allowing for a calculation of the fields in shadow regions. GTD is fast but the results are often fair to poor for complex geometries.

1.1.2.2 Physical Optics

Physical Optics (PO) [8] is a method for approximating the surface currents, allowing a boundary integration to be performed to obtain the fields. As we will see, PO and the MOM are closely related as they use the same equations to integrate the surface currents, however the MOM calculates the surface currents directly instead approximating them. While robust, the PO does not account for the fields diffracted by edges or those from multiple reflections, so supplemental corrections are typically added to it. The PO method is used extensively in high-frequency reflector antenna analysis, as well as many radar cross section prediction codes such as lucernhammer MT.

1.1.2.3 Physical Theory of Diffraction

The Physical Theory of Diffraction (PTD) [9, 10] is a means for supplementing the PO solution by adding to it the fields radiated by nonuniform currents along diffracting edges of an object. PTD is commonly used in high-frequency radar cross section and scattering analysis.

1.1.2.4 Shooting and Bouncing Rays

The Shooting and Bouncing Ray (SBR) method [11, 12] was developed to predict the multiple-bounce scattered fields from complex objects. It uses the ray-optics model to determine the path and amplitude of a ray bundle, but uses a physical optics-based scheme that integrates the surface currents induced by the ray at each bounce point. SBR is often used in scattering codes to account for multiple reflections on a surface or those inside a cavity, and as such it supplements the fields computed by PO and the PTD. SBR is also used to predict wave propagation and scattering in complex urban environments to determine the coverage for cellular telephone service.

¹In this context, it is also referred to as the Boundary Element Method (BEM).

References

• [1] D. E. Amos, “A subroutine package for Bessel functions of a complex argument and nonnegative order,” Tech. Rep. SAND85-1018, Sandia National Laboratory, May 1985.
• [2] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Artech House, third ed., 2005.
• [3] K. Kunz and R. Luebbers, The Finite Difference Time Domain Method for Electromagnetics. CRC Press, 1993.
• [4] J. Jin, The Finite Element Method in Electromagnetics. John Wiley and Sons, 1993.
• [5] J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics. IEEE Press, 1998.
• [6] J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Amer., vol. 52, pp. 116–130, February 1962.
• [7] R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE, vol. 62, pp. 1448–1461, November 1974.
• [8] C. A. Balanis, Advanced Engineering Electromagnetics. John Wiley and Sons, 1989.
• [9] P. Ufimtsev, “Approximate computation of the diffraction of plane electromagnetic waves at certain metal bodies (i and ii),” Sov. Phys. Tech., vol. 27, pp. 1708–1718, August 1957.
• [10] A. Michaeli, “Equivalent edge currents for arbitrary aspects of observation,” IEEE Trans. Antennas Propagat., vol. 23, pp. 252–258, March 1984.
• [11] H. Ling, S. W. Lee, and R. Chou, “Shooting and bouncing rays: calculating the RCS of an arbitrarily shaped cavity,” IEEE Trans. Antennas Propagat., vol. 37, pp. 194–205, February 1989.
• [12] H. Ling, S. W. Lee, and R. Chou, “High-frequency RCS of open cavities with rectangular and circular cross sections,” IEEE Trans. Antennas Propagat., vol. 37, pp. 648–652, May 1989.

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Preface to the Third Edition
• Preface to the Second Edition
• Preface
• Acknowledgments
• About the Author
• 1 Computational Electromagnetics
• 2 The Method of Moments
• 3 Radiation and Scattering
• 4 Solution of Matrix Equations
• 5 Thin Wires
• 6 Two-Dimensional Problems
• 7 Bodies of Revolution
• 8 Three-Dimensional Problems
• 9 Adaptive Cross Approximation
• 10 Multi-Level Adaptive Cross Approximation
• 11 The Fast Multipole Method
• 12 Integration
• A Scattering Using Physical Optics
• Index