Chapter1
Computational Electromagnetics
Before the digital computer was developed,the analysis and design of electromag-netic devices and structures were largely experimental.Once the computer and nu-merical languages such as FORTRAN came along,people immediately began using them to tackle electromagnetic problems that could not be solved analytically.This led to aflurry of development in afield now referred to as computational electromag-netics(CEM).Many powerful numerical analysis techniques have been developed in this area in the last50years.As the power of the computer continues to grow, so do the nature of the algorithms applied as well as the complexity and size of the problems that can be solved.
While the data gleaned from experimental measurements are invaluable,the entire process can be costly in terms of money and the manpower required to do the required machine work,assembly,and measurements at the 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 the engineer to engage in levels of customization and optimization that would be painstaking or even impossible if done experimentally.CEM also helps to provide fundamental i
nsights into electromagnetic problems through the power of computation and computer visualization,making it one of the most important areas of engineering today.
1.1COMPUTATIONAL ELECTROMAGNETICS ALGORITHMS
The extremely wide range of electromagnetic problems has led to the development of many different CEM algorithms,each with its own benefits and limitations. These algorithms are typically classified as so-called“exact”or“low-frequency”and “approximate”or“high-frequency”methods and further sub-classified into time-or frequency-domain methods.We will quickly summarize some of the most commonly used methods to provide some context in how the moment methodfits in the CEM environment.
1
2The Method of Moments in Electromagnetics
1.1.1Low-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 comput
ation time and system memory.Though computers continue to grow more powerful and solve problems of ever increasing size,this nomenclature will likely remain common in the literature.
1.1.1.1Finite Difference Time Domain Method
Thefinite difference time-domain(FDTD)method[1,2]uses the method offinite differences to solve Maxwell’s Equations in the time domain.Application of the FDTD method is usually very straightforward:the solution domain is typically discretized into small rectangular or curvilinear elements,with a“leap frog”in time used to compute the electric and magneticfields from one another.FDTD excels at analysis of inhomogeneous and nonlinear media,though its demands for system memory are high due to the discretization of the entire solution domain,and it suffers from dispersion issues as well and the need to artificially truncate the solution boundary.FDTDfinds applications in packaging and waveguide problems,as well as in the study of wave propagation in complex dielectrics.
1.1.1.2Finite Element Method
Thefinite element method(FEM)[3,4]is a method used to solve frequency-domain boundary valued electromagnetic problems by using a variational form.It can be used with two-and three-dimensional ca
nonical elements of differing shape, allowing for a highly accurate discretization of the solution domain.The FEM is often used in the frequency domain for computing the frequencyfield distribution in complex,closed regions such as cavities and waveguides.As in the FDTD method, the solution domain must be truncated,making the FEM unsuitable for radiation or scattering problems unless combined with a boundary integral equation approach [3].
1.1.1.3Method of Moments
The method of moments(MOM)is a technique used to solve electromagnetic bound-ary or volume integral equations in the frequency domain.Because the electromag-netic sources are the quantities of interest,the MOM is very useful in solving ra-diation and scattering problems.In this book,we focus on the practical solution of boundary integral equations of radiation and scattering using this method.
1.1.2High-Frequency Methods
Electromagnetic problems of large size have existed long before the computers that could solve them.Common examples of larger problems are those of radar cross
Computational Electromagnetics3 section prediction and calculation of an antenna’s radiation pattern
when mounted on a large structure.Many approximations have been made to the equations of radiation and scattering to make these problems tractable.Most of these treat thefields in the asymptotic or high-frequency(HF)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.
2.1Geometrical Theory of Diffaction
The geometrical theory of diffraction(GTD)[5,6]uses ray-optics to determine electromagnetic wave propagation.The spreading,amplitude intensity and decay in a ray bundle are computed using from Fermat’s principle and the radius of curvature at reflection points.The GTD attempts to account for thefields diffracted by edges, allowing for a calculation of thefields in shadow regions.The GTD is fast but often yields poor accuracy for more complex geometries.
1.1.
2.2Physical Optics
Physical optics(PO)[7]is a method for approximating the high-frequency surface currents,allowing a boundary integration to be performed to obtain thefields.As we will see,the PO and the MOM are used to solve the same integral equation, though the MOM calculates the surface currents directly instead of approximating them.While robust,PO does not account for thefields diffracted by edges or those from multiple reflections,so supplemental corrections are usually added to it.The PO method is used extensively in high-frequency reflector antenna analyses,as well as many radar cross section prediction codes.
1.1.
2.3Physical Theory of Diffraction
The physical theory of diffraction(PTD)[8,9]is a means for supplementing the PO solution by adding the effects of nonuniform currents at the diffracting edges of an object.PTD is commonly used in high-frequency radar cross section and scattering analyses.
1.1.
2.4Shooting and Bouncing Rays
The shooting and bouncing ray(SBR)method[10,11]was developed to predict the multiple-bounce backscatter from complex objects.It uses the ray-optics model to determine the path and amplitude of a ray bundle,but uses a PO-based scheme that integrates surface currents deposited by the ray at each bounce point.The SBR method is often used in scattering codes to account for multiple reflections on a surface or that encountered inside a cavity,and as such it supplements PO and the PTD.The SBR method is also used to predict wave propagation and scattering
4The Method of Moments in Electromagnetics
in complex urban environments to determine the coverage for cellular telephone service.
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[4]J.L.V olakis,A.Chatterjee,and L.C.Kempel,Finite Element Method for
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[5]J.B.Keller,“Geometrical theory of diffraction,”J.Opt.Soc.Amer.,vol.52,
116–130,February1962.
[6]R.G.Kouyoumjian and P.H.Pathak,“A uniform geometrical theory of diffrac-
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[7]C.A.Balanis,Advanced Engineering Electromagnetics.John Wiley and Sons,
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[8]P.Ufimtsev,“Approximate computation of the diffraction of plane electro-
magnetic waves at certain metal bodies(i and ii),”Sov.Phys.Tech.,vol.27, 1708–1718,August1957.
[9]A.Michaeli,“Equivalent edge currents for arbitrary aspects of observation,”
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[10]S.L.H.Ling and R.Chou,“Shooting and bouncing rays:Calculating the
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[11]S.L.H.Ling and R.Chou,“High-frequency RCS of open cavities with
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