The electrical engineering handbook CH045

The electrical engineering handbook

Miller, E.K “Computational Electromagnetics”

The Electrical Engineering Handbook

Ed Richard C Dorf

Boca Raton: CRC Press LLC, 2000

Computational Electromagnetics

45.1 Introduction45.2 Background DiscussionModeling as a Transfer Function • Some Issues Involved in Developing a Computer Model

45.3 Analytical Issues in Developing a Computer ModelSelection of Solution Domain • Selection of Field Propagator45.4 Numerical Issues in Developing a Computer ModelSampling Functions • The Method of Moments

45.5 Some Practical ConsiderationsIntegral Equation Modeling • Differential Equation Modeling • Discussion • Sampling Requirements45.6 Ways of Decreasing Computer Time45.7 Validation, Error Checking, and Error AnalysisModeling Uncertainties • Validation and Error Checking45.8 Concluding Remarks

45.1 Introduction

The continuing growth of computing resources is changing how we think about, formulate, solve, and interpretproblems In electromagnetics as elsewhere, computational techniques are complementing the more traditionalapproaches of measurement and analysis to vastly broaden the breadth and depth of problems that are nowquantifiable Computational electromagnetics (CEM) may be broadly defined as that branch of electromagneticsthat intrinsically and routinely involves using a digital computer to obtain numerical results With the evolu-tionary development of CEM during the past 20-plus years, the third tool of computational methods has beenadded to the two classical tools of experimental observation and mathematical analysis

This discussion reviews some of the basic issues involved in CEM and includes only the detail needed toillustrate the central ideas involved The underlying principles that unify the various modeling approaches used

in electromagnetics are emphasized while avoiding most of the specifics that make them different Listedthroughout are representative, but not exhaustive, numbers of references that deal with various specialty aspects

of CEM For readers interested in broader, more general expositions, the well-known book on the momentmethod by Harrington [1968]; the books edited by Mittra [1973, 1975], Uslenghi [1978], and Strait [1980];the monographs by Stutzman and Thiele [1981], Popovic, et al [1982], Moore and Pizer [1984], and Wang[1991]; and an IEEE Press reprint volume on the topic edited by Miller et al [1991] are recommended, as isthe article by Miller [1988] from which this material is excerpted

This chapter is excerpted from E K Miller, “A selective survey of computational electromagnetics,” IEEE Trans Antennas Propagat., vol AP-36, pp 1281–1305, ©1988 IEEE.

E.K Miller

Los Alamos National Laboratory

45.2 Background Discussion

Electromagnetics is the scientific discipline that deals with electric and magnetic sources and the fields thesesources produce in specified environments Maxwell’s equations provide the starting point for the study ofelectromagnetic problems, together with certain principles and theorems such as superposition, reciprocity,equivalence, induction, duality, linearity, and uniqueness, derived therefrom [Stratton, 1941; Harrington, 1961].While a variety of specialized problems can be identified, a common ingredient of essentially all of them is that

of establishing a quantitative relationship between a cause (forcing function or input) and its effect (the response

or output), a relationship which we refer to as a field propagator, the computational characteristics of whichare determined by the mathematical form used to describe it

Modeling as a Transfer Function

The foregoing relationship may be viewed as a

gener-alized transfer function (see Fig 45.1) in which two

basic problem types become apparent For the

anal-ysis or the direct problem, the input is known and the

transfer function is derivable from the problem

spec-ification, with the output or response to be

deter-mined For the case of the synthesis or inverse

problem, two problem classes may be identified The

easier synthesis problem involves finding the input,

given the output and transfer function, an example

of which is that of determining the source voltages

that produce an observed pattern for a known

antenna array The more difficult synthesis problem

itself separates into two problems One is that of finding the transfer function, given the input and output, anexample of which is that of finding a source distribution that produces a given far field The other and stillmore difficult is that of finding the object geometry that produces an observed scattered field from a knownexciting field The latter problem is the most difficult of the three synthesis problems to solve because it isintrinsically transcendental and nonlinear

Electromagnetic propagators are derived from a particular solution of Maxwell’s equations, as the causementioned above normally involves some specified or known excitation whose effect is to induce some to-be-determined response (e.g., a radar cross section, antenna radiation pattern) It therefore follows that the essence

of electromagnetics is the study and determination of field propagators to thereby obtain an input–outputtransfer function for the problem of interest, and it follows that this is also the goal of CEM

Some Issues Involved in Developing a Computer Model

We briefly consider here a classification of model types, the steps involved in developing a computer model, thedesirable attributes of a computer model, and finally the role of approximation throughout the modeling process

Classification of Model Types

It is convenient to classify solution techniques for electromagnetic modeling in terms of the field propagatorthat might be used, the anticipated application, and the problem type for which the model is intended to beused, as is outlined in Table 45.1 Selection of a field propagator in the form, for example, of the Maxwell curlequations, a Green’s function, modal or spectral expansions, or an optical description is a necessary first step

in developing a solution to any electromagnetic problem

Development of a Computer Model

Development of a computer model in electromagnetics or literally any other disciplinary activity can bedecomposed into a small number of basic, generic steps These steps might be described by different names but

FIGURE 45.1 The electromagnetic transfer function relates the input, output, and problem.

would include at a minimum those outlined in Table 45.2 Note that by its nature, validation is an open-endedprocess that cumulatively can absorb more effort than all the other steps together The primary focus of thefollowing discussion is on the issue of numerical implementation.

Desirable Attributes of a Computer Model

A computer model must have some minimum set of basic properties to be useful From the long list of attributesthat might be desired, we consider: (1) accuracy, (2) efficiency, and (3) utility the three most important assummarized in Table 45.3 Accuracy is put foremost because results of insufficient or unknown accuracy haveuncertain value and may even be harmful On the other hand, a code that produces accurate results but atunacceptable cost will have hardly any more value Finally, a code’s applicability in terms of the depth andbreadth of the problems for which it can be used determines its utility

The Role of Approximation

As approximation is an intrinsic part of each step involved in developing a computer model, we summarizesome of the more commonly used approximations in Table 45.4 We note that the distinction between anapproximation at the conceptualization step and during the formulation is somewhat arbitrary, but choose touse the former category for those approximations that occur before the formulation itself

Field Propagator Description Based on

Integral operator Green’s function for infinite medium or special boundaries

Differential operator Maxwell curl equations or their integral counterparts

Modal expansions Solutions of Maxwell’s equations in a particular coordinate system and expansion

Optical description Rays and diffraction coefficients

Application Requires

Radiation Determining the originating sources of a field and patterns they produce

Propagation Obtaining the fields distant from a known source

Scattering Determining the perturbing effects of medium inhomogeneities

Problem type Characterized by

Solution domain Time or frequency

Solution space Configuration or wave number

representation Numerical implementation Transforming into a computer algorithm using various numerical techniques

Computation Obtaining quantitative results

Validation Determining the numerical and physical credibility of the computed results

45.3 Analytical Issues in Developing a Computer Model

Attention here is limited primarily to propagators that use either the Maxwell curl equations or source integralswhich employ a Green’s function, although for completeness we briefly discuss modal and optical techniques as well

Selection of Solution Domain

Either the integral equation (IE) or differential equation (DE) propagator can be formulated in the timedomain, where time is treated as an independent variable, or in the frequency domain, where the harmonic

Attribute Description

Accuracy The quantitative degree to which the computed results conform to the mathematical and physical reality being

modeled Accuracy, preferably of known and, better yet, selectable value, is the single most important model attribute

It is determined by the physical modeling error ( eP) and numerical modeling error ( eN).

Efficiency The relative cost of obtaining the needed results It is determined by the human effort required to develop the computer

input and interpret the output and by the associated computer cost of running the model.

Utility The applicability of the computer model in terms of problem size and complexity Utility also relates to ease of use,

reliability of results obtained, etc.

Approximation Implementation/Implications

Conceptualization

Physical optics Surface sources given by tangential components of incident field, with fields subsequently

propagated via a Green’s function Best for backscatter and main-lobe region of reflector antennas, from resonance region (ka > 1) and up in frequency.

Physical theory of diffraction Combines aspects of physical optics and geometrical theory of diffraction, primarily via use of

edge-current corrections to utilize best features of each.

Geometrical theory diffraction Fields propagated via a divergence factor with amplitude obtained from diffraction coefficient.

Generally applicable for ka > 2–5 Can involve complicated ray tracing.

Geometrical optics Ray tracing without diffraction Improves with increasing frequency.

Compensation theorem Solution obtained in terms of perturbation from a reference, known solution.

Born–Rytov Approach used for low-contrast, penetrable objects where sources are estimated from incident

field.

Rayleigh Fields at surface of object represented in terms of only outward propagating components in a

modal expansion.

Formulation

Surface impedance Reduces number of field quantities by assuming an impedance relation between tangential E

and H at surface of penetrable object May be used in connection with physical optics Thin-wire Reduces surface integral on thin, wirelike object to a line integral by ignoring circumferential

current and circumferential variation of longitudinal current, which is represented as a filament Generally limited to ka < 1 where a is the wire radius.

Computation

Deviation of numerical model

from physical reality

Affects solution accuracy and relatability to physical problem in ways that are difficult to predict and quantify.

Nonconverged solution Discretized solutions usually converge globally in proportion to exp(–AN x) with A determined

by the problem At least two solutions using different numbers of unknowns N xare needed to estimate A.

time variation exp(jwt) is assumed Whatever propagator and domain are chosen, the analytically formalsolution can be numerically quantified via the method of moments (MoM) [Harrington, 1968], leadingultimately to a linear system of equations as a result of developing a discretized and sampled approximation

to the continuous (generally) physical reality being modeled Developing the approach that may be best suited

to a particular problem involves making trade-offs among a variety of choices throughout the analyticalformulation and numerical implementation, some aspects of which are now considered

Selection of Field Propagator

We briefly discuss and compare the characteristics of the various propagator-based models in terms of theirdevelopment and applicability

Integral Equation Model

The basic starting point for developing an IE model in electromagnetics is selection of a Green’s functionappropriate for the problem class of interest While there are a variety of Green’s functions from which tochoose, a typical starting point for most IE MoM models is that for an infinite medium One of the morestraightforward is based on the scalar Green’s function and Green’s theorem This leads to the Kirchhoff integrals[Stratton, 1941, p 464 et seq.], from which the fields in a given contiguous volume of space can be written interms of integrals over the surfaces that bound it and volume integrals over those sources located within it.Analytical manipulation of a source integral that incorporates the selected Green’s function as part of itskernel function then follows, with the specific details depending on the particular formulation being used.Perhaps the simplest is that of boundary-condition matching wherein the behavior required of the electricand/or magnetic fields at specified surfaces that define the problem geometry is explicitly imposed Alternativeformulations, for example, the Rayleigh–Ritz variational method and Rumsey’s reaction concept, might be usedinstead, but as pointed out by Harrington [in Miller et al., 1991], from the viewpoint of a numerical imple-mentation any of these approaches lead to formally equivalent models

This analytical formulation leads to an integral operator, whose kernel can include differential operators aswell, which acts on the unknown source or field Although it would be more accurate to refer to this as anintegrodifferential equation, it is usually called simply an integral equation Two general kinds of integralequations are obtained In the frequency domain, representative forms for a perfect electric conductor are

(45.1a)

(45.1b)

where E and H are the electric and magnetic fields, respectively, r, r¢are the spatial coordinate of the observationand source points, the superscript inc denotes incident-field quantities, and j(r,r¢) = exp[–jk*r – r¢*]/*r – r¢* isthe free-space Green’s function These equations are known respectively as Fredholm integral equations of thefirst and second kinds, differing by whether the unknown appears only under the integral or outside it as well[Poggio and Miller in Mittra, 1973]

Differential-Equation Model

A DE MoM model, being based on the defining Maxwell’s equations, requires intrinsically less analyticalmanipulation than does derivation of an IE model Numerical implementation of a DE model, however, candiffer significantly from that used for an IE formulation in a number of ways for several reasons:

1 The differential operator is a local rather than global one in contrast to the Green’s function upon which

the integral operator is based This means that the spatial variation of the fields must be developed from

sampling in as many dimensions as possessed by the problem, rather than one less as the IE model

permits if an appropriate Green’s function is available

2 The integral operator includes an explicit radiation condition, whereas the DE does not

3 The differential operator includes a capability to treat medium inhomogeneities, non-linearities, and

time variations in a more straightforward manner than does the integral operator, for which an

appro-priate Green’s function may not be available

These and other differences between development of IE and DE models are summarized in Table 45.5, with

their modeling applicability compared in Table 45.6

Modal-Expansion Model

Modal expansions are useful for propagating electromagnetic fields because the source-field relationship can

be expressed in terms of well-known analytical functions as an alternate way of writing a Green’s function for

special distributions of point sources In two dimensions, for example, the propagator can be written in terms

of circular harmonics and cylindrical Hankel functions Corresponding expressions in three dimensions might

involve spherical harmonics, spherical Hankel functions, and Legendre polynomials Expansion in terms of

analytical solutions to the wave equation in other coordinate systems can also be used but requires computation

Differential Form Integral Form

On object Appropriate field values specified on

mesh boundaries to obtain stairstep, piecewise linear, or other approximation

to the boundary

Appropriate field values specified on object contour which can in principle be a general, curvilinear surface, although this possibility seems to be seldom used

Sampling requirements

No of space samples N xµ (L/ DL)D N xµ (L/ DL)D–1

No of time steps N tµ (L/ DL) »cT/ dt N tµ (L/ DL) »cT/ dt

Sparse, but larger Dense, but smaller In this comparison, note that we

assume the IE permits a sampling of order one less than the problem dimension, i.e., inhomogeneous problems are excluded.

Dependence of solution time on highest-order term in (L/ DL)

distinction is important because when an appropriate Green’s function is available, the source integrals are usually one dimension

less than the problem dimension, i.e., d = D – 1 An exception is an inhomogeneous, penetrable body where d = D when using an

IE We also assume for simplicity that matrix solution is achieved via factorization rather than iteration but that banded matrices

are exploited for the DE approach where feasible The solution-time dependencies given can thus be regarded as upper-bound

estimates See Table 45.10 for further discussion of linear-system solutions.

of special functions that are generally less easily evaluated, such as Mathieu functions for the two-dimensionalsolution in elliptical coordinates and spheroidal functions for the three-dimensional solution in oblate or prolatespheroidal coordinates.

One implementation of modal propagators for numerical modeling is that due to Waterman [in Mittra,1973], whose approach uses the extended boundary condition (EBC) whereby the required field behavior issatisfied away from the boundary surface on which the sources are located This procedure, widely known as

the T-matrix approach, has evidently been more widely used in optics and acoustics than in electromagnetics.

In what amounts to a reciprocal application of EBC, the sources can be removed from the boundary surface

on which the field-boundary conditions are applied These modal techniques seem to offer some computationaladvantages for certain kinds of problems and might be regarded as using entire-domain basis and testingfunctions but nevertheless lead to linear systems of equations whose numerical solution is required Fouriertransform solution techniques might also be included in this category since they do involve modal expansions,but that is a specialized area that we do not pursue further here

Modal expansions are receiving increasing attention under the general name “fast multipole method,” which

is motivated by the goal of systematically exploiting the reduced complexity of the source-field interactions astheir separation increases The objective is to reduce the significant interactions of a Green’s-function basedmatrix from being proportional to (Nx)2 to of order Nx log (Nx), thus offering the possibility of decreasing theoperation count of iterative solutions

Ö signifies highly suited or most advantageous.

~ signifies moderately suited or neutral.

x signifies unsuited or least advantageous.

Geometrical-Optics Model

Geometrical optics and the geometrical theory of diffraction (GTD) are high-frequency asymptotic techniqueswherein the fields are propagated using such optical concepts as shadowing, ray tubes, and refraction anddiffraction Although conceptually straightforward, optical techniques are limited analytically by the unavail-ability of diffraction coefficients for various geometries and material bodies and numerically by the need totrace rays over complex surfaces There is a vast literature on geometrical optics and GTD, as may be ascertained

by examining the yearly and cumulative indexes of such publications as the Transactions of the IEEE Antennas

and Propagation Society.

45.4 Numerical Issues in Developing a Computer Model

Sampling Functions

At the core of numerical analysis is the idea of polynomial approximation, an observation made by Arden andAstill [1970] in facetiously using the subtitle “Numerical Analysis or 1001 Applications of Taylor’s Series.” Thebasic idea is to approximate quantities of interest in terms of sampling functions, often polynomials, that arethen substituted for these quantities in various analytical operations Thus, integral operators are replaced byfinite sums, and differential operators are similarly replaced by generalized finite differences For example, use

of a first-order difference to approximate a derivative of the function F(x) in terms of samples F(x + ) and F(x –)leads to

dF x dx

h

2 2

0 22

a given number of samples This suggests the benefits that might be derived from using unequal sample sizes

in MoM modeling should a systematic way of determining the best nonuniform sampling scheme be developed

The Method of Moments

Numerical implementation of the moment method is a relatively straightforward, and an intuitively logical,extension of these basic elements of numerical analysis, as described in the book by Harrington [1968] anddiscussed and used extensively in CEM [see, for example, Mittra, 1973, 1975; Strait, 1980; Poggio and Miller,1988] Whether it is an integral equation, a differential equation, or another approach that is being used forthe numerical model, three essential sampling operations are involved in reducing the analytical formulationvia the moment method to a computer algorithm as outlined in Table 45.7 We note that operator samplingcan ultimately determine the sampling density needed to achieve a desired accuracy in the source–field rela-tionships involving integral operators, especially at and near the “self term,” where the observation and sourcepoints become coincident or nearly so and the integral becomes nearly singular Whatever the method usedfor these sampling operations, they lead to a linear system of equations or matrix approximation of the originalintegral or differential operators Because the operations and choices involved in developing this matrix descrip-tion are common to all moment-method models, we shall discuss them in somewhat more detail

When using IE techniques, the coefficient matrix in the linear system of equations that results is most oftenreferred to as an impedance matrix because in the case of the E-field form, its multiplication of the vector ofunknown currents equals a vector of electric fields or voltages The inverse matrix similarly is often called anadmittance matrix because its multiplication of the electric-field or voltage vector yields the unknown-current

vector In this discussion we instead use the terms direct matrix and solution matrix because they are more

generic descriptions whatever the forms of the originating integral or differential equations As illustrated inthe following, development of the direct matrix and solution matrix dominates both the computer time andstorage requirements of numerical modeling

In the particular case of an IE model, the coefficients of the direct or original matrix are the mutualimpedances of the multiport representation which approximates the problem being modeled, and the coeffi-cients of its solution matrix (or equivalent thereof) are the mutual admittances Depending on whether asubdomain or entire-domain basis has been used (see Basic Function Selection), these impedances and admittances

represent either spatial or modal interactions among the N ports of the numerical model In either case, these

Can use either subdomain or entire-domain bases Use of latter is generally confined to bodies of rotation Former is usually of low order, with piecewise linear or sinusoidal being the maximum variation used.

Equation via weight

Pointwise matching is commonly employed, using

a delta function For wires, pulse, linear, and sinusoidal testing is also used Linear and sinusoidal testing is also used for surfaces Operator Operator sampling for DE models is

entwined with sampling the unknown in terms of the difference operators used.

Sampling needed depends on the nature of the

integral operator L(s,s¢) An important

consideration whenever the field integrals cannot

be evaluated in closed form.

Solution of:

Z ij a j = g i for the a j Interaction matrix is sparse Time-domain

approach may be explicit or implicit In frequency domain, banded-matrix technique usually used.

Interaction matrix is full Solution via factorization

or iteration.

coefficients possess a physical relatability to the problem being modeled and ultimately provide all the mation available concerning any electromagnetic observables that are subsequently obtained.

infor-Similar observations might also be made regarding the coefficients of the DE models but whose multiportrepresentations describe local rather than global interactions Because the DE model almost always leads to alarger, albeit less dense, direct matrix, its inverse (or equivalent) is rarely computed It is worth noting thatthere are two widely used approaches for DE modeling, finite-difference (FD) and finite-element (FE) methods.They differ primarily in how the differential operators are approximated and the differential equations aresatisfied, i.e., in the order of the basis and weight functions, although the FE method commonly starts from avariational viewpoint, while the FD approach begins from the defining differential equations The FE method

is generally better suited for modeling problems with complicated boundaries to which it provides a piecewiselinear or higher order approximation as opposed to the cruder stairstep approximation of FD

Factors Involved in Choosing Basis and Weight Functions

Basis and weight function selection plays a critical role in determining the accuracy and efficiency of the resultingcomputer model One goal of the basis and weight function selection is to minimize computer time whilemaximizing accuracy for the problem set to which the model is to be applied Another, possibly conflicting,goal might be that of maximizing the collection of problem sets to which the model is applicable A third might

be to replicate the problem’s physical behavior with as few samples as possible Some of the generic combinations

of bases and weights that are used for MoM models are listed in Table 45.8 [Poggio and Miller from Mittra, 1973]

Basis Function Selection We note that there are two classes of bases used in MoM modeling, subdomain

and entire-domain functions The former involves the use of bases that are applied in a repetitive fashion oversubdomains or sections (segments for wires, patches for surfaces, cells for volumes) of the object being modeled.The simplest example of a subdomain basis is the single-term basis given by the pulse or stairstep function,which leads to a single, unknown constant for each subdomain Multiterm bases involving two or more functions

on each subdomain and an equivalent number of unknowns are more often used for subdomain expansions.The entire-domain basis, on the other hand, uses multiterm expansions extending over the entire object, forexample, a circular harmonic expansion in azimuth for a body of revolution As for subdomain expansions,

an unknown is associated with each term in the expansion Examples of hybrid bases can also be found, wheresubdomain and entire-domain bases are used on different parts of an object

Although subdomain bases are probably more flexible in terms of their applicability, they have a disadvantagegenerally not exhibited by the entire-domain form, which is the discontinuity that occurs at the domain

boundaries This discontinuity arises because an n s -term subdomain function can provide at most n s – 1 degrees

of continuity to an adjacent basis of the unknown it represents, assuming one of the n s constants is reserved

for the unknown itself For example, the three-term or sinusoidal subdomain basis a i + b i sin(ks) + c i cos(ks)

used for wire modeling can represent a current continuous at most up to its first derivative This providescontinuous charge density but produces a discontinuous first derivative in charge equivalent to a tripole charge

at each junction

Method jth Term of Basis ith Term of Weight

General collocation a j b j(r¢) d(r – ri)

Subsectional collocation U(r j)åajkb k(r¢) d(r – ri)

Subsectional Galerkin U(r j)åajkb k(r¢) U(ri)åbi(r)

r¢ and r denote source and observation points respectively; aj , a jk are unknown constants

associated with the jth basis function (entire domain) or the kth basis function of the jth

subsection (subdomain); U(r k ) is the unit sampling function which equals 1 on the kth

subdomain and is 0 elsewhere; b j(r¢) is the jth basis function; wi(r) is the ith testing function;

d(r – ri ) is the Dirac delta function; Q(r) is a positive-definite function of position; and e(r)

is the residual or equation error [from Poggio and Miller in Mitra (1973)].