time variation explor) is assumed. Whatever propagator and domain are chosen, the analytically formal solution can be numerically quantified via the method of moments(MoM)[Harrington, 1968], leading ultimately 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 analytical formulation 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 their develo Integral Equation Model The basic starting point for developing an IE model in electromagnetics is selection of a Green's function appropriate for the problem class of interest. While there are a variety of Greens functions from which to choose, a typical starting point for most IE MoM models is that for an infinite medium. One of the more straightforward is based on the scalar Greens function and Greens 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 in terms 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 its kernel 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 electric and/or magnetic fields at specified surfaces that define the problem geometry is explicitly imposed. Alternative formulations, for example, the Rayleigh-Ritz variational method and Rumsey's reaction concept, might be used instead, but as pointed out by Harrington [in Miller et al., 1991, from the viewpoint of a numerical impl mentation any of these approaches lead to formally equivalent models. This analytical formulation leads to an integral operator, whose kernel can include differential operators a well, which acts on the unknown source or field. Although it would be more accurate to refer to this as an integrodifferential equation, it is usually called simply an integral equation. Two general kinds of integral equations are obtained. In the frequency domain, representative forms for a perfect electric conductor are nxEim(r)=LnxvoHin'x H(ro)lo(r,r") (45.1a) n'·E(r,rvp(r,r)ds;r∈S n×Hxr)=2nxH(r)+n×[×Hxr)]×kVr,r)ds;r∈S(45.1b) where E and H are the electric and magnetic fields, respectively, r,r are the spatial coordinate of the observation and source points, the superscript inc denotes incident-field quantities, and o(r, r)=exp[-jkr-rlr-rlis the free-space Greens function. These equations are known respectively as Fredholm integral equations of the first and second kinds, differing by whether the unknown appears only under the integral or outside it as well d Miller in Mittra, 1973 Differential-Equation Model A DE MoM model, being based on the defining Maxwells equations, requires intrinsically less analytical anipulation than does derivation of an IE model. Numerical implementation of a DE model, however, can differ significantly from that used for an IE formulation in a number of ways for several reasons c 2000 by CRC Press LLC
© 2000 by CRC Press LLC time variation exp(jwt) is assumed. Whatever propagator and domain are chosen, the analytically formal solution can be numerically quantified via the method of moments (MoM) [Harrington, 1968], leading ultimately 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 analytical formulation 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 their development and applicability. Integral Equation Model The basic starting point for developing an IE model in electromagnetics is selection of a Green’s function appropriate for the problem class of interest. While there are a variety of Green’s functions from which to choose, a typical starting point for most IE MoM models is that for an infinite medium. One of the more straightforward 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 in terms 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 its kernel 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 electric and/or magnetic fields at specified surfaces that define the problem geometry is explicitly imposed. Alternative formulations, for example, the Rayleigh–Ritz variational method and Rumsey’s reaction concept, might be used instead, but as pointed out by Harrington [in Miller et al., 1991], from the viewpoint of a numerical implementation any of these approaches lead to formally equivalent models. This analytical formulation leads to an integral operator, whose kernel can include differential operators as well, which acts on the unknown source or field. Although it would be more accurate to refer to this as an integrodifferential equation, it is usually called simply an integral equation. Two general kinds of integral equations 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 observation and source points, the superscript inc denotes incident-field quantities, and j(r,r¢) = exp[–jk*r – r¢*]/*r – r¢* is the free-space Green’s function. These equations are known respectively as Fredholm integral equations of the first 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 analytical manipulation than does derivation of an IE model. Numerical implementation of a DE model, however, can differ significantly from that used for an IE formulation in a number of ways for several reasons: n E r n n H r r, r n E r, r r, r r ¥ = ¥ ¢ ¥ ¢ ¢ - ¢ × ¢ —¢ ¢ ¢ Œ Ú inc ( ) { [ ( )] ( ) [ ( ) ( )} ; 1 4p wm j j j ds S S n ¥ H r = n ¥ H r + n ¥ n¢ ¥ H r¢ ¥ —¢ r,r¢ ¢ r Œ Ú ( ) 2 ( ) [ ( )] ( )} ; 1 2 inc ds p j S S
TABLE 45.5 Comparison of IE- and DE-Field Propagators and Their Numerical Treatment Differential Form Integral Form Field propagator Maxwell curl equations Boundary treatment Local or global"lookback"to Green's function On object Appropriate field values specified on Appropriate field values specified on object contour mesh boundaries to obtain stairstep, hich can in principle be a general, curvilin piecewise linear, or other approximation rface, although this possibilit N2∞(LAD)° N2∞(L△L)21 (L△L)=cTδr No, of excitations Nh(L△D) N∞(LAL (right-hand sides) Sparse, but larger Dense, but smaller. In this comparison, note that w is no of problem than the problem dimension, ie, inhomogeneous T is observation tin AL is spatial resolution δ t is time resolution Dependence of solution time on highest-order term in(L/AL) Frequency domain Tr∞ NINThs=(L△L)p Tr∞ NaNHe=(△L1;0≤r≤1 Implicit Tr∞Nxmp=(LAL2,D=2,3;T=N=(L△D cNNN=(L△)2,D=1;0≤r≤1 Note that D is the number of spatial dimensions in the problem and is not necessarily the sampling dimensionality d. The 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 sim 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 1. The differential operator is a local rather than global one in contrast to the Greens function upon which Sap ntegral operator is based. This means that the spatial variation of the fields must be developed from ling in as many dimensions as possessed by the problem, rather than one less as the ie model permits if an appropriate Greens 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-linearit time variations in a more straightforward manner than does the integral operator, for which 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 Greens 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 c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 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 appropriate 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 TABLE 45.5 Comparison of IE- and DE-Field Propagators and Their Numerical Treatment Differential Form Integral Form Field propagator Maxwell curl equations Green’s function Boundary treatment At infinity (radiation condition) Local or global “lookback” to approximate outward propagating wave Green’s function 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 Nx µ (L/DL)D Nx µ (L/DL)D–1 No. of time steps Nt µ (L/DL) ª cT/dt Nt µ (L/DL) ª cT/dt No. of excitations Nrhs µ (L/DL) Nrhs µ (L/DL) (right-hand sides) Linear system L is problem size D is no. of problem dimensions (1, 2, 3) T is observation time DL is spatial resolution dt is time resolution 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) Frequency domain Tw µ Nx 2(D–1)/D+1 = (L/DL)3D–2 Tw µ Nx 3 = (L/DL)3(D–1) Time domain Explicit Tt µ NxNt Nrhs = (L/DL)D+1+r Tt µ Nx 2 NtNrhs = (L/DL)2D–1+r ; 0 £ r £ 1 Implicit Tt µ Nx 2(D–1)/D+1 = (L/DL)3D–2, D = 2, 3; Tt µ Nx 3 = (L/DL)3(D–1) µ NxNtNrhs = (L/DL)2+r , D = 1; 0 £ r £ 1 Note that D is the number of spatial dimensions in the problem and is not necessarily the sampling dimensionality d. The 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
TABLE 45.6 Relative Applicability of IE-and DE-Based Computer Models Time Domain Frequency Domain D Medium vv√vvvxx Time-varying Closed surface √ Penetrable volume Boundary Conditions v Exterior problem √ Nonlinear √ varying xx~ Number of unknowns Length of code ty for Hybridizing with Other: Numerical procedures nalytical procedures GTI v signifies highly suited or most advantageous. signifies moderately suited or neutral. of special functions that are generally less easily evaluated, such as Mathieu functions for the two-dimensional solution in elliptical coordinates and spheroidal functions for the three-dimensional solution in oblate or prolate oheroidal 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 is satisfied 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 computational advantages for certain kinds of problems and might be regarded as using entire-domain basis and testing functions but nevertheless lead to linear systems of equations whose numerical solution is required. Fourier transform 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 eir separation increases. The objective is to reduce the significant interactions of a Green's-function based matrix from being proportional to(N )2 to of order N, log(Nx), thus offering the possibility of decreasing the c 2000 by CRC Press LLC
© 2000 by CRC Press LLC of special functions that are generally less easily evaluated, such as Mathieu functions for the two-dimensional solution in elliptical coordinates and spheroidal functions for the three-dimensional solution in oblate or prolate spheroidal 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 is satisfied 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 computational advantages for certain kinds of problems and might be regarded as using entire-domain basis and testing functions but nevertheless lead to linear systems of equations whose numerical solution is required. Fourier transform 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 as their separation increases. The objective is to reduce the significant interactions of a Green’s-function based matrix from being proportional to (Nx)2 to of order Nx log (Nx), thus offering the possibility of decreasing the operation count of iterative solutions. TABLE 45.6 Relative Applicability of IE- and DE-Based Computer Models Time Domain Frequency Domain DE IE Issue DE IE Medium ÷ ÷ Linear ÷ ÷ ~ x Dispersive ÷ ÷ ÷ x Lossy ÷ ÷ ÷ ~ Anisotropic ÷ ÷ ÷ x Inhomogeneous ÷ x ÷ x Nonlinear x x ÷ x Time-varying x x Object ~ ÷ Wire ~ ÷ ÷ ÷ Closed surface ÷ ÷ ÷ ÷ Penetrable volume ÷ ÷ ~ ÷ Open surface ~ ÷ Boundary Conditions ÷ ÷ Interior problem ÷ ÷ ~ ÷ Exterior problem ~ ÷ ÷ ÷ Linear ÷ ÷ ÷ ÷ Nonlinear x x ÷ ÷ Time-varying x x ~ x Halfspace ~ ÷ Other Aspects ~ ~ Symmetry exploitation ÷ ÷ ~ ÷ Far-field evaluation ~ ÷ x ~ Number of unknowns ~ ÷ ÷ ~ Length of code ~ x Suitability for Hybridizing with Other: ~ ÷ Numerical procedures ÷ ÷ x ~ Analytical procedures ~ ÷ x ~ GTD x ÷ ÷ signifies highly suited or most advantageous. ~ signifies moderately suited or neutral. x signifies unsuited or least advantageous