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 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 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