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 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 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