Generalized method of moments for three-dimensional penetrable scatterers

Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A 1383

Generalized method of moments for three-dimensionalpenetrable scatterers

L. N. Medgyesi-Mitschang, J. M. Putnam, and M. B. Gedera

McDonnell Douglas Corporation, R0. Box 516, St. Louis, Missouri 63166

Received August 9, 1993; accepted October 25, 1993

We outline a generalized form of the method-of-moments technique. Integral equation formulations are devel-oped for a diverse class of arbitrarily shaped three-dimensional scatterers. The scatterers may be totally orpartially penetrable. Specific cases examined are scatterers with surfaces that are perfectly conducting, di-electric, resistive, or magnetically conducting or that satisfy the Leontovich (impedance) boundary condition.All the integral equation formulations are transformed into matrix equations expressed in terms of five generalGalerkin (matrix) operators. This allows a unified numerical solution procedure to be implemented forthe foregoing hierarchy of scatterers. The operators are general and apply to any arbitrarily shaped three-dimensional body. The operator calculus of the generalized approach is independent of geometry and basis ortesting functions used in the method-of-moments approach. Representative numerical results for a number ofscattering geometries modeled by triangularly faceted surfaces are given to illustrate the efficacy and the ver-satility of the present approach.

1. INTRODUCTION

The field of optics has spawned many theoretical develop-ments for scattering from three-dimensional (3-D) objects.The pioneering investigations of Rayleigh, Debye, Lorenz,and Mie have provided the foundation for subsequentdevelopments in the field.' The early studies focused onspherical or nearly spherical scatterers by using classicalanalysis.2 But in many applications the scatterer is anarbitrarily shaped 3-D body.3 5 For example, it is wellknown in meteorology that gravitational effects deformhailstones and raindrops. Aerosols are often character-ized as complex agglomerated scatterers. Laser scatter-ing from polymers and suspensions of various biologicalcells involve complex 3-D shapes. Finally, artificial di-electrics are often composed of nonspherical inclusionsenbedded in the host medium. The specific scatteringproperties of the individual particle need to be knownbefore an aggregation of them is analyzed. In these andmany other physical problems it is desirable to relax therequirement of sphericity.

Recent developments in the field of computationalelectromagnetics have greatly expanded the palette ofanalysis tools to include problems in which the boundaryconditions or the shape of the scatterer makes previousclassical mathematical approaches intractable. Promi-nent among these are the finite-difference methods ineither the time or the frequency domain,6'7 the extended-boundary-condition method,8'0 the generalized multipolemethod," and approaches that use high-frequency asymp-totic methods such as the geometrical and the physicaltheories of diffraction.12'5

Perhaps most widely used is the method of moments(MM).16"7 This approach is- based on solving volumeintegral equation or surface integral equation (SIE) for-mulations of a physical problem. It has been shown to berobust in the resonance and transresonance regions, in-corporating all specular and nonspecular phenomena.

The boundary and radiation conditions that arise for thevector electromagnetic fields require special considera-tion. They are generally complex, depending on both bodygeometry (shape) and material composition. Integralformulations of Maxwell's equations provide an attractive,rigorous approach for such scattering problems since thenecessary boundary conditions are easily incorporatedand the radiation condition is implicitly satisfied. Solu-tion methods based on the MM technique have achieved ahigh state of refinement.

The term method of moments was coined byKantorovich and Akilov.1' The method is also known invarious fields as the method of weighted residuals, themethod of projections, or the Petrov-Galerkin method.Kantorovich and Akilov,18 Petrov,'189 and Kantorovich andKrylov19 provided the rigorous mathematical under-pinnings for the originally heuristically based formulationproposed by Galerkin.2 0 In essence the MM providessolutions of functional equations. These solutions maybe conceptualized as projections to subspaces of func-tional spaces. Formal equivalence can be demonstratedwith the Rayleigh-Ritz variational approach2 ' and Rum-sey's reaction concept.2 2 Harrington was the first to pro-vide a systematic exposition of the use of the method inthe context of modern electromagnetics.' 6 His seminalcontributions have stimulated rapid advancements of thefield.

In this paper we develop a generalized form of the MMtechnique for a hierarchy of arbitrarily shaped 3-D scat-terers. Specifically, we examine the cases in which thescattering surface is either entirely or partially pene-trable, as shown in Fig. 1. [For clarity of illustrationthe 3-D geometries are shown generically as two dimen-sional (2-D).] The unified approach outlined here allowsall cases to be formulated as matrix equations expressedin terms of five general Galerkin (matrix) operators. Thisleads to a streamlined numerical solution procedure for allcases, as is illustrated by specific examples.

0740-3232/94/041383-16$06.00 C 1994 Optical Society of America

Medgyesi-Mitschang et al.

1384 J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994

(a)

(b)

(c)

Fig. 1. Classes of penetrable or partially penetrable arbitrary3-D scatterers analyzed with the present method. Objects areshown generically in two dimensions for clarity: (a) connecteddiscrete regions, (b) multiple homogeneous coatings, (c) genericcombination of cases (a) and (b).

2. FORMULATION

A. Fundamental ConceptsFor clarity of presentation we begin at first principleswith Maxwell's equations written in symmetric form for ahomogeneous, isotropic region with electric and magneticsources23:

sequent boundary conditions conform to a separable coor-dinate system. An alternative approach that is free ofthis restriction recasts the scattering problem in terms ofvolume integral equation or SIE formulations with Green'sfunctions. The latter formulation is often preferred forhomogeneous or discretely inhomogeneous scatterers be-cause of the smaller memory requirements for 3-D prob-lems if they are solved by the MM technique. With theSIE approach the dimensionality of a problem is reducedfrom a 3-D volume to a 2-D surface. When the SIE islinear, as is the case here, such formulations permit thesuperposition of effects from several causes (or sources),either discrete or distributed. Even complex boundaryconditions resulting from discontinuities in the surface ofthe scatterer or its physical constituents can be incorpo-rated into the SIE formulation. The versatility of thisapproach is illustrated below.

B. Surface Integral Equation Formulation forPenetrable ScatterersLet us consider the problem of a penetrable body repre-sented by the surface S with an outward normal vector 7i,

as shown in Fig. 2(a). Starting with Eqs. (5) and (6) andusing the vector Green's theorem, we can show that thetotal electric and magnetic fields at an arbitrary point r inregion R, are 23

O(r)El(r) = Einc(r) - f {jwcu(iV x Hl)(

- (0 X El) X V'([l - (7P * El)V'(cl}ds', (7)

R1 : All_ _H

V E = -opH - M,

V x H =jcosE + J,

V E = p/c,

V H = m/,,

(1)

(2)

(3)

(4)

where p and m are the electric and magnetic charges andJ and M are the electric and magnetic currents in theregion. Sinusoidal time variation of exp(jcot) is implied;E and u are the complex permittivity and permeability ofthe medium, respectively. Magnetic charges and cur-rents are introduced here as a mathematical artifice toyield symmetric forms of the subsequent integral equa-tions for the fields and thus to allow duality principles tobe invoked. Equations (1)-(4), written in terms of thevector Helmholtz wave equation for the electric field, yield

V V E - k2E = -jwp.J - V X M (5)

or, by duality for the magnetic field,

V V H - k2H= -jwsM + V x J, (6)

where k = 2/A. The partial differential equations (5)and (6) have been solved by the method of separation ofvariables for scatterers in which the geometry and the sub-

_ inhc

I

I,e-_

(a)

R.: E. u,. .-'f I --

R2 : E,/ .tI

E2 4 = 2 ....

(b)

R.: , 2

E= = o

R'12: 4, 21 M2J

p2' 2 --S.--.. -- --'(c)

Fig. 2. Generic penetrable 3-D scatterer: (a) original problem,(b) equivalent problem exterior to S, (c) equivalent problem in-terior to S.


Ii


and, again by duality,

O1(r)Hl(r) = Hinc(r) + f {jws,(i X El)

+ ( x H,) x V'cF, + ( Hl)VkI,}ds'. (8)

In Eqs. (7) and (8), O,(r) is the Heaviside function thatprovides the jump condition at S and is given by

(l for r E R6,(r)= 1/2 forrEaRl,

t0 otherwise

(9)

where Rl denotes the boundary of region R. The term(Di in Eqs. (7) and (8) is the Green's function for the arbi-trary 3-D region RI:

4),(r - r) exp(-jklir - rl)47rlr - rI

(10)

where r is the field point and S is the surface enclosing thescatterer. The nature of the surface S and the associatedboundary conditions are specified below. Equations (7)and (8) are a form of the Stratton-Chu equations.24 Theprimed quantities are associated with the source point r',and the dependence on r' for the terms under the integralsis implied. We can write a similar set of equations for thetotal fields in R2 , replacing the subscripts 1 - 2 and not-ing that Einc(r) = Hinc(r) = 0 if r E R2.

Significant simplifications occur if one recasts theoriginal problem by using the equivalence principle.Figure 2(b) shows a problem equivalent to that given inFig. 2(a) and external to S, and Fig. 2(c) shows an equiva-lent problem internal to S, where the equivalent currentsare given by

Ji = ni X HiaRi, (11)

Mi = E, X i I Ri (12)

for i = 1, 2 and where Ei and Hi are evaluated on S, de-noted by aRt, which is the boundary between the externalregion R, and the interior of the body R2. The unit nor-mal vector ni points into region Ri (i.e., 7i1 = ni and n2 =

- n, as is shown in Fig. 2). The relationships between thenormal and the tangential field components, deduced fromMaxwell's equations, are

n*Ei=- V *( x H = - V *J, (13)

Jw~S ~ J()flop. jwp.

Substituting Eqs. (11)-(14) into Eqs. (7) and (8), one ob-tains the Stratton-Chu integral equations in terms of theinduced equivalent electric and magnetic currents.

It is convenient to write the Stratton-Chu equations interms of generic linear operators. Adhering to the nota-tion in Fig. 2, one obtains the fields in region R, in termsof the equivalent surface currents J+- J, M - M, andintegrodifferential operators L, and K, as

01(r)EI(r) = Einc(r)- LJ(r) + KM'(r), (15)

01(r)Hl(r) = Hinc(r) - KJ+(r)- -LM+(r). (16)21

Likewise, in region R2 (assuming that there are nosources in R2) the total fields are given by

02(r)E2(r) = -L 2J-(r) + K2 M(r),

02 (r)H2(r) = -K 2 J-(r)- 2L2M-(r),r12

(17)

(18)

where J = J2, M a M2. The linear integrodifferen-tial operators Li and Ki (i = 1, 2) are defined as

L IX(r) = jwp.X(r') + - VV'* X(r')] i(r - r')ds',

(19)

KiX(r) = fi X) X VcDi(r - r')ds', (20)

where the integrals are taken as Cauchy principal values.The vector function X is defined on the boundary of regionRi (aR), and the kernel (i is the Green's function; si, i,and (Di are defined in region Ri, where s = sosri, / =poA,i, 7i = 1o(pri/Srj)

1 1 2 and so and puo are thepermittivity and the permeability of free space, respec-tively. If there are sources within region R2 , they wouldbe added to the right-hand sides of Eqs. (17) and (18) asEinc and Hinc in Eqs. (15) and (16). The boundary condi-tions on the tangential field components,

n~ x (El - E 2)s° 0,

n~ X (H, - H2 )Is = 0,

(21)

(22)

relate the interior and the exterior fields and currents.Letting J = J = -J- and M = = -M- and equat-ing the tangential components of Eqs. (15) and (17) andEqs. (16) and (18), we obtain a coupled set of integral equa-tions for the unknown electric and magnetic currents, i.e.,

Einc(r) tan = (Li + L 2)J(r)ltan - (K, + K2)M(r)ltan, (23)

Hinc(r)Itan = (K, + K2)J(r)ltan

+ I ?L + 2 L2 M(r)?II2 7122 tan

(24)

Equations (23) and (24) constitute the formulation byPoggio, Miller, Chang, Harrington, and Wu (thePMCHW formulation) for a dielectric boundary.25 26 Thisformulation has been shown to yield a unique solution atinternal resonances associated with the correspondingconducting body.

3. METHOD-OF-MOMENTS FORMULATION

Next we solve Eqs. (23) and (24), using the Galerkin formof the MM formulation. This can be done in two ways:the conventional approach and a generalized one. In theformer the MM expansion and testing of the SIE formula-tion are performed after the boundary conditions havebeen imposed on the surface of the body, i.e., one beginswith Eqs. (23) and (24). In the generalized approach theMM expansion and testing are first applied to the coupledintegral equations [Eqs. (15)-(18)] for the fields on eachsurface of the body, followed by application of the bound-ary conditions. As noted above, this approach provides a



compact numerical implementation for many classes ofscatterer. We illustrate these two approaches for thecase of the (totally) penetrable scatterer in Fig. 2(a). Sub-sequently, the generalized approach is explicated forscatterers with electrically conductive, resistive, magneti-cally conductive, and impedance (Leontovich) boundaryconditions.

A. Conventional ApproachIn the conventional approach the equivalent currents on Sare expanded in terms of a set of basis functions fr)spanning this surface, i.e.,

J`-(r) = Jn fn (r), r E S,

M+(r) = rqoEM.:1.(r)X r E S,

(25)

(26)

where ± denote the external and the internal currents onS, respectively. These expansions are substituted intoEqs. (23) and (24). The factor of 770 is required since theH-field equation is normalized by mqo. Next the complexinner product is formed with these equations, transform-ing the integral operators L and K into correspondingmatrix operators, denoted by 2e and N. Formally, thecomplex inner product between vector functions A and Bon a surface S is defined as

(A,B)s = J A* Bds, (27)

where * indicates the complex conjugate. In the Galerkinform of the MM solution, testing functions wn = fn* arechosen on each of the surfaces, and the inner products areformed between these testing functions and the integralequations. The E-field integral equation [Eq. (23)] istested with w, and the H-field integral equation [Eq. (24)]is tested with -qown. The original integral equations arethus transformed into a set of linear equations. Writtenin matrix form, they are

2 + 2 -K,-K2 J+F + 2.2] [7J e (28)

where we have used the fact that the tangential electricand magnetic fields are continuous across the interface Sand thus the equivalent interior and exterior current coef-ficients are related by J = -J+ and M- = -M+. Ex-plicit forms for the 2e and f operators are given in Appen-dix A. The column vectors W and We result from testingthe incident fields Einc and rjoHinc, respectively.

B. Generalized ApproachIn this case, in addition to the expansion of the surfacecurrents, we also expand the fields on S, i.e.,

E+(r) = qoEEnfn(r), r E S, (29)n

H±(r) = EH.±fn(r), r E S. (30)

where E + = E ,I tan(s) and E - = E2 1 n(S) and similarly forH+ and H-. Although rooftop basis functions are used

subsequently in the Galerkin expansions to handle arbi-trary 3-D bodies (triangular patch formulation), the dis-cussion that follows does not depend on the choice ofspecific basis functions. The details of this generalizedMM approach for 2-D and body-of-revolution (BOR)geometries are given in Ref. 27.

Substituting these expansions into Eqs. (15)-(18) andusing Galerkin testing transforms the four integral equa-tions into the following matrix equation:

P, YC0 0

cIC,0 0

0 0 E2 -72

0 0 G2 /f 22p.2

i -

iO

'0iO

0

0

0

0000V 0

0 g'

0'0]

(31)

where i and XJi are the matrix operators (see Appendix A)that result from testing the integral operators Li and Ki,respectively. The subscript specifies the region in whichthe Green's function is evaluated. The matrix operator WTresults from testing the total electric and magnetic fieldson the surface, and Z and We are the column vectors thatresult from testing the incident electric and magneticfields, respectively. Note that the form of Eq. (31) assumesthat the two H-field equations have been multiplied by -qo.

The next step is the imposition of the boundary condi-tions. To streamline the discussion it is convenient to re-write the generalized MM formulation in Eq. (31) inshorthand form as

ZZI + HI= V,

where I represents the four current unknowns (J:, M+) inEqs. (25) and (26) and I represents the four field un-knowns (E+, H+) in Eqs. (29) and (30). We now incorpo-rate the boundary conditions into this overdeterminedsystem by specifying equivalent relationships between thecorresponding unknown current and field coefficients.These relationships are used to eliminate unknowns fromthe overdetermined system as follows. Let

I [I, (32)

where I1 represents the unknown current coefficients thatwill be retained in the final solution and 12 represents thecoefficients to be eliminated by use of the boundary condi-tions. Next, the boundary conditions are incorporatedinto a matrix A so that I = Al,. Matrix A gives the rela-tionships between I, and I2 that result from an applicationof the boundary conditions. Finally, we use the boundaryconditions to eliminate the field unknowns I by specifyinga matrix B so that I = BI1. The overdetermined systemof equations is thus reduced to the system of equations

(ZA + B)I1 = V. (33)

At this point there may still be more equations than un-knowns. The field equations are combined to reduce thesystem of equations to a determined system. We combine


lI


the field equations by specifying a matrix operator C thatis applied to Eq. (33), i.e.,

C(ZA + ZB)I1 = CV. (34)

The exact form of the matrix C depends on the combined-field formulation to be solved. For a given boundary con-dition certain choices for the matrices A, B, and C resultin a symmetric system of equations. Examples are givenbelow for the dielectric, the conducting, the resistive, andthe impedance boundary conditions.

1. Dielectric BoundaryThe dielectric boundary (i.e., the case given in Fig. 2) wastreated above. The boundary conditions are used to re-late the interior and the exterior current coefficients (i.e.,J- = -J + and M- = -M) yielding I, and the matrix Agiven in Table 1. It can be shown that the field coef-ficients are related to the equivalent current coefficientsin a least-squares sense by H- = H+ - 'J+ and E- =E+ -- X'M+, resulting in matrix B. Using these rela-tionships, we obtain the respective transformation ma-trices I, A, and LB given in Table 1. We obtain thePMCHW formulation given in Eq. (28) by letting C =A,where t denotes the transpose operation.

2. Conducting BoundaryThe perfectly conducting scatterer is a subcase of thedielectric boundary. The interior fields are zero, and

Table 1. Transformation Matrices

Etan= 0 on surface S. This implies that I = J+, andM+ J- = M- = 0. In Table 1 the single nonzero entryin A corresponds to the fact that J+ is the only nonzerocurrent on the surface. The equivalent electric current isdefined as J+ = V x H'. This implies that the corre-sponding unknown coefficients in Eqs. (25) and (30) areapproximately related by J+ = XH+ or, equivalently,H = -X+, where X' = X-.

As shown in Ref. 27, the transformation matrix X has aparticularly simple form for 2-D and BOR geometries, inwhich case J+ = XH' is exact. This follows from the factthat for these geometries the surface currents and fieldsare expanded in terms of two orthogonal vector compo-nents. For currents and fields that are expanded interms of basis functions that have no inherent orthogonal-ity properties (as in the case of arbitrary 3-D bodies), thetransformation is approximate in a least-squares sense.In the formulations presented here that use the general-ized approach, the X matrix does not appear in the finalMM matrix equation.

There are three formulations for the conducting body:the electric-field integral equation, the magnetic-field in-tegral equation, and the combined-field integral equation(CFIE). We obtain the electric-field integral equation bysetting the matrix C = A, and substituting A, B, and Cinto Eq. (34) yields

-Y r = c. (35)

for Generalized Galerkin MethodMatrix

Boundary Condition I, A B

1, ~~~~~~~~~~00 x

Conductive [J+] 0 0

0 ~~~~~~~~~0

1 0, 0 -X'

0 1 X' 0Dielectric 1 0 0

0 - 1 XI0'

+ - 1 0 OR ° 0 Rs

Resistive M 1 0 XI 0 0Resistve 001 R8 0 Rs

.J_ ~~~~~~~~~- R. 0]0-1 0 0 X Rs °

Rs + 4G. ° Rs - 4Gs °

Resistive- J+10 0 0 G magnetically M+ 0 1 0 0 S 4R ° 4R,

conductive J_ 0 0 1 0

IM-i 0 0 0 1 Rs - 4G ° Rs + 4G °

4Rs 4R

Impedance 1 0 ?s 0

(Leontovich) [+] 0 1 0 1/7

00 0 0

~0 0j ~0 0



The magnetic-field integral equation formulation is devel-oped directly from the integral equation instead of fromthe generalized approach, which involves the transforma-tion X. Noting that on a perfectly conducting bodyM = 0 we can obtain the magnetic-field integral equa-tion formulation by testing Eq. (16) after taking the crossproduct with the surface normal vector tn, which yields

[.,f + ST]J+ = e (36)

where the operators xWi and ,X result from testing 7E X K,and -qo7E X Hinc, respectively. These operators are definedin Appendix A. Finally, the CFIE formulation is given by

[aLE + (1 - a)(.,% + WA]J+ = a + (1 - a),XC, (37)

where 0 a c 1 is an adjustable weighting parameter.

3. Resistive BoundaryFor a scatterer with a resistive surface the boundaryconditions are Etan = Etan- and iV X (H - H-) =[1/(Rs,71)]Etan+.8 These can be expressed in terms of theequivalent currents as M- = -M+ and RS77o(J + J-) =Etan+, respectively. The matrices I, and A are definedin Table 1. We obtain the matrix B by noting thatthe boundary conditions can also be written as Htan- =[-1/(RSrio)]M + Htan+, where Htan" = - X J+, andthen converting to the equivalent expressions relatingthe field and the current coefficients. Thus

(i.e., R2 = R), the above matrix equation reduces to

[g, + 2R,9f]J = %X (43)

where J now denotes the total current (i.e., J = J+ + J-).

4. Resistive-Magnetically Conductive BoundarySometimes it is convenient to represent the coating on ascatterer as a combination resistive-magnetically conduc-tive surface. Such a surface is characterized by theboundary conditions 8

n X (H - H-) = 1 Et. = 1 (E+ + E)tan ,R -s o 2 Rs -qo

- V X (E+ - E) =Ht - -7-(H+ + H)ta..Gs 2G,

(44)

(45)

These two equations can be rewritten in terms of theequivalent currents as

(J+ + J-) = I (E+ + E))tan ,2R -s q

(M+ + M-) = -O (H+ + H-)ta.-2G,

(46)

(47)

They can also be rewritten in the following form after across product with the surface normal C is performed:

Et..+ = RS77o(J+ + J-) -E+ = Rs(J+ + J-),

Etan = Etan+ E- = Rs(J+ + J-),

J+ = n x H+ - H+ = %,J+

Htan = Htan+ - M+- H- •J+ - -M+.Rs-qo Rs

(38)

(39)

(40)

(41)

The matrix B can now be written as is given in Table 1.Setting C = A results in the following MM matrix equa-tion for a scatterer with a resistive boundary separatingtwo different non-free-space regions:

_afl62 yy i y 1 -2f2 + -Ee + -O(p,2 p1. R,

Wf2

RK J 1 9

-2X2 M+ = e .

- 0

(H+ H)ttn = I (M+-MI,2R -s o

(E+-E-)t = 72G -J-)

(48)

(49)

Forming sums and differences of Eqs. (46) and (49) andEqs. (47) and (48), we obtain

E Itan (B8s + 1) oJ +

E|tan = s( 4Gs)

4G,

(Rs + I noj°4G,

t(n G+ I )M+ + (G,

(42)

The foregoing assumes that the resistance Rs is constantover the surface S. For a variable surface resistance thematrices Rf and (1/R,)If can easily be modified to in-clude this variation. For a resistive sheet in free space

H |tan = (Gs

(50)

(51)

(52)

(53)I I 71M +(G, + I M--

Using these relationships, we obtain A, LB, and C matrixentries in Table 1. The resulting MM matrix equation forthis class of scatterers becomes

0

(G - 1 T

1-2

-22 + Gs +1p.2 \4R,

.T + Rst

X/,

RSfIC

1 + 1

(R - 1f

(R - I i0

22 + (R + I5Ee, + G + 4

(Gs - 4R 2

0

J+ W

M+ X

M- 0M] 0]

(54)

- -


- I I M-,4R., 710


terer are modeled with flat triangular facets (see Fig. 3).An outward unit normal is associated with each facet, de-fined by the right-hand rule applied to the vertex pointsdefining each facet. For generality the scatterer geome-try may be composed of multiple surfaces, each of whichrepresents the interface between different dielectric andconducting regions of space (see Fig. 1). For each surfacethe electric currents J and/or magnetic currents M areexpanded in terms of the Rao-Wilton-Glisson rooftop ba-sis functions.32 Current continuity between intersectingsurfaces (i.e., across junction edges) is accomplished by in-clusion of a half-rooftop function on each side of the junc-tion edge. Although these half-rooftop functions are ondifferent surfaces, we enforce current continuity by equat-ing the unknowns associated with these junction edges.

Let S represent a triangularly faceted surface composedof Ni interior edges and Nj junction edges. The electricand magnetic currents on S are expanded in terms of roof-top functions fn as

0

I. /I

I

Fig. 3. Faceted 3-D scatterer with details of patch coordinatesused in rooftop expansion functions.

For a resistive-magnetically conducting sheet in freespace, this matrix equation reduces to

E 21 + 2GaF LMJ [X]-p. j

5. Impedance (Leontovich) BoundaryOften for lossy, penetrable, electrically large scatterers theimpedance boundary condition (IBC) can be used. Scat-tering from lossy dielectric spheres modeled with an IBCis discussed at length in Ref. 29. An impedance bound-ary is characterized by the boundary condition E'Itan =

,7s'7o17E X H', where the interior fields and currents arezero.30 This boundary condition can be implemented asan equivalent combination resistive-magnetically con-ducting boundary where R, = -qs/2 and G, = 1/(2ris).Substituting these expressions for R, and G. into Eqs. (44)and (45) reduces the combination resistive-magneticallyconducting boundary condition to a Leontovich boundarycondition. The associated MM matrix equation is givenby Eq. (55). The limits of this approach are discussed atlength in Ref. 31.

4. THREE-DIMENSIONAL PATCHREPRESENTATION

Next we specialize the MM developments to arbitrary 3-Dscatterers satisfying any of the boundary conditions dis-cussed above. We assume that all the surfaces on the scat-

Ni+NjJ(r) E Infn(r),

n=1

Ni+NjM(r) -rio E Knfn(r),

n=1

(56)

(57)

where

[ i+ (r- n

fn(r) = I(r - Vn),2An

to

r E Tn+

r E Tn (58)

otherwise

The - is included in the magnetic-current expansion sothat the resulting matrix equations are symmetric. Theinclusion of the sign requires that the matrix equationsdeveloped in Section 3 be modified to include a - on thematrices that multiply the magnetic-current coefficients.

With respect to Fig. 3, the two triangular facets form-ing the nth edge are denoted as Tn+ and Tn, with vertexpoints n+ and vn-, which are opposite the edge. Thelength of the edge is given by In, and the areas of the tri-angular facets Tn' and Tn- are given by An' and An, re-spectively. The half-rooftop functions at a junction edgeare defined on only one of the two triangular facets. Thesurfaces forming the junction edge will use either the Tn+facet or the Tn- facet to define the half-rooftop function.Figure 4 illustrates some of the junction (intersecting)edges that can be modeled with the present formulation.In essence, this is a numerical realization of the Meixnercondition for all these cases.3 3 A discussion of this classi-cal approach with an application to BOR geometries isgiven in Ref. 34.

A. Galerkin OperatorsThe current expansions, relations (56) and (57), are substi-tuted into the integral equations, and the Galerkin testingprocedure is used to transform the integral equations intoa matrix equation. The integral operators L, K, 7E X L,and i! X K are transformed into corresponding matrix op-erators, denoted by .f, If, x2, and K.X Thus the MM solu-tion for a general 3-D body based on any CFIE formulation

k I ' I n

% I

I

2E + 2Rixi


I

II

II

t

- - - - - - - - -I

I


PEC

R-card

P EC P

Dingy PEC BC

I Dielectic

Dielec

Rrcard

LT(S 2 , Si; R) = tL(S1, S2'; R),

at(S 2, Si; R) = - t'(Sl, S2 i; R),

(61)

(62)

where the surfaces Si and S2 i are the images across theplane of symmetry of the surfaces Si and S2, respectively.It is assumed that the image surfaces Si and S2i have theedges ordered in the same manner as the original surfacesSi and S2. When Si = S2 the matrix L is symmetric andthe matrix X is antisymmetric. These relationships canbe extended to multiple planes of symmetry.

C. Geometric Symmetry RelationshipsWhen the geometric model contains a plane of symmetry,the system matrix can be decoupled into two smaller sys-tem matrices, each with approximately half the numberof unknowns as the original matrix equation. Assuminga faceted model with right-left symmetry, the edgeunknowns on all the surfaces can be ordered so that thePMCHW-electric-field integral equation symmetric sys-tem matrix can be written in terms of the Galerkin matrixoperators L and X as

differentdielectricsurfaces

Fig. 4. Some representative surface-junction conditions imple-mented in the 3-D patch formulation.

can be obtained in terms of five generalized Galerkin (ma-trix) operators. These operators correspond to the fourintegral operators Li, Ki, ri X Li, and 7n X Ki, along withone that results from testing the current directly. Oncethese operators are defined it is a simple matter to proceeddirectly from the SIE to the MM matrix equation, whichcan be solved directly by inversion or iteration. For brev-ity we omit discussion of this step.

B. Galerkin-Operator Symmetry RelationshipsThe Galerkin matrix operators L and at have certain sym-metry relationships that can be exploited to save bothmemory and matrix fill time. If Si and S2 represent twofaceted surfaces on the boundary of region R, then thefollowing relationships can be established:

LrrE xrrE

g/ H ,H

,YIl E %f, Ea'rs lrs

~rsH LH

erlT E E ]Jr

atcri1H _TriH M

y E If E L 6E G E y, E Jf, E

af, H Es H f, H H a 1 H y, H

~1irgT) E

_ actr

i Er

Hr

2e Eatis

1h 1E 1 yE

Xf, H -Y HJ,

ErHr

EsH,

El

_H1

(63a)

(63b)

(63c)

(63d)

(63e)

(63f)

where the subscripts r, s, and I refer to edges on the right-hand side, those in the plane of symmetry, and those onthe left-hand side, respectively. The double subscripts onthe operators 2 and X specify the location of the test edgesand the source edges, respectively. The superscripts Eand H on the operators refer to the operators that resultfrom testing the E-field and the H-field integral equations,respectively.

If the left-hand edge unknowns are ordered in the sameway as the right-hand edge unknowns, then the followingmatrix relationships can be obtained from the matrixsymmetry described in Subsection 4.B:

L(S2 ,Si; R) = t (Sl, S2; R),

X(S 2, Si; R) = %S1, S2; R),

(59)

(60)

where t is the matrix transpose operation. When Si = S 2

the two matrices L and X are symmetric. These relation-ships, in conjunction with the - in the magnetic-currentexpansion, result in a symmetric system of equationswhenever the electric-field integral equation, the PMCHW,the resistive boundary condition, the magnetically conduc-tive boundary condition, and/or the IBC formulations areused. If the CFIE formulation is used, the matrix is notsymmetric; however, the above relations can still be usedto reduce the matrix fill time.

Relationships can also be established for scatterers withone plane of geometric symmetry. These relationships are

2L1 = Lrr

ir = rX21e = -- r.es Lrs ,

1si = Eesr

aCII = Wfrr,

aIlr = -a6l

ach, = alrs

Xsl = afsr-

(64)

(65)

If we take the sum and the difference of Eqs. (63a) and(63e) and Eqs. (63b) and (63f) and use the above relation-ships, the matrix equation can be rewritten as



2(LrrE - Lr E)

2(Wrr H- 9fr H)

2(WfrrE + Crl E)

2 (2rrH + LriH)

22r E Es 0

0

0

0

0

0

2LsrE 2 f rE yE 0 s

0 0 0 2(LrrE + LrE) 2(9frrE - rlE) 2 aIlrsE

0 0 0 2 (StrrH + If rlH) 2 (YrrH - rH) 2 LrsH

0 0 HassH I 2rf H -yRH

In the above equation it can be shown that the matricesSSE and SCssH are identically zero. Hence the original ma-

trix equation decouples into two matrix equations that canbe solved separately, in turn reducing the memory re-quirements by a factor of 4 and the solution times by afactor of 2-4 for large problems. These two matrices arealso symmetric since the original matrix was symmetric.

For geometries with resistive or IBC surfaces the aboveprocedure still results in a decoupled system, since the WZoperator has the same right-left relationships as the Loperator. Again the matrices are symmetric. The CFIEformulation also results in a decoupled system, since inthis case the ,X~ and the IL operators have the same right-left relationships as the L and the Xf operators, respec-tively. However, the matrices are no longer symmetric.

5. NUMERICAL RESULTS

The foregoing generalized MM formulation was applied tosome representative cases of 3-D scatterers. The calcula-tions were performed with the CARLOS-3D code,35 3 6 whichimplements this formulation. As an initial benchmark weapplied the analysis to spherical scatterers for which theclassical Mie solutions provide a comparison. In Fig. 5 thebistatic scattering cross sections of a penetrable sphere, ahollow spherical shell, and a coated perfectly electricallyconducting (PEC) sphere are compared with the corre-sponding Mie results. The sensitivity of the solutions asa function of facetization is also shown, where the numberof triangles (denoted by A) on each of the surfaces isgiven in Fig. 5(a). The complex dielectric constant chosen(Sr = 1.75 - j.3) corresponds to a representative aerosolat visible wavelengths.

The monostatic cross section of a cylindrical body, halfconducting and half penetrable (r = 2.6), is depicted inFig. 6. Superimposed are the calculations performedwith a MM formulation for axially inhomogeneous BOR's.34

In Fig. 7 we depict the bistatic scattering from an oblatespheroid that is entirely penetrable, together with twopartially penetrable cases, one with a hollow core and asecond with a PEC core, both centered within the dielec-tric spheroid. We show the facetization sensitivity for thefully penetrable case. The convergence for the other twocases is similar in both principal polarizations. The re-sults for various facet densities are omitted for figure clar-ity, with the results presented corresponding to the 320Acase. As expected, the scattering cross section is higherfor the case with the PEC core. In Fig. 8 the correspond-

ing results are depicted for the same three variants of theprolate spheroid. Not unexpectedly, the scattering crosssections for these cases are smaller than are those for theoblate case shown in Fig. 7. The cross section with thePEC core is again the largest.

In Figs. 9 and 10 we compare the bistatic cross sectionsfor the oblate and the prolate spheroids, each having a cen-tered or an offset spherical PEC core. In the oblate casethere is minimal difference resulting from placing the coreoff center. In the prolate case, however, the offset corecauses destructive interference that results in a dramaticdecrease in the backscatter. This effect can be enhancedor diminished with other choices of complex dielectric con-stants. Brevity precludes further discussion of this effect.

Figure 11 depicts the scattering from different agglom-erated arrangements of spherical PEC scatterers. Forthis example the calculations assume that the scatterersare only radiatively closely coupled. As can be seen, thethree configurations result in very different bistatic crosssections with major shifts of the scattering nulls. InFigs. 12 and 13 the corresponding results are depicted forspheres that are penetrable. Again there are significantdifferences in the scattering patterns for these cases.

In the foregoing numerical examples we have attemptedto show the flexibility of the generalized MM formulationas implemented for arbitrary, triangularly faceted 3-D ge-ometries. The individual or aggregated scatterers consid-ered in the above examples could also have been partiallycoated with resistive and/or magnetic films. A furtherextension for all these cases by use of the IBC is also pos-sible. If the penetrable scatterers are electrically verylarge, the IBC approximation is very effective.2 9

-3' In the

interest of brevity we have omitted a discussion of thesecases here.

6. SUMMARY

In this paper a generalized form of the method of momentshas been developed with application to arbitrarily shaped3-D scatterers. The formulation was based on surfaceintegral representations of Maxwell's equations. TheGalerkin form of the method of moments was used. Weshowed that a broad class of penetrable or partially pene-trable scatterers can be analyzed with this approach. Allthese cases could be cast into matrix equations expressedin terms of five general Galerkin (matrix) operators. Thegeneral expressions for these are given in Appendixes Aand B. They are amenable to streamlined numerical im-

JrJi2

M,- Ml2

Js

Jr + J2

Mr-Ml2

Er -El

Hr + Hi

E.,

Er + El

Hr-HI

H

(66)



10

-0-

'r--10-

-20-

-30(a)

9 (deg)

(b)

Fig. 6. Monostatic scattering cross section of a partially coated(e, = 2.6) right circular cylinder ( = 1.22A, a = 0.455A): (a) 00polarization, (b) d4 polarization. MM BOR solution versus 3-Dpatch MM formulation.

0 30 60 90 120 150 1806 (deg)(C)

Fig. 5. Bistatic scattering cross section of spherical penetrable(Sr = 1.75 - j.3) scatterers. Comparison of Mie and MM solu-tions: (a) dielectric sphere, a = 0.2A; (b) hollow spherical shell,a = 0.2A, b = 0.18A; (c) coated PEC sphere, a = 0.2A, b = 0.18A.

plementation. Finally, we used representative results forvarious 3-D scattering configurations to illustrate the gen-eral efficacy and versatility of this approach.

APPENDIX A: EXPRESSIONS FORGALERKIN OPERATORS

1. L OperatorWith the notation given in Fig. 3, the Galerkin operatorL(S1 , S 2; R) is the matrix that we obtain from the integraloperator L in Eq. (19) by forming inner products betweentesting functions on Si and L operating on expansionfunctions on S2. An element of the matrix can be writtenexplicitly as

L2kl(Si, S 2; R) = (fk, L(f1 )) s - (Al)

The index k refers to the kth rooftop function spanningthe kth edge on surface Si; correspondingly, the index refers to the th rooftop function on surface S2. Notethat in this notation R specifies the region in which , pland the Green's function 4D are defined, and hence thesubscript i has been omitted from the integral operator L.The integration in the inner product is over the test sur-

-1

~MNMOR /--- CARLS-3DT 4

sOE .6 E

(a)

(b)

.. .. . . . ..-. . . . . .-- -- -

............................................



* I I I I I . I I I .-- .---- 48 A ----- 204A

- 116 A ---- 320 A

. .---------... ,_. ....... _

- * -t? ~ Polarization

-9 Polarization

. I , I . . I ..I . . I .

-10

-20

e-4

-40

-50j

-60

(a)

-~~~~~~ ~ Polarization _____

- N

-\ /

\ ,

\'II

- 99 PolarizationII

I 8

_ I a I a I . I a I -

(b)

I I I I I I I I I I I I I I I- - -- 204A ------ 320A -

- 99 ~Polarization

0 Polarization I ,

I �

(a)

in ---1n.L . . . . I

-201-

-40

-SO

-60

(b)

A * * * * * * * * * * * . . .

- 48A---- 116A

-20;-

-40

-50

-60

-70L) 30 60 90 120 150 1809 (deg)(C)

Fig. 7. Bistatic scattering cross section of penetrable (r =1.75 - j.3) oblate spheroidal scatterers (a = 0.2A, b = 0.1A):(a) dielectric spheroid illustrating sensitivity of MM solution withfacetization, (b) penetrable spheroidal shell with spherical void(r = 0.05A), (c) penetrable spheroid with spherical conducting core(r = 0.05A).

_1

-20-p

-40

-50

-60

0 30 60 909 (deg)

(C)

120 150 180

Fig. 8. Bistatic scattering cross section of penetrable (r =1.75 - j.3) prolate spheroidal scatterers (a = 0.1A, b = 0.2A):(a) dielectric spheroid illustrating sensitivity of MM solution withfacetization, (b) penetrable spheroidal shell with spherical void(r = 0.05A), (c) penetrable spheroid with spherical conducting core(r = 0.05A).

-10

-20

-40

-50

-60

-1c

-20

"'a

-40

-50

-60

-70

0 Polarization _-

'II99 Polarization I;

'I

I . I . . I . . I . I I .

- _ _ -_ % _ " Polariation _ _ _ _-_

- N /~~~~~~~~99 Polarization 1

-_ I a a I Iv I I I

*+ Polarization- - - - - - - - -- - - - -

P n /

99O Polarization1

'

. z~~~~~~~~I

-70r I I I I I I I I I I I I 1 -

-. (] - - - - - - -

1). . . . . . . . . | . . . . . . . .

I I I I I I I I I | I I 8 g I | @

i I I . . . . . . . . . . . . . . .


-"'L .

-v

I


(a)

I -,

I, I Iu 1 - [ I I I . . I -

Cn...e-ed-Spe-----Offset SphereCentered Sphere

:.

. 1, I, - . 1 1,,-2CL

60 120 180 2409 (deg)

300 360

(b)Fig. 9. Bistatic scattering cross section of penetrable (, =1.75 - j.3) oblate spheroid (a = 0.2A, b = 0.1A): (a) 00 polariza-tion, (b) Aft polarization. Spherical conducting core (r = 0.05A)centered and offset by 2r = 0.1A.

face S1. The elements of Tf are given as

Tkl ( ) = ilk ds I ds'{copufk(r) f(r')(D(r - r')

+ -[fk(r) * PD (r-r')]V' f(01,co iS

0,

With respect to the definition for fn in Eq. (58), it can beshown that

In

An+

V f(r) = Ale'A-

0

r E T

r E T,- (A4)

otherwise

Substituting the expressions for the testing and the ex-pansion functions into Eq. (A3) and splitting each surfaceintegral into an integral over T+ and an integral over T-,we obtain

Eel ( ) = E E ,pq if ds ds'4 p=± q=± AkPA1 I Lk T1q

x [.(r - vP) (r' - vq) -- 4 (r - r').

(AS)

(a)

'-p

(A2)

where Sk = Tk' U Tk- and Tk: are the two triangularfacets (Fig. 3) that form the kth edge. Similarly, SI =T+ U TF. The form of the rooftop functions fk allowsone to transfer the differentiation from the Green's func-tion (D to the test function fk. The _T operator can now berewritten in a symmetric form as

SkI( ) = ilf ds I ds{wpfk (r) f1(r')

- -[V fk(r)][V' f(r')]}c1(r - r'). (A3)COS

-301-

S 0 60 120 1809 (deg)

(b)

240 300 360

Fig. 10. Bistatic scattering cross section of penetrable (Sr =1.75 - j0.3) prolate spheroid (a = 0.1A, b = 0.2A): (a) 06 polar-ization, (b) /it polarization. Spherical conducting core(r = 0.05A) centered and offset by 2r = 0.1A.

I

..........------- Offset SpherecCentered Sphere

l I I I l I I a I I a I l a I l

l t J K . . . .

Eve

. . . . . . . . . . . . . . . . .

-Z . - - - - - - - - - - - - - - - - -. I I I I I I I I . I I .

.1 ------------------------------------


_

-15 _

l)

I

IA1


The indices k and I refer to the kth and the Ith edges onthe surfaces Si and S2, respectively. The factor of -q isincluded to account for the factor that is in the magnetic-current expansion and the factor that multiplies theH-field integral equations. The elements of axC) are givenas follows:

Xkl( = oJffJ dsif ds'fk(r) f 1(r') X PD(r - r'),

where

VcD(r - r) = (1 + jk Jr - r) -),4,Tr- r 3 ( 'ep-k

= - (r - r')"I(r - r').

Substituting for VF, we obtain

14--

(-44

0 30 60 90 120 150 180* (deg)

)Fig. 11. Bistatic scattering cross section of multiple agglomer-ated conducting spheres (r = 0.4A): (a) 0 polarization, (b) +0polarization. MM solutions incorporating radiative coupling.

The double surface integrals over pairs of triangularfacets can be written in terms of generic integrals,35 whichcan be integrated numerically. The singular part of theGreen's function can be extracted and integrated analyti-cally.37 For a given pair of triangular facets these genericintegrals can be used to compute the £ matrix contri-bution to nine pairs of edges forming the sides of thetriangular facets. This facet-based approach is used tominimize the number of double surface integrals thatmust be computed.

2. C OperatorThe Galerkin operator x((S,, S2; R) is the matrix that weobtain from the integral operator K in Eq. (20) by forminginner products between testing functions on Si and K op-erating on expansion functions on S2. An element of thematrix can be written explicitly as

Wlfkz(S1, S2; R) = 77o(fk, K(fD)), (6-

(a)

(-4

-0 30 60 90 120 150 180* (deg)(b)

Fig. 12. Bistatic scattering cross section of multiple agglomer-ated penetrable spheres (r = 1.75 - j.3, r = 0.4A): (a) 0 po-larization; (b) 4a polarization. MM solutions incorporatingradiative coupling.

1-

--

(A7)

(a)

(A8)


(A6)


(a)

-1

-0 30 60 90 120 150 180* (deg)O

Fig. 13. Bistatic scattering cross section of multiple agglomer-ated penetrable spheres ( = 1.75 - jO.3,g2 = 2.25 - jO.5,r = 0.4A): (a) 66 polarization; (b) d4 polarization. MM solu-tions incorporating radiative coupling.

Xk0l() = -Lko i dsIf ds'fk(r) fi(r') (r - r')T(r - r')

- 174 Z i A^P~z9 JTpf ds f ds'p= q- AkA k I1

X (r - vkP) (r' - v1q) X (r - r')P(r - r'). (A9)

The above double surface integrals can be written in termsof some simple generic integrals3 5 if the triple product isrewritten as

(r - r') - (kP x vjq) + (r X r') - (kP - vlq).

3. a OperatorThe Galerkin operator XP(Sl, S2; R) is the matrix ob-tained from testing the integral operator 7: x L thatarises in a CFIE formulation involving coated conductors.The vector A: is a unit normal vector to the surface Si.An element of the matrix can be written as

TkIL(S1, S2 ; R) = (fk, A xL(fi))s,

= - ( x fk,L(fC)s,. (All)

The indices k and I refer to the kth and the th edges onsurfaces Si and S2 , respectively. The elements of x.kl()

are as follows:

x.kI ( ) = -j L ds I ds{olA X f(r) f(r')CP(r - r')

+ -[ii X f(r) * VD(r - r')]V'- fir)Cos

(A12)

Splitting the surface integrals into two triangular facetintegrals and substituting for the testing and the expan-sion functions, we have

k2kl( ) = 'jk ' E E pq r ds ds4 p=± q=± AkPA * ( lq -

X {(Opql X (r - v) (r - Vjq)(D(r - r')

+ -[n X (r - v?) V(r - 0)].Cos

(A13)

It can be shown that, for a given triangular face T,

f fdsn X (r - v) VL = -adt - (r - v)cF, (A14)

where the line integral is around the boundary of T with aunit vector talong the boundary. Therefore the xf opera-tor can be written as

- jIkil E ~pq[ 1q t dsi ds'71/k -= 4 =± Ak 1 [ 'JJTP -' T~qx (r - VkP) (r' - vlq)(r - r')

-2 J dt ds'T (r - v?)cF(r - r)]. (A15)

The first term can be computed in terms of the samegeneric integrals as the 2 operator. The second term,however, requires that additional integrals be computed.

4. ,JC OperatorThe Galerkin operator 1f(S1 , S 2; R) is the matrix ob-tained from testing the integral operator n: x K, whicharises in a CFIE formulation. The unit vector n: is nor-mal to surface Si. An element of the matrix can be writ-ten as

(A10)

Note that the double surface integral above is zero whenTI'P = T q. Alternative forms of the W operator can be de-veloped in terms of surface and line integrals that allowthe singular part to be extracted.

x.Wk(S1, S2; R) = ?lo(fk, x K(f)s,

= -71o(n X fkK(f))s 1 . (A6)

The indices k and refer to the kth and the th edgeson surfaces Si and S2 , respectively. The elements of



xXkl() are as follows:

xffkl() = -710 ff dsff ds'[R X f(r)] f(r') X VF(r - r')

= 710ff dsff ds'[A X f(r)]

-fi(r') X (r - r')T(r - r')

= 71otktz E pq dsf ds'

X X (r - kf)] (r'- vq) X (r - r')T(r - r'). (A17)

For the numerical implementation it is advantageous torewrite the triple product as a sum of four terms, with thevertex points being extracted from the r and the r' depen-dence. The triple product is written as

[A X (r - V)] ( - vq) X (r - r')=n* r (r X r') - [(: X vkP) X Vlq] (r - r')

- ( X v) (r X r') - q [(A X r) X (r - r')].(A18)

The double surface integrals can be computed in terms ofthe same generic integrals as the X operator, along withsome additional integrals.

5. OperatorThe Galerkin operator MF(S) is the matrix obtained fromtesting the current on surface S. This operator arises ina CFIE formulation and when resistive or impedance sur-faces are present. An element of the matrix can be writ-ten as

2fk(S) = 2 (fi, fl~s -2

(A19)

The indices k and I refer to the kth and the th edges onthe surface S. Note that 2Ckl(S) will be nonzero only whenthe kth and the Ith edges border a common triangularfacet on the surface. The elements of 2/(S) are given asfollows:

2/k ( ) = ns dsfk(r) f(r)

aOEkl pq i8 p=± q=± A PA I nTq

RskI*k() = if dsR8 (r)fk(r) f(r)2 S, s

= 770lkl1 E pq Rlf ds8 p=± q=± AkpA1 q JJl T, q

X (r - vk) (r - vjq), (A21)

where Rlj is the value of R, on triangle Tkp. The matrixGASH is obtained from the above equation, with R? beingreplaced by G'.

APPENDIX B: EXPRESSIONS FOREXCITATION VECTORS

In an MM solution we obtain the right-hand-side vector ofthe matrix equation by forming inner products betweenthe testing functions and the incident fields on a surfaceS. We define the column vectors %(S; k, a) and 2(S; k, a)to be the vectors obtained from testing the incident fieldsEinc and 71oHinc, respectively, on the surface S. The vectork is the incident plane-wave propagation vector, and aspecifies the polarization of the incident E field in spheri-cal coordinate components (i.e., a = 0 or *. Elements ofthese column vectors can be written explicitly as

W (S; k, a) = (fk, Ea ) S

(Bi)= (fk(r), & exp(-jk r))s,XJk(S;k, a) = (fk,71oHafc)S

= (fk(r),f iexp(-jk- r))s,

where the subscript k refers to the kth rooftop function(edge) on the surface S. The vector f3 in the H-field innerproduct is the polarization of the incident H field (i.e.,k = IkIJ& X 1A). Note that the W vectors can be written interms of the W vectors as

W2(S; k, 0) = - (S; k, ), (B3)

W2(S; k, ) = W (S; k, 0). (B4)

Substituting for the test function in Eq. (Bi), we can write

Wk(S;k, a) =k 2 P2 p=± Al

X . * [ff ds(r - vP)exp(-jk- r)J, (B5)

ds(r - VkP) (r - . where p refers to the two faces forming the edge (seeFig. 3). The polarization vector and the propagation

(A20) vector k can be written out in terms of the sphericalangles (, ¢), which specify the incident direction

(B2)

The surface integration over the triangular facet can ei-ther be approximated by evaluation of r at the centroid ofthe facet or be evaluated analytically.

For resistive and/or impedance surfaces the variables R8and G, in Eq. (54) are assumed to be constant over eachtriangle. The matrices RS5 and GC are simply formalrepresentations for the matrix that we obtain by includingR, or G, in the surface integration. If R, or Qs were con-stant over the entire surface, then the elements of SC wouldall be multiplied by R, or G,. In the general case we have

0 = cos(0)[cos(ck)x + sin(O)yj - sin(0);6,

= -sin(k)2 + cos((P)Y7,

k = k[sin(0)cos(44^ + sin(0)sin(Gy)9 + cos(0)SI, (B8)

so that

exp(-jk r) = exp(jk{sin(6)[x cos(4 + y sin(o)]

+ z cos(0)}). (B9)

(B6)

(B7)



The surface integral over a triangular facet in Eq. (B5)can either be approximated with the centroid of the facetor be evaluated analytically. 3 1

Solutions that use a CFIE formulation involve testingA: X Einc or X Hin, where A: is the outward unit surfacenormal vector. We define the two corresponding columnvectors ., and ,X/ as follows:

.S(S; k, a) = (f%,A: X EalC)s,

./(S; k, a) = (fk,71oA: X H,!inc s.

(B10)

(B11)

These vectors can be computed with the same analyticintegrals as in the ' vector. Written explicitly,

.t(S; k, a) =2 E A P X )

*[ff ds(r - v)exp(-jk r)]. (B12)

The ,2X column vectors are related to the ,, column vec-tors as in Eqs. (B3) and (B4).

ACKNOWLEDGMENT

This research was conducted under the McDonnellDouglas Aerospace Independent Research and Develop-ment Program.

REFERENCES

1. M. Born and E. Wolf, Principles of Optics (Pergamon,New York, 1965).

2. N. A. Logan, "Survey of some early studies of the scatteringof plane waves by a sphere," Proc. IEEE 53, 773-785 (1965).

3. P. W Barber and C. Yeh, "Scattering of electromagnetic wavesby arbitrarily shaped dielectric bodies," Appl. Opt. 14, 2864-2872 (1975).

4. D. S. Wang and P. W Barber, "Scattering by inhomogeneousnonspherical objects," Appl. Opt. 18, 1190-1197 (1979).

5. P. W Barber and H. Massoudi, "Recent advances in light scat-tering calculations for nonspherical particles," Aerosol Sci.Technol. 1, 303-315 (1982).

6. M. A. Morgan, ed., Finite Element and Finite DifferenceMethods in Electromagnetic Scattering (Elsevier,Amsterdam, 1990).

7. A. Taflove and K. R. Umashankar, "Review of FDTD numeri-cal modeling of electromagnetic wave scattering and radarcross-section," Proc. IEEE 77, 682-699 (1989).

8. P. C. Waterman, "Scattering by dielectric obstacles," AltaFreq. 38, 348-352 (1969).

9. M. F. Iskander, S. C. Olson, R. E. Benner, and D. Yoshida,"Optical scattering by metallic and carbon aerosols of highaspect ratio," Appl. Opt. 25, 2514-2520 (1986).

10. A. Lakhtakia, M. F. Iskander, and C. H. Durney, 'An iterativeextended boundary condition method for solving the absorp-tion characteristics of lossy dielectric objects of large aspectratio," IEEE Trans. Microwave Theory Tech. MTT-31, 640-647 (1983).

11. A. C. Ludwig, "The generalized multipole method," Comput.Phys. Commun. 68, 306-314 (1991).

12. R. C. Hansen, ed., Geometric Theory of Diffraction (Instituteof Electrical and Electronics Engineers, New York, 1981).

13. P. Y. Ufimtsev, 'Approximate computation of the diffractionof plane electromagnetic waves at certain metal bodies," Soy.Phys. Tech. Phys. 27, 1708-1718 (1957).

14, P. Y. Ufintsov, "Mothod of edgo waves in physical thoory of

diffraction," Foreign Tech. Div. Doc. ID FTD-HC-23-259-71,(U.S. Air Force System Command, Wright-Patterson AirForce Base, Ohio, 1971).

15. N. A. Logan, "General research in diffraction theory," Reps.LMSD-288087 and LMSD-288088, (Lockheed Aircraft Cor-poration, Burbank, Calif., 1959).

16. R. F. Harrington, Field Computation by Moment Methods,2nd ed. (Krieger, Malabar, Fla., 1982).

17. E. K. Miller, L. N. Medgyesi-Mitschang, and E. H. Newman,eds., Computational Electromagnetics Frequency DomainMethod of Moments (Institute of Electrical and ElectronicsEngineers, New York, 1991).

18. L. V Kantorovich and G. P. Akilov, Functional Analysis inNormed Spaces (Pergamon, Oxford, 1964), pp. 586-587.

19. L. V Kantorovich and V I. Krylov, Approximate Methods ofHigher Analysis (Wiley, New York, 1964).

20. B. G. Galerkin, Vestn. Inzh. Tekh. 19, 897 (1915).21. Ref. 16, p. 18.22. V H. Rumsey, "The reaction concept in electromagnetic

theory," Phys. Rev. B 94, 1483-1491 (1954).23. A. J. Poggio and E. K. Miller, "Integral equation solutions

of three-dimensional scattering problems," in ComputerTechniques for Electromagnetics, R. Mittra, ed., 2nd ed.(Hemisphere, New York, 1987), pp. 159-264.

24. J. A. Stratton, Electromagnetic Theory (McGraw-Hill,New York, 1941), Chap. 8, pp. 424-481.

25. J. R. Mautz and R. F. Harrington, "H-field, E-field, and com-bined field solutions for conducting bodies of revolutions,"Arch. Elek. Ubertrag. 32, 157-164 (1978).

26. P. L. Huddleston, L. N. Medgyesi-Mitschang, and J.M.Putnam, "Combined field integral equation formulation forscattering from dielectrically coated conducting bodies,"IEEE Trans. Antennas Propag. AP-34, 510-520 (1986).

27. J. M. Putnam, "General approach for treating boundaryconditions on multi-region scatterer using the method ofmoments," presented at the Seventh Annual Review ofProgress in Applied Computational Electromagnetics,Monterey, Calif., March 1991.

28. T. B. A. Senior, 'Approximate boundary conditions," IEEETrans. Antennas Propag. AP-29, 826-829 (1981).

29. R. J. Garbacz, "Bistatic scattering from a class of lossy dielec-tric spheres with surface impedance boundary conditions,"Phys. Rev. A 133, 14-16 (1964).

30. K. M. Mitzner, 'An integral equation approach to scatteringfrom a body of finite conductivity," Radio Sci. 2, 1459-1470(1967).

31. D. S. Wang, "Limits and validity of the impedance boundarycondition for penetrable surfaces," IEEE Trans. AntennasPropag. AP-35, 453-457 (1987).

32. S. M. Rao, D. R. Wilton, and A. W Glisson, "Electromagneticscattering by surfaces of arbitrary shape," IEEE Trans.Antennas Propag. AP-30, 409-418 (1982).

33. J. Meixner, "The behavior of electromagnetic fields at edges,"IEEE Trans. Antennas Propag. AP-20, 442-446 (1972).

34. L. N. Medgyesi-Mitschang and J. M. Putnam, "Electro-magnetic scattering from axially inhomogeneous bodies ofrevolution," IEEE Trans. Antennas Propag. AP-32, 797-806(1984).

35. J. M. Putnam L. N. Medgyesi-Mitschang, and M. B. Gedera,"CARLOS-3D : Three-dimensional method of momentscode," McDonnell Douglas Aerospace Rep. (McDonnellDouglas Aerospace, Saint Louis, Mo., 1992), Vols. 1 and 2.

36. J. M. Putnam and M. B. Gedera, "CARLOS-3D TM: A general-purpose three-dimensional method-of-moments scatteringcode," IEEE Antennas Propag. Mag. 35, 69-71 (1993).

37. D. R. Wilton, S. M. Rao, A. W Glisson, D. H. Schaubert,0. M. Al-Bundak, and C. M. Butler, "Potential integrals foruniform and linear source distributions on polygonal andpolyhedral domains," IEEE Trans. Antennas Propag. AP-32,276-281 (1984).

38. K. McInturff and P. S. Simon, "The Fourier transform oflinearly varying functions with polygonal support," IEEETrans. Antonnas Propag. 39, 1441-1443 (1991).


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Generalized method of moments for three-dimensional penetrable scatterers

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