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Fast Radar Cross Section (RCS) Computationvia the Fast Multipole Method

Levent GürelDept. of Electrical and Electronics Engineering

Bilkent University

TR-0653 Bilkent, Ankara, Turkey19urel@ee.bilkent.edu.tr

1. SUMMARY of these geometries may generate extremely large nu-

Tl f t It . I th d (FMM) . d r th merical problems when translated into the languageýe M mu ýpo e me o IS use ýar e . .

f t ad t . (RCS) t t . f I of computatýonal electromagnetýcs. For these rea-as r ar cross sec ýan compu a ýan o arge . ... .

. -,. t d . . -,. t .. b th h sons, ýt lS ýmperatýve to develop efficýent and fastcanOnlC<L! wo- ýmensýon<L! geome rýes ýn o omo-, .'

d I d d . N , aL It ' f a.lgorýthms for computatýona.l electromagnetýcs,geneous an ayere me ýa. umerýc so u ýan o

electromagnetic scattering problems are invaluable in Maxwell's equations and the electromagnetic wavemany real-life applications, However, real-life prob- equation derived therefrom can be solved using ei-lems transiate into very large numerical problems ther partial-differential-equation (PDE) or integral-and when accurate solutions are desired, the size equation (lE) techniques, The most popular PDEof the problem that can be solved using traditional solvers of the computationa.l electromagnetics are thetechniques is limited by the given computational re- finite element method (FEM) and the finite-differencesources. The FMM overcomes this difficulty with time-domain (FDTD) method, PDE solvers gen-its reduced computational complexity and memory erally require that the problem space be finite, al-requirement, Thus, larger problems can be solved though several extensions have been devised in theusing the same computational resources and without form of absorbing boundary conditions (ABCs) andsacrificing the accuracy of the solution, The FMM hybrid methods to be able to use the PDE solvers inemployed here uses diagonalized translation opera- problems that are defined in unbounded media, Ontors for the two-dimensional Helmholtz equation, the other hand, lE techniques, such as the method

It ' h th t th FMM o

O(N l.5 ) ofmoments (MoM), fast multipole method (FMM)ýs s own a e requýres opera- d ' T . I .

h ( ), o, ,an recursýve -matrix a gorýt ms RTMAs, havetýons per ýteratýon as opposed to the matrýx-vector th ad , t ' d ' t . b ' It ' t th ' f 1 t O

1 ' 1' t ' b ' O(N 2 ) t ' Ot e r ýa ýan can 1 ýan uý ýn o elf or mu a ýonsmu týp ýca ýan eýng an opera ýan per 1 era- . . . .

t . d th d . t 1 t . h . O(N 3 ) and, therefore, can easýly sýmulate scatterýng and ra-ýan an e ýrec so u ýan avýng compu .

t . al L O I add .t O t th t . t dýatýon ýn unbounded medýa.ta ýan comp exýty. n 1 ýan o e es ýma es

of computational complexities and memory require-ments of these methods, their actual performancesare alsa compared and discussed. Solutions of sev-

eral large two-dimensional canonical conductor ge- (a)ometries are used in these comparisons, where theexcitation is a TM-polarized incident plane wave. ...

Through these comparisons, it is demonstrated thataccurate RCS computations of large geometries are

possible using the FMM.

2. INTRODUCTION (b)

Numerical simulation and analysis of the electromag- Fig. 1. (a) One-dimensional and (b) two-dimensional

netic scattering phenomena are used in several engi- clustering of strips.

neering applications to gain important informationabout a system belüre it is built, thus saving time In this work, electromagnetic scattering from twO-and other resources. For instaýýce, simulations of dimensional canonical conducting strip geometries inelectromagnetic scattering from stealth vehicles are homogeneoýýs and layered media will be analyzed. As

by far faster and cheaper than first manýýfacturing examples of canonical geometries, Figs. l(a) and l(b)these vehicles and then performing scattering mea- display the one-dimensional and two-dimensional clus-surements on them. terings of conducting strips in a homogeneous medium.

During a design process, it is desiralýle, and in some Canonical problems are important since tlýey dis-cases necessary, to numerically simulate and analyze play the performances of the solution techniques on

several different geometries. Furthermore, each one the class of geometries theyare representing, with-

Paper presented at the AGARD SPP Sympasium on "Radar Signature Analysis and lmaging of Military Targets",held in Ankara., Turkey, 7-10 Dctaber 1996, and published in CP-583.

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out Iýaving to apply the solution technique under ory requirement exhaust computational resoýýrces be-t.est to ea.ch member geometry of the class. Arbi- fare the relatively larger and more interesting prob-trary geometries can be viewed as special cases of lems can be solved. A general solution scheme withtlýe müre general canonical configurations. For in- reduced computational complexity and memory re-stanc;e, if a c:omputational technique can handIe the quirement is essential for problems comprising verycase of touching strips as in Fig. 2(a) and tilted strips large numbers of unknowns.as in Fig. 2(b), then it can be argued that the same .t ] . I dý b.t t . d The number of unknowns can ýndeed become verycc ýnýque can ýan c ar ý rary geome nes ma e up. .; ý:: .f t h. t ' lt d t . . F. 2( ) C . I large for real-Iýfe applýcatýons. Assume that a. Li) m

o ouc ýng ý e s nps as ýn ýg. c. anornca . . . .

l I .11 b d t th ffi . . d long conductýng cylýnder wýth a radýus of 3 m mod-proo ems wý e use o compare e e cýencýes an . . .th . f . I t . I . th els the fuselage of an aýrcraft. if thýs geometry ýs

e accuracýes o vanolis so u ýon a gün ms. '"dýscretýzed usýng square patches, whose sides are 0.1wavelength, and two unknowns are defined on each

--, patch t() model the two orthogonal components ofthe surface current, then the number of unknowns

(a) exceeds 300,000 at 1 GHz. If the frequency is in-creased to 20 GHz, then the number of unknowns

////// increases by 400 times to exceed 125,000,000! Thereis a clear need for algorithms with reduced compu-tationa! complexities and reduced memory require-

(b) ments in order to be able to solve this kind of largeproblems.

As a partial solution to the computational-complexityand the rnemory-requirement problems referred toabove, two recursive T-matrix algorithms (RTMAs)were developed earlier. [1-5] The RTMAs are recur-siye algorithms, however, they can be considered asdirect solvers for IEssince they share some of the nice

(c) properties of the direct solvers, such as generatingsollitiofis that are valid for multiple excitations, i.e.,

Fig. 2. (a) Touçhing strips forming a larger strip. right-hand sides. In the arena of iterative solvers,(b) Canonical configuration of tilted strips. (c) A the FMM has recently been develored. [6-10] Inmore complicated two-dimensional geometry composed this work, the FMM is applied to the solution ofof touching tilted strips. electromagnetic scattering problems involving two-

dimensional canailÝca! conducting strip geometries.The canonica! problem s illustrated in Figs. l(a),l(b), Both the induced current densities and the RCS ofand 2(b) are alsa important on their own account these structures are computed using FMM.since they or a slight modification thereof can be .considered as finite-size frequency-selective surfaces The efficýency and the accuracy of the FMM are(FSS). demonstra~ed thro~gh com~arisons w~th two r~fer-

en ce solutýons. Fýrst, a dýrect solutýon technýqueThe class of problems that are of interest in this employing Gaussian eliminatÝaD is used to sol ve thework can usua!ly be formulated using integral equa- resulting matrix equation. Then, the same matrixtions (lE) and solved by matrix solvers af ter con- equation is solved by using an iterative scheme, name-verting the integral equations to matrix equations, ly, the conjugate gradient squared (CGS) method,e.g., via the method of moments (MoM). The matrix in which every iteratÝaD involves two matrix-vectorbolvers can be eith-er direct solvers, e.g., Gaussian multiplications.eliminatÝaD, or iterative solvers, e.g., conjugate gra-dient (CG) method.

3. INTEGRAL-EQUATION FORMULATIONA problem comprising N unknowns requires O(N3)operations with a direct solver and O(N2) opera- Consider, for simplicity, that an Ey-polarized planetions per iteration with an iterative solver. If the wave is incident on the two-dimensional geometriesnumber of iterations of the iterative solvers grows as of Figs. 1 and 2. Since the ta.ngential component ofO(N), then the overall computational complexity of the total electric field must vanish on the conductors,th~~ itc-!rative solvers becomes O(N3), same as the di- we have

rect solvers. Both direct and iterative solvers require J SO(N2) memory locations for a problem of size N. Ey(p) + Ey (p) = O, P E S, (1)

Despite all of the advantages of lE solutions, their where S denotes the combined surface of all the con-O(!f3) computational complexity and O(N2) mem, ducting strips in the problem. The scattered field is

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given by operaticýýs, The FMM works in such an iterativek . ~he~e and replaces the matrix-vector multiplication

E;(p) = -~ 1 dp' H~l)(klp - pll)Jy(p'), (2) Z.a(ý) with a ~ore intelligent algorithm that requires4 s less than O(N2) operations per iteration.

where Jy(p) is the induced current densityon thestrips, This current density can be expanded using 4. MULTIPOLE EXPANSIONS ANDN bclSis functions bn(PJ and Nunknown coefficients ADDITION THEOREMS

,', '(I'n, ~,'e":, Cylindrical and spherical harmonic wave functions

;; N are solutions of the wave equation in two dimensionsi' Jy(p) = L bn(p)an' (3) and three dimensions, respectively. Thus, any solu-,: n=l tion of the wave equation cp(r), can be expanded in"Combining Eqs. (1)-(3), weobtain terms of a vector of outgoing wave functions'f/J(r) as

k N cp(r) = 'f/Jt(r) . f, (ll)

~ L f dp' H&l)(klp - p'l)bn(p')an = E;(p), , . . .4 n=l ls provýded that r ýs outs1de the smallest sphere (cýr-

pES. (4) cle) cont~ining the sources, if r is inside the largestsphere (cýrcle) that extends out to the sources, then

Equation (4) isa representation of the boundary con- cp(r) can be expanded in terms of the regular part ofdition that needs to be satisfied on the conductor 'f/J(r) assurf~ces. In order to impose the boundary condition teverywhere on the strips, testing functions t,n(P) can cp(r) = !Rg'f/J (r) . e. (12)

be d~ned on the strips and inner products of these '.'" .t t .' f . . h b h . d f E (4 ) b The multýpole expansýons gýven m the above are valýdes ýng unctýons wýt ot Si es o q. can e ., ..

c' d b ' for both two-dýmensýonal and three-dýmensýonal waveconstructe to o tam. . ., functýons ýf we mterpret r as

c, N..

~~L 1 dptm(p) 1 dp' H~l)(klp-p'l)bn(p')an r = xx + yy +.ýz in three-dimensional space,n=l S S (13)

=hdPtný(P)E~(P)' m=1,2".. ,N. (5) r=p=xx+zz intwo-dimensionalspace. (14)

The above can be expressed as a matrix equation The vectors 'f/J(r) and !Rg'f/J(r) of the cylindrical har-

Z . a - e (6) monic wave fullttiofis in the two-dimensional spaceNxN Nx1 - Nx1, 1. . I . bare exp ýcýt y gýven y

where'f/J(r) !Rg'f/J(r)

Zm,ý = ~4'1J fdpt"ý(P) f dp'H&l)(klp-p'l)bn(p') 1 1ls ls (7)

and H~ld(kp) e-i2.p J_2(kp) e-i2.p

em = f dptm(p)E~(p). (8) H~ll(kp) e-ic/ý J-ý(J.:p) e-i.pls '

The N x N lÝnear system ofequations in Eq. (6) can H~l)(kp) Jo(kp) (15)

be solved using a direct or an iterative method. The (ý). i.p . i.psolutions through direct inversion, i.e., Hý (kp) e Jý (kp) e

--1 H(I) (kp) ei24> J, (kp) ei2<f>a = Z . e, (9) 2 2

and through direct factorization of the matrix Z re-quire O(N3) operations. Iterative solutions are based

th ... t . f ' d I t We define a global coordinate system in which a co-on e mýnýmýza ýon o a resý ua error vec or ordinate vector is represented by r. Furthermore,

r(i) = e - Z. a{i) (10) assuming that there are N scatterers (or Nsubscat-terers) in the geometry, a local coordinate system for

through several iterations, where a(i) represents the each of the N scatterers is defined. Figure 3 showsith guess to the solution, Thus, the iterative so- only two of the N scatterers, namely, the 'ith and thelutions employ one or more matrix-vector multipli- ith ones. Also shown in Fig. 3 is the global coordi-cations per iteration, each of which requires O(N2) nate system,

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given by.\:;:::::::;;::::==:: ~ rj ' H(l) (kp) einý~j = ", ,1 ri P/2

\;::::::::;;::::~~.- .,..",,","ccc L J,,(kpi) ei"tPi H,~;~n(kaij) ei("ý-,ý)Vi,j

r O. O ,ý=-P/2 if P . < a'i j . '.1 ,.

P/2

Z L H~l)(kpi) ei,ý"'i .Jm-,ý(kaij) ei(ný-1')Vi;j

y Lx ,ý=-P/2 if Pi > ~;~;

Fig. 3. Global coordinate system and two of the N local J"ý(kPj) eimtPj =

coordinate systems. P/2

L Jn(kpi) eintPi J",-n(kaij) ei(1n-n)1/Jij. (24)

n=-P/2A coordinate vector associated with the ith scatterer

is represented by ri. This vector can alsa be written Comparing Eqs. (24) an~(25) to Eqs. (21)-(23), the

in terms of r and rj as element s of the aij and fJij matrices are found as

[ a' , ] = H(l) ( ka , ) ei(1n-n}1/Jij ( 25 )ri = r + aDi (16) 'J nm m-n 'J ,

= rj + aji ,

. [:l:i"ij] = Jm-,ý(kaij)ei(m-n)Viij. (26)where aDi and aji are translation vectors from the nm

ith local coordinate system to the global and the jth

local coordinate systems, respectively. Note that the summations in the addition thearemg,

which are infinitely long in principle, are truncated

Translational additiontheorems [11-13] for the cylin- to keep approximately P terrngo

drical and spherical wave functions will be used to

shift the origin from one scatterer to another. The

translation vector from the jth scatterer to the ith 5. FAST MULTIPOLE METHOD

~atterer is denoted by aij. We will use aij and The FMM is a relatively new method that is devised

fJij matrices as the translation matrices that express for the fast solution of integral equations in general

the expansion of the wave functions in the jth co- and partial' differential equations (PDEs) converted

ordinate system in terms of the ith coordinate sys- into integral equations. The FMM is a fast algorithm

tem. In other warrls, if 1jJ(ri) and ~g1jJ(ri) are the to compute the fields due to an arbitrary source dis-

column ve ct or s containing the cylindrical (or sp her- tribution at a set of predetermined points in space.

ical) wave functions and their regular parts, respec- By formulating the solution of an integral equation

tively, then the translation relations are as follows in an iterative scheme, where one or mor e such field

(fo~ rj = ri + aij): calculations are performed at each iteration, it be-

comes possible to reduce the complexity of the solu-

1jJt(rj) = ~g1jJt(ri) . aij if Iril < laij!, tion compared to traditional techniques. [6-8]

(17) An integral equation may be converted to an N x N

= 1jJt(ri) . :l:i"ij if Iril > laij I, matrix equation by applying a discretization method,

(18) e.g., method of moments. The resultinglinear system

~i1jJt(r ) = ~g1jJt(ri) .:Bi .. (19) of equations may be solved us ing a direct method,.Y J J e.g., Gaussian elimination, at the expense of O(N3)

F h h r ll . 1 t . . t th operations or an iterative technique, e.g., conjugateurt ermore t e ýO owýng re a ýons exýs among e. . .

-. -' . - . gtadýents, that requýres one or more matrýx-vector

o,) and fJ'J matrýces (for a,) - aýk + aký).L . 1 .. f 1 . t O(N 2 ) t h . tmu týp ýcatýons o comp exý y a eac i era-

- tion. The direct matrix-vector multiplication(s) atai,] = (JiJ., . aký if laikl < lakjl, (20) each iteration may be replaced by the FMM, which

= "iÝ"ik . :BkJ if laik! > lakjl, (21) has a complexity lower than O(N2). Both computa-

:Bi = :Bik .:Bk . (22) tional and memory-requirement complexities of the.1 J various FMM implementations are between O(N)

and O(N3/2), depending on the physical problem to

In two dimensions, the addition theorems (17)-(19) be solved and the specific variant of the FMM used.

for cylindrical harmonic wave functions are explicitly The FMM achieves this performance by grouping all

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the unknowns into clusters formed on the basis of Note that the vector Pk represents the observationphysical (geometrical) proximity and by manipulat- point and is defined in the local coordinates of "theing the fields of cl us ters instead of the fields of in- kth substrip in the Kth cluster" Similarly, th~ vec-dividual unknowns. In this context, the terms un- tür P; represents the source point and is defined inknowns, subscatterers, substrips, and basis functions the local coordinates of the ith substrip iff.the Lthare used analogously and interchangeably. This is be- cluster. Then, we havecause a scatterer (strip) caff be decomposed into sev-eral~maller subscatterers (sub~trips). Asubscatterer H~l)(klp - p'r) =can ";be made just as big to define a basis function H(l) lk l( + d. , ) + d, + (d - '

)1] ( 30). d hb ' f .. . td .th o Pk kI\ I\L LI Pý ,on 16 an eac asýs unctýon ýs assocýa e Wl an

,unkqown coefficient. Alternatively, large and com- where the vector dkK points from the center of theplica;ted geometries can be thought of as being con- ](th cluster to the reference point of the kth substripstrU<;ited from smaller pieces, where each smaIl piece in that cluster. Similarly, dLI points from the refer-is associate d with an unknown. en ce point of the Ith substrip in the Lth cluster to

the center of the same cluster. In order to simplifyClus/er L Cluster K the derivation, two new vectors will be defined as

~~~~ ~~ ~~ ~~O:'~~ ~~~=/ ~~" " /'-::;~;:;~~~ ~", PkK = Pk + dkK, '(31), A -- ,

i i , - - - , I d I

(32)i L d i' K ' PLI= LI-Pý.i~ KL i' ~ i i' "" k i

\ ý:ý . l?::l-"'"" = \ I d -~ R '" d - 'i kK ký. .. .", , u --:- ,", -~ ~ii Usýng the addýtýon theorems twýce, we have

~~~,~.,.- ~~~~~~' ;-.::~~~=~=~~~ H~l)(klp - p'l) = !Rg'f/Jt(Pk](). <Ý'](L. !Rg'f/J(pu),

\- p' ./"""'~~~-- r (33)" O y@-x

~!ic where

Fig. 4: Two of the many possible clusters in a repre- [ R9'f/J(PkK )] = J",(kpkK )ein,q,kX (34)sentatýve geometry. '"

The N unknowns in the problem are clustered in and

such a way that a total of M clusters are formed and [<Ý'K L] = (-1)m+n H(l) (kd]( L)e-i(m+n)q,K L.the number of unknowns in the Lth cluster is n(L). mn m+" (35)

Thus,M The regular wave functions in the above can be re-

N = L n(L). (27) placed by their integral representations

L-l .lo- Jrn(kPkK )e\m'l'kK =

Althoug~ the number of unknow~s in each cluster 1 2". ~can be dýfferent, as the problem sýze N gets larger, - ( dct> e\kpkK COS(q,-q,kK )+\n,(q,- 2) (36)

each cluster's population tends to be N/M. Figure 4 271" lo

depicts two of the many possible clusters in a rep- andresentative geometry:' The two clusters are labelledas ]( and L and the vectors PK and PL point to the Jm(kpu)eirnq,~ý =centers of these clusters. AIso shown in Fig. 4 are 2"the A:th unknown (subscatterer) in the ](th cluster 2-1 dct>' eikp~ý cos(q,'-q,~ý)+im(q,'-t) (37)and the ith unknown (substrip) in the Lth cluster. 271" o

Central to the success of the FMM is the use of the Substituting first (36) and (37) in (34), and then (34)addition theorems (see the previous section) to trans- and (35) in (33), replacing ýn + n with p, and using

'Iate the fields from one cluster to another. Referring the identityto Fig. 4, the field at point P due to aline source at 00

point p' is defined as the Green's function and given L eim(4>-q,') = 271"c(ct> - ct>'), (38)

by m=-oo

G(p, p') = -TH6l)(klp - p'I), (28) \ýle obtain

2"where H~l) (kip - plI) = 2-1 dct> eikpkK COS(q,-ýPkK)

I d d d I 271" oP - P = Pk + kI( + ]( L + LI - Pt i i

(29) Q](L(k)e\kPLýcos(q,-q,LI), (39)

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where Substituting (42) in the above, we obtain

pal\.L(k); L H(l)(kdKL)eip(4>-4>KL+j), (FF) - k1} '\:"""' '~( 1 i iJ=-P p (40) YKk - S; L., L., l,Ý;dPk tl(k(P;',) ,dpý bLI(PI)

i LE:F(I() 1=1 S ::;

Note that the infinite summation over p is truncated. 12" d ik, (P +d . )- ( k. (d - P' )A.e k kl\ G:I .L k )e' Li i X (48)If we define tlýe wavevector k as 'i' \ . LI,

o

k ; k( x cas ci> + z sin ci» ( 41) w hich can be rearranged to read

and use the definitions of PkK and Ptl as given in (FF) - k1} {2" (f' ik.p )(31) and (32), then Eq. (39) can be rewritten as YKk - S; 10 dcl> S dPk tl\.k(Pk)e ~

1 (2" ,,(L)H~1)(klp-p'I);2;lo dcl>e,k'(Pý+dkK) e,k.dkK LOKL(k)Le'k'dLI

o LE:F(I() 1=1

OI(L(k)eik'(dLI-P;) (42) ,(i dp; bLý(pi)e-ik'PI) XLI, (49)

As depicted in Eq. {lO), iterative solution techniquesemploy one or more matrix-vector multiplications per If BLý(k) and TKk(k) are defined as the Fourier trans-iteration. In electromagnetic problems, these matrix- forms of bLI(P) and tl(k(p), respectively, then thevector multiplications can be expressed as above can be rewritten as '

- k 12" Y = Z. x, (43)Y(FF) ; -.!J.. dA.T*. (k )eik'dkKKk 87r 'i" I\k

- owhere the matrix Z results from the discretization ,,(L)of an electromagnetic problem as in Eq. (7) and the L OKL(k) L eik.dLI BLý(k)XLI. (50)

vector x is an arbitrary coefficient vector. Using the LE:F(K) 1=1clustering concept, the above can be rewritten moreexplicitly as . .

It has been shown that the computat10nal complexýtyM ,,(L) of the FMM as presented in the above is O(N M) +

YKk = L L ZKk,LýXLI, O(N2 IM). [6-8] This complexity is minimized byL=1 1=1 choosing M cx VN, so that the computational com-

K;1,2,.."M, k;1,2"..,n(K). (44) plex~tyoftheFMMbeco~esO(N3/2).,Thememoryrequýrement of the FMM ýs alsa O(N3/2).

Let F(K) denote the set of all cl us ters that are in thefar-field region of cluster K. Then, the summation 6. COMPUTATIONAL RESULTS FORin (44) can be split into far-field and near-field parts HOMOGENEOUS-MEDIA PROBLEMSas In this section, in order to demanatýate the accu-

n(L) n(L) racy and the efficiency of the FMM, we will presentYlýk ; L L ZKk,LIXLI + L L ZKk,LIXLI some sample RCS results, The term RCS will be

LE:F(K) 1=1 Li:F(K) 1=1 used for the relevant bistatic scattering coefficient,- . ~ ~ - ..~ ~ which is O'yy(cI» in the case of TM scattering fromFar Fýeld (FF) Near F1eld (NF) two-dimensional geometries. This bistatic scattering

(45) coefficient is defined as

In the above, the summation in the near-field partI Es (p -+ 00, CI» 1 2 .

is performed as the multiplication ofasparse matrix O'yy«/»; 27rp Y E! for EI; yE;.

with the coefficient vector. On the other hand, the Y (51)

far-field part can be rewritten using Eq. (7) ci.') Th RCS I f h. . b . d re res u ts o t ýs sectýon are o ta1ne ýor ancý) TM (to y) plane-wave illumination. This TM plane

y}{"F); L L ZI(!."LýXLl (46) wave is incident at 45° as measured from the X axis,LE:F(I() 1=1 The RCS results presented in this section are ob-

krý n(L) ( tained using three different schemes: direct solution= 4 L L LÝ; dptKk(P) with Gaussian elimination, iterative solution with

LE:F(K) 1=1 S ordinary matrix-vector multiplication, and iterativej ' i (1) i i (47) solution with the FMM. The numerical results ob-

s dp Ho (kip - P l)bLI(p )XLI' tained using these three different schemes agree with

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50.0 50.0

40.0 40.0

30.0 30.0

~ 20.0 ~ 20.0

tl) ci)~ 10.0 ~ 10.0

OJ) 0.0

.1~O -10.0

.20:0 .20.0; 0.0 30.0 6~.0 90.0 120.0 150.0 180.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0; Angle or Renection Angle or Reneclion"

Fig. 5. RCS of a single 10-). strip. Fig. 7. RCS of five 10-). strips that are vertically sepa-rated by 10 )..

50.0ID).. ID).. ID).. ID).._ý -. -+ =1 05 '\ 40.0t 5 ).. -- ).. =-. i\.

lO).. - ==1 -+ 30.0- - -~ 20.0- ~

'o, ci)

--;'!tý:- - ~ 10.0

,~",.' 0.0

--r..", .10.0

l'!- .20.0i 0.0 30.0 60.0 90.0 120.0 150.0 180.0

Angle or Renection

Fig. 6. Five 10-). strips that are vertically separated by10 )., 5 )., )., 0.5 ).. and 0.1 ). (not shown). Fig. 8. RCS of five 10-), strips that are vertically sepa-

rated by 5 )..

each other for several digits and, therefore, are indis- 50.0tiguishable on the RCS plots, thus testifying to theaccuracy of the FMM. 40.0

Figure 5 shows the RCS of a single 10-). strip. Il- 30.0

lustrated in Fig. 6 are different confýgura.tions of ver- ~ 20.0

tically spaced fýve 10-). strips. Figure 7 shows the ~~ 10.0

RCS of fýve 10-). strips that are separated by 10 ).in the vertical direction. In this case, the largest 0.0dimension of the geometry is 40 ). and the RCS of -10.0this structure (Fig. 7) is dramatically different thanthat of a single 10-). strip (Fig. 5). Similarly, Fig. 8 .20.00.0 30.0 60.0 90.0 120.0 150.0 180.0shows the RCS of fýve 10-), strips that are separated Angle of Renectionby 5 ).. Once again, in this case, the vertical di men-sion of the geometry, which is 20 )., is mu ch larger Fig. 9. RCS of five 10-). strips that are vertically sepa-than the horizontal dimension, which is 10).. Fig- rated by 1 )..ures 9, 10, and ll, on the other hand, show the RCSof structures whose vertical dimensions are 4 )., 2 ).,and 0.4 )., respectively. Therefore, these RCS resuls to RCS computation of larger problems. Figure 12bear some resemblance to the RCS of a single 10-), illustrates fýve geometries that are made up of onestrip, especially in the neighborhood of the specular to fýve 100-), touching strips. Figures 13-17 showreflection directian, which is 135°. Particularly, the the RCS of these geometries. These RCS results areRCS of fýve 10-). strips that are separated by 0.1). also obtained using the refence solution techniques(Fig. ll) is al most the same as the RCS of a single as explained earlier in this section. It is seen thatstrip (Fig. 5), except near the grazing directions. the FMM results agree with the refence solutions for

. several digits.In order to stretch the test for the accuracy and theefficiency of the FMM, we have applied the FMM Figure 18 depicts the comparison between the com-

6-8

50.0 70,0

40.0 60.0

30.0 50.0

~ 20.0 ~ 40,0

'" '"~ 10.0 ~ 30.0

0.0 20.0

-10.0 10.0

-20.0 0.00.0 30.0 60.0 90.0 120.0 ]50.0 180.0 0.0 30,0 60.0 90.0 120.0 150.0 ]80.0

Angle of Renection Angle of Renection

Fig. 10. RCS of five 10-), strips that are vertically sep- Fig. 13. RCS of a single 100-), wide strip.arated by 0.5)..

70.0

50.060.0

40.050.0

30.0 ~~ 40.0~ ~

~ 20.0 (j~ ~ 30.0'"~ ]0.0

20.0

0.0]0.0

-10.00.0

0.0 30,0 60.0 90.0 120.0 150.0 ]80.0-20.0 A I f R n I '0.0 30.0 60,0 90.0 120.0 ]50.0 ]80.0 ng e o e ec ýon

Angle or Renection

Fig. 14, RCS of two touching 100-), wide strips.Fig. 11. RCS of five 10-), strips that are vertically sep-arated by 0,1 ).. 70.0

60.0

] OOA. 50.0LL

~ 40.0

'"~ 30.0

i 20.0

]0.0

0.00.0 30.0 60.0 90,0 120.0 ]50.0 ]80.0

] OOA. ] OOA. ] OOA. ] OOA. lOOA. Angle of Renection

F ' 12 G ' f f ' h. 100 \ ' d Fig. 15. RCS of three touching 100-), wide strips,ýg, , eometrýes o one to ýve touc ýng -/\ WL e

strips,

t . . d b th FMM d th d . computationaI complexity for the MVM and FMM,pu er týmes requýre y e an e or mary t . I. I . I. . (MVM) 1" h .t respec ýve y.matrýx-vector mu týp ýcatýon ýar eac ý era-tion of the iterative method, which is the conjugate Figure 19 shows the comparison among the memorygradient squared (CGS) method in this case. Both requirements of the CGS employing the FMM, CGSthe horizontaI and the vertical axes of Fig. 18 are employing the ordinary matrix-vector multiplication,scaled Iogarithmically, Thus, the sIope of the curve and the direct solution. It is seen that the sIope ofthat is plotted with respect to these axes is equaI to the FMM curve is smailer than both the MVM andthe ord er of computationaI complexity of these meth- the direct-solu tion curves. This is in agreement withods. Clearly, the FMM curve has asmaller slope the predictions of O(N3/2) memory requirement forthan the that of the MVM curve. This is in agree- the FMM and O(N2) memory requirements for thement with the predictions of O(N2) and O(N3/2) MVM and the direct solution,

,6-9

70.0 . CGS-FMM O CGS-MVM . Dir. Inv.60.0 -, 1000 ::::::~::::::::::::::

.. ,.", "' . ... ..."'.".'.".'."SO.O

~ 40.0

tl)~ 30.0 11\

~1.0.0 ~

~ 100-:10.0 ~" 11\

:11.0 50.0 30.0 60.0 90.0 120.0 lS0.0 180.0 ~

Angle of Renection :-;:

~Fig. 16. RCS of four touching 100-), wide strips.

70.0 1 O ::::.~::::::.:::~:.:.:::::::.

60.0 .8 .Q.Q 8 8 8 8 888 oo o o o o o o 000 oso.o ~ N M.qo u") <o ý- 000)0 o

~ N

~ Number of Unknowns~ 40.0tl)~ ~0.0 Fig. 19. Comparison of the memory requirements of the"io.o CGS employing the FMM, CGS employing the ordinary

:(' matrix-vector multiplication.. and the direct solution.;16.6...'";0.0i: 0.0 30.0 60.0 90.0 120.0 lS0.0 180.0 . CGS-FMM o CGS-MVM . Dir. Inv.,: Angle of Renection 100000 ::::::::::::::::::::::::

Fig. 17. RCS of five touching 100-..\ wide strips. 10000 ~::~~

1000 ::::::::::::::::::::::.::~~~::i:::::~::=::ý:: ::::::::::::::::::::::::: i :'~~~:.:.'.:

::::::::::::::::::::::~::::::::::::::!::::::::::i:::::::::i:::::t:::j:::+:::j:::: :::::::::::::::::::::::::: ~ ; ;...;...; ;... ,.. {1)

t"..!.""r..ii...j.i.y'" "; 1000 i 1~ :::~ ~ 100 --=

! :::::::::: :~:~::::::::::::

~ ~ 10 :::::::::.::

~ 10 .Q o o o o o o 0° 8 8~ ~ o 8.Q o o 8 8~U :::::::::::::::::: - Ri M:q: g g ý- 00a;8 8- N

::::::::::::::: Number of Unknowns

Fig. 20. Comparison of the total solution times required1 by ~he CGS ~mploying the. F~M., CGS emplo~ing the

.Q 8 8.Q 8° 8 888 8 ordýnary matrýx-vector multýplýcatýon, and the dýrect so-o 0000000 o I . F.ii'.' .

Idd- N M.qo u") <o ý- ooc>~ Ri utýon. i ýng týme ýs not ýnc u e .

Number of Unknowns

Fig. 18. Comparison of the solution times per iteration Finally, Fig. 20 shows the comparison among totalrequiJed by the FMM and the ordinaJY matrix-vector solution times taken by these three solution tech-multiplication. niques. Matrix filling times are not included for

any of the three methods. The FMM has the small-est slope, thus the lowest computational complexity,among the three solution techniques.

6-10

7. LAYERED-MEDIA PROBLEMS L~

Numerical solution of electromagnetic radiation and f, Jlscattering problem s involving layered media have gain- Y x

ed popularity due to the need to computationally - -analyze variolis important geometries, e.g., absorber fý, JlI'

coatings to reduce RCS and structures placed on orin material subs~rates, such as microwave integrated f2, Jl2

circuits (MICs), printed circuit boards (PCBs), and

the vast class of microstrip-like structures. Numer-ical analysis and simulation of these structures are f_" Jl-,

needed for both functional considerations and electro- 'PEC""""""""""""""""'"

magnetic-compatibility (EMC) issues.

The formulation of layered-media problems have tra- (a)

ditionally been carried out in the spectral domaindue to the availability of the Green's functions in

L., closed form s [14,15], Recently, a series of techniques f, Jl Y x

have been developed to obtain closed-form Green's

functions (CFGFs) for layered media in the spatial ~--~ domain {16,17], The use of the CFGFs in a method- " " . .of-moments (MoM) formulation replaces the numer- .. ..

ical computation of the improper integrals in the f, Jl

spectral domain with numerical integ rations over fi- .. "

. " '. .nite region s in the spatial domain, Furthermore, the

spatial-domain integrals can be evaluated analyti- .., , . .,cally in some cases [18]. Thus, this approach reduces .. o ,

the matrix-filling time by several orders of magni-

tude compared to the spectral-domain formulation, (b)

However, it does not reduce the computational com-. . . . .1 Ot f th t . fill ' t O d th Fýg. 21. (a) Orýgýnal problem ýn a layered medýum.

p exý y o e ma rýx- ýng ýme an e memary . "

. t h , h b th O(N 2 ) M t O (b) Equývalent problem wýth DCIs ýn a homogeneousrequýremen , w ýc are o o os ýmpor- d .t tl d . h .. h ' fill ' me rum.an y, espýte t e great savýngs ýn t e matrýx- ýng

time, the solution of the N x N dense matrix equa-

tion remains, which requires O(N3) operations in a substrate below this plane,

direct scheme or O(N2) operations per iteration in U o b h . b Ak [ ] ( )' t t ' h sýng a ro list tec nýque y sun 18, Eq. 52 canan ý efa ýve sc erne.

be converted to a closed-form expressian given by

By obtaining a closed-form expressian for the spatial-

domain Green's function for an arbitrarily layered G( i) = -~[ H(ý)(kl - ii)medium and by interpreting each ter m of the expres- P, P 4 o P P

sion as a discrete complex image, the FMM can be Nýapplied to the solution of layered-media problems. + L anH~ý)(klp - p~ý)

] , (53)Thus, a fast solution technique for the layered-media n=l

problems can be obtained and the applicability of theFMM can be extended from homogeneous-medium where ip - p~i = y'(x - X')2 + (z + z' + ion)2 and

problem s to layered-medium problems, an and An are complex constants for n = 1,2, . . . ,Ný.

The FMM is based on the expansion of the Green's8. FMM FOR LAYERED-MEDIA function using the addition theorems and no such

PROBLEMS expansions exist for the layered-media Green 's func-

The y-directed electric field at point p = xx + zz ti~ns given in Eqs, (52). and (53). ~owever, we can

due to a y-directed line sourc~ with unit amplitude stilI ,emplay the ,~MM ~n the solution of a layer:d-

located at point p' = xx' + Iz' is given by medrum problem ýf we ýnterpret Eq. (53) as the lýn-

ear superposition of the field due to a source at p'G(p,p') = -~ j (X) dk", eik"(x-,,,') and the fields due to Ný discrete complex images

. 47r -(X) (DCIs) located at p~ = xx' + z( -z' - ian)' The

.-!-.[eik.ýz-z'l + flTEeik.(Z+Z/) ] , (52) DCIs are located at co~~lex coordinates and, there-

kz fare, we need to use addýtion theoTeýýýs for wave func-L : R-TE' h I , d fL ' ffý ' d tions with complex arguments.

,w ýere ýs t e genefa ýze re ectýon coe cient e-

fined at the z = O plane due to an arbitrarily layered With the DCI interpretation, an eq~ivalent prob-

6-11

lem is set up in a homogeneolis medium. In this implementation of the FMM. The numerical resultsequivalent problem, which is illustrated in Fig. 21(b), obtained using these three different schemes agreeNý image sources in a'"homogeneous medilim are de- with each other for several digits and, therefore, arefined corresponding to each originaÝ. source in the indistiguishable on the plots presented in Fig. 23.layered-medium problem [Fig, 21{a)J.. Thus, if Ntesting functions are.defined. on the origina! cond uc- c -ð :

~~ U ~ U ~ ~ U ~ U ~ ~~ ~ ~ U U tors, N (J\Tý + 1) basýs functýons aredefined on the ~'2 2~""

origiq:al conductors and their imag~s. U~ i

, oTh '~ t t . f h fi Id f N(N 1) b . 0.0 P .. o.s J O. i Se ~ompu a ýon o t e e s o 1+ asý~ ""ýtýon on the A,'rtly (Wtývelength,,) .

functfons on the N testing functions is carried out (a)

using': the FMM and repeated several times in an .. H

~ U U U U U U U U U U U U U U U Yiterailve scheme. Since N i is a constant, this spe- ~~ :. ~""cific i'mplementation of the FMM for layered-media U~ 2problems has O( N3(2) computationa! complexity per (~).o P'. o.s 1.0 1..'1iteratÝaD and O(N3(2) memory requirement as its o"ýtýon on th(~I.rtlY (WtIVeleI1gth,,)

homogeneous-medium counterpart. c-ð:

~ ~ U V U VUU U U~ ~~ ~UU U

~.22

~""

EO, Jl,o U~ JO

0.0.. o.s 1.0 ý.sPo"'t'on "'0 the Aý.ray (Wavelength,,)

(a) (C)

" K

~ UUUUUUUUUUUUUUUy :{' c -ð 6 ,,'.1 ...~ 4

tL,. !3 6j,

E~; i'i; U~2

. O,~7""""""""",,"""""""'" 0.0 P .. o.s h A W 1.0 I.Sri "'"ýtý,.,n on t e rray ( avelength,,)

~EC (h) (d)

~;EO Jl,o Fig. 23. Current distributions on the array correspond-

, ing to the example structures of Fig, 22.

Eý, = 4, Jl,oEO, Jl,o (c) The magnitude of the curr~n~ distribution ~btained

on the array when the array ýs ýn free spa,ce [Fýg. 22( a)]E Jl, is shown in Fig. 23(a). An infinitely large conductingo, -~- -- - - -- - - - - - - -- plane placed Ao/100 away from the array [Fig. 22(b)]

Eý, = 4, Jl,o causes the magnitude of the current distribution to"""""""""""""""""""" increase as see n in Fig. 23(b). When the array is

PEC (cl) placed on a Ao/100 dielectric slab with Er = 4 as

in Fig. 22(c), the magnitude of the current distribu-Fig. 22. Examples of layered structures. tion, shown in Fig. 23(c), is seen to be modified and

increased, but not as much as that of the conducting-plane case. If the dielectric slab is backed by a co n-

9. COMPUTATIONAL RESULTS FOR ducting plane as depicted in Fig. 22(d), the currentLAYERED-MEDIA PROBLEMS magnitude becomes higher as seen in Fig. 23(d). In-I d t d t t th d th ffi deed, Fig. 23(d) can be compared to Fig. 23(b) ton or er o emons ra e e accuracy an e e-. . .. f thl d d;' I t t . f th con{:lude that the conductýng plane ýs more domýnant

cýencyo e ayere -me ýa ýmp emen a ýan o e .FMM t d . th ' I t . f t t than the relatývely thýn dýelectrýc slab ýn determýn-

presen e ýn ýs e ter, a serýes o s ruc uýes . . . . .( . 11 t d . F . 22) h. h h h d ýng the current dýstrýbutýon. However, by comparýngas ý us rate ýn ýg. , to w ýc t e met o can. . . . .b I. d h b d . d C t II th Fýgs. 23(a) and (c), ýt ýs easy to see that the dýelectrýc

e app ýe ave een esýgne. ommon o a ese . . .

'... slab has a sýgnýficant effect on the current dýstribu-

structures ýs an ýrregular, finýte and planar array of t . . th b f h d . It . h. h h II t t f 1 5 \ PL ýan ýn e a sence o t e can uctýng p ane.s rýps, w ýc as an overa ex en o . 1\0. anewaves, whose electric fields are polarized in the y di- The discretization of the conducting array shown inrection and have unit amplitudes, are incident on Fig. 22(a) results in 105 basis and testing functions.the structures at 45° as measured from the posi- This array is duplicated many times in ~he layeredtive .T axis. Electromagnetic scattering problem s for geometry of Fig. 22( d) to obtain problems that areall four structures are solved using three different ten times as large. Separating the solution and fill-schemes: direct solution with Gaussian eliminatÝaD, ing times, we have compared the solution times of theiterative solution with ordinary matrix-vector multi- FMM and the traditional solution techniques. Fig-plication, and iterative solution with layered-medium tire 24(a) compares thesolution time of the FMM

6-12

';J;' 200 3. L. Gürel and W. C. Chew, "A recursive T-matrix algo-"o / rithm for strips and patches," Radio,Sci., vol. 27 pp .387-

c auss. ým. '

8 150 . FMM 401, May-June 1992.~ 100 4. W. C. Chew, L. Gürel, Y. M. Wang, G. atta, R. Wag-Q) ner, and Q. H. Liu, "A generalized recursive algoritlým.5 50 for wave"scattering solutions in two dimensions," lEEEfoo; Trans. Microwave Theory Tech., vol. MTT-40, pp. 716-

~ O 723, April 1992.U O 200 400 600 800 1000 1200 5. W. C. Chew, Y. M. Wang, and L. Gürel, "Recursive

Number of Un known s algorithm for wave-scattering solutions usiýýg windowed

(a) addition theorems," J. of Electromagn. Waves Appl.,vol. 6, no. ll, pp. 1537-1560, Nov. 1992.

,-. 4 O 6. V. Rokhlin, "Rapid solution of integral equations of scat-~ 3'

5 tering theory in two dimensions," J. Comput. Phys.,c _.o 3 O vol. 86, pp. 414-439, Feb. 1990u ..~ 2.5 7. R. CaifIDan, V. Rokhlin, and S. Wandzura, "The fast';;' 2.0 multipole method for the wave equation: a pedestrian

.5 :.g prescription," lEEE Antennas and Propagation Maga-~ 0:5 zine, vol. 35, no. 3, pp. 7-12, June 1993.

ýl.. 0.0 8. C, C. Lu and W. C. Chew, "Fast algorithm for solvingU O 200 400 600 800 1000 1200 hybrid integral equations,' Proc, lEE, vol. 140, Part H,

Number of Unknowns pp. 455-460, Dec. 1993.

(b) 9. R. L. Wagner and W. C. Chew, "A ray-propagationfast multipole algorithm," Microwave Dpt. Tech. Lett.,

Fig. 24. (a) Comparison of the solution times of vol. 7, no. 10, pp. 435-438, July 1994.

the layered-medium implementation of the FMM and 10. C. C. Lu and W. C. Chew, "A multilevel algorithm forthe Gaussian elimination. (b) Comparison of the per- solving boundary integral equation of scattering," Mi-.t t . I . . f h I d d. . I crowave Apý. Tech. Lett., vol. 7, no. 10, pp, 466-470,i eya ýon so utýon týmes o teayere -me ýum ýmp e- July 1994.mentation of the FM M and the ordinary matrix-vector .

i . I. . ll. B. Frýedman and J. Russek, "Addition theorems for spher-mu týp ýcatýon. ical waves," Quart. Appl. Math., vol. 12, no. 1, pp. 13-

23,1954.

12. M. Danos and L. C. Maximon, "Multipole matrix ele-to the Gaussian elimination and Fig. 24(b) compares ments of of the translation operator," J. Math. Phys.,the CPU time required during a single iteration of vol. 6, pp, 766-778, May 1965.

the FMM to the ordinary matrix-vector multiplica- 13. W. C. Chew, "Recurrence relations for three-dimensionaltion, respectively. As for the filling time, since only scala~ ad~ition theorem," J. Electromagnetic Waves and

. f fi Id . .. fill d Applýcatýons, vol. 6, no. 2, pp. 133-142, 1992.a sparse matrIx O the near- e ýnteractýons ýs e. th FMM d t iili . N N d 14. T. ýtolý, "Spectral domain immittance approach for dis-m e as oppose o mg an x ense . lý .. f I. d . . I. ".. . . .. persýon c aracterýstýcs o genera ýze transmýssýon. ýnes,matrIx ýn the dýrect solutýon, iilIýng týme of the FMM lEEE Trans. Microwave Theory Tech., vol. MTT-28,

is always lower. pp. 733-736, July 1980.

15. W. C. Chew and L. Gürel, "Reflection and transmissionoperators for strips or disks embedded in homogeneous

ýo. CONCLUSIONS and layered rnedÝa," lEEE Trans. Microwave ..Theory

Th FMM . I. d h f RCS . f Tech., vol. MTT-36, pp. 1488-1497, Nov. 1988.e ýs app ýe to t e asý computatýon o

I . al t d. . I t . A . 16. Y. L. Chow, J. J. Yang, D. F. Fang, and G.. E. Howard,arge canonýc wo- ýmeýýsýona geome rýes. n m- " A I d f t ' al G ' f t . f h h. k. .. cose - orm spa i reens unc ýon or t e t ýc

tegral equatýon based on the two-dýmensýonal scalar microstrip substrate," lEEE Trans. Microwave TheoryHelmholtz equation is solved to compute the RCS of Tech., vol. MTT-39, pp. 588-592, Mar. 1991.sample geometries of conducting strips in both homo- 17. M. i. Aksun, " A robust approach for the derivation of

geneolis andlayered media in order to demonstrate closed-formGreen'sfunctions," lEEE Trans. Microwavethe accuracy and the efficiency of the FMM. The Theory Tech., in press.

FMM is shown to have reduced computational com- 18. L. Alatan, M. i. Aksun, K. Mahadevan, and T. Birand,plexity and memory requirement. "Analytical eval~ation of the MoM matrix elements,"

lEEE Trans. Mýcrowave Theory Tech., ýn press.

REFERENCESACKNOWLEDGEMENTS

1. L. Giirel and W. C. Chew, "Scattering solution of three-dimensional array ofpatches using the recursive T-matrix This work was supported in part by NATO's Sci-algoritlýms," lEEE Microwave and Guided Wave Lett., entific Affairs Division in the framework of the Sci-vol. 2, pp. 182-184, May 1992 en ce for Stability Programme and in part by the

2. L. C~ürel. and W, C: Chew,. ':Recursive T~matrix algo- Scientific and Technical Research Council of Turkeyrý.thms wýth reduced comple.xýtýes for scatterýng from three- (TUBITAK) under contract EEEAG-163.dýmensýonal patch geometrýes," lEEE Trans. AntennasPropagat., vol. AP-41, pp. 91-99, Jan. 1993.~