1550 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 11, NOVEMBER 2001
On The Tes ing o he Magne ic Field In eg al
Equa ion Wi h RWG Basis Func ions in Me hod o
Momen s
Juan M. Rius, Membe , IEEE, Edua d Úbeda, and Josep Pa ón
Abs ac —Fo elec omagne ic analysis using Me hod o Mo-
men s (MoM), h ee-dimensional (3-D) a bi a y conduc ing su -
aces a e o en disc e ized in Rao, Wil on and Glisson basis unc-
ions. The MoM Gale kin disc e iza ion o he magne ic ield in e-
g al equa ion (MFIE) includes a ac o

0
equal o he solid angle
ex e nal o hesu acea he es ingpoin s,which is
2
e e ywhe e
on he su ace o he objec , excep a edges o ips ha cons i u e
a se o ze o measu e. Howe e , he s anda d o mula ion o he
MFIE wi h

0
=2
leads o inaccu a e esul s o elec ically
small sha p-edged objec s. This pape p esen s a co ec ion o he

0
ac o ha , using Gale kin es ing in he MFIE, gi es accu acy
compa able o heelec ic ieldin eg alequa ion(EFIE), whichbe-
ha es e y well o small sha p-edged objec s and can be aken as
a e e ence.
Index Te ms—Elec omagne ic sca e ing, in eg al equa ions,
nume ical analysis, ada c oss sec ions.
I. INTRODUCTION
ELECTRIC and magne ic ield in eg al equa ions [2] a e
widely used in conjunc ion wi h Me hod o Momen s
(MoM) disc e iza ion [1] o he nume ical analysis o elec-
omagne ic adia ion and sca e ing. The MoM Gale kin
disc e iza ion o he elec ic ield in eg al equa ion (EFIE) o
h ee-dimensional (3-D) a bi a y su aces was p esen ed by
Rao, Wil on and Glisson (RWG) in he well-known e e ence
[3]. The linea iangle basis unc ion p esen ed in [3] will be
deno ed he e as RWG basis unc ion. An analogous o mula-
ion o he magne ic ield in eg al equa ion (MFIE) o 3-D
a bi a y su aces was la e de eloped [4].
As will be shown la e , he MFIE o mula ion includes a
ac o , equal o he solid angle ex e nal o he su ace a
he ield e alua ion poin . Since he Gale kin es ing in e-
g a ion poin s a e placed inside iangles, is equal o
a all he es ing poin s. The esul ing o mula ion p o ides
excellen esul s o smoo h geome ies [4]. Howe e , o small
sha p-edged objec s, a signi ican e o appea s in he MFIE
when compa ed o he MoM-EFIE. The las can be aken as a
e e ence since i is known o wo k well o hose objec s. The
Manusc ip ecei ed Augus 29, 2000; e ised Janua y 9, 2001. This wo k
was pa ially suppo ed by he Spanish Comisión In e minis e ial de Ciencia y
Tecnología (CICYT) unde P ojec TIC 98-1037. E. Úbeda and J. Pa ón we e
suppo ed by he Gene ali a de Ca alunya, Comissiona pe a Uni e si a s i Re-
ce ca, unde G an s 1997 FI 00747 and 1997 FI 00679.
The au ho s a e wi h he Depa men Teo ia del Senyal i Comunicacions
(TSC), Uni e si a Poli ècnica de Ca alunya, 08034 Ba celona, Spain.
Publishe I em Iden i ie S 0018-926X(01)07641-4.
aim o his pape is o dec ease he e o o he MoM-MFIE
o small sha p-edged objec s. Al hough his e o would be
smalle wi h a be e choice o he es ing unc ions se [6], [7],
we will es ic ou a en ion he e o Gake kin es ing unc ions.
Fo sha p-edged objec s o elec ically small dimensions, he
con ibu ion o basis unc ions loca ed a edges is pa icula ly
impo an , since he bounda y o many basis unc ions is on ob-
jec edges. The ac ha a ips and edges he solid angle ex-
e nal o he su ace is clea ly di e en han sugges s us ha
a be e alue o could be used o he es ing poin s on ian-
gles neighbo o edges. On he o he hand, o elec ically la ge
sha p-co ne ed objec s; 1) di ac ion a edges is less impo an
han e lec ion a su aces and 2) he numbe o basis unc ions
neighbo o edges is ela i ely smalle and as a consequence,
he sensibili y o he sca e ing compu a ions o he solid angle
choice is unno iceable.
In his pape a en ion is limi ed o sha p-edged objec s o
small elec ical dimensions. The aim is o de elop a co ec-
ion o he s anda d e sion o he MoM-MFIE by modi ying
he alue o he solid angle in o de o ob ain esul s simila o
he MoM-EFIE ones. Al hough he imp o emen will be shown
he e only o objec s o small elec ical dimensions, he alidi y
o he co ec ion is ex endable o all kinds o su aces, including
elec ically la ge ei he sha p-edged o smoo h- a ying ones.
In hese cases, since he esul s om he s anda d MoM-MFIE
we e al eady good, he co ec ion u ns ou impe cep ible.
II. FORMULATION: EFIE AND MFIE
Le us conside an a bi a ily shaped 3-D closed conduc ing
objec , wi h i s su ace disc e ized in nono e lapping iangles.
Fo RCS compu a ions, he inciden ield in he sca e ing
p oblem is an impinging plane wa e. The induced cu en is
expanded in RWG basis unc ions [3] as
(1)
whe e a e he unknown cu en coe icien s o be de e mined
in he solu ion o he p oblem.
The EFIE in a pe ec elec ic conduc o (PEC) su ace o ces
he elec ic ield bounda y condi ion
(2)
0018–926X/01$10.00 © 2001 IEEE
RIUS e al.: ON THE TESTING OF THE MAGNETIC FIELD INTEGRAL EQUATION 1551
whe e is he inciden ield and is he sca e e ed ield,
ha is, he ield due o he induced cu en s
(3)
whe e deno es he ee-space G een’s unc ion and he ime
dependence is implici .
The EFIE (2) is disc e ized using MoM [1] as a sys em o
linea equa ions ha can be exp essed in ma ix o m as [3]
(4)
whe e , . He e
deno es he Hilbe inne p oduc .
Simila ly, he MFIE is de i ed om he magne ic ield
bounda y condi ion
(5)
whe e deno es he uni no mal ec o o su ace. Again,
e e s o he inciden ield and he sca e ed ield is due o
he induced cu en s
(6)
Fo obse a ion poin s on he su ace o he sca e e , he su -
ace in eg a ion in (6) in ol es an imp ope in eg al ha mus
be spli in o wo de ini e pa s [2]
(7)
whe e deno es he Cauchy p incipal alue in eg a ion and
s ands o he solid angle ex e nal o he su ace, as de-
pic ed in Fig. 1.
The need o co ec ion in (7) is no appa en he e, since
e e ywhe e on he conduc ing objec excep a an
edge o a ip. Edges and ips ha e no su ace a ea and he e-
o e hey do no con ibu e o he su ace in eg al ha a ises in
he es ing p ocedu e. In addi ion, al hough he solid angle ex-
e nal o an edge o a ip is well de ined, he alidi y o (7) is
ques ionable a an edge o a ip because he cu en could
be ill beha ed and e en i was well beha ed, he quan i y
would be s ill di icul o e alua e [6].
A e he MoM disc e iza ion o (5) and (6), allowing o [4]
and de ining , and
, he MFIE (5) becomes
(8)
Fig. 1. Local solid angle.
The assump ion is o en made and (8) is simpli-
ied o he s anda d MFIE [4]
(9)
III. COMPUTATION
A. Sou ce In eg a ion
In o de o achie e he maximum accu acy in sou ce in e-
g als, he singula e ms o bo h ope a o s ( o EFIE and
o MFIE) a e analy ically in eg a ed ollowing [4]
and [5]. A e ex ac ion o he singula e ms, he emaining
onesa e slow- a yingand canbe nume icallycompu edconsid-
e ing a ou poin quad a u e ule on each iangle. Fu he mo e,
in he MoM-MFIE o mula ion he c oss-p oduc can be mo ed
ou o hein eg alassugges edin[4].Thisaccu a ecompu a ion
is e y impo an when analyzing elec ically small geome ies
in o de o ensu e unbiased alues o and .
B. Field In eg a ion
Whene e he ield and sou ce iangles a e he same, he
Gale kin es ing in eg als ha e been ca ied ou h ough a ou -
poin Gaussquad a u e ule. When he ieldand sou ce iangles
a e di e en , ei he ou o one poin s a e used in he quad a u e
summa ion o all iangle pai s, leading o wo se s o esul s o
di e en accu acy.
Figs. 3–7, espec i ely, show he bis a ic RCS o a
cubewi hsideo 0.1 and48 ace s,acubewi hsideo 0.2 and
48 ace s, a ec angula basis py amid wi h side 0.07 and 72
ace s, a egula oc ahed on wi h side 0.07 and 72 ace s, and a
cone wi h basis adius o 0.05 , heigh o 0.04 and 288 ace s.
In Fig. 7, he cone is o e disc e ized along he pola di ec ion
in o de o cancel he in luence o a coa se disc e iza ion o he
cu a u e. In all cases he impinging plane wa e has axial
incidence and -pola ized elec ic ield.
One ema ks on he disag eemen o he MoM-MFIE esul s
wi h he MoM-EFIE ones despi e he imposed high accu acy
o he compu ed sca e ed ields. Since he condi ioning o he
linea sys ems (4) and (8) is good, he only easons o he dis-
ag eemen mus be; 1) he inaccu acy o es ing in eg als and 2)
he imp ope choice o in he MFIE. The MoM-EFIE is e y
1552 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 11, NOVEMBER 2001
Fig. 2. Local solid angle es ima e.
Fig.3.
H
planebis a icRCS o a pe ec ly conduc ing cube wi hsideo 0.1

.
Fig. 4.
E
plane bis a ic RCS o a pe ec ly conduc ing cube wi h side o 0.2

.
Fig. 5.
E
plane bis a ic RCS o a pe ec ly conduc ing py amid wi h side o
0.07

.
Fig. 6.
H
plane bis a ic RCS o a pe ec ly conduc ing oc ahed on wi h side
o 0.07

.
obus sincei s esul s a e e ysimila when a ying henumbe
o ield es ing in eg a ion poin s and, u he mo e, he e is no
solid angle in he EFIE o mula ion.
On he o he hand, he MoM-MFIE esul s show a signi ican
dependence on he numbe o ield es ing in eg a ion poin s.
Indeed, when inc easing he o de o he Gauss quad a u e ule,
he MoM-MFIE esul s app oach slowly o he MoM-EFIE e -
e ence, bu s ill an impo an e o emains.
IV. MOM-MFIE CORRECTION
Ino de o educe heMoM-MFIE e o ,we p oposea co ec-
ion on he MoM-MFIE ope a o by sligh ly modi ying he solid
angle alue. By ial and e o , we ha e been able o de ine he
new equi alen solid angle alue any iangle as he weigh ed
RIUS e al.: ON THE TESTING OF THE MAGNETIC FIELD INTEGRAL EQUATION 1553
Fig. 7.
H
-plane bis a ic RCS o a pe ec ly conduc ing cone wi h basis adius
o 0.05

and heigh o 0.04

.
a e age o he local solid angles o e each edge o he iangle
(Fig. 2)
(10)
whe e , and a e he leng hs o he iangle edges.
One can eadily ob ain he new ope a o by eplacing by
in (8). This co ec ion app oaches he MoM-MFIE esul s
o he MoM-EFIE ones o any geome y.
Themodi iedMoM-MFIEope a o is e yaccu a e o all he
geome ies es ed as long as one quad a u e in eg a ion poin is
used in MoM Gale kin es ing. One can assess i s good beha io
o sha p-edged geome ies in Figs. 3–7. The accu acy is com-
pa able o he MoM-EFIE wi h ou quad a u e poin s, bu he
modi ied MoM-MFIE excels hanks o i s simplici y and as
compu a ion, since only one quad a u e in eg a ion poin is e-
qui ed.
V. CONCLUSION
The beha io o he MFIE and he EFIE o he analysis o
elec ically small sha p-edged objec s has been s udied. The
inaccu acy o he MFIE o his kind o objec s is assumed
o be caused by an inapp op ia e choice o he ex e nal solid
angle alue a ield e alua ion poin s inside iangles neighbo
o edges. In his pape , we p opose a co ec ion o he solid
angle alue ha allows o he ex e nal solid angles a iangle
edges. The esul ing app oach needs only one ield e alua ion
poin a each iangle and is as accu a e as he EFIE wi h ou
quad a u e in eg a ion poin s in Gale kin es ing.
REFERENCES
[1] R. F. Ha ing on, Field Compu a ion by Momen Me hods. New Yo k:
MacMillan, 1968.
[2] N. Mo i a, N. Kumagai, and J. R. Mau z, In eg al Equa ion Me hods o
Elec omagne ics. No wood, MA: A ech House, 1990.
[3] S. M. Rao, D. R. Wil on, and A. W. Glisson, “Elec omagne ic sca e ing
by su aces o a bi a y shape,” IEEE T ans. An ennas P opaga ., ol.
AP-30, pp. 409–418, May 1982.
[4] R. E. Hodges and Y. Rahma -Samii, “The e alua ion o MFIE in eg als
wi h heuseo ec o ianglebasis unc ions,”Mic owa eOp .Technol.
Le ., ol. 14, no. 1, pp. 9–14, Janua y 1997.
[5] D. R. Wil on, S. M. Rao, A. W. Glisson, D. H. Schaube , O. M.
Al-Bundak, and C. M. Bu le , “Po en ial in eg als o uni o m and
linea sou ce dis ibu ions on polygonal and polyhed al domains,”
IEEE T ans An ennas P opaga ., ol. AP-32, pp. 276–281, Ma . 1984.
[6] Re iewe ’s commen s.
[7] E. Úbeda, “Con ibu ion o he imp o emen o in eg al equa ion
me hods o pene able sca e e s,” Ph.D. disse a ion, Uni e si a
Poli ècnica de Ca alunya, Jan. 2001.
Juan M. Rius (M’89) ecei ed he Ingenie o de
Telecomunicación deg ee and he Doc o Ingenie o
deg ee om Poli echnic Uni e si y o Ca alunya
(UPC), Ba celona, Spain in 1987 and 1991, espec-
i ely.
In 1985, he joined he Elec omagne ic and
Pho onic Enginee ing g oup a UPC, whe e he
cu en ly is Associa e P o esso . Since 1989, he
has been engaged in he esea ch o new and
e icien me hods o nume ical compu a ion o
elec omagne ic sca e ing and an enna adia ion. He
de eloped he G aphical Elec omagne ic Compu a ion (GRECO) app oach
o high- equency RCS compu a ion, he In eg al Equa ion o mula ion o
he Measu ed Equa ion o In a iance (IE-MEI) and he Mul ile el Ma ix
Decomposi ion Algo i hm in 3D (MLMDA-3D). He has been a Visi ing
P o esso and CLUSTER Chai a EPFL (Lausanne); and a Visi ing Fellow
a Ci y Uni e si y o Hong Kong, China. He has 25 publica ions in e e eed
in e na ional jou nal pape s and 55 in in e na ional con e ences.
Edua d Úbeda was bo n in Ba celona, Spain, in
1971. He ecei ed he Telecommunica ion Enginee
deg ee and he Doc o Ingenie o deg ee om
he Poli echnic Uni e si y o Ca alunya (UPC),
Ba celona, Spain, in 1995 and 2001, espec i ely.
In 1996, he was wi h he Eu opean Commission
Join Resea ch Cen e , Isp a, I aly. F om 1997 o
2000, he was a Resea ch Assis an a he Elec o-
magne ic and Pho onic Enginee ing g oup a UPC.
Since 2001, he has been wi h he Elec ical Engi-
nee ing Depa men , Pennsyl ania S a e Uni e si y,
Uni e si y Pa k. He is au ho o h ee pape s in in e na ional jou nals and 11
in in e na ional con e ence p oceedings. His main esea ch in e es s include
nume ical compu a ion o sca e ing and adia ion using in eg al equa ions.
Josep Pa ón was bo n in Sabadell (Ba celona),
Spain in 1970. He ecei ed he elecommunica ion
enginee deg ee om he Poli echnic Uni e si y o
Ba celona, Spain, in 1994.
In 1997, he became a Ph.D. s uden in he Signal
Theo yandCommunica ion Depa men , Poli echnic
Uni e si y o Ba celona, and since 2000, he has been
an Assis an P o esso . He is au ho o i e pape s in
in e na ional jou nals and 12 in in e na ional con e -
encep oceedings. His main esea chin e es s include
as algo i hms o compu a ional elec omagne ics
and hei applica ion o ac al an ennas, mul ilaye ed mediums, and supe con-
duc o s.