Ad . Radio Sci., 22, 9–15, 2024
h ps://doi.o g/10.5194/a s-22-9-2024
© Au ho (s) 2024. This wo k is dis ibu ed unde
he C ea i e Commons A ibu ion 4.0 License.
A Nume ical Al e na i e o 3D Addi ion Theo ems Based
on he Bilinea Fo m o he Dyadic G een’s Func ion and
he Equi alence P inciple
Giacomo Gianne i1,zand Ludge Klinkenbusch2
1Depa men o In o ma ion Enginee ing, Uni e si y o Flo ence, 50139 Flo ence, I aly
2Ins i u e o Elec ical and In o ma ion Enginee ing, Kiel Uni e si y, 24143 Kiel, Ge many
zFo his wo k, Giacomo Gianne i ecei ed he Young Scien is Awa d om he Ge man URSI membe commi ee a he
Kleinheubache Tagung 2023.
Co espondence: Giacomo Gianne i ([email p o ec ed])
Recei ed: 31 Ma ch 2024 – Re ised: 30 July 2024 – Accep ed: 12 Augus 2024 – Published: 6 Sep embe 2024
Abs ac . A nume ical me hod based on he equi alence
p inciple and he dyadic G een’s unc ion is p esen ed. I can
be used o compu e he sphe ical-mul ipole ampli udes wi h
espec o an o igin in a subdomain 2 due o sou ces in a
dis inc subdomain 1. As an example, conside ha subdo-
main 1 con ains a ho n an enna ha is sol ed nume ically
using a comme cial ull-wa e simula o . The adia ed ield
se es as he inciden ield o subdomain 2 which con ains
he sca e e , in ou example a lossless dielec ic sphe e. The
p oposed me hod is based on he equi alence cu en s on a
Huygens su ace enclosing he an enna and uses he ee-
space dyadic G een’s unc ion o compu e he elec ic and
magne ic ields on a sphe e enclosing he sca e e . F om his
elec omagne ic ield on he sphe ical su ace, he sphe ical-
mul ipole ampli udes o he inciden ield wi h espec o he
cen e o he sphe e enclosing he sca e e a e ob ained nu-
me ically and can be u he p ocessed. The esul s ob ained
wi h his me hod a e compa ed o he esul s solely compu ed
by he nume ical ull-wa e simula o .
1 In oduc ion
Compu a ional Elec omagne ics plays a c ucial ole in elec-
ical enginee ing, wi h many applica ions in mic owa e
echniques, an ennas, and p opaga ion, among o he s (Da id-
son, 2010; Sumi h a and Thi ipu asunda i, 2017). Se e al
me hods ha e been de eloped o sol e elec ically la ge p ob-
lems, whe e he ypical size o he s uc u e exceeds ens o
e en hund eds o wa eleng hs. In his con ex , he case o ob-
jec s sepa a ed by a homogeneous backg ound medium is o
pa icula in e es and will be add essed in he ollowing.
A ypical app oach o sol ing such p oblems is based on
a physical domain decomposi ion, ha is, o i s sol e he
elec omagne ic p oblem in each o he subdomains sepa-
a ely and o subsequen ly ind he solu ion o he en i e p ob-
lem by le ing he subdomain solu ions in e ac h ough a
sui able coupling p ocedu e. The elec omagne ic p oblem in
each o he subdomains and he coupling can be sol ed by a
specialized nume ical me hod.
An example o a ela ed me hod can be ound in a e-
cen wo k by Losenicky e al. (2021), whe e he Me hod o
Momen s (MoM) and he T-ma ix app oach a e combined.
In pa icula , he e he MoM is used o sol e he adia o ,
an elec ic dipole, and he ields a e p ojec ed on a sphe e
enclosing he adia o o ob ain a mul ipole ep esen a ion.
This is hen used o ansla e he mul ipoles and sol e he
sca e ing p oblem in ano he subdomain. Howe e , his ap-
p oach equi es ha he subdomains a e enclosed by wo
non-in e sec ing sphe es; he e o e, his me hod does no al-
low ha ing he an enna and he sca e e in close p oximi y.
This aspec is mo e limi ing when he aspec a io o he an-
enna o he sca e e is la ge, e.g., an elec ic dipole which
is la ge in one dimension and small in he o he wo. O he
ins ances o ela ed app oaches a e ound in Alian and O aizi
(2018, 2019), whe e he addi ion heo em is combined wi h
he equi alence p inciple algo i hm (EPA) (Li e al., 2006;
Li and Chew, 2007) o sol e sca e ing p oblems wi h mul i-
Published by Cope nicus Publica ions on behal o he URSI Landesausschuss in de Bundes epublik Deu schland e.V.
10 G. Gianne i and L. Klinkenbusch: A Nume ical Al e na i e o 3D Addi ion Theo ems
ple PEC objec s. Again, sphe es enclosing he objec s which
mus no in e sec a e needed o apply he me hod. Addi ion-
ally, in Alian and O aizi (2018, 2019) a ields a e analyzed;
hence no adia o is included in he analysis.
In he p esen wo k, we ex end he 2D mul ipole app oach
we in oduced in Gianne i and Klinkenbusch (2023) o he
3D case. The me hod is based on he equi alence p inciple
and he ee-space dyadic G een’s unc ion. Two subdomains
a e conside ed: one con ains an an enna, and he o he a sca -
e e . The classical app oach is based on he addi ion heo-
em o ec o sphe ical-mul ipole unc ions (VSMFs) and
equi es he Huygens su ace enclosing he adia o o be a
sphe e. The me hod desc ibed he e allows one o enclose he
adia o wi h a Huygens su ace o a bi a y shape. Hence, i
is mo e lexible and he dis ances be ween he di e en sub-
domains can be educed. The ield adia ed by he sca e e
can hen be compu ed using a sphe ical-mul ipole expansion
cen e ed a subdomain 2 and a sui able sol e . The sca e e
can ha e an a bi a y geome y – hough he analysis na u-
ally simpli ies o a sphe ical sca e e , e.g. he model o he
human head as used in Losenicky e al. (2021).
The pape is o ganized as ollows. The p oblem is ou lined
in Sec . 2 while he p oposed me hod is desc ibed in Sec . 3.
Fi s nume ical esul s and some conclusions a e shown in
Sec s. 4 and 5, espec i ely.
2 Fo mula ion o he P oblem
Figu e 1 shows he p oblem and he used no a ions. The sca -
e e is enclosed by a sphe ical su ace o adius Rs. In he
phaso domain and a a ime ac o e+jω , he o al ield o
0≥Rsis spli in o an inciden and a sca e ed pa . The mul-
ipole expansions o he co esponding elec ical ields a e
gi en by (Klinkenbusch, 2008)
Ei( 0)=X
n,m
ai
n,mN(1)
n,m( 0)+Z
jX
n,m
bi
n,mM(1)
n,m( 0)(1)
Es( 0)=X
n,m
as
n,mN(2)
n,m( 0)+Z
jX
n,m
bs
n,mM(2)
n,m( 0), (2)
espec i ely. He e, Z=√µ/ε is he in insic wa e
impedance o he medium ( acuum is conside ed in he ol-
lowing) and Pn,m =PN
n=1Pn
m=−n,ai(s)
n,m and bi(s)
n,m ep e-
sen he mul ipole coe icien s o he inciden and sca e ed
elec omagne ic ields, espec i ely. The ec o sphe ical-
mul ipole unc ions (VSMF) N(q)
n,m( )and M(q)
n,m( )a e de-
ined by (Klinkenbusch, 2008)
M(q)
n,m( )=z(q)
n(k )mn,m(θ,φ) (3)
N(q)
n,m( )=− z(q)
n(k )
k n(n +1)Yn,m(θ,φ)ˆ
+w(q)
n(k )nn,m(θ,φ). (4)
Figu e 1. De ini ion o he p oblem: subdomain 1 (an enna), sub-
domain 2 (sca e e ), and co esponding no a ions.
He e, w(q)
n(k ) =− 1
k
d
d  z(q)
n(k ),ˆ is he uni ec o in
he adial di ec ion, and k=ω√εµ is he wa enumbe o he
homogeneous medium. The supe sc ip s used he e, (q) =(1)
and (q) =(2), indica e ha he adial dependence is gi en by
sphe ical Bessel unc ions o he i s kind (z(1)
n=jn) o by
sphe ical Hankel unc ions o he second kind (z(2)
n=h(2)
n),
espec i ely. No e ha sphe ical Bessel unc ions o he i s
kind a e egula e e ywhe e and mus be used o ep esen
egula ields a he o igin ( =0), while a he gi en ime
ac o only Hankel unc ions o he second kind comply wi h
he adia ion condi ion o →∞.
The ans e se sphe ical mul ipole unc ions (TSMFs)
mn,m(θ,φ) and nn,m(θ,φ) a e de ined as
mn,m(θ,φ) =− 1
sin(θ)
∂Yn,m(θ,φ)
∂φ ˆ
θ+∂Yn,m(θ,φ)
∂θ ˆ
φ(5)
nn,m(θ,φ) =∂Yn,m(θ,φ)
∂θ ˆ
θ+1
sin(θ)
∂Yn,m(θ,φ)
∂φ ˆ
φ, (6)
whe e ˆ
θand ˆ
φa e he uni ec o s along θand φ, espec-
i ely, and whe e he su ace sphe ical ha monics Yn,m(θ,φ)
a e de ined by
Yn,m(θ,φ) =s(n −m)!
(n +m)!
2n+1
4πPm
n(cos(θ))ejmφ.(7)
He e, Pm
n(cos(θ)) deno es an associa ed Legend e unc ion
o he i s kind. No e ha Yn,−m(θ,φ) =(−1)mY∗
n,m(θ,φ)
holds, wi h he as e isk indica ing he complex conjuga e.
In he absence o a sca e e , he sca e ed ield anishes
and he o al ield is iden ical o he inciden ield, which is in
his case also alid o < Rs.
In case a sca e e is p esen , he mul ipole coe icien s o
he inciden and sca e ed ields a e ela ed by he sca e ing
ma ix, which ully cha ac e izes he sca e e . Fo he sim-
ple case o a homogeneous iso opic dielec ic sphe e wi h
wa enumbe ksand in insic wa e impedance Zs, he sca e -
ing ma ix is diagonal, and he ela ions be ween he mul i-
pole coe icien s o he sca e ed and inciden ields a e ound
Ad . Radio Sci., 22, 9–15, 2024 h ps://doi.o g/10.5194/a s-22-9-2024
G. Gianne i and L. Klinkenbusch: A Nume ical Al e na i e o 3D Addi ion Theo ems 11
as
as
n,m
ai
n,m =−bs
n,m
bi
n,m
=w(1)
n(kRs)jn(ksRs)−Z
Zsw(1)
n(ksRs)jn(kRs)
w(2)
n(kRs)jn(ksRs)−Z
Zsw(1)
n(ksRs)jn(kRs)
.(8)
Mo eo e , he ield inside he dielec ic sphe e can be ex-
panded using he sphe ical-mul ipole expansion
Ein( 0)=X
n,m
ain
n,mN(1)
n,m( 0)+Zs
jX
n,m
bin
n,mM(1)
n,m( 0), (9)
wi h he mul ipole ampli udes
ain
n,m =Zs
Z
ai
n,mjn(kRs)+as
n,mh(2)
n(kRs)
jn(ksRs)
bin
n,m =bi
n,mw(1)
n(kRs)+bs
n,mw(2)
n(kRs)
w(1)
n(ksRs)
.(10)
3 P oposed me hod
Fi s , we apply he equi alence p inciple wi h a Huygens su -
ace ha comple ely encloses he an enna (Fig. 1). The e,
he equi alen cu en s, ha comple ely ep esen he an enna
ou side i s subdomain, a e gi en by (Balanis, 2012)
Jeq =ˆ
n×Ha(0), Meq =−ˆ
n×Ea(0), (11)
whe e 0is he Huygens su ace, and ˆ
n he uni ec o di-
ec ed ou wa ds (Fig. 1). The equi alen cu en s Eq. (11) a e
calcula ed om he ields deli e ed by a ull-wa e simula o .
The elec omagne ic ield ou side o he Huygens su ace
can be exp essed wi h espec o he coo dina e sys em o
subdomain 2 acco ding o (Li e al., 2006; Al a ez e al.,
2007; Quijano e al., 2011; Balanis, 2012)
E( 0)=−jkZLJeq( 00)−KMeq( 00)(12)
H( 0)=−jkY LMeq( 00)+KJeq( 00),(13)
whe e he in eg al ope a o s Kand La e de ined by
KIeq( 00)=Z0
∇0g( 0, 00)×Ieq( 00)dS00 (14)
LIeq( 00)=Z0
G 0, 00·Ieq  00dS00.(15)
He e, Ieq is ei he Meq o Jeq, and 0and 00 a e he obse -
a ion and sou ce poin s, espec i ely, desc ibed in he co-
o dina e sys em o he sca e e (Fig. 1). In Eq. (14), he ∇0
ope a es on 0, and he scala ee space G een’s unc ion is
gi en by
g 0, 00=e−jk| 0− 00|
4π| 0− 00|.(16)
The ee-space dyadic G een’s unc ion G( 0, 00)in Eq. (15)
is
G 0, 00=I+1
k2∇0∇0g(R)
= 3
k2R2+3j
kR −1ˆ
Rˆ
R
+1−j
kR −1
k2R2Ig(R), (17)
whe e I ep esen s he uni dyadic, R= 0− 00,R=|R|, and
ˆ
R=R/R is he uni ec o poin ing om he sou ce poin o
he obse a ion poin . Addi ionally, we ha e
∇0g 0, 00=−jk −1
Rg(R) ˆ
R.(18)
E alua ing Eq. (12) on he sphe ical su ace enclosing he
sca e e , we ge Ei( 0) 0=Rs
, and he mul ipole coe icien s
o he inciden ield ai
n,m and bi
n,m a e calcula ed om hei
ans e sal (i.e., non- adial) componen s h ough exploi ing
he o hogonali y o he VSMFs and TSMFs (Klinkenbusch,
2008)
ai
n,m =− 1
n(n +1)
1
1
kRs
d
d ( jn(k )) 0=Rs
·
2π
Z
0
π
Z
0
Ei( 0) 0=Rs·n∗
n,m(θ0,ϕ0)sin(θ0)dθ0dϕ0(19)
Z
jbi
n,m =1
n(n +1)
1
jn(kRs)·
2π
Z
0
π
Z
0
Ei( 0) 0=Rs·m∗
n,m(θ0,ϕ0)sin(θ0)dθ0dϕ0.(20)
Al e na i ely, he coe icien s ai
n,m and bi
n,m can be ound
om he adial componen s o he elec ic and magne ic
ields (Klinkenbusch, 2008):
ai
n,m =− 1
n(n +1)
kRs
jn(kRs)·
2π
Z
0
π
Z
0
Ei( 0) 0=Rs· ˆ 0Y∗
n,m(θ0,ϕ0)sin(θ0)dθ0dϕ0(21)
bi
n,m =− 1
n(n +1)
kRs
jn(kRs)·
2π
Z
0
π
Z
0
Hi( 0) 0=Rs· ˆ 0Y∗
n,m(θ0,ϕ0)sin(θ0)dθ0dϕ0.(22)
Fo compa ison, he inciden elec ic ield in Eqs. (19) and
(20) o he inciden elec ic and magne ic ields in Eqs. (21)
and (22) can be ob ained di ec ly by a ull-wa e simula o .
h ps://doi.o g/10.5194/a s-22-9-2024 Ad . Radio Sci., 22, 9–15, 2024
12 G. Gianne i and L. Klinkenbusch: A Nume ical Al e na i e o 3D Addi ion Theo ems
4 Nume ical esul s
4.1 Se ings
In he ollowing, a single equency 0=2 GHz is ixed,
which co esponds o a wa eleng h in acuum o λ≈
150 mm. Howe e , he p oposed me hod can be ex ended o
a equency ange, by epea edly applying he me hod o a
se o equency poin s, and subsequen ly o he ime-domain
by applying an in e se Fou ie ans o m.
The sca e e is a lossless iso opic dielec ic sphe e wi h
adius Rs=30 mm and a ela i e dielec ic pe mi i i y ε =
2.2. A ho n an enna, op imized o wo king a 0, is consid-
e ed and analyzed by he comme cial CST © ime-domain
sol e . In Fig. 2, he an enna and i s echnical d awings a e
shown. The e u n loss a 0in he ee space is 20.7 dB. Fo
a g aphical ep esen a ion, a possible posi ion o he sca e e
is also depic ed as a g een ci cle in Fig. 2a. Addi ionally, he
Huygens su ace o he p oposed me hod is also d awn: i is
a pa allelepiped and is called enclosing box in Fig. 2.
The pa allelepiped has i s sides o hogonal o ei he ix0,iy0,
o iz0, i s cen e in he an enna e e ence sys em is loca ed a
(xa=0,ya=0,za=−31)mm and he leng hs o he x-,y-,
and z-sides o he pa allelepiped a e 325.1 mm =2.17λ,
270.5 mm =1.80λ, and 692.0 mm =4.62λ, espec i ely.
The ma ix ep esen a ions o he in eg al ope a o s
Eqs. (12), (13), (19), and (20) a e e alua ed nume ically by
applying he poin ma ching me hod (Chew, 1995). The dis-
ance be ween he ma ching poin s is λ/20 ≈7.5 mm and
hence he o al numbe o poin s o e which he ields a e
expo ed is 18 322. Fo he alue o he unca ion limi N
in Eq. (1), he ollowing ule o humb is applied (Hansen,
1988)
N=dksRs+10e=12.(23)
Fo compa ison wi h he classical app oach based on he
addi ion heo em (S ein, 1961) o VSMFs (Alian and O aizi,
2018, 2019; Losenicky e al., 2021), he enclosing sphe e
wi h adius Ra=385 mm is also d awn in Fig. 2. No e ha
he enclosing sphe e is mo e ex ensi e han he enclosing
box, hus limi ing he minimum dis ance be ween he an-
enna’s and sca e e ’s subdomains.
Two posi ions o he sca e e wi h espec o he e e ence
sys em o he an enna a e conside ed:
–P1: O0=(15,30,700)mm, 0=700.8 mm;
–P2: O0=(0,0,375)mm, 0=375.0 mm;
whe e 0is he dis ance be ween he o igins o he wo e -
e ence sys ems (Fig. 1). The elec ical dis ances k 0a e 29.4
and 15.7 o P1 and P2, espec i ely. No e ha o P1, bo h
he p oposed me hod and he classical one based on he addi-
ion heo em o VSMFs can be applied. Howe e , P2 can be
analyzed only wi h he me hod p oposed he e, since in his
Figu e 2. Ho n an enna: (a) model in CST; (b) yz-c oss sec ion (no
in scale); (c) xz-c oss sec ion (no in scale). The dimensions o he
eeding ec angula wa eguide a e hose o he WR430 s anda d. In
(b) and (c), he ape angles a e in deg ees, while he o he dimen-
sions a e in millime e s.
Figu e 3. Equi alence p inciple in CST wi h sca e e in P2: max-
imum magni ude o he elec ic ield on xa=0 mm (Fig. 2b); HS
s ands o Huygens’ su ace while SC o sca e e .
case he sphe e enclosing he an enna and he one enclosing
he sca e e in e sec .
A coupling be ween he sca e e and he an enna is ex-
pec ed, bu his is no ye modeled by he p oposed me hod.
The e o e, o ob ain CST esul s ha a e compa able wi h
Ad . Radio Sci., 22, 9–15, 2024 h ps://doi.o g/10.5194/a s-22-9-2024
G. Gianne i and L. Klinkenbusch: A Nume ical Al e na i e o 3D Addi ion Theo ems 13
Figu e 4. Coe icien s o he mul ipole expansions: P1 o P2 indica es he posi ion o he sca e e ; a is o he magni ude, b he phase, and c
he e o Eq. (24); S, i p esen , indica es ha he sca e e is conside ed. The legend o (P1a) is he same o all he g aphs in he i s wo
columns [wi hou (wi h) sca e e ˜
Z=Z(˜
Z=Zs)]; he legend o (P1c) is he same o all he g aphs in he hi d column.
hose o he p oposed me hod, he CST e e ence solu ion is
compu ed using he equi alence p inciple as ollows:
–s ep 1: he subdomain o he an enna is sol ed and he
equi alen cu en s on he bounding box a e expo ed;
–s ep 2: he equi alen cu en s a e loaded in CST as
nea - ield sou ces and he an enna is eplaced by he
Huygens su ace (Fig. 3).
No e ha he ield inside he Huygens su ace in Fig. 3 does
no anish, which means ha , as expec ed, he e is an in e -
ac ion be ween he an enna and he sca e e .
Once he p oposed me hod is alida ed and no e e ence
solu ion is u he needed, i is su icien o sol e in CST only
he subdomain o he an enna and o expo he equi alen
cu en s on he bounding box (s ep 1 o he a o emen ioned
bulle lis ). The equi alen cu en s a e hen used as he inpu
o he p oposed me hod, which sol es he sca e ing p ob-
lem in he subdomain o he sca e e .
4.2 Compa isons
The mul ipole coe icien s deli e ed by he p oposed me hod
and CST a e now compa ed. Simila ly o Hansen (2012), he
ela i e e o o his me hod (TM) and o he CST esul s is
de ined as
c
n,m =20 ·log10 
c(s) TM
n,m −c(s) CST
n,m ·Fn
maxn,m c(s) CST
n,m ·Fn
(24)
Fn=1
√n1
nn
(25)
whe e cis ei he ao b, and (s) is ei he (i) o he inciden
ield o (in) o he ield inside he sca e e . In Eq. (24), n
and m ange om 1 o Nand om −n o n, espec i ely,
as o he double summa ions in Eq. (1). The weigh ing ac-
o Fnin Eq. (25) is necessa y o accoun o he ac ha
he e alua ion o he se ies Eq. (1) is con e gen wi h n, bu
i is no o he se ies o med by he mul ipole ampli udes
cn,m. The eason o his can be ound in he beha iou o he
sphe ical Bessel unc ion jn(k ) which con e ge e y as o
inc easing n. Hence, we obse e o he limi o n→∞ o
he sphe ical Bessel unc ion jn(x):
jn(x) ≈1
2√xnex
2nn
o n→∞ (26)
which hus makes Eq. (25) a sui able no maliza ion ac o in
Eq. (24). On he o he hand, Fnis close o one o small
h ps://doi.o g/10.5194/a s-22-9-2024 Ad . Radio Sci., 22, 9–15, 2024

14 G. Gianne i and L. Klinkenbusch: A Nume ical Al e na i e o 3D Addi ion Theo ems
Figu e 5. P ojec ion o he inciden elec ic ield on he su ace 0=
Rs, 0 ≤ϕ0≤π/2, 0 ≤θ0≤π o he sca e e in P1: (a) p oposed
me hod; (b) CST.
alues o nand i is exac ly one o n=1. This means ha
he no maliza ion ac o Fnacco dingly educes he impac
o he no malized mul ipole ampli udes o inc easing alues
o n.
As an example, we i s conside he mul ipole expansion
in P1, wi h and wi hou he sca e e , and second he mul i-
pole expansion in P2 wi hou he sca e e .
Fo P1, he mul ipole coe icien s o he inciden elec o-
magne ic ield ai
n,m and (Z/j)bi
n,m a e shown in Fig. 4 ( i s
ow) while hose o he ield inside he sca e e ain
n,m and
(Zs/j)bin
n,m a e shown in Fig. 4 (second ow). The esul s
ag ee well, excep o sligh disc epancies be ween he wo
me hods ha occu when he magni ude o he coe icien s
is less han 100V m−1, i.e., become less ele an . The max-
imum alue o he ela i e e o Eq. (24) o ai
n,m (bi
n,m) is
−30.5 dB (−28.1 dB) and i is −25.2 dB (−31.3 dB) o ain
n,m
(bin
n,m).
Fo P1, elec ic ields a e also quali a i ely compa ed. The
elec ic ield o he impinging wa e on a sphe e wi h adius
Rsis depic ed in Fig. 5. We obse e in bo h cases a simila i y
o he esul s om CST and he p oposed me hod.
Fo P2, he mul ipole coe icien s o he inciden ield a e
shown in Fig. 4 ( hi d ow). Fo his case oo, he esul s
om he wo me hods ag ee well, apa o mul ipole ampli-
udes wi h a magni ude less han 2 ×10−1V m−1. The max-
imum alue o he e o Eq. (24) o ai
n,m (bi
n,m) is −47.8 dB
(−36.0 dB).
In Fig. 4 ( hi d column), he e o dec eases as o in-
c easing mul ipole o de o he e m nnin he denomina o
o he ac o Fn. Due o his, he maximum alue o he de-
nomina o in Eq. (25) is ob ained o n=1 and m=±1 in
he examples analyzed. In addi ion, he ela i ely la ge al-
ues o he e o s may de i e om he limi ed accu acy o he
CST e e ence solu ion. To suppo his, he maximum alues
o he e o s a e lowe o he sca e e in P2, ha is, o he
sca e e close o he an enna and hence a smalle solu ion
domain in CST. The e o s may dec ease when sol ing he
adia o wi h a dedica ed sol e .
5 Conclusions
We ha e in oduced a me hod based on he dyadic G een’s
unc ion and he equi alence p inciple o ep esen an elec-
omagne ic ield in a coo dina e sys em di e en om he
o iginal one. The me hod wo ks e en when he classical ap-
p oach based on VSMF ansla ion o mulas ails. This ea-
u e also allows one o educe he dis ance be ween he an-
enna and he sca e e , pa icula ly o elonga ed an ennas.
The p oposed me hod has been compa ed o he nume ical
esul s pu ely ob ained om he ull-wa e simula o CST,
showing good ag eemen .
As he nex s ep, he au ho s in end o sol e he ull sca -
e ing p oblem by including he elec omagne ic in e ac ion
be ween di e en subdomains.
Code and da a a ailabili y. The code and he da a ha suppo he
indings o his s udy a e a ailable om he co esponding au ho ,
Giacomo Gianne i, upon easonable eques .
Au ho con ibu ions. GG was in ol ed in Concep ualiza ion, Fo -
mal Analysis, Me hodology, So wa e, Valida ion, Visualiza ion,
W i ing – o iginal d a , and W i ing – e iew & edi ing; LK was
esponsible o Supe ision, P ojec adminis a ion, and W i ing –
e iew & edi ing.
Compe ing in e es s. A leas one o he (co-)au ho s is a mem-
be o he edi o ial boa d o Ad ances in Radio Science. The pee -
e iew p ocess was guided by an independen edi o , and he au ho s
also ha e no o he compe ing in e es s o decla e.
Disclaime . Publishe ’s no e: Cope nicus Publica ions emains
neu al wi h ega d o ju isdic ional claims made in he ex , pub-
lished maps, ins i u ional a ilia ions, o any o he geog aphical ep-
esen a ion in his pape . While Cope nicus Publica ions makes e -
e y e o o include app op ia e place names, he inal esponsibili y
lies wi h he au ho s.
Special issue s a emen . This a icle is pa o he special issue
“Kleinheubache Be ich e 2023”. I is a esul o he Klein-
heubache Tagung 2023, Mil enbe g, Ge many, 26–28 Sep embe
2023.
Ad . Radio Sci., 22, 9–15, 2024 h ps://doi.o g/10.5194/a s-22-9-2024
G. Gianne i and L. Klinkenbusch: A Nume ical Al e na i e o 3D Addi ion Theo ems 15
Re iew s a emen . This pape was edi ed by Romanus Dyczij-
Edlinge and e iewed by wo anonymous e e ees.
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