Enginee ing Analysis wi h Bounda y Elemen s 149 (2023) 86–91
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Pa ial-induc ance e a ded pa ial coe icien s: Thei exac compu a ion
based on he Cagnia d–DeHoop echnique
Ma in S ump a, Fab izio Lo e ob,∗, Giuseppe Pe anicec, Giulio An oninib
aLe ch Labo a o y o EM Resea ch, Depa men Radio Elec onics, FEEC, B no Uni e si y o Technology, B no, Czech Republic
bDepa men o Indus ial and In o ma ion Enginee ing and Economics, Uni e si à degli S udi dell’Aquila, L’Aquila, I aly
cDepa men o Enginee ing and In o ma ion Sciences and Ma hema ics, Uni e si à degli S udi dell’Aquila, L’Aquila, I aly
ARTICLE INFO
Keywo ds:
Pa ial Elemen Equi alen Ci cui me hod
Cagnia d–DeHoop echnique
Compu a ional elec omagne ics
Pa ial induc ances
Time-domain modeling
ABSTRACT
The Pa ial Elemen Equi alen Ci cui (PEEC) me hod is a well ecognized in eg al-equa ion (IE) echnique o
sol e Maxwell’s equa ions. Simila ly o he me hod o momen s (MoM), he elec omagne ic (EM) in e ac ions
be ween cu en s and be ween cha ges a e desc ibed in e ms o in eg als. In con as o he s anda d MoM,
he PEEC me hod keeps he elec ic and magne ic coupling phenomena sepa a e, which leads o di e en
in e ac ion in eg als o be compu ed. These in eg als admi simpli ied solu ions o he case o he s a ic
ee-space G een’s unc ion and o hogonal geome ies bu hei applicabili y is limi ed o elec ically small
p oblems only. When he ull-wa e ee-space G een’s unc ion is conside ed, he in eg als a e ypically
compu ed in he equency domain (FD) by eso ing o Gaussian quad a u e schemes. The accu acy and
e iciency o such schemes is a delica e issue. The e o e, ecen wo ks ha e in es iga ed he possibili y
o applying he Cagnia d–DeHoop (CdH) echnique o calcula e he in e ac ion in eg als o ze o- hickness
elemen a y domains. In his pape , we close he loop and shall apply he CdH echnique o calcula e he
pa ial-induc ance be ween wo elemen a y b icks as p esc ibed by he PEEC echnique exac ly in he ime
domain (TD). The analy ical app oach is demons a ed on he in e ac ion be ween wo b icks as i occu s
in he modeling o he magne ic ield coupling be ween olume ic cu en s. The accu acy o he p oposed
app oach is (success ully) es ed o wo ep esen a i e cases.
1. In oduc ion
The no ion o pa ial induc ance, as in oduced by D . A. E. Ruehli
in 1972 [1], is a undamen al concep on which he PEEC me hod
lays i s ounda ions. The PEEC me hod is an IE me hod capable o
analyzing an EM sca e ing p oblem by means o an equi alen ci cui
ep esen a ion [2]. The PEEC me hod can be o mula ed wi h he aid
o s anda d EM con as -sou ce in eg al ep esen a ions [3, Sec. 28.9],
in which he G een’s unc ions apply o he ( ypically homogeneous,
iso opic and loss- ee) medium o he embedding. Hence, he equi -
alen ci cui can be di ec ly associa ed wi h a disc e ized sca e e ,
whe e elemen a y olumes and su aces a e assumed o adia e in he
backg ound medium (e.g. he ee-space). The PEEC me hod has been
p ima ily employed in he solu ion o FD p oblems and, in his con ex ,
in e ac ions in eg als ha e been compu ed ei he making quasi-s a ic
assump ions (e.g., neglec ing he e a da ion e ms in he in eg als) [4]
o by eso ing o quad a u e schemes in he equency domain (FD) [5]
o by means o he Taylo expansion o he G een’s unc ion [6]. Simi-
la lines o easoning a e commonly ollowed o ex ac he singula i y
∗Co esponding au ho .
E-mail add ess: [email p o ec ed] (F. Lo e o).
o G een’s unc ion in s anda d in eg al-equa ion o mula ions (e.g., [7–
9]). Owing o he e e inc easing in e es in accu a e TD simula ions
in he ield o Elec omagne ic Compa ibili y (EMC), howe e , ull-
wa e TD o mula ions o he pa ial induc ance a e becoming mo e and
mo e impo an . Acco dingly, his issue has been ecen ly add essed
analy ically o ela i ely simple 2-D ze o- hickness pa ches [10,11].
In his a icle, we shall apply he CdH echnique [12] o calcu-
la e a pa ial-induc ance e a ded PEEC coe icien exac ly in he TD.
The p esen ed analy ical esul s apply o he in e ac ion be ween wo
b icks (= igh pa allelepipeds). The CdH echnique is a join - ans o m
me hod ha has been o iginally de eloped o analy ically analyze
he seismic-wa e p opaga ion in ho izon ally laye ed media (e.g. [13,
14]). Mo e ecen ly, i has been demons a ed ha his sophis ica ed
in e sion me hodology can also be use ul o cons uc ing pu ely nu-
me ical solu ions. Indeed, he CdH echnique is a key ing edien in he
Cagnia d–DeHoop me hod o momen s, a no el TD-IE echnique o he
TD analysis o EM adia ion and sca e ing p oblems [15,16]. Mo e-
o e , ou ini ial s udies (see [10,11]) analyzing ze o- hickness PEEC
h ps://doi.o g/10.1016/j.enganabound.2023.01.008
Recei ed 9 Augus 2022; Recei ed in e ised o m 8 Janua y 2023; Accep ed 8 Janua y 2023
Enginee ing Analysis wi h Bounda y Elemen s 149 (2023) 86–91
87
M. S ump e al.
Fig. 1. Two in e ac ing b ick elemen s.
coe icien s ha e demons a ed ha he CdH in e sion is a p omising
s a egy o achie ing hei analy ical exp essions in he TD. Acco d-
ingly, his pape epo s on he ecen esul s o ou e o s o de elop
a quasi-s a ic-app oxima ion- ee PEEC sol e ha is equipped wi h
exac , CdH-based TD olume ic pa ial-induc ance. I is an icipa ed
ha he p esen ed app oach can be employed along wi h he p e-
co ec ed as Fou ie ans o m (FFT) [17] and he FFT-based app oach
o accele a e he ma ix– ec o p oduc s [18–20].
The pape is o ganized as ollows: in Sec ion 2 he p oblem o mu-
la ion is p esen ed. Consequen ly, he TD analy ical solu ion is gi en in
Sec ion 3, which is supplemen ed wi h Appendix. Finally, in Sec ion 4,
h ee nume ical applica ions a e p esen ed and success ully alida ed.
2. P oblem o mula ion
A PEEC model is ep esen ed h ough a se o pa ial elemen s,
he alue o which is ound upon e alua ing spa ial in eg als o e he
su aces/ olumes o in e ac ing disc e iza ion elemen s. In his wo k
we shall analyze he in e ac ion o wo b icks (= igh pa allelepipeds)
(see Fig. 1). In pa icula , wi h e e ence o [21, Eq. (4)], we shall s udy
a e a ded pa ial po en ial coe icien exp essed h ough a double
in eg al

𝐿𝑚𝑛(𝑠) = 𝜇0
𝑚𝑛∫𝒓∈𝑚
d𝑉∫𝒓′∈𝑛
𝑔(𝒓−𝒓′, 𝑠)d𝑉′,(1)
whe e 𝑚= {−𝛥𝑚
𝑥∕2 < 𝑥 −𝑥𝑚< 𝛥𝑚
𝑥∕2,−𝛥𝑚
𝑦∕2 < 𝑦 −𝑦𝑚< 𝛥𝑚
𝑦∕2,−𝛥𝑚
𝑧∕2 <
𝑧−𝑧𝑚< 𝛥𝑚
𝑧∕2} and 𝑛= {−𝛥𝑛
𝑥∕2 < 𝑥 −𝑥𝑛< 𝛥𝑛
𝑥∕2,−𝛥𝑛
𝑦∕2 < 𝑦 −𝑦𝑛<
𝛥𝑛
𝑦∕2,−𝛥𝑛
𝑧∕2 < 𝑧 −𝑧𝑛< 𝛥𝑛
𝑧∕2}, whe e 𝛥𝑚,𝑛
𝑥>0,𝛥𝑚,𝑛
𝑦>0and 𝛥𝑚,𝑛
𝑧>0
deno e he spa ial disc e iza ion s eps in he 𝑥-, 𝑦- and 𝑧-di ec ion,
espec i ely. Fu he mo e, 𝑠is he Laplace- ans o m pa ame e wi h
Re(𝑠)>0, and 𝑚,𝑛 a e c oss sec ions o he olumes 𝑚,𝑛, espec-
i ely, ha a e pe pendicula o he co esponding elec ic-cu en
lows. Nex ,
𝑔(𝒓, 𝑠) = exp(−𝑠|𝒓|∕𝑐)
4𝜋|𝒓|(2)
is he ee-space G een’s unc ion o he 3-D scala modi ied Helmhol z
equa ion and 𝑐= (𝜀𝜇)−1∕2 >0deno es he pe inen ( eal- alued and
posi i e) EM wa e speed.
3. P oblem solu ion
The e a ded pa ial po en ial coe icien as exp essed h ough
Eq. (1) will be nex ans o med o he TD analy ically wi h he
aid o he CdH echnique. Pu suing his app oach and assuming ha
Fig. 2. Two in e ac ing cube elemen s.
|𝑧𝑚−𝑧𝑛|>(𝛥𝑚
𝑧+𝛥𝑛
𝑧)∕2, one may exp ess he TD o iginal o Eq. (1),
u he deno ed by 𝐿𝑚𝑛(𝑡)(in hen y/second = ohm), as ollows:
𝐿𝑚𝑛(𝑡)=(𝜇0∕𝑚𝑛)[𝐽(|𝑧𝑚−𝑧𝑛|+𝛥𝑚𝑛+
𝑧, 𝑡)
−𝐽(|𝑧𝑚−𝑧𝑛|+𝛥𝑚𝑛−
𝑧, 𝑡) − 𝐽(|𝑧𝑚−𝑧𝑛|−𝛥𝑚𝑛−
𝑧, 𝑡)
+𝐽(|𝑧𝑚−𝑧𝑛|−𝛥𝑚𝑛+
𝑧, 𝑡)],(3)
whe e
𝐽(𝑧, 𝑡) = 𝐼(𝑥𝑚−𝑥𝑛+𝛥𝑚𝑛+
𝑥, 𝑧, 𝑡)
−𝐼(𝑥𝑚−𝑥𝑛+𝛥𝑚𝑛−
𝑥, 𝑧, 𝑡) − 𝐼(𝑥𝑚−𝑥𝑛−𝛥𝑚𝑛−
𝑥, 𝑧, 𝑡)
+𝐼(𝑥𝑚−𝑥𝑛−𝛥𝑚𝑛+
𝑥, 𝑧, 𝑡),(4)
and
𝐼(𝑥, 𝑧, 𝑡) = 𝐾(𝑥, 𝑦𝑚−𝑦𝑛+𝛥𝑚𝑛+
𝑦, 𝑧, 𝑡)
−𝐾(𝑥, 𝑦𝑚−𝑦𝑛+𝛥𝑚𝑛−
𝑦, 𝑧, 𝑡)
−𝐾(𝑥, 𝑦𝑚−𝑦𝑛−𝛥𝑚𝑛−
𝑦, 𝑧, 𝑡)
+𝐾(𝑥, 𝑦𝑚−𝑦𝑛−𝛥𝑚𝑛+
𝑦, 𝑧, 𝑡),(5)
whe e we used
𝛥𝑚𝑛±
𝑥,𝑦,𝑧 = (𝛥𝑚
𝑥,𝑦,𝑧 ±𝛥𝑛
𝑥,𝑦,𝑧)∕2,(6)
espec i ely. He e, 𝐾(𝑥, 𝑦, 𝑧, 𝑡) ep esen s he TD o iginal o he gene ic
slowness in eg al, he de ini ion and in e sion o which is p esen ed
in Appendix. Finally, we emphasize ha Eq. (3) applies o he con-
igu a ion whe e |𝑧𝑚−𝑧𝑛|> 𝛥𝑚𝑛+
𝑧. The case |𝑧𝑚−𝑧𝑛|< 𝛥𝑚𝑛+
𝑧mus be
analyzed sepa a ely.
4. Nume ical examples
In his sec ion, nume ical examples ela ed o h ee di e en ge-
ome ies a e p esen ed. Fo alida ion pu poses, he esul s ob ained
h ough he p oposed me hod a e compa ed wi h hose ob ained
h ough he nume ical-in e sion o he Laplace ans o m (NILT) ap-
p oach [22–24].
4.1. Two in e ac ing cubes
As a pa icula applica ion, he esul ing TD exp ession (3) has been
implemen ed in MATLAB®and applied o he case o wo in e ac ing
cubes (see Fig. 2). The e alua ions a e pe o med in he ini e ime
window {0 ≤𝑐𝑡∕𝑅𝑚𝑛 ≤2}, whe e 𝑅𝑚𝑛 = [(𝑥𝑚−𝑥𝑛)2+ (𝑦𝑚−𝑦𝑛)2+ (𝑧𝑚−
𝑧𝑛)2]1∕2 ep esen s he cen e - o-cen e dis ance be ween wo (iden ical)
cubes loca ed a
•(𝑥𝑚, 𝑦𝑚, 𝑧𝑚) = (0,0,0),
Enginee ing Analysis wi h Bounda y Elemen s 149 (2023) 86–91
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M. S ump e al.
Fig. 3. The TD coe icien o he wo iden ical cubes: compa ison be ween he
p oposed echnique and he dNILT2 - He mi e echnique..
Fig. 4. Spec um compa ison o he in e ac ion be ween wo iden ical cubes.
•(𝑥𝑛, 𝑦𝑛, 𝑧𝑛) = (2𝛥𝑥,2𝛥𝑥,2𝛥𝑥).
In he p esen example we ake 𝛥𝑥=𝛥𝑚,𝑛
𝑥=𝛥𝑚,𝑛
𝑦=𝛥𝑚,𝑛
𝑧= 1.0 mm.
The esul ing pulse shape o he TD coe icien is shown in 3. He e, o
he sake o alida ion, he esul s ob ained h ough he p oposed CdH-
based and he ( e e en ial) dNILT2 - He mi e echnique a e p esen ed.
Fo a de ailed desc ip ion o he e e en ial me hodology we e e he
eade o [24]. The ci cle poin s a e he ini ial samples needed o
he in e polan building. Finally, he FD coun e pa o he wo TD
esponses is ske ched in Fig. 4. As can be seen, he compu ed esul s
show good co espondence wi h dissimila i ies occu ing om 400 GHz
up. These disc epancies a e, howe e , i ually negligible in echnical
applica ions.
4.2. In e ac ions o a sys em o cubes
As a u he example we conside he geome y depic ed in Fig. 5,
whe e he mu ual pa ial induc ance be ween he cube wi h he cen e
loca ed a he axes o igin and he o he s a e conside ed. All he cubes
ha e sides 𝛥𝑥=𝛥𝑦=𝛥𝑧= 1.0 mm. The cubes sys em is composed by
27 elemen s, h ee o each dimension. The cen e coo dina es span a
ange [2𝛥𝑥− 8𝛥𝑥]. In Fig. 6 a e ske ched all he TD coe icien s ela ed
o he sys em, each one compu ed h ough he p oposed echnique
and compa ed o he dNILT2 - He mi e echnique [24]. The la e
Fig. 5. Cubes sys em geome y.
Fig. 6. TD coe icien s o he cubes sys em geome y.
echnique is based on he cons uc ion o an accu a e in e pola o ,
s a ing om he knowledge o he alues o he unc ion and i s i s
highe o de de i a i es a he s a ing poin s, enci cled in ed in he
igu e (gene ally up o he i h o de is su icien ).
4.3. Induced ol age on a cube by he cu en s lowing in ou b icks
In Fig. 7 a e shown ou iden ical pa allelepipeds: 1,2,3,4, wi h
sides: 𝛥𝑥= 1.5 mm, 𝛥𝑦= 0.5 mm, 𝛥𝑧= 0.25 mm, abo e a cube 0,
wi h sides: 𝛥0
𝑥=𝛥0
𝑦=𝛥0
𝑧= 1.0 mm, and cen e loca ed a he axes
o igin. The pa allelepipeds a e conside ed a he same heigh and a e
colloca ed unsymme ically in he 𝑥–𝑦plane, wi h espec o he cube.
In pa icula , he coo dina es o he pa allelepipeds a e:
•(𝑥1, 𝑦1, 𝑧1) = (−2.5 mm,0.5 mm,3𝛥0
𝑥),
•(𝑥2, 𝑦2, 𝑧2) = (−2.5 mm,5 mm,3𝛥0
𝑥),
•(𝑥3, 𝑦3, 𝑧3) = (11 mm,0.5 mm,3𝛥0
𝑥),
•(𝑥4, 𝑦4, 𝑧4) = (11 mm,5 mm,3𝛥0
𝑥).
The o e all induced ol age on he cube 0by he sys em o pa al-
lelepipeds can be compu ed h ough a combina ion o ou con olu ion
in eg als as:
𝑣𝐿0(𝑡) =
4
∑
𝑛=1 ∫𝑡
0
𝐿𝑝0,𝑛 (𝑡−𝜏)d𝑖𝑛(𝜏)
d𝜏d𝜏(7)
Enginee ing Analysis wi h Bounda y Elemen s 149 (2023) 86–91
89
M. S ump e al.
Fig. 7. Geome y o he compu a ion o he induced ol age on a cube by he cu en s
lowing in he b icks sys em.
Fig. 8. Induced ol age on he cube by he cu en s lowing in he sys em o b icks.
whe e 𝑖𝑛(𝑡)is he imp essed cu en on each pa allelepiped, 𝑛= 1,
⋯,4, lowing in he 𝑥di ec ion. The imp essed cu en 𝑖𝑛(𝑡), o
each pa allelepiped, is assumed o exhibi a windowed-powe (WP)
wa e o m [25]:
𝑖𝑛(𝜏, 𝑡) = 𝑡′𝜏(2 − 𝑡′)𝜏H(𝑡′)H(2 − 𝑡′)(8)
whe e H(𝑡)is he Hea iside uni -s ep unc ion (H(𝑡)=0i 𝑡 < 0,
H(0) = 1∕2,H(𝑡) = 1 i 𝑡 > 0), 𝑡′=𝑡∕𝑡 ,𝑡 being he pulse ise ime.
We choose 𝜏= 2 and 𝑡 = 4.6ps. The induced ol age on he cube 0is
depic ed in Fig. 8, whe e i is obse ed an excellen ag eemen be ween
he CdH me hod and he NILT-based me hod.
5. Conclusions
The PEEC me hod equi es ha in e ac ion in eg als desc ibing he
magne ic ield coupling be ween elemen a y olume ic egions be
compu ed, namely pa ial induc ances. In he equency domain, his
is usually done by eso ing o quad a u e schemes. In he TD, he
use o o e -simpli ying assump ions on he p opaga ion delay leads
o app oxima e esul s wi h a nega i e impac on physical p ope ies
such as he causali y and s abili y o he model. In his wo k, quasi-
closed- o m o TD e a ded pa ial induc ances ha e been de i ed
using he Cagnia d–DeHoop (CdH) echnique. A pe inen in eg a ion
pa h de o ma ion in he complex slowness plane allows o ob ain semi-
analy ical o ms o he ansien in e ac ion in eg als o a pai o
o hogonal b icks as hey occu in he PEEC me hod using Manha an-
ype meshes o oxelliza ion echniques. The p oposed app oach has
been es ed o ep esen a i e es cases by compa ison wi h o he
nume ical me hods, always exhibi ing a e y good ag eemen .
Fig. 9. Complex slowness planes. (a) 𝜎-plane wi h he CdH-pa h o 𝑦 < 0; (b) 𝜅-plane
wi h he CdH-pa hs o 𝑥 < 0.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no con lic o in e es .
Da a a ailabili y
The p esen ed da a a e a ailable upon eques om he au ho s.
Acknowledgmen s
The esea ch o Ma in S ump was suppo ed by he Czech Science
Founda ion unde G an No. 20-01090S.
Appendix. The gene ic in eg al
The in eg al ep esen a ion o be ans o med o TD has he ollow-
ing o m

𝐾(𝑥, 𝑦, 𝑧, 𝑠) = (𝑠
2i𝜋)2∫𝜅∈K0
exp(𝑠𝜅𝑥)
𝑠2𝜅2d𝜅
×∫𝜎∈S0
exp{−𝑠[−𝜎𝑦 +𝛤(𝜅, 𝜎)𝑧]}
𝑠2𝜎2
d𝜎
2𝑠3𝛤3(𝜅, 𝜎)(9)
o 𝑥∈R,𝑦∈R,{𝑧∈R;𝑧≥0} and {𝑠∈R;𝑠 > 0}, whe e K0and
S0a e he in eg a ion pa hs ex ending along Re(𝜅)=0and Re(𝜎)=0,
espec i ely, ha a e inden ed o he igh wi h semi-ci cula a cs wi h
cen e s a he o igins and anishingly small adii (see Fig. 9). Finally,
𝛤(𝜅, 𝜎), being he slowness pa ame e along he 𝑧-di ec ion, is de ined
as
𝛤(𝜅, 𝜎) = (1∕𝑐2−𝜅2−𝜎2)1∕2 wi h Re(𝛤)≥0.(10)
The gene ic in eg al will nex be ans o med o he TD wi h he
aid o he CdH echnique (see [12] and [16, Ch. 2]). To ha end,
he in eg a ion con ou in he complex 𝜎-plane, S0, is by i ue o
Jo dan’s lemma and Cauchy’s heo em [3, p. 1054] de o med in o a
CdH pa h, say ∪∗(he e ∗deno es he complex conjuga e), along
Enginee ing Analysis wi h Bounda y Elemen s 149 (2023) 86–91
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M. S ump e al.
which −𝜎𝑦 +𝛤(𝜅, 𝜎)𝑧=𝑢𝑑𝛺(𝜅) o {1 ≤𝑢 < ∞} wi h 𝑑2=𝑦2+𝑧2
and 𝛺(𝜅) = (1∕𝑐2−𝜅2)1∕2 is sa is ied (see Fig. 9a). Upon combining
he con ibu ions om and ∗, he inne in eg al wi h espec o
𝜎can be cas in o he in eg al wi h espec o he ( eal- alued and
posi i e) pa ame e s 𝑢. In addi ion, he con ibu ion om he (double)
pole singula i y a 𝜎= 0 mus be o 𝑦 > 0accoun ed o . The hus
exp essed inne in eg al is subsequen ly subs i u ed back in Eq. (9),
which yields

𝐾(𝑥, 𝑦, 𝑧, 𝑠) = 
𝑀(𝑥, 𝑦, 𝑧, 𝑠) + 
𝑁(𝑥, 𝑦, 𝑧, 𝑠),(11)
whe e

𝑀=1
2𝜋i
𝑑4
2𝜋𝑠3∫∞
𝑢=1
𝑦2𝑧2−𝑢2(𝑢2− 1)(𝑦4− 6𝑦2𝑧2+𝑧4)
(𝑢2𝑑2−𝑦2)2(𝑢2𝑑2−𝑧2)2
×d𝑢
(𝑢2− 1)1∕2 ∫𝜅∈K0
exp{−𝑠[−𝜅𝑥 +𝛺(𝜅)𝑢𝑑]} d𝜅
𝑠2𝜅2𝛺4(𝜅)(12)
and

𝑁=1
2𝜋i
𝑦H(𝑦)
2𝑠2∫𝜅∈K0
exp{−𝑠[−𝜅𝑥 +𝛺(𝜅)𝑧]} d𝜅
𝑠2𝜅2𝛺3(𝜅),(13)
whe e H(𝑦)has again he meaning o he Hea iside uni -s ep unc ion,
i.e. H(𝑦)=0i 𝑦 < 0,H(0) = 1∕2,H(𝑦)=1i 𝑦 > 0.
Fi s , we shall desc ibe he ans o ma ion o 
𝑀as gi en by Eq. (12).
Fo his pu pose, he in eg a ion con ou in he complex 𝜅-plane, K0, is
de o med in o a CdH pa h, say ∪∗, along which −𝜅𝑥+𝛺(𝜅)𝑢𝑑 =𝜏 o
{𝑅(𝑢)∕𝑐≤𝜏 < ∞} wi h 𝑅(𝑢)=(𝑥2+𝑢2𝑑2)1∕2 >0is sa is ied (see Fig. 9b).
In he esul ing exp ession, we combine he con ibu ions om and
∗and change he o de o he in eg a ions acco ding o, symbolically
∫∞
𝑢=1
d𝑢∫∞
𝜏=𝑅(𝑢)∕𝑐
d𝜏→∫∞
𝜏=𝑅(1)∕𝑐
d𝜏∫𝑈(𝑐𝜏)
𝑢=1
d𝑢(14)
whe e 𝑈(𝑐𝜏) = (𝑐2𝜏2∕𝑑2−𝑥2∕𝑑2)1∕2. Upon ca ying ou he in eg a ion
wi h espec o 𝑢, Eq. (12) can be cas in o he ollowing o m

𝑀=𝑐6
2𝜋2𝑠5∫∞
𝜏=𝑅(1)∕𝑐
exp(−𝑠𝜏)(𝑥, 𝑦, 𝑧, 𝑐𝜏)d𝜏
+
𝑃(𝑥, 𝑦, 𝑧, 𝑠)(15)
whe e 
𝑃a ises om he (double) pole singula i y a 𝜅= 0. Bo h e ms
on he igh -hand side o Eq. (15) ha e he o m ha allows hei
s aigh o wa d ans o m o he o iginal domain. In his p ocedu e,
Le ch’s uniqueness heo em applying o he eal- alued and posi i e
Laplace- ans o m pa ame e is an essen ial esul ha we ely on [26,
Appendix].
The ans o ma ion o 
𝑁(see Eq. (13)) ollows simila lines o
easoning. Indeed, he o iginal in eg a ion con ou , K0, is i s eplaced
wi h a new CdH pa h along which −𝜅𝑥+𝛺(𝜅)𝑧=𝜏is me o all {𝜌∕𝑐≤
𝜏 < ∞}, whe e 𝜌2=𝑥2+𝑧2. Combining again he con ibu ions om he
hype bolic a cs in he lowe and uppe hal es o he complex 𝜅-plane,
we end up wi h an in eg al wi h espec o 𝜏 ha can be exp essed as

𝑃(𝑦, 𝑥, 𝑧, 𝑠)(c . Eq. (15)). Rep esen ing u he he con ibu ion om he
(double) pole singula i y a 𝜅= 0 by 
𝑄(𝑥, 𝑦, 𝑧, 𝑠), we a i e a

𝑁=
𝑃(𝑦, 𝑥, 𝑧, 𝑠) + 
𝑄(𝑥, 𝑦, 𝑧, 𝑠).(16)
Upon subs i u ing Eqs. (15) wi h (16) in (11) and ans o m he
esul o he TD, we inally ge
𝐾(𝑥, 𝑦, 𝑧, 𝑡) = 𝑐
48𝜋2∫𝑐𝑡
𝑣=𝑅
(𝑐𝑡 −𝑣)4(𝑥, 𝑦, 𝑧, 𝑣)d𝑣
+𝑃(𝑥, 𝑦, 𝑧, 𝑡) + 𝑃(𝑦, 𝑥, 𝑧, 𝑡) + 𝑄(𝑥, 𝑦, 𝑧, 𝑡).(17)
The unc ion behind he in eg al sign is gi en by
(𝑥, 𝑦, 𝑧, 𝑣) = ∫𝜋∕2
𝜓=0
𝑓(𝑥, 𝑦, 𝑧, 𝑣, 𝜓)
×𝑝2−𝑞2(𝑈2− 1) sin2(𝜓)[cos2(𝜓) + 𝑈2sin2(𝜓)]
{(𝑧2∕𝑑2) cos2(𝜓) + [(𝑣2−𝑟2)∕𝑑2] sin2(𝜓)}2
×d𝜓
{(𝑦2∕𝑑2) cos2(𝜓) + [(𝑣2−𝜌2)∕𝑑2] sin2(𝜓)}2,(18)
wi h 𝑟2=𝑥2+𝑦2,𝑝2=𝑦2𝑧2∕𝑑4,𝑞2=𝑦4∕𝑑4− 6𝑝2+𝑧4∕𝑑4,𝑈2=
𝑣2∕𝑑2−𝑥2∕𝑑2and
𝑓=𝑣∕𝑑2
𝑈3[(𝑥2∕𝑑2) sin2(𝜓)+(𝑣2∕𝑑2− 1) cos2(𝜓)]2
×{(3𝑥2
𝑑2+𝑣2
𝑑2)[cos2(𝜓) + 𝑈2sin2(𝜓)]3
+(4𝑥4
𝑑4−11𝑥2𝑣2
𝑑4−𝑣4
𝑑4)[cos2(𝜓) + 𝑈2sin2(𝜓)]2
−(𝑥6
𝑑6+5𝑥4𝑣2
𝑑6− 10 𝑥2𝑣4
𝑑6)[cos2(𝜓) + 𝑈2sin2(𝜓)]
− 2 𝑥8
𝑑8+ 7 𝑥6𝑣2
𝑑8− 5 𝑥4𝑣4
𝑑8}.(19)
The emaining e ms in he inal TD esul (17) can be exp essed as
ollows
𝑃(𝑥, 𝑦, 𝑧, 𝑡) = 𝑐𝑥𝑑3H(𝑥)
12𝜋{6|𝑦|𝑧𝑐𝑡
𝑑3
×{ an−1 [|𝑦|(𝑐2𝑡2−𝑑2)1∕2
𝑧𝑐𝑡 ]
+ an−1 [𝑧(𝑐2𝑡2−𝑑2)1∕2
|𝑦|𝑐𝑡 ]}
− 3 𝑧
𝑑
𝑐2𝑡2+𝑦2
𝑑2 an−1 [(𝑐2𝑡2−𝑑2)1∕2
𝑧]
− 3 |𝑦|
𝑑
𝑐2𝑡2+𝑧2
𝑑2 an−1 [(𝑐2𝑡2−𝑑2)1∕2
|𝑦|]
+(2𝑐2𝑡2
𝑑2+ 1)(𝑐2𝑡2
𝑑2− 1)1∕2},(20)
and, inally,
𝑄(𝑥, 𝑦, 𝑧, 𝑡) = 𝑐𝑥𝑦H(𝑥)H(𝑦)
4(𝑐𝑡 −𝑧)2H(𝑐𝑡 −𝑧).(21)
Re e ences
[1] Ruehli AE. Induc ance calcula ions in a complex in eg a ed ci cui en i onmen .
IBM J Res De 1972;16(5):470–81.
[2] Ruehli AE, An onini G, Jiang L. Ci cui o ien ed elec omagne ic modeling using
he PEEC echniques. John Wiley & Sons, Inc., Hoboken, New Je sey; 2017.
[3] de Hoop AT. Handbook o adia ion and sca e ing o wa es. London, UK:
Academic P ess; 1995.
[4] An onini G, O landi A, Ruehli A. Analy ical in eg a ion o quasi-s a ic po en ial
in eg als on non-o hogonal coplana quad ila e als o he PEEC me hod. IEEE
T ans Elec omagn Compa 2002;44(2):399–403.
[5] Ruehli AE, An onini G, Esch J, Ekman J, Mayo A, O landi A. Non-o hogonal
PEEC o mula ion o ime and equency domain EM and ci cui modeling. IEEE
T ans Elec omagn Compa 2003;45(2):167–76.
[6] Lomba di L, An onini G, Ruehli AE. Analy ical e alua ion o pa ial elemen s
using a e a ded Taylo se ies expansion o he G een’s unc ion. IEEE T ans
Mic ow Theo y Tech 2018;66(5):2116–27.
[7] Wil on D, Rao S, Glisson A, Schaube D, Al-Bundak O, Bu le C. 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 opag 1984;32(3):276–81.
[8] Jä enpää S, Taskinen M, Ylä-Oijala P. Singula i y ex ac ion echnique o
in eg al equa ion me hods wi h highe o de basis unc ions on plane iangles
and e ahed a. In e na J Nume Me hods Eng g 2003;58(8):1149–65.
[9] Jä enpää S, Taskinen M, Ylä-Oijala P. Singula i y sub ac ion echnique o high-
o de polynomial ec o basis unc ions on plana iangles. IEEE T ans An ennas
P opag 2006;54(1):42–9.
[10] Š ump M, An onini G, Ruehli AE. Cagnia d-Dehoop echnique-based com-
pu a ion o e a ded pa ial coe icien s: The coplana case. IEEE Access
2020;8:148989–96.
[11] Š ump M, Lo e o F, Pe anice G, An onini G. Cagnia d–DeHoop echnique-based
compu a ion o e a ded ze o- hickness pa ial elemen s. Eng Anal Bound Elem
2022;137:56–64.
[12] de Hoop AT. A modi ica ion o Cagnia d’s me hod o sol ing seismic pulse
p oblems. Appl Sci Res 1960;B(8):349–56.

Enginee ing Analysis wi h Bounda y Elemen s 149 (2023) 86–91
91
M. S ump e al.
[13] de Hoop AT. La ge-o se app oxima ions in he modi ied Cagnia d me hod o
compu ing syn he ic seismog ams: a su ey. Geophys P ospec 1988;36(5):465–
77.
[14] de Hoop AT. Re lec ion and ansmission o a ansien , elas ic, line-sou ce
exci ed SH wa e by a plana , elas ic bonding su ace in a solid. In J Solids
S uc 2002;39(21):5379–91.
[15] Š ump M. Time-domain elec omagne ic ecip oci y in an enna modeling.
Hoboken, NJ: IEEE P ess–Wiley; 2019.
[16] Š ump M. Me asu ace elec omagne ics: The Cagnia d-DeHoop ime-domain
app oach. London, UK: IET; 2022.
[17] Phillips JR, Whi e JK. A p eco ec ed-FFT me hod o elec os a ic analysis o
complica ed 3-D s uc u es. IEEE T ans Compu -Aided Des In eg Ci cui s Sys
1997;16(10):1059–72.
[18] Polime idis AG, Villena JF, Daniel L, Whi e JK. S able FFT-JVIE sol e s
o as analysis o highly inhomogeneous dielec ic objec s. J Compu Phys
2014;269:280–96.
[19] Yucel AC, Geo gakis IP, Polime idis AG, Bağci H, Whi e JK. VoxHen y: FFT-
accele a ed induc ance ex ac ion o oxelized geome ies. IEEE T ans Mic ow
Theo y Tech 2018;66(4):1723–35.
[20] To chio R, Lucchini F, Schanen J-L, Chadebec O, Meunie G. FFT-PEEC: A as
ool om CAD o powe elec onics simula ions. IEEE T ans Powe Elec on
2021;37(1):700–13.
[21] Lomba di L, Tao Y, Nou i B, Fe an i F, An onini G, Nakhla MS. Pa ame e ized
model o de educ ion o delayed PEEC ci cui s. IEEE T ans Elec omagn Compa
2019;62(3):859–69.
[22] Lomba di L, Lo e o F, Fe an i F, Ruehli A, Nakhla MS, Tao Y, e al. Time-
domain analysis o e a ded pa ial elemen equi alen ci cui models using
nume ical in e sion o Laplace ans o m. IEEE T ans Elec omagn Compa
2021;63(3):870–9.
[23] Lo e o F, Romano D, S ump M, Ruehli AE, An onini G. Time-domain compu a-
ion o ull-wa e pa ial induc ances based on he modi ied nume ical in e sion
o Laplace ans o m me hod. IEEE T ans Signal Powe In eg 2022;1:32–42.
[24] Lo e o F, Pe anice G, An onini G, Gad E, Nakhla MS, Tao Y, e al. Modi ied
nume ical in e sion o Laplace ans o m me hods o he ime-domain analysis
o e a ded pa ial elemen s equi alen ci cui models. IEEE T ans Elec omagn
Compa 2021;64(6):2179–88.
[25] Lage IE, an Be kel SL. Fini e empo al suppo pulses o EM exci a ion. IEEE
An ennas Wi el P opag Le 2017;16:1659–62.
[26] Š ump M. Elec omagne ic ecip oci y in an enna heo y. Hoboken, NJ: IEEE
P ess–Wiley; 2018.