Valida ions agains SINBAD benchma k o he adia ion anspo module,
a de e minis ic sol e o he neu on anspo equa ion.
Alejand o Soba1,2, Mau icio E. Cazado2, Guillaume Houzeaux1, Albe Gu ie ez-Milla1,
Me i J. Man sinen1,3 and Xa ie Saez1
1 Ba celona Supe compu ing Cen e , Ba celona, Spain.
2 CNEA-CONICET, Buenos Ai es, A gen ina.
3 ICREA, Ba celona, Spain.
Keywo ds: Radia ion anspo , neu on-wall in e ac ion, usion eac o s shielding, SINBAD.
Abs ac
We p esen signi ican imp o emen s and alida ions o e a de e minis ic neu on anspo
code (NEUTRO) dedica ed o sol ing he Bol zmann T anspo Equa ion. The code is
in eg a ed as a module in he Alya sys em de eloped by he Ba celona Supe compu ing Cen e
which uses he Disc e e O dina es Me hod o e an angula po ion, mul i-g oup o ene gy
disc e iza ion and FEM o e uns uc u ed meshes o ea special complex domains. Ma e ial
aniso opy o he sca e ing medium is in oduced in o he sca e ing ke nel using eal base
exp essions o sphe ical ha monics. In o de o build he o al c oss-sec ion and he espec i e
g oup ma ix o he elas ic c oss-sec ion, we use he NJOY code. We es he sol e using
di e en geome ies, o de s o in eg a ion o he angula disc e iza ion and numbe o ene gy
g oups. Finally, we compa e some esul s agains benchma ks ob ained om an IAEA da abase,
basically measu ing leakage e ec s o se e al ma e ials.
1. In oduc ion
In his pape , we p esen some imp o emen s and alida ions o a de e minis ic adia ion
anspo code based on he Bol zmann T anspo Equa ion and in eg a ed in o he Alya sys em
a Ba celona Supe compu ing Cen e [1], which is in ended o mul i-physics applica ions in
he nuclea usion ield [2, 3]. Imp o ing he unde s anding o his ma e is essen ial o he
de elopmen o u u e usion eac o s, which equi es highly demanding simula ions o he
neu on anspo p ocesses. In pa icula , sol ing his kind o adia ion anspo equa ions in
o de o p edic he beha iou o neu on damage and adia ion-ma e in e ac ion demands high
memo y equi emen s and a high compu a ional complexi y [4].
The main idea behind his de elopmen has a double impac by in eg a ing wo complex
esea ch ields. On he one hand, i will be aligned wi h he Eu opean Fusion Roadmap [5] o
he ealiza ion o usion ene gy, being ocused on he i ium sel -su iciency om he Mission
4 and he design o he Tes Blanke Module (TBM). I will p o ide a cu ing-edge
compu a ional ool o add essing his complex mul i-physics p oblem bo h in exis ing and
u u e usion de ices which is in inc easing demand wi hin he usion communi y. On he o he
hand, i has he po en ial o become a d i e o u u e usion eac o design challenges gi en
i s High-Pe o mance Compu ing (HPC) capabili ies which make he analysis o di e en
design a ia ions easible.
This new modelling ool o usion is based on he pa allel compu a ional mechanics code Alya
[6, 7]. Alya is a amewo k c ea ed o sol e di e en physical phenomena in a coupled way in
la ge scale supe compu e s, accomplishing high pa allelism s anda ds and scalabili y. The code
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Final e sion a ailable ia: h ps://doi.o g/10.1016/j. usengdes.2021.112497
is o ganized in a modula way including se ices, ke nel and modules. The ke nel is he co e o
Alya and implemen s inpu /ou pu ope a ions, ou ines o he execu ion o he main loop and
all ope a ions ela ed o he mesh geome y. Se ices cons i u e he unc ionali ies o e ed o
he execu ion, o which he mos impo an is he se ice ha deals wi h he di ision o he wo k
and is esponsible o he communica ion be ween nodes. Alya’s modules, as in he case o he
NEUTRO module c ea ed o neu on anspo , implemen he se o Pa ial Di e en ial
Equa ions (PDEs) which de ine he physics o he p oblem. The modules a e coded and
compiled independen ly and can be easily coupled oge he . Using FEM in ol ing millions o
billions o elemen s equi es gene al spa se sys ems o equa ions sol ed using i e a i e
me hods. Mode n supe compu ing a chi ec u es consis o independen compu ing nodes,
ypically mul ico e and he e ogeneous a chi ec u es, in e connec ed by a sha ed communica ion
ne wo k. Alya uses he Message Passing In e ace (MPI) lib a y o implemen communica ions
be ween nodes and Open Mul i-P ocessing (OpenMP) o sha ed-memo y pa allelism. De ailed
desc ip ions o he s uc u e o he Alya code can be ound in e e ences [1, 6-9]. The isual
pos -p ocessing o he esul s was pe o med wi h Pa a iew so wa e [10].
The sol e uses he Disc e e O dina es Me hod [11-13] o e an angula po ion, a mul i-g oup
me hod o he ene gy disc e iza ion and he Fini e Elemen Me hod (FEM) [9, 14] o e
uns uc u ed meshes o ea special complex domains. We use he NJOY code [15] o build he
o al c oss-sec ion and he espec i e g oup ma ix o he elas ic c oss-sec ion. NJOY eads he
da a om an ENDF/B da abase [16] p o ided by Los Alamos Na ional Secu i y o ice. We es
he sol e in wo-dimensional geome ies (slabs o di e en dep hs), and h ee-dimensional
geome ies using laye s and sphe es. Quad a u es o se e al o de s in he equal-le el disc e e
o dina e se a e a emp ed [17, 18] and due o he sca e ing o high ene gy 14 MeV usion
neu ons being highly aniso opic, a new ype o quad a u e based in he FEM disc e iza ion
[19] is conside ed. In addi ion, we ha e es ed he sol e using di e en ene gy g oups
disc e iza ions, om 3 o 256, and he esul s ob ained we e alida ed agains o he codes and
he SINBAD (Shielding In eg al Benchma k A chi e Da abase) [20] benchma ks, ob ained
om an IAEA da abase, in which he leakage e ec s o se e al ma e ials we e measu ed.
Al hough he alida ions included in his wo k a e compa ed wi h he neu on anspo esul s,
he code is p epa ed o sol e p oblems o gamma adia ion anspo as well [21, 22].
Valida ions o he la e will be p esen ed in u u e wo ks.
I is impo an o men ion ha al hough he e a e many codes in he wo ld ha sol e he
adia ion anspo equa ion [23-26], none o hem is in eg a ed wi h compu a ional mechanics
amewo ks o mul i-physics coupled p oblems and hey ha e been de eloped as s and-alone
so wa e. In his sense, he in eg a ion o he NEUTRO module in o he Alya sys em has an
inno a i e aspec wi h espec o o he be e -known solu ions. The inal in en ion o his
de elopmen is o achie e a mul i-physics ea men o high- olume en i onmen s composed
o mul iple ma e ials, being able o sol e hei he mal, mechanical and neu onic beha iou a
he same ime, using he coupling capabili ies p o ided by he Alya sys em. In his espec , we
made a deep scalabili y s udy o he sol e , showing he beha iou o he code and he quali y
o he esul s as we inc ease he numbe o p ocesso s in ol ed in he p oblem sol ing. Some
o hese scalabili y measu es a e included in sec ion 3.
This pape is o ganized as ollows: in sec ion 2, we b ie ly desc ibe he sol ing amewo k o
he NEUTRO module included in Alya. In sec ion 3, we include some o he es s ca ied ou
wi h his module and we show some nume ical d awbacks ha had o be sol ed in o de o
achie e a good quali y in he inal esul s. In addi ion, we p esen some scalabili y esul s
including he beha iou o he code when a la ge numbe o p ocesso s a e in ol ed in he
calcula ions. In sec ion 4, we show he compa isons agains some selec ed expe imen s o he
SINBAD da abase. Finally, in sec ion 5, we p esen ou conclusions.
2. Nume ical adia ion anspo equa ion
The gene al s a iona y adia ion anspo equa ion sol ed a he momen in he NEUTRO
module included in Alya has he ollowing exp ession [7-10]:
Ω∙∇φ+Σ$(%,&,Ω)'=* * Σ-(%,&.→&,Ω.→Ω)'0Ω.0&.
12
3
4
(1)
Whe e
Ω: Solid angle o he possible di ec ion in whe e a pa icle a els.
E: Pa icle ene gy.
: Gene al spa ial a iable.
φ: Flux as a unc ion o , E and Ω.
Σ$( ,E,Ω): To al mac oscopic c oss-sec ion.
Σ-( ,E.→E,Ω): To al mac oscopic sca e ing c oss-sec ion.
The ene gy mul i-g oup app oxima ion [9, 27, 28] and he disc e iza ion o angula dependence
we e applied o equa ion (1). Fo he la e , he Me hod o Disc e e O dina es (equal-le el se ,
Sn) [11-14] o quad a u es se based on discon inuous ini e elemen s [19] may be chosen. The
mul i-g oup me hodology consis s o di iding he neu on ene gy ange in se e al g oups (G)
and e alua e all he ene gy-dependen pa ame e s in ela ion o hose ene gy g oups G.
Disc e iza ion o e he angula (di ec ional) a iable is usually pe o med by means o
mechanical quad a u e. The con inuous angula a iable Ω is eplaced by a se o disc e e
di ec ions, each o which has a ela i e weigh . The di ec ions may be ep esen ed as poin s on
he uni sphe e.
Using a mul i-g oup app oxima ion and an angula disc e iza ion, he in eg al-di e en ial
anspo equa ion is con e ed in o a sys em o G×M di e en ial equa ion as ollows:
78∙∇φ98 +Σ$;%,&9<φ98 = > > Σ-9?,9 8?,8(%)φ9?8?
@
8?AB
8.C8
D
9?AB
9.C9
(2)
Wi h F =1,…,I; K=1,…,L
Ma e ial aniso opy o he sca e ing medium is in oduced in o he sca e ing ke nel using eal-
based exp essions o sphe ical ha monics. Combining his app oxima ion wi h he ene gy
g oup di ision and disc e e o dina e me hods, he ollowing exp ession o he sca e ing e m
is ob ained:
    4
2!+1Y"#"(Ω#)
"
#"%&"
'
"%*
,
-.%/
0
3.%/ Y
5"#"(Ω#6) Σ#"3.,3 -6,-(:)φ36-6(:)
(3)
( he o de o he sphe ical ha monics was changed o ‘ml’ o a oid con usion). In his
exp ession, a maximum alue o L is chosen o cu he expansion. I L=0, he ke nel exp ession
in equa ion (2) is eco e ed. Any alue o L g ea e han ze o in oduces angula aniso opy.
The mos gene al bounda y condi ion o he anspo equa ion is he Albedo condi ion [1,9],
which consis s o :
φ<=( ,E,Ω!")= φ!" + αφ( , E, Ω%)
(4)
whe e φ!" is a ixed lux, α akes he alue 0 o he acuum bounda y condi ion and α=1 o
he gene al e lexi e condi ion. Gene ally, o model a e lec o su ounding he co e he alue
o α is 1. O e all, he Albedo coe icien α may a y wi h he e lec o p ope ies and should
also cap u e angula and ene ge ic edis ibu ion o he e lec ed neu ons due o hei di usion
h ough he e lec o .
The e a e wo main s eps when implemen ing mi o bounda y condi ions. Fi s ly, i is
necessa y o calcula e he no mal ec o o he su ace on o which he ays a e e lec ed.
Secondly, he calcula ion o he e lec ion angle is equi ed. Since we a e wo king wi h disc e e
o dina es we need o p ojec he ajec o y on o one o he disc e ized di ec ions ha we ha e.
In each bounda y whe e he e lec ing condi ion is u ned on, he algo i hm de eloped
de e mines which di ec ion is he closes o he cu en di ec ion o e lec ion o use i in he
building o he bounda y con ibu ion. In he disc e e o dina es me hod, gi en a di ec ion used
in he angula disc e iza ion, i is necessa y o e alua e he co esponding specula e lec i e
di ec ion a e he neu on a i ed a he e lec i e medium. To do so, a special unc ion was
p og ammed in o de o de e mine a he beginning o he simula ion he co esponding
di ec ion o each no mal p esen in he limi s o he domain o ou p oblem. Using his
p ocedu e, bo h angula di ec ion and e lec i e di ec ion a e e alua ed o e each bounda y
whe e a e lec i e condi ion is equi ed.
In o de o build he o al, cap u e and elas ic sca e ing c oss-sec ions and he espec i e g oup
ma ix o elas ic and ission c oss-sec ion, he NJOY code [15] was used. NJOY eads and
p ocesses da a om ENDF/B da abase [16]. F om SINBAD benchma k da abase, some
exe cises des ined o analyse he leakage e ec o se e al ma e ials we e selec ed. Ini ially, he
inpu iles wi h he o al and sca e ing ma ix o i on, nickel, aluminium, ungs en, manganese
and silicon wi h NE-ene gy g oups we e c ea ed. We es ed wi h NE = 238, 118 and 59 g oups
o compa e wi h he benchma k cases. Fo mos o he shielding expe imen s aken om he
SINBAD da abase, we used 59 ene gy g oups. In addi ion, o es ing coa se de ini ions, 29-
ene gy g oups and 3-ene gy g oups we e used.
3. Nume ical d awbacks and pe o mance analysis
3.1 Angula disc e iza ion: Once he de elopmen o he mul i-g oup equa ion was comple ed
and be o e going o he alida ion s age, we ca ied ou a se ies o nume ical expe imen s o
e i y he unc ioning o his new e sion o he code wi h espec o wha was p e iously
p og ammed, as well as i s in e ac ion wi h he Alya sys em. In he i s place, he solu ion o a
sphe e wi h an in e nal hole wi h an inne lux in he adial di ec ion and acuum in he ex e nal
bounda y was deeply analysed. We used a e ahed al mesh o linea elemen s o disc e ize he
eal space. As he in en ion was o analyse he solu ion wi h he di e en quad a u e o mulas
implemen ed in he code, a h ee-ene gy g oups disc e iza ion was used. Fo compa ison, we
analysed he solu ion wi h di e en app oxima ions o he equal le el me hods and he
implemen ed angula disc e iza ion based on he ini e elemen me hod wi h 512 poin s (called
in e nally in he code FE64). This me hod consis ed o he disc e iza ion o he medium plane
o he sphe e using a iangula mesh and he p ojec ion o e he sphe e su ace o each iangle
wi h a well-de ined weigh [19]. In Figu e 1, we p esen a gene al iew o he code capaci ies.
Figu e 1 a) shows a domain di ision o be dis ibu ed among 16 nodes, used o sol e he eal
space using FEM. Figu e 1 b) p esen s he neu onic cu en in he z di ec ion showing he
symme y o he solu ion ob ained. Figu e 1 c) is a con ou zone o one ene gy g oup as an
example solu ion o he domain. All hese solu ions we e ob ained using he mo e accu a e
angula disc e iza ion (FE64). Las ly, in Figu e 1 d), we plo he o al lux in all he nodes on
he su ace o he sphe e o he di e en angula disc e iza ions implemen ed in he code. I
can be app ecia ed how he solu ions ob ained wi h he equal le el me hod Sn4 (24 poin s) and
Sn8 (80 poin s) p esen ed a g ea dispe sion and nega i e alues o he solu ion. When we
passed o Sn10 (120 poin s) and Sn12 (168 poin s) posi i e alues and less dispe sion in he
solu ion we e ob ained. Finally, o he quad a u e based on FE he esul s achie ed show he
leas dispe sion, ob aining he uni o mi y ha he symme y o he solu ion p oposes. Le us
highligh ha he Sn12 me hod has 168 in eg a ion poin s in all he sphe e while he FE64 has
512 in eg a ion poin s, inc easing he compu a ion ime o he p oblem p opo ionally. Bu he
mo e impo an esul is ha using he highes quad a u e solu ion, we a oid he spu ious
nega i e alues in he lux solu ion, a ecu ing p oblem in he use o disc e e me hods o sol e
he anspo equa ion.
Figu e 1: a) Pa i ioned mesh o pa allelize he solu ion o he eal space. b) Neu onic cu en in z
di ec ion. c) Flux solu ion o one selec ed ene gy g oup d) Dispe sion o he su ace solu ion o di e en
modes o angula disc e iza ion.
3.2 Compa ison be ween 2D and 3D solu ions: An impo an and e ec i e way o accele a e a
p oblem om he nume ical poin o iew is o educe he dimensions o he domain o be

analysed when some ypes o symme y a e p esen . This educ ion, in he case o anspo
p oblems, in ol es educing he physical domain bu also he angula disc e iza ion by omi ing
ou oc an s o angula di ec ions. On he one hand, his sys em inc eases he esolu ion speed,
bu a he same ime enla ges he e o o he solu ion. In his sub-sec ion, we compa e a solu ion
ob ained on a h ee-dimensional laye and a slab o equal wid h, bu wo-dimensional. Bo h
cons i u ed by he same ma e ial, and wi h he same inne lux om he adial di ec ion (x in
he slab2d and z in he 3D laye ). A acuum bounda y condi ion is conside ed in he ou pu
di ec ion and e lec i e bounda ies in he es o he limi s. Figu e 3 shows he o al lux
ob ained on he ex e nal ace as a unc ion o one o he signi ican coo dina es. The solu ion is
di ided by he maximum ob ained in he h ee-dimensional case, eaching a di e ence o 14%
in he esul be ween bo h simula ions.
Figu e 2: Solu ion o a wo-dimensional slab compa ed wi h a h ee-dimensional laye o he
same wid h
3.3 Pe o mance Analysis: Be o e discussing he pe o mance analysis, a ew ema ks should
be made abou he pa alleliza ion in Alya. Fi s o all, he implemen ed pa alleliza ion s a egy
is he main-seconda y echnique. This means ha when he pa alleliza ion akes place, he main
p ocess is in cha ge o dis ibu ing he da a and coo dina ing he o he p ocesses (seconda y).
The seconda ies a e he ones ca ying ou he calcula ions. All p ocesses communica e among
hemsel es h ough messages using he MPI s anda d, which p o ides synch oniza ion and
communica ion unc ionali ies be ween a se o p ocesses wi h dis ibu ed memo y (meaning
ha no all p ocesses can access he same memo y space and he e o e need o exchange da a
among hem). The Alya sys em is w i en in FORTRAN language: MPI is implemen ed as a
lib a y wi hin he code om which we can call he unc ions we may need once we ha e
included i , as we would do wi h any o he lib a y.
Ano he impo an ea u e o he Alya implemen a ion is he possibili y o mul iplying he ini ial
mesh. Using his op ion each ini e elemen is di ided in o up o 8 ini e elemen s o he same
cha ac e is ics: his implies an inc ease in he numbe o elemen s o he inpu mesh. In o de
o es he pe o mance, a h ee-dimensional sphe e o Fe56 ma e ial wi h an in e io sou ce in
he adial di ec ion using 59-ene gy g oups and Sn12 o he angula quad a u e (see igu e 1)
we e used. Mul iplying by 2 and by 3, meshes o 77984 and 623872 ini e elemen s a e de ined,
espec i ely. In Figu e 3 he speed up o each case is p esen ed. In he case o 77984 elemen s,
a 95% e iciency wi h 256 p ocesso s is achie ed, while in he case o 623872 elemen s, an
82% e iciency wi h 1024 p ocesso s and 70% wi h 2048 p ocesso s a e ob ained. Bo h cases
show sa u a ion due o he inc ease in communica ion. No e ha his e iciency is measu ed
wi h espec o he un on 16 ( o he small case) and 128 ( o he big case) MPI p ocesses.
Despi e he sa u a ion men ioned, he esul s show good pe o mance wi h mo e han wo
housand p ocesso s.
Fo all he calculus we used a gene alized minimal esidual algo i hm (GMRES) [29] o sol e
he nonsymme ic linea sys em p oduced by his ype o p oblem. We used a diagonal
p econdi ione , which esul ed in he mos sui able op ion o each con e gence in less ime.
Figu e 3: Speedup o wo cases, using wo meshes, 77984 elemen s wi h an a e age olume o
6.73x10-6 and 623872 elemen s wi h 8.4x10-7 a e age olume.
4. SINBAD es ing
In o de o alida e he de eloped NEUTRO module, we selec ed a numbe o benchma k cases
o es ing pu poses om SINBAD, an OECD/NEA Da a Bank (O ganiza ion o Economic
Coope a ion and De elopmen , Nuclea Ene gy Agency Da a Bank) and ORNL/RSICC (Oak
Ridge Na ional Labo a o y, Radia ion Sa e y In o ma ion Compu a ional Cen e) join p ojec
[20, 30]. The main objec i e o hese es s was o p oduce a da abase con aining an
in e na ionally es ablished se o adia ion shielding and dosime y da a ela i e o expe imen s
ele an o eac o shielding, usion blanke neu onics and accele a o shielding, o e ing a
s anda dized ool o alida ion o adia ion anspo compu e codes and nuclea da a lib a ies.
Ou pa icula analysis was based on he se o expe imen s ha ook place in Osaka Uni e si y
a he OKTAVIAN acili y. De ailed desc ip ions o he expe imen s, he de ec o sys em and
he sou ce spec um used can be ound in e e ences [31-36]. In summa y, his g oup o es s
used hollow sphe es wi h an inne neu on sou ce wi h a p onounced peak in he ange o 14
MeV. The sphe es we e made o di e en ma e ials (Fe, Ni, Al, W, Mn, Si, among o he s) and
wi h di e en diame e s in a ange o 14 cm o 100 cm. The main pu pose o hese benchma ks
was o s udy he neu on and gamma- ay leakage spec a om he ou e su ace o he sphe e.
The e a e wo kinds o measu emen s ca ied ou depending on he posi ion o he de ec o . Fo
some ma e ials (Mn, Si, Al, W), he neu on de ec o was loca ed wi h a 55° angle wi h espec
o he cen al axis o he sphe e [35, 36]. In o he cases, such as o Fe and Ni, he de ec o did
no co e he en i e su ace o he sphe e bu he solid angle o 17.28° om he sphe e cen e
due o he p esence o he collima o s [32-34].
In o de o pe o m he es , wo ypes o models we e used: a wo-dimensional model, using
slabs wi h a wid h equal o he sphe ical shell hickness; a h ee-dimensional laye wi h a wid h
equal o he sphe ical shell hickness and a h ee-dimensional sphe e wi h an in e nal hole. Table
I p esen s a summa y o he es s pe o med and he ma e ials used.
Table I: Desc ip ion o he Benchma k used in his wo k [31,36]
Model
Ma e ial
Ex .
Radi
us
(cm)
In e nal hole
adius
(cm)
E ec i e
hickness (cm)
De ec o posi ion
Sphe e o 40
cm
Silicon 20 10 10
55º de ec o
Sphe e o 60
cm
Silicon 30 10 20
55º de ec o
Sphe e o 60
cm
Mangane
se
30 10 20
55º de ec o
Sphe e o 40
cm
Aluminium
20
10 10
55º de ec o
Sphe e o 40
cm
Tungs en 20 10 10
55º de ec o
Sphe e o 32
cm Nickel 16 2 14
17.28º solid angle
om he
sphe e
cen e
Sphe e o
100 cm I on 50 10 40
17.28º solid angle
in he
sphe e
cen e
In Figu e 4, compa isons be ween expe imen al esul s and se e al benchma ks un wi h Alya
using wo-dimensional slabs a e p esen ed. In each case, he wid h o he slabs ep esen s he
e ec i e hickness o he sphe e in each expe imen . Al hough o he cases o aluminium o
10 cm hickness (Figu e 4 a), manganese o 20 cm hickness (Figu e 4 b), i on o 40 cm
hickness (Figu e 4 c) and ungs en o 10 cm hickness (Figu e 4 d) he esul s show some
dispe sion in he ange o ene gy be ween 0.01-1 MeV, he solu ion ob ained using he
NEUTRO module shows a good ep esen a ion o he expe imen al da a endency.
Figu e 4: Expe imen al measu ed and Alya esul s o Leakage s. Ene gy o SINBAD es using
wo-dimensional slabs. Be ween pa en heses he wid h o he slabs conside ed, co esponding
o he hickness o he sphe e on which he expe imen s we e ca ied ou . 59 ene gy g oups
we e used o ene gy disc e iza ion.
Figu e 5 p esen s he compa ison be ween expe imen al da a and calcula ed alues o h ee-
dimensional domains, bo h laye s and sphe es. In each case, we used meshes o linea
hexahed a o he laye s and linea e ahed a o he sphe es. The Sn8 quad a u e o he laye
and FE64 o he sphe es we e used. 59 ene gy g oups we e used o he ene gy disc e iza ion.
The esul s ob ained o a silicon laye o 20 cm hickness (Figu e 5 a), a manganese laye o
20 cm hickness (Figu e 5 b), a ungs en sphe ical shell o 20 cm hickness (Figu e 5 c) and a
nickel sphe ical shell o 14 cm hickness (Figu e 5 d) show some dispe sion in di e en anges
o ene gy, depending on he expe imen s.
In gene al, he esul s o h ee-dimensional geome ies a e hose ha p esen g ea e dispe sion
wi h espec o he expe imen al measu emen s. And among hese, hose co esponding o
sphe ical geome ies a e he a hes om he measu ed alues. The e a e se e al easons ha
con e ge o hese esul s. The i s is he e o ha an uns uc u ed mesh on a ci cula geome y
in oduces, which in luences he en i e calcula ion om he in e nal bounda y condi ion, whe e
he incoming low is ixed, o he ou e edge o he domain. Secondly, he ex ension o he
domain in ol ed, which, al hough i is ea ed by a la ge numbe o elemen s, is s ill mo e
di icul o ep esen by means o a mesh o e eal space. This di icul y is pa ly a esul o
using wo-dimensional domains, which allow dense meshing wi hou aising he compu a ional
cos . Fu he mo e, as in he laye s, he slabs do no ha e a meshing e o and i is possible o
co ec ly ep esen he en i e analysed su ace.