Compu a ional Pa icle Mechanics
h ps://doi.o g/10.1007/s40571-020-00354-1
G and challenges o Smoo hed Pa icle Hyd odynamics nume ical
schemes
Rena o Vacondio1·Co ado Al oma e2,3 ·Ma hieu De Leffe4·Xiangyu Hu5·Da id Le Touzé6·S e en Lind7·
Jean-Ch is ophe Ma ongiu8·Sal a o e Ma one9·Benedic D. Roge s10 ·An onio Sou o-Iglesias11
Recei ed: 22 Decembe 2019 / Re ised: 16 July 2020 / Accep ed: 24 Augus 2020
© The Au ho (s) 2020
Abs ac
This pape p esen s a b ie e iew o g and challenges o Smoo hed Pa icle Hyd odynamics (SPH) me hod. As a meshless
me hod, SPH can simula e a la ge ange o applica ions om as ophysics o ee-su ace lows, o complex mixing p oblems
in indus y and has had no able successes. As a young compu a ional me hod, he SPH me hod s ill equi es de elopmen o
add ess impo an elemen s which p e en mo e widesp ead use. This e o has been led by membe s o he SPH Esea ch
and engineeRing In e na ional Communi y (SPHERIC) who ha e iden i ied SPH G and Challenges. The SPHERIC SPH
G and Challenges (GCs) ha e been g ouped in o 5 ca ego ies: (GC1) con e gence, consis ency and s abili y, (GC2) bounda y
condi ions, (GC3) adap i i y, (GC4) coupling o o he models, and (GC5) applicabili y o indus y. The SPH G and Challenges
ha e been o mula ed o ocus he a en ion and ac i i ies o esea che s, de elope s, and use s a ound he wo ld. The s a us
o each SPH G and Challenge is p esen ed in his pape wi h a discussion on he a eas o u u e de elopmen .
Keywo ds SPH ·Smoo hed Pa icle Hyd odynamics ·G and challenges ·Meshless ·Na ie –S okes equa ions ·Lag angian
BRena o Vacondio
ena o. acondio@unip .i
Co ado Al oma e
[email p o ec ed]
Ma hieu De Le e
ma hieu.de-le e@nex low-so wa e.com
Xiangyu Hu
[email p o ec ed]
Da id Le Touzé
da[email p o ec ed]
S e en Lind
s e en.lind@manches e .ac.uk
Jean-Ch is ophe Ma ongiu
[email p o ec ed]
Sal a o e Ma one
sal a o e.ma one@cn .i
Benedic D. Roge s
benedic . oge s@manches e .ac.uk
An onio Sou o-Iglesias
[email p o ec ed]
1Depa men o Enginee ing and A chi ec u e, Uni e si y o
Pa ma, Pa co A ea delle Scienze 181/A, 43121 Pa ma, I aly
1 In oduc ion
The smoo hed-pa icle hyd odynamics (SPH) nume ical
me hod was o iginally in oduced in 1977 o as ophys-
ical simula ions [41,60]. Since hen, SPH has p og essed
2Uni e si a Poli écnica de Ca alunya - Ba celonaTech, Ca e
Jo di Gi ona 1-3, 08034 Ba celona, Spain
3Ghen Uni e si y, Technologiepa k Zwijnaa de 60, 9052
Zwijnaa de, Belgium
4Nex low So wa e, 1 ue de la Noë, 44321 Nan es, F ance
5Depa men o Mechanical Enginee ing, Technical Uni e si y
o Munich, 85748 G aching, Ge many
6Ecole Cen ale Nan es, LHEEA Lab. (ECN and CNRS), 1 ue
de la Noë, 44300 Nan es, F ance
7School o Enginee ing, The Uni e si y o Manches e ,
Manches e M13 9PL, UK
8ANDRITZ Hyd o, Rue des Deux Ga es 6, 1800 Ve ey,
Swi ze land
9CNR-INM, INs i u e o Ma ine Enginee ing, Rome, I aly
10 School o Enginee ing, The Uni e si y o Manches e ,
Manches e M13 9PL, UK
11 CEHINAV, DACSON, ETSIN, Uni e sidad Poli écnica de
Mad id (UPM), 28040 Mad id, Spain
123
Compu a ional Pa icle Mechanics
signi ican ly and i is now a nume ical echnique adop ed
in nume ous di e en ields om as ophysics, o enginee -
ing applica ions o biological lows. I s meshless Lag angian
na u e, whe e he pa icles mo e acco ding o he go e ning
dynamics, has enabled i o be applied ela i ely easily o
a la ge ange o a eas. I s pa icle–pa icle in e ac ions wi h
compac suppo mean ha i is well sui ed o pa allelisa-
ion o accele a ion [17,25,75,86,88,106]. This has led o
he de elopmen and elease o nume ous SPH simula ion
codes ha a e now widely used. Wi h i s basis in Lag angian
and Hamil onian mechanics, he meshless o mula ion has
enabled p og ess in i s undamen al ma hema ical analy-
sis [99]. Despi e his, SPH can be s ill conside ed a young
nume ical me hod and i p esen ly su e s o some d aw-
backs in compa ison wi h classical Eule ian mesh-based
schemes such as Fini e Di e ence Me hod (FDM), Fini e
Elemen Me hod (FEM) o Fini e Volume Me hod (FVM).
These d awbacks include comple e p oo s o con e gence,
s anda disa ion o echniques, and use o pa ame e s o un
simula ions. Wi h SPH using smoo hing ke nels, and mul i-
ple o mula ions o ep esen media such as luids and solids
( o example, om weakly comp essible o incomp essible),
he me hod has mul iple ea u es ha equi e in ensi e in es-
iga ion.
The SPH Esea ch and engineeRing In e na ional Com-
muni y (SPHERIC), h ps://sphe ic-sph.o g, was ounded in
2005 wi h he aim o os e ing collabo a ion and o push
he de elopmen o he SPH me hod p o iding a ne wo k
o esea che s and indus ial use s a ound he wo ld as a
means o communica e and collabo a e. Since hen, i has
con inually s i ed o de elop he undamen al basis o SPH,
discuss cu en and new concep s, os e communica ion
be ween esea ch and use s, p o ide access o exis ing so -
wa e and me hods, de ine benchma k es cases, and o
iden i y he u u e needs o SPH. The annual in e na ional
wo kshops, a ended by o e 130 delega es, ha e equen ly
been he e en s ha ha e highligh ed he gaps in ou unde -
s anding and de elopmen needs. I is om hese e en s
ha an awa eness o key challenges in SPH has eme ged.
Concei ed in 2012, he SPHERIC S ee ing Commi ee o -
mula ed i e g and challenges (GCs) h ps://sphe ic-sph.o g/
g and-challenges o ocus he a en ion o esea che s, de el-
ope s and use s a ound he wo ld.
The SPH G and Challenges we e ini ia ed o b ing he
SPH communi y’s a en ion o a eas o SPH ha p e en
i s mo e widesp ead de elopmen and use. The GCs, and
his pape speci ically, do no aim o co e all ields whe e
esea ch in SPH is needed, o example ields such as u bu-
lence modelling, mul iphase lows (including he ea men o
sha p in e aces) clea ly need u he in es iga ion. Ins ead,
he issues highligh ed by he SPH G and Challenges a e gen-
e al and mus be add essed o SPH o compe e wi h mo e
es ablished me hods, such as FDM, FEM and FVM, whose
heo e ical ounda ions ha e been secu ed and whose s a e-
o - he-a simula ion packages a e ma u e.
SPHERIC has de ined he SPH G and Challenges as:
– GC1: Con e gence, consis ency and s abili y
– GC2: Bounda y Condi ions
– GC3: Adap i i y
– GC4: Coupling o o he me hods
– GC5: Applicabili y o indus y.
I is essen ial ha he SPH communi y a ound he wo ld
collabo a es and add esses hese SPH G and Challenges.
Wi hou being able o demons a e cha ac e is ics, beha iou
and applicabili y ha a e undamen al o any nume ical
me hod, SPH will con inue o be o e looked by some sci-
en i ic and use communi ies. Wi h he eno mous ange o
applica ions, his is unaccep able. In he pas decade, SPH
has made massi e p og ess, and his is e idenced by he
inc easing in e es and up ake o he me hod, by de elope s
and use s in bo h indus y and esea ch and he expo-
nen ially inc easing numbe o publica ions. In he yea s
2016–2019, he e ha e been 5 e iew pape s on SPH alone
[44,83,102,105,110]. The SPH G and Challenges ha e he e-
o e been o mula ed o ocus he wo ldwide de elopmen al
e o s in aking SPH o a poin whe e he undamen al he-
o y and p ac ical use a e ma u e so ha SPH akes i s igh ul
place in he ange o me hods a he disposal o scien is s and
enginee s.
To incen i ise his p ocess, he SPHERIC S ee ing Com-
mi ee inaugu a ed The Monaghan p ize, h ps://sphe ic-sph.
o g/joe-monaghan-p ize, named in honou o P o . Joseph
Monaghan, who has played such a key ole h oughou he
en i e li e o SPH. The Monaghan P ize has been ins iga ed
o highligh and ewa d ou s anding wo k ha helps add ess
and p og ess he SPH G and Challenges. The i s wo Mon-
aghan P izes we e awa ded in 2015 o Colag ossi e al. [23]
o hei pape on ee-su ace bounda y condi ions and in
2018 o Ma one e al. [62] o hei 2012 pape on de elop-
ing he densi y di usion echnique now so widely used.
Despi e p og ess, he e is s ill much wo k o do. Hence,
he SPHERIC S ee ing Commi ee conside ed i imely o
ask he leade s and leading igu es o each GC o summa ise
he cu en s a e o he a in hei espec i e challenge. This
pape p esen s a p ecis o each SPH G and Challenge iden-
i ying p og ess, and mos impo an ly he challenges ha we
ace and mus sol e. Resea che s and de elope s a e s ongly
encou aged o ocus a en ion on helping his collabo a i e
e o .
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Compu a ional Pa icle Mechanics
2 G and Challenge 1: Con e gence,
consis ency and s abili y (Lind, Hu)
The no ions o con e gence, consis ency and s abili y a e
undamen al and unde pin all nume ical me hods, wi h hese
concep s easie o o malise in some me hods han o he s.
SPH is a me hod whe e he e emains a signi ican lack o
unde s anding and o malism conce ning all h ee and, qui e
igh ly, add essing his is a G and Challenge. This iew-
poin men ions some ecen wo ks in he li e a u e ha shine
mo e ligh on hese issues in SPH, as well as posing a ew
philosophical ques ions o s i deba e. The abo e 3 p ope -
ies a e o cou se in e linked, a e all he Lax Equi alence
heo em p o es ha consis en ini e di e ence schemes o
well-posed linea p oblems a e s able i and only i hey a e
con e gen : a me hod may be s able, bu no con e ge; i may
also be consis en o some le el bu no con e ge as expec ed.
Rega ding s abili y, we ha e always been o una e in
SPH in compa ison wi h o he me hods by being able o
ob ain physically meaning ul esul s o ime s eps o eso-
lu ions whe e o he me hods o en b eak down. His o ically,
he pai ing and ensile ins abili ies ha e been a conce n, bu
ou unde s anding has much imp o ed in ecen yea s. Fo
example, conside he pai ing ins abili y and he bene i s o
using he Wendland ke nels [107] wi h nonnega i e Fou ie
ans o ms [30]. Simila ly, he use o a backg ound p essu e
is bene icial in p e en ing he ensile ins abili y, al hough
excessi e nume ical dissipa ion can a ise. No e he e y ac
ha adding a cons an backg ound p essu e a ec s SPH a all
ela es o issues a ound conse a ion and consis ency, which
we will men ion sho ly.
Clea ly, pa icle dis ibu ion is key o main aining s abili y
and addi ional nume ical ea men s ha imp o e dis ibu-
ions, such as pa icle numbe -densi y cons ain [48], pa -
icle shi ing [57,69,108] and anspo o mula ion [2,111],
ha e inc eased in popula i y in ecen yea s gi en hei e i-
cacy and ela i e ease o implemen . P ac ically speaking,
in weakly comp essible SPH (WCSPH) s abili y can also be
main ained h ough di usion (physical o nume ical), and
ollowing he ea lies uses o a i icial iscosi y, we now
ha e some sophis ica ed app oaches including, o example,
del a-SPH [62] and i s mo e ecen a ian del aplus-SPH
[90], which combines di usi e e ms in he conse a ion o
mass equa ion wi h shi ing o imp o ed pa icle dis ibu-
ions. Indeed, o mula ions inco po a ing a i icial iscosi y,
del a-SPH [5,62], and Riemann sol e s [98] can all be seen
as di e en s abilisa ion al e na i es o he explici spa ially
cen ed SPH scheme. An al e na i e SPH o mula ion is
he so called Incomp essible SPH (I-SPH), which is based
on a di e gence- ee p ojec ion [27] o he eloci y ield,
[48,51,57,84]. I-SPH models gene a e smoo he p essu e
ields, a oiding he in oduc ion o addi ional explici di -
usi e e ms. We a e s ill a long way om o malising much
o his—impo an headway is being made ega ding s a-
bili y in ime s epping in weakly comp essible SPH [100]
and in incomp essible SPH [49,101]—bu a con inued goal
should be he de e mina ion o well-de ined s abili y egions
wi h bounds ha ha e a known dependence on disc e isa ion
and ke nel pa ame e s, physical pa ame e s, and nume i-
cal ea men pa ame e s (e.g. shi ing coe icien s, del a
pa ame e s). The oppo uni y o u he inpu om ma h-
ema icians/nume ical analys s he e is g ea .
Like s abili y, con e gence depends c i ically on pa icle
dis ibu ions. Fo example, Quinlan e al. [79]ha ep o-
ided impo an guidance on con e gence, wi h dependence
seen on smoo hing leng h, pa icle spacing, ke nel smoo h-
ness, and pa icle diso de . Two key con ibu ions o he
e o include he e o due o he smoo hing ope a ion and
he nume ical in eg a ion (o disc e isa ion) e o (due o
he spli ing o ou domain in o pa icles). The o me is
commonly second o de in smoo hing leng h, and he la -
e can be quan i ied i we spli ou in eg al in o equi-spaced
ec angula pa icles as pe he ec angle o apezoid ules.
Consequen ly, as we e ine and dec ease smoo hing leng h,
he numbe o neighbou s should also be inc eased app o-
p ia ely. Howe e , o p ac ical easons, his is o en no
done, esul ing in he smoo hing e o e en ually becoming
sa u a ed. I we ake ca e in e inemen o e uni o m (e.g.
Ca esian) a ays o pa icles, SPH can be shown o con-
e ge in nume ical expe imen s wi h a es o con e gence
ma ching heo e ical e o measu es ex emely well. E e s
e al. [32] de i ed he a e o con e gence o SPH nume ical
scheme using he leas ac ion p inciple. F anz & Wendland
ha e ecen ly p o ided a ma hema ical p oo o con e gence
o SPH o a speci ic ba o opic luid and unde ce ain p op-
e ies o he unde lying ke nel [39]. Howe e , as soon as
some le el o pa icle diso de is in oduced, hings become
a mo e di icul . E o s and con e gence a es a e much
mo e di icul o quan i y, wi h con e gence la -lining, e en
di e ging, once pa icles become su icien ly diso de ed—
no ideal when you pa icles a e Lag angian.
This close dependence o con e gence on pa icle dis-
ibu ion seems o ha e mo i a ed a g owing numbe o
esea che s o explo e A bi a y Lag angian Eule ian (ALE)
o mula ions o SPH [74,98]. The ully Eule ian SPH me hod
can con e ge eadily and o high o de s o spa ial accu acy
[56] (see Fig. 1), while ALE-SPH ( o example, [74]) pe mi s
s udy o a g ea e class o lows while also allowing con ol
o e pa icle dis ibu ions in o de o imp o e accu acy and
con e gence. The e is some eally p omising ongoing wo k
he e [7,47,71,112], and his is an encou aging pa hway, a e
all, e en i one s ongly alues he Lag angian na u e, o clas-
sical SPH, a legi ima e ques ion is whe he he de e mined
pa icle eloci y is indeed he Lag angian eloci y. O cou se,
ma hema ical o malism is lacking he e also, and quan i ica-
123
Compu a ional Pa icle Mechanics
Fig. 1 High-o de con e gence o an SPH g adien o di e en ke nels
see [56] o mo e in o ma ion
ion o e o and con e gence a es o i egula dis ibu ions
in pa icula should be a key goal.
Consis ency and con e gence a e closely linked, and while
consis en o mula ions may be cons uc ed o a bi a y pa -
icle dis ibu ions, his can be cos ly and con e gence is no
necessa ily ideal. The SPH disc e isa ion o de i a i es has
wo ypical o mula ions: he an i-symme ic and symme ic
o mula ions. The nume ical e o s due o hese wo o -
mula ions a e qui e complex and a e s ongly dependen on
pa icle dis ibu ion [79]. Wi h he an i-symme ic o mula-
ion, he SPH disc e isa ion o compu ing p essu e o ces
on a pa icle implies ha wi h momen um conse a ion o
he pa icle sys em we canno es ima e co ec ly he an-
ishing g adien o a cons an scala ield, in p ac ice he e
is a non- anishing o al o ce ac ing on a pa icle in a ield
wi h cons an p essu e. On he o he hand, wi h he symme -
ic o mula ion, he SPH disc e isa ion o compu ing he
densi y a ia ion o a pa icle p o ides ze o-o de consis-
ency, and a uni o m eloci y leads o a anishing densi y
a ia ion. One may expec o cancel inconsis ency e o o
he p essu e ield by applying he symme ic o mula ion o
he disc e ised momen um equa ion. The dilemma is ha he
conse a ion o momen um, one o he mos impo an p op-
e ies o he o iginal SPH me hod [66,67], is no sa is ied
any mo e. Again, pa icle dis ibu ions emain key he e, and
ecen in es iga ions ha e ocused on i e a i e edis ibu ion
p ocedu es based on anspo eloci ies ( [58]) o shi ing
[52,93]. Such app oaches ha e shown p omise in eco e ing
consis ency wi hou co ec ion o SPH schemes which may
also wan o e ain conse a ion. Impo an ly, wi h bo h con-
sis ency and conse a ion in place, he e could be a ou e o
o malising con e gence in SPH ia he Lax–Wend o he-
o em, wi h con e gen conse a i e schemes o hype bolic
equa ions p o iding a leas weak solu ions.
The modynamic consis ency o SPH nume ical schemes
has been analysed by di e en au ho s, showing ha
Hamil onian-consis en o mula ions ensu e also o al ene gy
conse a ion [78]. Fo weakly comp essible SPH, An uono
e al. [6] ha e shown how di e en ene gy e ms e ol e du -
ing he nume ical simula ion, and he same analysis has been
ex ended o luid–solid in e ac ion in [18]. Khayye e al. [53]
ha e also in es iga ed he ene gy conse a ion in incomp ess-
ible SPH schemes showing ha be e ene gy conse a ion
is achie ed when co ec ed SPH in e pola ion is adop ed.
In summa y, a key goal o his g and challenge emains
in imp o ing he ma hema ical o malism a ound quan i i-
ca ion o e o , con e gence, and s abili y. Hence, he e a e
signi ican challenges going o wa d:
1. The inal objec i e o GC1 is o de elop a igo ous ame-
wo k whe e we unde s and he nume ical mechanisms
in SPH, he heo e ical easons explaining how SPH
wo ks, i s limi a ions and he need o modi ica ions o
he me hodology and accompanying analysis.
2. This analysis is made ex emely di icul by he low
being Lag angian, as well as by he ac ha pa icle ol-
umes ha e no explici spa ial shape (no aces, cells) and
do no o m a pa i ion o uni y du ing he ime e olu ion.
Ne e heless, u he esea ch on hese opics will enable
us o un in o med simula ions wi h con idence, and will
inspi e con idence in SPH in ex e nal ields and in indus-
y. Howe e , we should also no be a aid o pose ques ions
and o highligh nuance. Fo example, wha do we mean by
con e gence? I we a e sol ing a pa ial di e en ial equa-
ion, assuming he e is a solu ion, hen con e gence becomes
meaning ul. I , howe e , we a e wo king a he mesoscale,
whe e many ashionable p oblems eside and whe e he con-
inuum hypo hesis s a s o b eak down, he disc e e pa icle
sys em ( ha was always unde lying) becomes appa en , and
ou usual no ion o con e gence loses meaning (i.e. we do
no wan x o go o 0!). O cou se, i is in such examples o
he e sa ili y and lexibili y o SPH ha we ind he easons
o he me hod’s g ea appeal.
3 G and Challenge 2: Bounda y condi ions
(Sou o-Iglesias)
In o de o close he luid dynamics equa ions, ini ial (ICs)
and bounda y condi ions (BCs) a e necessa y. BC includes
solid bounda ies ( ee slip, no slip, p essu e no mal de i a-
i e), ee su ace, inle /ou le (aka open BCs—OBCs), s ess
condi ions in s uc u al mechanics, hose ela ed o he cou-
pling wi h o he models, e c., and ICs a e included in his
123
Compu a ional Pa icle Mechanics
challenge since hey usually equi e special ea men in SPH,
e.g. when a hyd os a ic condi ion is needed. This need a ises
mos ly due o un easibili y o exac ly link mass and olume
in SPH. Due o he meshless na u e o he me hod, impos-
ing bounda y condi ions is a om i ial in SPH, leading
o in ense ela ed esea ch since he i s applica ions o he
me hod o bounded lows by Monaghan [64] in he nine ies.
I is ele an o men ion ha in ecen SPH e iew pape s
[42,43,65,67,102], he e a e speci ic e iew sec ions on BCs.
The in luen ial e iew by P ice [78] does no , howe e , con-
ain any e e ence o BCs as, in as ophysics, hey a e less o
an issue han in ypical enginee ing scales.
To include ICs and BCs in SPH, esea che s use a ious
echniques. The e a e a numbe o key issues ha emain o
be ully add essed, such as:
1. How o include BCs wi hou loosing in insic SPH con-
se a ion p ope ies?
2. How o include BCs consis en ly and wi hou comp o-
mising s abili y? This is di ec ly ela ed wi h he ole o
bounda y in eg als.
3. How o include solid wall BCs o ac ual geome ies wi h
complex shapes (2D, 3D)?
4. How o p o ide an ini ial dis ibu ion o pa icles which
a oids he onse o shocks once he ime-in eg a ion
s a s?
5. How o ea con ac lines be ween ee su aces and solid
bounda ies?
6. How o ea back lows (aka eci cula ion) when imple-
men ing OBCs?
7. How o implemen BCs in he in e ace be ween subdo-
mains sol ed wi h di e en me hods?
8. How o accu a ely impose BCs in Incomp essible SPH
(ISPH) in complex lows?
9. How o accu a ely impose BCs when pa icle shi ing
(wi hin a consis en ALE amewo k o no ) is used?
Some ecen in e es ing e e ences ha e looked in o hese
ques ions: Ni e al. [70] implemen ed a wa e lume wi h
SPH using OBCs bu did no look in o eci cula ion issues.
Along he same line, Bouscasse e al. [11] used OBCs o
simula ing he iscous low a ound a subme ged cylinde .
In o de o a oid back low, hey had o signi ican ly ex end
he low domain ups eam and downs eam, as well as limi -
ing he simula ion ime (see Fig. 2). Back low is held in
FVM-VOF me hods by indica ing he physical p ope ies
o he incoming luid, applying o i he local low p ope -
ies ( eloci y, empe a u e, e c.), bu i is no clea how o
implemen i a Lag angian app oach. Ta uni e al. [91]ha e
ecen ly ex ended OBC algo i hms o he popula GPU HPC
implemen a ion DualSphysics, and Wang e al. [104]ha e
p oposed a no el OBC implemen a ion based on he me hod
o cha ac e is ics using imeline in e pola ions.
Long- ime-du a ion simula ions o ee-su ace lows
ha e been adi ionally an issue in SPH due o he onse o s a-
bili y p oblems. Howe e , G een and Pei ó [45] ha e ecen ly
been able o ca y ou long and accu a e simula ions o
lows inside anks by using ixed/p esc ibed mo ion dummy
pa icles de eloped by Adami e al. [1], and by pe o m-
ing a good selec ion o simula ion pa ame e s. Ex ending
low ields ou side o he bounda ies o o ce BCs has been
ecen ly in es iga ed by Fou akas e al. [38]. They claim hei
locally uni o m s encil-based o mula ion is able o model
solid bounda y condi ions in complex 2-D and 3-D geome-
ies, wi h imp o emen s o e exis ing echniques based on
dummy pa icles (e.g. [1,26]) pa ially achie ed by using
δ−SPH [62] o educe spu ious p essu e oscilla ions. How-
e e , alida ion wi h non-o hogonal geome ies was no ye
pu sued. The low ield ex ension echniques ha e also been
ecen ly used in hea ans e applica ions by Wang e al.
[103].
Rega ding BCs a ec ing consis ency o he ope a o s,
Fouge on and Aub y [36] ha e p oposed a no el me hod
based on non-bounda y i ed clouds o poin s; hey ede ine
he Lag angian na u e o he model by c ea ing a se o nodes
on he bounda y, which hen use o app oxima e he di e -
en ial ope a o s. They use his app oach in ellip ic equa ions,
and hough appealing ideas can be ound, he applica ion o
ypical SPH p oblems, such as wa e-body in e ac ions, is no
e iden o us.
In insic good conse a ion p ope ies a e an asse o he
SPH me hod. How hese a e a ec ed by BCs has been in es-
iga ed by Ce cos-Pi a e al. [18] in he p esence o luid–solid
in e ac ions, when hese a e modelled using ghos pa icles.
They showed ha due o he solid BCs, he ene gy equa ion
o he pa icle sys em con ains some ex a e ms ha end o
anish when he spa ial esolu ion is inc eased ( e y slowly),
and ha a ec he ene gy conse a ion o he sys em. Based
on he es cases hey un, hey conjec u ed ha he con ibu-
ion is dissipa i e, bu no igo ous p oo was p o ided.
As o bounda y in eg als (see [35] o a undamen al
e e ence on his kind o BC implemen a ion, whe e o -
mulae o i s and second de i a i es wi h a semi-analy ic
o mula ion wi h bounda y in eg als a e p oposed and ali-
da ed), hey p o ide consis en o mula ions and a e a i s
choice in ex emely agmen ed lows, such as hose ound
in hyd oplaning simula ions [19]. Fo his ype o echnique,
Calde on e al. [13] ha e ecen ly de eloped a o mula ion
ha imp o es he compu a ion o he eno malisa ion ac o
in wo and h ee dimensions. One main p oblem o bounda y
in eg als is ha he in insic good conse a ion p ope ies o
SPH a e a ec ed by he use o eno malised ope a o s.
Looking in o incomp essible SPH and BCs, Takahashi e
al. [92] p o ided an in e es ing discussion on he di icul ies
o imposing Di ichle and Neumann BCs, including some
imp o emen s. Rega ding ALE o mula ions, Oge e al. [74]
123

Compu a ional Pa icle Mechanics
Fig. 2 Flow a ound a cylinde in he p esence o a ee su ace a
Reynolds numbe equal o 180 (see Bouscasse e al. [11] and Cola-
g ossi e al. [24] o mo e de ails on his ype o pa icula lows). The
colo code ep esen s he s eak-lines by iden i ying he e ical posi-
ion in he unpe u bed inle . The ho izon al and e ical coo dina es a e
made non-dimensional wi h he cylinde diame e . (Colo igu e online)
epo ed he need o emo e shi ing when close o he ee
su ace, de ining in u n he ghos luid p ope ies wi hou
equi ing any speci ic ALE- ela ed co ec ion and Khayye
e al. [52] applied he i e a i e shi ing, o iginally p oposed in
[93] o mul iphase and ee-su ace lows in he ISPH ame-
wo k.
Looking ahead, he e a e some clea challenges going o -
wa d:
1. Iden i ying and alida ing BCs ha a e obus o a bi-
a ily complex non-o hogonal geome ies o he as
ange o SPH applica ions.
2. Ex ending he beha iou o SPH BCs o possess highe -
o de con e gence p ope ies.
3. Main aining he in insic conse a ion p ope ies o SPH
while e aining he consis ency o ope a o s.
4. Supplemen ing he eme ging p oo s o con e gence o
GC1 wi h he added complica ion o BCs.
4 G and Challenge 3: Adap i i y (Vacondio,
Roge s)
Adap i i y is he capabili y o a nume ical scheme o use a
domain disc e isa ion based on elemen s wi h di e en size.
Fo Eule ian mesh-based me hods such as ini e olume,
ini e elemen s o ini e di e ences hose elemen s a e he
g id cells, whe eas in Lag angian meshless-based nume ical
me hods hey a e he compu a ional nodes ha mo e wi h he
luid eloci y. Adap i i y is a c ucial ea u e o nume ical
schemes. I allows us o inc ease he numbe o compu a-
ional nodes (cells o pa icles) only in he po ions o he
domain whe e he low ea u es equi e highe esolu ion.
In his way, he o al numbe o compu a ional nodes (and
so he compu a ional cos o he simula ion) used o dis-
c e ise a domain can be d ama ically dec eased, o a gi en
le el o e o . In mesh-based me hods, a iable esolu ion
is a common ea u e and i has been in oduced in se e al
di e en ways. O en e e ed o as Adap i e Mesh Re ine-
men (AMR), he mos common app oaches a e uns uc u ed
g ids o quad ee g ids. Mo eo e , se e al di e en algo-
i hms ha e been used success ully o dynamically adjus
he mesh esolu ion, acco dingly o some measu es o he
disc e isa ion e o o smoo hness indica o s o he nume -
ical solu ions (see, o example, [31,50]). Despi e he need
o in oduce a iable esolu ion in SPH nume ical schemes
o luids, almos all SPH codes a e based on uni o m eso-
lu ion and his p e en s he use o SPH models o simula e
all enginee ing p oblems which a e inhe en ly mul iscale.
Fo comp essible luids and as ophysical simula ions, a
consis en o mula ion which conside s he space a iabili y
o he smoo hing leng h has been de i ed many yea s ago [41,
46,78], and in his app oach, he conse a ion o undamen al
p ope ies is ensu ed and he esolu ion implici ly inc eases
in high-densi y egion (and dec eases i in low-densi y one).
E ec i ely, his c ea es pa icles wi h di e en olume bu
cons an masses. Un o una ely, he same app oach canno
be used o weakly comp essible (o s ic ly incomp essible)
luids whe e densi y emains (app oxima ely) cons an and
so pa icles wi h di e en olumes ha e o ha e also di e en
masses. Simila o as ophysical applica ions, in enginee ing
he Lag angian cha ac e is ics o SPH can lead o spa se o
condensed dis ibu ions o pa icles, which can be add essed
by me ging/spli ing pa icles o p ese e a good in e pola-
ion accu acy. When he compe ing demands o adap i i y
ac oss phases wi h di e en dis ibu ions o pa icles a e con-
side ed, one phase wi h a di e en dis ibu ion o pa icles
migh gene a e e o s o a g ea e magni ude and he e o e
can ha e he opposi e e ec o he uni ied goal o a ge ing a
local e inemen and minimised e o .
Ini ial e o s ha e been made o weakly comp essible
SPH models by in oducing egions wi h di e en esolu-
ion a he beginning o he simula ions [9,10,72,76,77].
A e wa ds, wi h he aim o dynamically a ying he pa -
icle esolu ion, some au ho s p oposed some p ocedu es
o dynamically inc ease and educe he pa icle esolu ion
123
Compu a ional Pa icle Mechanics
[8,80,94,95]. Ve y ecen ly, Sun e al. [89] simula ed low
pas di e en bodies in he p esence o a ee su ace by
using he Adap i e Pa icle Re inemen (APR) me hodology
p oposed in [21]. Sp eng e al. [87] p oposed a c i e ion o
au oma ically adjus he pa icle esolu ion acco dingly o
some measu e o he SPH spa ial disc e isa ion e o . Despi e
he p og esses in de eloping dynamic pa icle adap i i y, we
hink ha some majo challenges ha e s ill o be add essed
in o de o ob ain a me hodology ha is su icien ly obus
o be adop ed by p ac i ione s and indus y. Looking a in o
he u u e, om he use s’ pe spec i e, dynamic adap i i y
should be ully au oma ed and ac i a ed only when needed.
Full au oma ion equi es c i e ia o be de eloped ha con ol
he ac i a ion. A ques ion hen a ises as o wha hese c i e-
ia should be and how hey should ope a e? While his has
been well in es iga ed in adap i e mesh e inemen (AMR),
he same concep s do no necessa ily apply in SPH since he
na u e o he disc e isa ion is di e en . Mos impo an ly, i is
p esen ly unclea wha is he bes gene al app oach, and his
equi es (i) a ocused esea ch e o om he SPH commu-
ni y and (ii) an unde s anding om use s ha implemen ing
and using adap i i y in SPH aces some key challenges and
is a om s aigh o wa d. Howe e , i is al eady clea ha
he e a e a leas h ee key objec i es:
1. E o minimisa ion: i is impossible o a oid he in o-
duc ion o e o , bu any o m o SPH adap i i y should
gua an ee ha he e o has been minimised. To da e,
limi ed a en ion has been gi en o his [33,94,95]. Too
o en, schemes simply spli pa icles in o an a bi a y
numbe ( o example, 4) o so-called daugh e pa icles
(mo i a ed by simplici y o ease-o -coding) wi h li le
conside a ion o he e o and how i p opaga es h ough-
ou he solu ion. Simila o ma u e AMR schemes, e o
minimisa ion is a na u al candida e as a c i e ion o APR.
2. Uni o m e o dis ibu ion o a gi en esolu ion: he
dynamic adap a ion o pa icles should no gene a e addi-
ional e o o inconsis encies due o he iola ion o
conse a ion p ope ies, in compa ison wi h a uni o m
pa icle dis ibu ion con igu a ion wi h he same esolu-
ion
3. Robus schemes o all applica ions: due o i s lexibil-
i y, he ange o SPH applica ions is huge wi h highly
complex p ocesses. This na u ally p esen s a challeng-
ing ques ion—how o de elop pa icle adap i i y ha is
widely applicable and obus ? I ce ain ypes o adap-
i i y only wo k o a es ic ed numbe o ype o
applica ions, his calls in o ques ion he alidi y o he
app oach—in p ac ice his means ensu ing consis ency
and con e gence.
In addi ion o he heo e ical conside a ions and de elop-
men s, he e a e mul iple challenges going o wa d:
1. Implemen a ion wi h HPC and eme ging echnology:
E en wi h APR, wi h i s disc e isa ion SPH will need
some o m o ha dwa e accele a ion o he o eseeable
u u e. In he pas decade, he e has been a unda-
men al shi om as e clock speeds o di e en ypes
o pa allelism. Fo adap i i y, his poses he challenge
o implemen a ion. Wi h di e en ypes o ha dwa e
con inually appea ing, de eloping implemen a ions o
adap i i y ha a e u u e-p oo ed will a oid cos ly ecod-
ing.
2. Mul i-phase implemen a ions: Applica ions in ol ing
mul iple phases can be ex ao dina ily complex, and o
da e, only simple cases o applica ions ha e been simu-
la ed in SPH. De eloping obus adap i i y schemes o
mul i-phase lows whose p ope ies can e ol e ep esen s
a o midable challenge.
5 G and Challenge 4: Coupling o o he
models (Ma one, Al oma e, Le Touzé)
The SPH me hod is na u ally able o esol e mul i-mechanics
p oblems and include di e en physical models in i s mesh-
less o malism. As wi h o he Lag angian meshless me hods,
SPH is e y accu a e and e icien when dealing wi h mo -
ing bounda ies and complex in e aces, which a e gene ally
add essed wi h di icul ies by con en ional nume ical me h-
ods (e.g. FVM, FEM). Howe e , o p oblems whe e he
la e me hods a e cu en ly used and well es ablished SPH
is gene ally less e ec i e and, o he same le el o a ained
accu acy, esul s a e mo e cos ly.
In se e al con ex s, i can be much mo e e ec i e o couple
an SPH sol e o ano he nume ical sol e , hus enhancing he
capabili ies o bo h me hods wi hin hei speci ic applica ion
ields. In his way, a wide ange o p oblems is e icien ly
add essed. The coupling algo i hm and he ela ed imple-
men a ion complexi y can la gely a y depending on se e al
aspec s:
1. One-way (o line) o wo-way coupling;
2. He e ogenei y o he modelled physics (e.g. po en ial
low/Na ie -S okes, luid/solid, comp essible/
incomp essible, e c.);
3. Lag angian o Eule ian app oach adop ed in he me hod
coupled o SPH;
4. Disc e e coupling in e aces be ween sol e s (mesh/
meshless, sha p in e ace/blending egion, e c.);
5. Time s epping and s abili y o he coupled algo i hm (e.g.
explici /implici ime in eg a ion, mul iple ime s ep);
6. P ese a ion o conse a i e quan i ies by he coupling.
Besides, he complexi ies ela ed o he coupling o e y
di e en sol e s can be coun e balanced by imp essi e gains
123
Compu a ional Pa icle Mechanics
in e ms o e iciency [20]. Mos o he wo ks ega ding
SPH coupling add ess luid–s uc u e in e ac ion (FSI) p ob-
lems o which he solid s uc u e is gene ally sol ed by
Fini e Elemen Me hods (FEM) and Disc e e Elemen Me h-
ods (DEM). The Lag angian cha ac e o hose model has
allowed a qui e as de elopmen o his kind o coupling and
has been a ge ed in he i s a emp s o coupling he SPH
me hod (see A away e al. 1994). In pa icula , SPH-FEM
coupling is eaching ma u i y and has been used in se e al
ecen wo ks add essing hyd o-elas ici y p oblems (see, e.g.
[37,55,59,109]) p o ing ha his coupling pa adigm can be
highly compe i i e in FSI p oblems [85].
SPH-DEM coupling has been mos ly used o p oblems
in which se e al solid igid bodies in e ac wi h a luid low
[15,81] including g anula lows [16,61]. Ve y ecen ly cou-
pling wi h open sou ce mul i-mechanics lib a ies has been
implemen ed o simula e luid-mechanism in e ac ions by
modelling ic ional and mul i- es ic ion-based beha iou s
[14].
Fu he mo e, SPH coupling has been la gely de eloped
o coas al enginee ing pu poses. In his case, SPH is cou-
pled wi h non-linea shallow wa e equa ion models [3,4]o
po en ial low sol e s in he o m o spec al me hods [73]o
ini e di e ence [96] o sol ing wa e p opaga ion in he a
ield and es aining SPH in he egion whe e wa es uc u e
in e ac ions and wa e-b eaking a e expec ed. In Fig. 3, one
example o a coupling scheme be ween OceanWa e3D and
DualSPHysics [25] is shown. This includes he simula ion
o ship mo ions and he associa ed sloshing dynamics in he
in e nal anks as ecen ly done in [82] and Bulian and [12].
Finally, a ecen and g owing b anch is he coupling be ween
Fini e Volume Schemes (FVM) and SPH (see, o example,
Fig. 4). In his case, he coupling s a egy aims a low simu-
la ions in which he accu acy and he abili y o g id s e ching
o he FVM can be use ully coupled wi h he SPH p ope ies
in modelling complex in e aces [34,54,63,68].
To summa ise, coupling SPH models wi h o he nume ical
sol e s is a clea e ec i e s a egy o expand he in insic
capabili ies o SPH-based models o sol e complex physics
and hyd odynamics, while educing he compu a ional cos
ela ed o he mesh ee na u e o he me hod.
1. Coupling algo i hms a e o complex implemen a ion and
gene alisa ion due o he di e en na u e o he coupled
models: om one side a ully Lag angian SPH me hod,
om he o he FEM, DEM, FVM, o ini e di e ence
schemes.
2. In addi ion o he di e ences in o mula ions, he e is he
addi ional challenge o coupling me hodologies ha a e
sui ed, o ha e been highly op imised, o e y di e en
ypes o ha dwa e accele a ion and coding cons uc s.
This is non- i ial.
Fig. 3 P inciple o 2D coupling be ween OceanWa e3D and Dual-
SPHysics a ound a s uc u e unde wa e ac ion om Ve b ugghe e al.
[96]. The op pa shows he comple e domain in OceanWa e3D. The
bo om pa illus a es he DualSPHysics zone
Fig. 4 Coupled SPH-FVM simula ion o a sloshing low in a ank wi h
a co uga ed bo om om Chi on e al. [20]. Top: SPH pa icles (blue)
and FVM g id (black). Bo om: a ime ins an o he e olu ion showing
o ici y con ou s and he ee su ace p o ile c ossing he coupling
in e ace. (Colo igu e online)
No e, howe e , ha he coupling ask is eased by he meshless
na u e o he SPH me hod compa ed o couplings be ween
he e ogeneous mesh-based me hods (e.g. FVM wi h FEM)
whe e mesh in e pene a ion is a di icul issue. The achie ed
e iciency and i s encou aging esul s jus i y he inc easing
use o coupling algo i hms o p ac ical applica ions and eal
enginee ing p oblems
6 G and Challenge 5: Applicabili y o
indus y (de Leffe, Ma ongiu)
Indus y has been slow o accep he SPH me hod as a
“se ious” CFD me hod. Apa om some e y speci ic
applica ions, such as bi d s ike o high-p essu e wa e je s
123
Compu a ional Pa icle Mechanics
impac ing pel on u bine blades, i is only e y ecen ly ha
we can no e a g owing in e es in he SPH me hod in he
indus ial wo ld. The main easons o his ecen change a e
he esea ch p og ess by he scien i ic communi y on G and
Challenges 1 and 2 has con inced enginee s o he abili y o
he SPH me hod o sol e applica ions wi h highly dis o ed
complex in e aces wi h applica ions such as gea box o i e
aquaplaning becoming mo e equen .
As he doo o he indus y begins o open, i is essen ial
ha he me hod con inues o p og ess o maximise oppo -
uni ies o demons a e i s sui abili y o u u e applica ion.
One o he i s ques ions asked by an indus ialis ha is keen
o use SPH o a speci ic applica ion is ela ed o he elapsed
ime o he simula ion. The p og ess in High-Pe o mance
Compu ing (HPC) in accele a ing SPH so wa e on di -
e en a chi ec u es (CPU o GPUs), enables SPH o be
compe i i e wi h con en ional mesh-based me hods. How-
e e , me hods such as FVM and FEM ha e also p og essed
in cap u ing complex in e aces, so he challenge emains
open and he ields whe e he SPH me hod is mo e e icien
could be u he educed. Two undamen al cha ac e is ics
make SPH inhe en ly mo e expensi e han classical mesh-
based me hods: (i) he much la ge numbe o neighbou s
o a gi en compu a ional poin , and (ii) he smalle com-
pu a ional ime s ep ha has o be adop ed due o he
weakly comp essible explici o mula ion. Fo he i s poin ,
o da e a ma u e echnical solu ion has no ye eme ged.
Howe e , wo k has been done o inc ease he o de o con-
e gence o SPH schemes o a gi en numbe o neighbou s
(see G and Challenge 1). Ne e heless, his gene a es addi-
ional calcula ion, and he gain in e ms o accu acy is no
ye demons a ed o indus ial applica ions. Fo he sec-
ond poin , an impo an wo k has been done o de elop
semi-implici incomp essible SPH (ISPH) schemes based on
di e gence- ee p ojec ion [27]. The GPU implemen a ion
o ISPH, as epo ed in [22], will p obably ein o ce i s e i-
cacy.
The gain on he ime s ep aises in e es ing ques ions i
he e is a loss o accu acy on he desc ip ion o he ee
su ace. A i al poin o no e he e is ha p og ess in HPC
should no be pu sued o he de imen o he accu acy o
he nume ical scheme. Fo example, when po ing on GPU,
he emp a ion o in oducing simpli ica ions in he adop ed
nume ical scheme is o u he inc ease i s e iciency, losing
he in e es o ha d-won gains in G and Challenges 1 and
2. I he SPH me hod is no able o p og ess on he HPC
objec i es compa ed o o he me hods, SPH should be used
in po ions o he domain cha ac e ised by s ong dynam-
ics and complex in e aces. The comple e simula ion can be
ob ained by coupling SPH wi h o he nume ical me hods (see
G and Challenge 4) [20].
The second ques ion asked by an indus ialis is he abil-
i y o he SPH me hod o simula e phenomena cha ac e ised
by complex physics as u bulence, bounda y laye , phase
change, he mal di usion and con ec ion, su ace ension,
e c. Indus ial SPH codes canno simula e all he a o emen-
ioned phenomena (wi h he excep ion o he mal ones).
I is now c ucial o he SPH me hod o include addi-
ional physical p ocesses o simula e he ull complexi y
o indus ial cases. This is bes illus a ed wi h an exam-
ple: he ocke o sa elli e ank in mic og a i y. The liquid
phase is subjec ed o an impo an sloshing wi h a com-
plex in e ace. The case he e o e seems e y p omising o
he SPH me hod. Excep ha he e a e compe ing dom-
ina ing e ec s o su ace ension wi h he con ac angle
and he mal physics due o he sun’s adia ion. The uel
o oxidan is in equilib ium be ween i s gaseous phase and
liquid phase, causing signi ican phase changes. Ano he
example is he wa e impac du ing slamming o di ching
e en . The case is dynamic wi h a complex ee su ace.
The case he e o e seems also e y p omising o he SPH
me hod. Excep ha i he case has s ong dynamics ope -
a ing a di e en scales he e is he domina ing e ec he
gas phase, whe e he eal comp essibili y o he gas mus
be conside ed. In some ex eme cases, phenomena o ca i a-
ion may appea . The SPH me hod mus p og ess o p opose
obus physical models o simula e hese physical phenom-
ena.
Many exci ing challenges a e wai ing o he SPH me hod
whe he in HPC o in e ms o modelling complex physics, i
SPH wan s o con ince he indus y on a long- e m basis and
no emain con ined o a small applica ion co e. The p og ess
made by he adi ional olume-o - luid (VOF) me hod o
mo e ecen me hod such as La ice–Bol zmann Me hod
(LBM), Ma e ial Poin Me hod (MPM), Mo ing Pa icle
Simula ion (MPS), Pa icle Fini e Elemen Me hod (PFEM)
mus se e as a mo i a ion and a sou ce o inspi a ion o he
SPH communi y.
The ecen con ibu ions om he SPH esea ch com-
muni y ha e b ough signi ican p og ess likely o os e
he adop ion o SPH among indus y. The appea ance o
ools wi h G aphical Use In e aces (GUIs) o he p e-
and pos -p ocessing o SPH simula ions is no iceable (see,
o example, Figu e 5). DesignSPHysics [97] and Visual-
SPHysics [40] p o ide a comple e simula ion ool chain
dedica ed o SPH simula ions. An al e na i e has been de el-
oped based on Pa aView [29]. Ad anced analysis o low
ea u es s ill elies mainly on he p ojec ion o he pa i-
cles da a on o a g id. Fo he c ea ion o he ini ial pa icle
dis ibu ion in complex geome ies, he pa icle packing algo-
i hm has gained popula i y as in [28]. The ease o use o he
me hod will p obably bene i om he ecen imp o emen s
o he dynamic and adap i e pa icle e inemen echniques.
Signi ican con ibu ions in his ield ha e been gi en by
[94] and [21]. A u he de elopmen o hese echniques
will elie e simula ion enginee s om he bu den o se -
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