Coupling Pa icle Simula ions
Challenges, S a egies, and he ON-DEM Vision
Tho n on, Weinha , Sc ase, Schneide , Pos , Ba e o, e al
6 h p eCiCE Wo kshop, 11 Sep embe 2025
Open-sou ce coupling wi h
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G ains in indus y
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G ains in na u e
Teph a expulsion Flow h ough con ac ion
Seg ega ion Elonga ed uno Finge ing ins abili y
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The o ce model
•Disc e e pa icle model go e ned by New onian mechanics:
mi
d2xi
d 2=
i
•Con ac o ces and body o ces:

i=X
j

ij +
bi,
•Con ac o ce model:

ij = n
ijn +
ij
,
n
ij =kδij+γ n
ij,
ij =−min(µ n
ij, k δ
ij+γ
ij)
(Luding, 2008, En i o. and Ci il. Eng.)
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The o ce model
•Disc e e pa icle model go e ned by New onian mechanics:
mi
d2xi
d 2=
i
•Con ac o ces and body o ces:

i=X
j

ij +
bi,
•Con ac o ce model:

ij = n
ijn +
ij
,
n
ij =kδij+γ n
ij,
ij =−min(µ n
ij, k δ
ij+γ
ij)
(Luding, 2008, En i o. and Ci il. Eng.)
4 / 30
The o ce model
•Disc e e pa icle model go e ned by New onian mechanics:
mi
d2xi
d 2=
i
•Con ac o ces and body o ces:

i=X
j

ij +
bi,
•Con ac o ce model:

ij = n
ijn +
ij
,
n
ij =kδij+γ n
ij,
ij =−min(µ n
ij, k δ
ij+γ
ij)
(Luding, 2008, En i o. and Ci il. Eng.)
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Pa icle simula ions wi h Me cu yDPM
•Fas
Fas con ac de ec ion o polydispe se simula ions
•Flexible
Complex walls and bounda y condi ions
•Accu a e
Coa se-g aining echnique o ex ac con inuum ields
•Open-sou ce
A ailable a www.me cu ydpm.o g
•Coupled/in e aced wi h , ,
, ,
G ainLea ning
y [m]
x [m]
0.02 0.04 0.06 0.08 0.1 0.12 0.14
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0 mm
1 mm
2 mm
3 mm
4 mm
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Fas compu a ions, op imal con ac de ec ion
T adi ional Me cu yDPM Speed-Up
Linked Cell Hie a chical G id o high polydispe si y
#checks: 1000 #checks: 89 up o 200x quicke !
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Coupling me hods
Concu en mul i-scale modeling o pa icle-con inuum in e ac ions 3
(a) (b)
(c)
Fig. 1: (a) 2D illus a ion o he disc e e pa icle and ini e
elemen domains WDE and WFE, coupled on he su ace
GC=∂WFE, and p ojec ion o he su ace coupling o ce
on o he ini e elemen using (b) he con en ional and (c)
CG-en iched app oaches (adap ed om [10]).
2.1 Go e ning equa ions
2.1.1 The mic oscopic pa icle model
The Disc e e Elemen Me hod models g anula ma e ials as
assemblies o igid pa icles ha in e ac ia bina y
in e -pa icle con ac o ces and o ques. Simpli ied, igid
geome ic objec s (walls) can be added ha also in e ac
wi h he pa icles ia bina y con ac o ces and o ques.
Ex e nal o ces and o ques can be applied o he pa icles
as well. The mac oscopic beha io o he pa icle assembly
is hen ob ained by esol ing he ansla ional and
o a ional mo ions o all pa icles indi idually acco ding o
New on’s second law. Conside pa icle a ha has Nand
Nwexis ing con ac s om he neighbo ing pa icles and
walls. Fo gi en ini ial and bounda y condi ions, he ime
e olu ion o he eloci y a, posi ion xa, angula eloci y
wa, and o ien a ion qao a pa icle Pais gi en by
d a
d =1
ma⇣N
Â
b=1
ab +
Nw
Â
g=1
w
ag + b
a⌘
| {z }
:= a
,(1)
dxa
d = a,(2)
(3)
dwa
d =I1
a⇣N
Â
b=1
lab ⇥ ab +
Nw
Â
g=1
lag ⇥ w
ag⌘
| {z }
:= a
,(4)
dqa
d =C(qa)wa.(5)
whe e maand Iaa e pa icle’s mass and ine ia enso ,
C(qa)is he ans o ma ion ma ix ha allows an e icien
handling o i s o ien a ion qaas qua e nion, b
aa e he body
o ces, and aand aa e he o al o ces and o ques ac ing
a he pa icle posi ion xa. Eqs. (3) and (5) a e closed using
so-called con ac laws ha de ine in e -pa icle con ac
o ces ab as a unc ion o he o e lap 1be ween a pai o
in e ac ing pa icles Paand Pb. Ano he con ac law,
simila o, o di e en om he in e -pa icle con ac law,
can be used o compu e pa icle-wall in e ac ion o ces w
ag.
The b anch ec o lab =xc
ab xaconnec s he pa icle
cen e xawi h he con ac poin xc
ab, and he c oss p oduc
lab ⇥ ab con ibu es a o que ha a ec s he pa icle
o a ion and angula eloci y [12]. In i s simples o m, a
linea -elas ic o ce-displacemen law ela es he con ac
o ce in he di ec ion no mal o he con ac a ea and he
in e -pa icle o e lap un( esembling de o ma ion) by a
linea sp ing cons an , namely, k n
abk=knun. Simila
o ce-displacemen laws a e o en aken in he angen ial
di ec ion wi h a Coulomb yield c i e ion, k s
abkµk n
abk,
in o de o cap u e i eco e able plas ic de o ma ion, using
a ic ion coe icien µ.
The abo e di e en ial equa ions a e sol ed nume ically
using he Veloci y-Ve le algo i hm. Using a highe -o de
accu a e ime in eg a ion scheme would no inc ease
accu acy, because mos con ac laws used in he DEM ha e
discon inuous de i a i es. The open-sou ce DEM code
Me cu yDPM [11] is used o he pa icle side o ou new
DEM-FEM coupled code.
2.1.2 The mac oscopic con inuum model
We assume ha he de o mable body beha es as a
h ee-dimensional elas ic solid and desc ibe i s beha io
using ini e s ain heo y. A ini e-s ain amewo k is
essen ial o co ec ly handle he impac o g anula
ma e ials, in pa icula o de o mable s uc u es ha ha e
la ge displacemen s and/o geome ical nonlinea i y. The
ec o s xand X, a ached o he body, a e used o de ine he
ma e ial’s unde o med and de o med con igu a ions. A
s ain measu e, he G een s ain enso , is de ined as
e=1
2(FT·F1), whe e F =1+—u is he de o ma ion
g adien w. . . x,u=Xx he displacemen ield, 1 he
iden i y ma ix, and (·)T he anspose.
Le us conside a solid body subjec ed o a su ace
ac ion on a subse G o he body’s bounda y ∂WFE,a
body o ce densi y b ac ing on he domain WFE, and a
displacemen bounda y condi ion, p esc ibed by X(BC)on
ano he subse o he bounda y GX. Using he s ess and
s ain measu es sand e, he de o ma ion is go e ned by
1No e ha he inc emen al shea displacemen compu ed om he
ela i e eloci y be ween wo pa icles a he con ac poin is used o
compu e he angen ial in e ac ion o ce.
Su ace Coupling
a smoo h and con inuous unc ion w(x), x2⌦C,TW: You de ine w on X o x which110
mono onically inc eases om ze o on @⌦DE ⌦C o one on @⌦FE ⌦C, as shown in Fig. 1a.111
The i ual wo k done in he DEM model WDE is weigh ed by 1 w(x)such ha he112
coupling weigh s on he FEM and DEM sides sum o uni y a any gi en loca ion. The113
coupled go e ning equa ions also need o be adap ed by he espec i e coupling weigh s, as114
demons a ed in Sec. 2.3.115
(a) (b)
Figu e 1: (a) 2D illus a ion o olume coupling be ween disc e e pa icle and ini e elemen domains ⌦FE
and ⌦DE, in he o e lapping olume ⌦C=⌦
FE ⌦DE =SNe
e=1 ⌦C
e, and (b) mapping o pa icle eloci y ↵
on o FEM nodes.
2.2. Homogeniza ion using coa se-g aining116
Be o e applying cons ain s on he a iables (e.g., displacemen o eloci y) a di↵e en 117
scales, a compu a ional homogeniza ion is needed o b ing hem on o he same scale, as118
illus a ed in Fig. 1b. The ac ha FEM and DEM gi e solu ions a he pa icle and119
con inuum scales, espec i ely, makes hei olume coupling non- i ial: he leng h scale120
(ei he mic oscopic o mac oscopic) on which he a iables should be cons ained is no well121
de ined. A no el app oach we ake he e is CG-en iched homogeniza ion, using smoo h ke nel122
unc ions whose leng h scale can be chosen independen o pa icle and ini e elemen sizes.123
In he ollowing, we will demons a e how a compu a ional homogeniza ion is en iched wi h124
CG and in which limi his gene alized o mula ion educes o he con en ional one.125
Le us now conside Nppa icles ha eside wi hin a ini e elemen in he coupling domain126
⌦C
e, as illus a ed in Fig. 2. Using coa se-g aining, we can de ine a homogenized eloci y ield127
6
Scale Coupling
Timeline Table ing Mul i- esol ed coupling Unde - esol ed simula ions Change o plans
Coupling pa icles o luid
Fully- esol ed Mul i- esol ed Unde - esol ed
IT ade-o be ween speed and de ail o in o ma ion.
IFully- esol ed: L 1
8⇡dp,min
IUnde - esol ed: L 3dp,max
8
Pa icle-Fluid Coupling
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The Moomph amewo k – coupling disc e e and con inuum sol e s
Moomph coupling amewo k
mic o-
da a
mac o-
da a
hook
hook
Mul iscale coupling
Pa icle- luid coupling
Pa icle-solid coupling
5
124 5. ATHERMO-MECHANICALLY COUPLED MULTI-SCALE MODEL OF GRANULAR MEDIA
Hea ans e model + sin e ing
Figu e 5.21: Mul i-scale amewo k o simula e he lase sin e ing p ocess using Me cu yDPM and oomph-lib.
solid ini e elemen domain, he e a e a o al o 1500 olume coupling elemen s. Each
coupling elemen con ains a single disc e e pa icle. The dimensions o he domain
ha e been chosen o be ep esen a i e, based on he a e age size o PA12 pa icles,
which is app oxima ely 125 µm. Fo a isual ep esen a ion o his con igu a ion, see
Figu e 5.22.
Figu e 5.22: Coupled domain o simula e lase sin e ing. Disc e e elemen s (pa icles) a e modelled using
Me cu yDPM, ini e elemen s a e modelled using oomph-lib. The domains a e coupled using olume
elemen s, con aining one disc e e elemen pe olume.
The ma e ial pa ame e s a e lis ed in Table. 5.6
A empe a u e and p essu e-dependen isco-elas ic-plas ic model [2] is u ilized o
5.4. RESULTS AND DISCUSSION
5
117
Table 5.4: Pa ame e s o simula ion
Pa ame e Uni s Value
Leng h - ly[m] 25.0
Heigh - lz[m] 0.5
Wid h - lx[m] 0.5
Flexu al igidi y - M/EI [N/m2] 1.0£10°38
Conduc i i y - Ø[W/mK] 1.0
The mal expansion - Æ[1/±C] 0.0°7.1£10°3
G a i y - g[m/s2] 0.0
T| =0=T0[±C] 0.0
T|z=ly=T18 >0[
±C] 1.0
T|z=0=T28 >0[
±C] 0.0
gene alized Hooke’s law, and he Poisson e ec is neglec ed. This co esponds o a
s eady hea ans e p oblem, o which is assumed ha he he mal equilib ium is
eached once he simula ion s a s. Two empe a u es, T1=1.0 ±C and T2=0.0 ±C, a e
se o he op (z=lz) and bo om (z=0) su aces o he domain, espec i ely. The beam
is ully cons ained a lx=0. Fig. 5.12 shows he can ile e beam and he he mal
de lec ion.
Figu e 5.12: The mal de lec ion o a can ile e beam. The beam o leng h lyis ully cons ained a x=0. We
assume no in e nal hea gene a ion H=0, and no hea lux o hea ans e occu ing ac oss he bounda ies o
he olume V.
Fig. 5.12 shows he can ile e beam de lec ing. The displacemen esul s o he nodes
along he °yaxis, loca ed a z=lz/2, wi h di e en he mal expansion coe icien s Æ,
a e p esen ed in Fig. 5.13.
The can ile e beam de lec s highe acco ding o he inc emen o i s he mal
Mul iphysics coupling
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Pa icle- luid coupling
•Fully esol ed: Accu a e and well-de ined, bu compu a ionally expensi e.
Valid o h < 1
8πdmin.
•Unde - esol ed: Fas , bu equi es closu e ela ions, modelling assump ions.
Valid o h > 3dmax.
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Fully esol ed pa icle- luid coupling
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Unde - esol ed pa icle- luid coupling
We apply a wo-phase low model (Ande son & Jackson):
•Each pa icle expe iences a d ag o ce,

d
p=β(u −up).
•The luid phase is go e ned by:
∂(ϵρ)
∂ +∇·(ϵρu ) = 0,
ϵDu
D =−ϵ∇p+µ∇2u +ϵg −
d.
•Coupled ia coa se-g ained d ag o ce and po osi y (Rui & Heng):

d=X
p

d
pϕ( − p), ϵ =X
p
Vpϕ( − p)
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Towa ds as e simula ions
Challenges
•Mapping be ween pa icles and
luid mesh is expensi e
•Op imal coa se-g aining mesh
o en di e s om he luid mesh
P oposed Solu ion
Use in e pola ion meshes
ha s o e mapping in o ma ion
and e ine au oma ically
Fluid
sol e
Pa icle
sol e
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Towa ds as e simula ions
Challenges
•Mapping be ween pa icles and
luid mesh is expensi e
•Op imal coa se-g aining mesh
o en di e s om he luid mesh
P oposed Solu ion
Use in e pola ion meshes
ha s o e mapping in o ma ion
and e ine au oma ically
Fluid
sol e
Pa icle
sol e
In e pola ed
luid eloci y
Coa se-g ained
po osi y/d ag
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Example: Pa icles se ling in a lid-d i en ca i y
Mesh
Field
Fluid eloci y Po osi y
.
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Valida ion: Pa icles se ling in a lid-d i en ca i y
∼10 imes speedup
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Code de elopmen by Me cu yLab
This ea u e was de eloped by Me cu yLab, a spin-o suppo ing Me cu yDPM.
Wha Me cu yLab o e s
•Access o Me cu yCloud (a -cos o academics)
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•Code suppo and main enance
Lea n mo e a me cu ylab.o g o mee us a Pow ech.
All Me cu yLab-de eloped ea u es a e con ibu ed back o he open-sou ce codebase.
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Wha is ON-DEM
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•p omo e ex ensi e use o open-sou ce DEM codes
•p oduce scien i ic ad ances enabling DEM simula ion o mo e ealis ic p oblems
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ON-DEM Pa ne s
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Mo e ?
Au opas
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Join he Ac ion!
Cos Ac ion: Open Ne wo k on DEM Simula ions
1Passing h ough ime & space scales
GPU, mul iscale, ML, coupled
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Valida ion & calib a ion
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La ge da a, common ile o ma s
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Benchma king, alida ion, guidelines
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u ilisa ion
T aining, UI/UX, Cloud compu ing
The e is a sho age o specialis s:
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ON-DEM Acknowledgemen s
This alk and my pa icipa ion in his con e ence was unded by he COST ac ion
ON-DEM
Check ou ON-DEM.o g o scan he QR code o join
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Conclusions and Ou look
Conclusions
•Su ace, mul i-scale and pa icle- luid coupling implemen ed in Moomph and
(coming) wi h p eCICE.
•Applied o a ious applica ions (wall impac , dosing in o elas ic pouch,
he mal expansion, sedimen a ion).
•Coa se-g aining imp o es coupling by smoo hing he coupled ields
(less oscilla ions, imp o ed ene gy conse a ion).
30 / 30
Conclusions and Ou look
Conclusions
•Su ace, mul i-scale and pa icle- luid coupling implemen ed in Moomph and
(coming) wi h p eCICE.
•Applied o a ious applica ions (wall impac , dosing in o elas ic pouch,
he mal expansion, sedimen a ion).
•Coa se-g aining imp o es coupling by smoo hing he coupled ields
(less oscilla ions, imp o ed ene gy conse a ion).
P esen and Fu u e de elopmen s
•Compa e and alida e pa icle- luid coupling echniques.
•Valida e wi h expe imen s o o a ing d um case.
•Add p eCICE adap o o oomph-lib.
•Include he mal and chemical e ec s.
•Ex end mul i-scale modelling o p ocess scale, using model o de educ ion.
30 / 30
Conclusions and Ou look
P esen and Fu u e de elopmen s
•Compa e and alida e pa icle- luid coupling echniques.
•Valida e wi h expe imen s o o a ing d um case.
•Add p eCICE adap o o oomph-lib.
•Include he mal and chemical e ec s.
•Ex end mul i-scale modelling o p ocess scale, using model o de educ ion.
B oade Impac (open-sou ce ecosys em o coupled simula ions, ON-DEM)
•ON-DEM is looking o wide membe s.
•Dedica ed o open-sou ce.
•Please conside joining.
•Follow-up ne wo k on coupled p oblems is being w i en.
30 / 30
Choosing he igh coa se-g aining scale
(Weinha e al., 2013, Phys. Fl.)
1 / 39
Coa se g aining unc ions
Essen ial p ope ies o ϕ:
1Mass conse a ion: V=RR3ϕ( )d = 1,
2Posi i i y: ϕ( )≥0∀ ∈R3,
3Locali y: ϕ( ) = 0 o | |> c.
Commonly used unc ions:
1Cu o Gaussian
2Hea iside
3Lucy polynomial
| |
φ3
G
φ3
˜
G
φ3
˜
H
φ3
˜
L
0 1 23
0
0.2
0.4
CG unc ions o wid h w= 1
6 / 39

Coa se g aining unc ions
Essen ial p ope ies o ϕ:
1Mass conse a ion: V=RR3ϕ( )d = 1,
2Posi i i y: ϕ( )≥0∀ ∈R3,
3Locali y: ϕ( ) = 0 o | |> c.
Commonly used unc ions:
1Cu o Gaussian
2Hea iside
3Lucy polynomial
| |
φ3
G
φ3
˜
G
φ3
˜
H
φ3
˜
L
0 1 23
0
0.2
0.4
CG unc ions o wid h w= 1
6 / 39
Coa se g aining: mass balance
We de ine eloci y us. . mass balance,
∂ρ
∂ +∇·(ρu)=0,
is sa is ied:
u=j
ρ,j=
n
X
i=1
mi iϕ( − i).
7 / 39
Coa se g aining: momen um balance
Simila ly, we de ine s ess σbased on momen um balance:
∂ρu
∂ +∇·(ρuu) = −∇·σ+ρg.
σ=σk+σc, σk=Xn
i=1 mi ′
i ′
iϕ( − i),
σc=Xi,j ijbij Z1
0
ϕ( − i+sbij)ds
0
3
Magni ude o s ess |σ|2
8 / 39
Ad an ages
CG
+P oduces con inuum ields (no disc e e alues)
+Sa is ies local mass/momen um conse a ion o all w,N.1
+Gene al: Accoun s o bounda y in e ac ions3, g anula mix u es4,
empo al smoo hing4, non-sphe ical pa icles5, e c
+Applicable o bo h DPM and expe imen al da a6
+A ailable open-sou ce a Me cu yDPM.o g
+Used o wo-way coupling, e.g. wi h oomph-lib7
1Goldhi sch, G anula Ma e (2010) 12(3), 239–252
2Weinha e al, Physics o Fluids (2013) 25(7), 070605
3Weinha e al, G anula Ma e (2012) 14(2), 289-294
4Tunugun la e al, Compu . Pa icle Mechanics (2016) 3(3), 349-365
5Weinha e al, Powde Technology (2016) 293, 138-148
6Roy e al, EPJ E (2019) 42: 14
7Cheng e al, CMAME, 403, 115651 (2023)
9 / 39
Con inuum modelling wi h oomph-lib
•Fini e elemen sol e de eloped a
Uni e si y o Manches e .
•A ailable open-sou ce a
oomph-lib.ma hs.man.ac.uk
•Focus on luid-s uc u e
in e ac ions.
•Suppo s mesh e inemen .
•Simila code base as Me cu yDPM
100 Compu Mech (2008) 43:91–101
Fig. 6 Flow pas a cylinde wi h an a ached lag o he pa ame e
alues o es case “FSI2” in Re . [10]. aTime- ace o he x2-coo dina e
o he lag’s ip. bSnapsho o he low ield (ins an aneous s eamlines
and p essu e con ou s) a =6.04
Fig. 7 Flow pas a cylinde wi h an a ached lag o he pa ame e
alues o es case “FSI3” in Re . [10]. aTime- ace o he x2-coo dina e
o he lag’s ip. bSnapsho o he low ield (ins an aneous s eamlines
and p essu e con ou s) a =3.615
ime o 75.3 s o he monoli hic New on sol e a each
imes ep.
Figu e 7shows he co esponding esul s o he es case
“FSI3” whe e luid and wall densi ies a e equal and Re =
200,S =0.05,Q=3.571 ×10−6and Λ2=1.7855 ×
10−6.Compa ed o hecondi ionsin es case“FSI2”, he
sys em pe o ms oscilla ions o much highe equency and
smalle ampli ude. Ou compu a ion was pe o med wi h a
ixed imes ep o ∆ =0.005 and esul ed in oscilla ions
wi h a pe iod o ≈0.19 and an ampli ude o he e ical ip
displacemen o ≈0.01±0.36.Theinc easein equencyand
Reynolds numbe leads o he de elopmen o hinne bound-
a y and shea laye s which equi e a ine spa ial esolu ion,
in ol ing an a e age o 84,000 deg ees o eedom. This had
e y li le e ec on he GMRES con e gence a es. Wi h P
P
P1
used as p econdi ione an a e age o 45.8 i e a ions we e
equi ed o ob ain a con e ged solu ion o he linea sys-
ems,while hesolu iono he nonlinea sys emsbyNew on’s
me hod now equi ed an a e age o 104.4 s.
Fig. 8 Uns eady low h ough a collapsible ube o Q=10−7,S =
1. aTime- ace o he adial posi ion o a con ol poin loca ed a 70% o
he ube leng h, a he mos s ongly collapsed poin in he c oss sec ion,
o a ious Reynolds numbe s. bSnapsho o he axial eloci y p o iles
a =181.26 o he Re =100 case
4.2.4 Sel -exci ed oscilla ions o a 3D collapsible ube
In he inal example we e alua e he pe o mance o he FSI
p econdi ione in p elimina y 3D simula ions o uns eady
ini e-Reynolds numbe lows in a collapsible ube, mod-
elled as a ci cula cylind ical shell o adius L,leng h10L
and wall hickness h/L=1/20, and a Poisson a io o
ν=0.49, moun ed on wo sho igid ubes o leng h L.
We p esc ibe Poiseuille low wi h a e age eloci y Ua he
a downs eam end and impose pa allel, axially ac ion-
ee in low ups eam. Figu e 8shows he ime- ace, c l( ),
o he adial posi ion o a con ol poin on he ube wall,
loca ed owa ds he downs eam end o he ube a 70% o
i s leng h, o Q=10−7,S =1andΛ=0a a ious
Reynolds numbe s. In all cases he simula ion was s a ed
om he s eady solu ion wi h c l =0.55, which equi ed
di e en ex e nal p essu es o each di e en Reynolds num-
be . A =0we educedpex o a alue co esponding o an
equilib ium con igu a ion in which c l =0.6 and ollowed
he sys em’s e olu ion. The ime- ace shows ha he sys-
em oscilla es abou he new equilib ium posi ion. A small
Reynolds numbe s he oscilla ions decay bu o su icien ly
la ge Reynolds numbe hey g ow in ampli ude, as obse ed
in physical expe imen s; see e.g. [8].
The esul s p esen edhe e a ep elimina y in hesense ha
he assessmen o hei mesh and imes ep independence has
no ye beencomple ed.Howe e , heplo o he eloci y ield
in Fig. 8sugges s ha e en o a ela i ely coa se disc e isa-
ion wi h 57,486 deg ees o eedom, he eloci y ield is well
esol ed. The Jacobian ma ix a ising om he monoli hic
disc e isa ion o 3D p oblems is signi ican ly mo e dense
123
10 / 39

The Moomph amewo k – coupling disc e e and con inuum sol e s
Moomph coupling amewo k
mic o-
da a
mac o-
da a
hook
hook
Mul iscale coupling
Pa icle- luid coupling
Pa icle-solid coupling
5
124 5. ATHERMO-MECHANICALLY COUPLED MULTI-SCALE MODEL OF GRANULAR MEDIA
Hea ans e model + sin e ing
Figu e 5.21: Mul i-scale amewo k o simula e he lase sin e ing p ocess using Me cu yDPM and oomph-lib.
solid ini e elemen domain, he e a e a o al o 1500 olume coupling elemen s. Each
coupling elemen con ains a single disc e e pa icle. The dimensions o he domain
ha e been chosen o be ep esen a i e, based on he a e age size o PA12 pa icles,
which is app oxima ely 125 µm. Fo a isual ep esen a ion o his con igu a ion, see
Figu e 5.22.
Figu e 5.22: Coupled domain o simula e lase sin e ing. Disc e e elemen s (pa icles) a e modelled using
Me cu yDPM, ini e elemen s a e modelled using oomph-lib. The domains a e coupled using olume
elemen s, con aining one disc e e elemen pe olume.
The ma e ial pa ame e s a e lis ed in Table. 5.6
A empe a u e and p essu e-dependen isco-elas ic-plas ic model [2] is u ilized o
5.4. RESULTS AND DISCUSSION
5
117
Table 5.4: Pa ame e s o simula ion
Pa ame e Uni s Value
Leng h - ly[m] 25.0
Heigh - lz[m] 0.5
Wid h - lx[m] 0.5
Flexu al igidi y - M/EI [N/m2] 1.0£10°38
Conduc i i y - Ø[W/mK] 1.0
The mal expansion - Æ[1/±C] 0.0°7.1£10°3
G a i y - g[m/s2] 0.0
T| =0=T0[±C] 0.0
T|z=ly=T18 >0[
±C] 1.0
T|z=0=T28 >0[
±C] 0.0
gene alized Hooke’s law, and he Poisson e ec is neglec ed. This co esponds o a
s eady hea ans e p oblem, o which is assumed ha he he mal equilib ium is
eached once he simula ion s a s. Two empe a u es, T1=1.0 ±C and T2=0.0 ±C, a e
se o he op (z=lz) and bo om (z=0) su aces o he domain, espec i ely. The beam
is ully cons ained a lx=0. Fig. 5.12 shows he can ile e beam and he he mal
de lec ion.
Figu e 5.12: The mal de lec ion o a can ile e beam. The beam o leng h lyis ully cons ained a x=0. We
assume no in e nal hea gene a ion H=0, and no hea lux o hea ans e occu ing ac oss he bounda ies o
he olume V.
Fig. 5.12 shows he can ile e beam de lec ing. The displacemen esul s o he nodes
along he °yaxis, loca ed a z=lz/2, wi h di e en he mal expansion coe icien s Æ,
a e p esen ed in Fig. 5.13.
The can ile e beam de lec s highe acco ding o he inc emen o i s he mal
Mul iphysics coupling
11 / 39
Pa icle-solid coupling (H Cheng)
Concu en mul i-scale modeling o pa icle-con inuum in e ac ions 3
(a) (b)
(c)
Fig. 1: (a) 2D illus a ion o he disc e e pa icle and ini e
elemen domains WDE and WFE, coupled on he su ace
GC=∂WFE, and p ojec ion o he su ace coupling o ce
on o he ini e elemen using (b) he con en ional and (c)
CG-en iched app oaches (adap ed om [10]).
2.1 Go e ning equa ions
2.1.1 The mic oscopic pa icle model
The Disc e e Elemen Me hod models g anula ma e ials as
assemblies o igid pa icles ha in e ac ia bina y
in e -pa icle con ac o ces and o ques. Simpli ied, igid
geome ic objec s (walls) can be added ha also in e ac
wi h he pa icles ia bina y con ac o ces and o ques.
Ex e nal o ces and o ques can be applied o he pa icles
as well. The mac oscopic beha io o he pa icle assembly
is hen ob ained by esol ing he ansla ional and
o a ional mo ions o all pa icles indi idually acco ding o
New on’s second law. Conside pa icle a ha has Nand
Nwexis ing con ac s om he neighbo ing pa icles and
walls. Fo gi en ini ial and bounda y condi ions, he ime
e olu ion o he eloci y a, posi ion xa, angula eloci y
wa, and o ien a ion qao a pa icle Pais gi en by
d a
d =1
ma⇣N
Â
b=1
ab +
Nw
Â
g=1
w
ag + b
a⌘
| {z }
:= a
,(1)
dxa
d = a,(2)
(3)
dwa
d =I1
a⇣N
Â
b=1
lab ⇥ ab +
Nw
Â
g=1
lag ⇥ w
ag⌘
| {z }
:= a
,(4)
dqa
d =C(qa)wa.(5)
whe e maand Iaa e pa icle’s mass and ine ia enso ,
C(qa)is he ans o ma ion ma ix ha allows an e icien
handling o i s o ien a ion qaas qua e nion, b
aa e he body
o ces, and aand aa e he o al o ces and o ques ac ing
a he pa icle posi ion xa. Eqs. (3) and (5) a e closed using
so-called con ac laws ha de ine in e -pa icle con ac
o ces ab as a unc ion o he o e lap 1be ween a pai o
in e ac ing pa icles Paand Pb. Ano he con ac law,
simila o, o di e en om he in e -pa icle con ac law,
can be used o compu e pa icle-wall in e ac ion o ces w
ag.
The b anch ec o lab =xc
ab xaconnec s he pa icle
cen e xawi h he con ac poin xc
ab, and he c oss p oduc
lab ⇥ ab con ibu es a o que ha a ec s he pa icle
o a ion and angula eloci y [12]. In i s simples o m, a
linea -elas ic o ce-displacemen law ela es he con ac
o ce in he di ec ion no mal o he con ac a ea and he
in e -pa icle o e lap un( esembling de o ma ion) by a
linea sp ing cons an , namely, k n
abk=knun. Simila
o ce-displacemen laws a e o en aken in he angen ial
di ec ion wi h a Coulomb yield c i e ion, k s
abkµk n
abk,
in o de o cap u e i eco e able plas ic de o ma ion, using
a ic ion coe icien µ.
The abo e di e en ial equa ions a e sol ed nume ically
using he Veloci y-Ve le algo i hm. Using a highe -o de
accu a e ime in eg a ion scheme would no inc ease
accu acy, because mos con ac laws used in he DEM ha e
discon inuous de i a i es. The open-sou ce DEM code
Me cu yDPM [11] is used o he pa icle side o ou new
DEM-FEM coupled code.
2.1.2 The mac oscopic con inuum model
We assume ha he de o mable body beha es as a
h ee-dimensional elas ic solid and desc ibe i s beha io
using ini e s ain heo y. A ini e-s ain amewo k is
essen ial o co ec ly handle he impac o g anula
ma e ials, in pa icula o de o mable s uc u es ha ha e
la ge displacemen s and/o geome ical nonlinea i y. The
ec o s xand X, a ached o he body, a e used o de ine he
ma e ial’s unde o med and de o med con igu a ions. A
s ain measu e, he G een s ain enso , is de ined as
e=1
2(FT·F1), whe e F =1+—u is he de o ma ion
g adien w. . . x,u=Xx he displacemen ield, 1 he
iden i y ma ix, and (·)T he anspose.
Le us conside a solid body subjec ed o a su ace
ac ion on a subse G o he body’s bounda y ∂WFE,a
body o ce densi y b ac ing on he domain WFE, and a
displacemen bounda y condi ion, p esc ibed by X(BC)on
ano he subse o he bounda y GX. Using he s ess and
s ain measu es sand e, he de o ma ion is go e ned by
1No e ha he inc emen al shea displacemen compu ed om he
ela i e eloci y be ween wo pa icles a he con ac poin is used o
compu e he angen ial in e ac ion o ce.
(a) Simula es pa icle in e ac ions wi h an elas ic solid.
(b) Requi es coupling ia su ace ac ion ield, =P− αγδ(x−xc
αγ)
(c) Fo upscaled pa icles, smoo h su ace ac ion: CG =P− αγϕ(x−xc
αγ).
Thus, solid esponds o bulk a he han indi idual pa icles.
Cheng e al, Compu e Me hods in Applied Mechanics and Enginee ing, 403, 115651 (2023)
12 / 39
Pa icle-solid coupling (H Cheng)
Concu en mul i-scale modeling o pa icle-con inuum in e ac ions 3
(a) (b)
(c)
Fig. 1: (a) 2D illus a ion o he disc e e pa icle and ini e
elemen domains WDE and WFE, coupled on he su ace
GC=∂WFE, and p ojec ion o he su ace coupling o ce
on o he ini e elemen using (b) he con en ional and (c)
CG-en iched app oaches (adap ed om [10]).
2.1 Go e ning equa ions
2.1.1 The mic oscopic pa icle model
The Disc e e Elemen Me hod models g anula ma e ials as
assemblies o igid pa icles ha in e ac ia bina y
in e -pa icle con ac o ces and o ques. Simpli ied, igid
geome ic objec s (walls) can be added ha also in e ac
wi h he pa icles ia bina y con ac o ces and o ques.
Ex e nal o ces and o ques can be applied o he pa icles
as well. The mac oscopic beha io o he pa icle assembly
is hen ob ained by esol ing he ansla ional and
o a ional mo ions o all pa icles indi idually acco ding o
New on’s second law. Conside pa icle a ha has Nand
Nwexis ing con ac s om he neighbo ing pa icles and
walls. Fo gi en ini ial and bounda y condi ions, he ime
e olu ion o he eloci y a, posi ion xa, angula eloci y
wa, and o ien a ion qao a pa icle Pais gi en by
d a
d =1
ma⇣N
Â
b=1
ab +
Nw
Â
g=1
w
ag + b
a⌘
| {z }
:= a
,(1)
dxa
d = a,(2)
(3)
dwa
d =I1
a⇣N
Â
b=1
lab ⇥ ab +
Nw
Â
g=1
lag ⇥ w
ag⌘
| {z }
:= a
,(4)
dqa
d =C(qa)wa.(5)
whe e maand Iaa e pa icle’s mass and ine ia enso ,
C(qa)is he ans o ma ion ma ix ha allows an e icien
handling o i s o ien a ion qaas qua e nion, b
aa e he body
o ces, and aand aa e he o al o ces and o ques ac ing
a he pa icle posi ion xa. Eqs. (3) and (5) a e closed using
so-called con ac laws ha de ine in e -pa icle con ac
o ces ab as a unc ion o he o e lap 1be ween a pai o
in e ac ing pa icles Paand Pb. Ano he con ac law,
simila o, o di e en om he in e -pa icle con ac law,
can be used o compu e pa icle-wall in e ac ion o ces w
ag.
The b anch ec o lab =xc
ab xaconnec s he pa icle
cen e xawi h he con ac poin xc
ab, and he c oss p oduc
lab ⇥ ab con ibu es a o que ha a ec s he pa icle
o a ion and angula eloci y [12]. In i s simples o m, a
linea -elas ic o ce-displacemen law ela es he con ac
o ce in he di ec ion no mal o he con ac a ea and he
in e -pa icle o e lap un( esembling de o ma ion) by a
linea sp ing cons an , namely, k n
abk=knun. Simila
o ce-displacemen laws a e o en aken in he angen ial
di ec ion wi h a Coulomb yield c i e ion, k s
abkµk n
abk,
in o de o cap u e i eco e able plas ic de o ma ion, using
a ic ion coe icien µ.
The abo e di e en ial equa ions a e sol ed nume ically
using he Veloci y-Ve le algo i hm. Using a highe -o de
accu a e ime in eg a ion scheme would no inc ease
accu acy, because mos con ac laws used in he DEM ha e
discon inuous de i a i es. The open-sou ce DEM code
Me cu yDPM [11] is used o he pa icle side o ou new
DEM-FEM coupled code.
2.1.2 The mac oscopic con inuum model
We assume ha he de o mable body beha es as a
h ee-dimensional elas ic solid and desc ibe i s beha io
using ini e s ain heo y. A ini e-s ain amewo k is
essen ial o co ec ly handle he impac o g anula
ma e ials, in pa icula o de o mable s uc u es ha ha e
la ge displacemen s and/o geome ical nonlinea i y. The
ec o s xand X, a ached o he body, a e used o de ine he
ma e ial’s unde o med and de o med con igu a ions. A
s ain measu e, he G een s ain enso , is de ined as
e=1
2(FT·F1), whe e F =1+—u is he de o ma ion
g adien w. . . x,u=Xx he displacemen ield, 1 he
iden i y ma ix, and (·)T he anspose.
Le us conside a solid body subjec ed o a su ace
ac ion on a subse G o he body’s bounda y ∂WFE,a
body o ce densi y b ac ing on he domain WFE, and a
displacemen bounda y condi ion, p esc ibed by X(BC)on
ano he subse o he bounda y GX. Using he s ess and
s ain measu es sand e, he de o ma ion is go e ned by
1No e ha he inc emen al shea displacemen compu ed om he
ela i e eloci y be ween wo pa icles a he con ac poin is used o
compu e he angen ial in e ac ion o ce.
Concu en mul i-scale modeling o pa icle-con inuum in e ac ions 3
(a) (b)
(c)
Fig. 1: (a) 2D illus a ion o he disc e e pa icle and ini e
elemen domains WDE and WFE, coupled on he su ace
GC=∂WFE, and p ojec ion o he su ace coupling o ce
on o he ini e elemen using (b) he con en ional and (c)
CG-en iched app oaches (adap ed om [10]).
2.1 Go e ning equa ions
2.1.1 The mic oscopic pa icle model
The Disc e e Elemen Me hod models g anula ma e ials as
assemblies o igid pa icles ha in e ac ia bina y
in e -pa icle con ac o ces and o ques. Simpli ied, igid
geome ic objec s (walls) can be added ha also in e ac
wi h he pa icles ia bina y con ac o ces and o ques.
Ex e nal o ces and o ques can be applied o he pa icles
as well. The mac oscopic beha io o he pa icle assembly
is hen ob ained by esol ing he ansla ional and
o a ional mo ions o all pa icles indi idually acco ding o
New on’s second law. Conside pa icle a ha has Nand
Nwexis ing con ac s om he neighbo ing pa icles and
walls. Fo gi en ini ial and bounda y condi ions, he ime
e olu ion o he eloci y a, posi ion xa, angula eloci y
wa, and o ien a ion qao a pa icle Pais gi en by
d a
d =1
ma⇣N
Â
b=1
ab +
Nw
Â
g=1
w
ag + b
a⌘
| {z }
:= a
,(1)
dxa
d = a,(2)
(3)
dwa
d =I1
a⇣N
Â
b=1
lab ⇥ ab +
Nw
Â
g=1
lag ⇥ w
ag⌘
| {z }
:= a
,(4)
dqa
d =C(qa)wa.(5)
whe e maand Iaa e pa icle’s mass and ine ia enso ,
C(qa)is he ans o ma ion ma ix ha allows an e icien
handling o i s o ien a ion qaas qua e nion, b
aa e he body
o ces, and aand aa e he o al o ces and o ques ac ing
a he pa icle posi ion xa. Eqs. (3) and (5) a e closed using
so-called con ac laws ha de ine in e -pa icle con ac
o ces ab as a unc ion o he o e lap 1be ween a pai o
in e ac ing pa icles Paand Pb. Ano he con ac law,
simila o, o di e en om he in e -pa icle con ac law,
can be used o compu e pa icle-wall in e ac ion o ces w
ag.
The b anch ec o lab =xc
ab xaconnec s he pa icle
cen e xawi h he con ac poin xc
ab, and he c oss p oduc
lab ⇥ ab con ibu es a o que ha a ec s he pa icle
o a ion and angula eloci y [12]. In i s simples o m, a
linea -elas ic o ce-displacemen law ela es he con ac
o ce in he di ec ion no mal o he con ac a ea and he
in e -pa icle o e lap un( esembling de o ma ion) by a
linea sp ing cons an , namely, k n
abk=knun. Simila
o ce-displacemen laws a e o en aken in he angen ial
di ec ion wi h a Coulomb yield c i e ion, k s
abkµk n
abk,
in o de o cap u e i eco e able plas ic de o ma ion, using
a ic ion coe icien µ.
The abo e di e en ial equa ions a e sol ed nume ically
using he Veloci y-Ve le algo i hm. Using a highe -o de
accu a e ime in eg a ion scheme would no inc ease
accu acy, because mos con ac laws used in he DEM ha e
discon inuous de i a i es. The open-sou ce DEM code
Me cu yDPM [11] is used o he pa icle side o ou new
DEM-FEM coupled code.
2.1.2 The mac oscopic con inuum model
We assume ha he de o mable body beha es as a
h ee-dimensional elas ic solid and desc ibe i s beha io
using ini e s ain heo y. A ini e-s ain amewo k is
essen ial o co ec ly handle he impac o g anula
ma e ials, in pa icula o de o mable s uc u es ha ha e
la ge displacemen s and/o geome ical nonlinea i y. The
ec o s xand X, a ached o he body, a e used o de ine he
ma e ial’s unde o med and de o med con igu a ions. A
s ain measu e, he G een s ain enso , is de ined as
e=1
2(FT·F1), whe e F =1+—u is he de o ma ion
g adien w. . . x,u=Xx he displacemen ield, 1 he
iden i y ma ix, and (·)T he anspose.
Le us conside a solid body subjec ed o a su ace
ac ion on a subse G o he body’s bounda y ∂WFE,a
body o ce densi y b ac ing on he domain WFE, and a
displacemen bounda y condi ion, p esc ibed by X(BC)on
ano he subse o he bounda y GX. Using he s ess and
s ain measu es sand e, he de o ma ion is go e ned by
1No e ha he inc emen al shea displacemen compu ed om he
ela i e eloci y be ween wo pa icles a he con ac poin is used o
compu e he angen ial in e ac ion o ce.
(a) Simula es pa icle in e ac ions wi h an elas ic solid.
(b) Requi es coupling ia su ace ac ion ield, =P− αγδ(x−xc
αγ)
(c) Fo upscaled pa icles, smoo h su ace ac ion: CG =P− αγϕ(x−xc
αγ).
Thus, solid esponds o bulk a he han indi idual pa icles.
Cheng e al, Compu e Me hods in Applied Mechanics and Enginee ing, 403, 115651 (2023)
12 / 39
Pa icle-solid coupling: Example
G anula je impac ing an elas ic beam
13 / 39
Mul iscale coupling: De i a ion
P inciple o i ual wo k:
•De i e a con inuum model by balancing he wo k done when applying
in ini esimally small displacemen s δX:
ZΩhσ:δϵ−(b−ρ¨
X)i·δXdV+ZΓ
·δXdA= 0
•Simila ly, we can w i e he disc e e model as
Xαh(mα¨xα− α)−Xβ αβi·δxα= 0
20 / 39

Mul iscale coupling: De i a ion
Now couple hese wo app oaches
ZΩh(1 −w)σ:δϵ−(1 −w)(b−ρ¨
X)− ci·δXdV+ZΓ
(1 −w) ·δXdA= 0
Xαhwα(mα¨xα− α) + c
α−Xβwαβ αβi·δxα= 0
wi h coupling weigh s wand coupling o ces
c=ϵ( FE − DE)
c
α=ϵΠα( FE − DE)
(a)
0
(b)
Figu e 18: (a) A egula g anula squa e la ice wi h one hi d o i s olume o e lapping wi h a FEM
mesh, and (b) injec ion o a comp essional P-wa e om he le end o he g anula sys em a ime = 0s.
The colo indica es he coupling weigh won he con inuum body, a ying om app oxima ely 1.0 on he
le mos bounda y o 0.0 on he igh mos bounda y o he coupling zone. No e, one disc e e pa icle pe
ini e elemen is shown he e only o he illus a i e pu pose; mo e pa icles pe elemen a e used o he
simula ions in Secs. 6.2.4–6.2.4. The ini ial eloci y o he pa icles is indica ed by he a ows in Figu e 18b.
Table 6: Penal y pa ame e o di↵e en ini e pa icle-elemen con igu a ions o olume coupling.
Tes cases
1P 2P 3P 4P
Num. o pa icles pe elemen n312
33343
Pa icle diame e d(mm) 400 200 133.¯
3 100
Penal y pa ame e ✏(Pa/mm2)1⇥1044⇥1049⇥10416 ⇥104
Penal y pa ame e ✏d2(Pa) 4 ⇥108
40
Cheng e al, Compu e Me hods in Applied Mechanics and Enginee ing, 403, 115651 (2023)
21 / 39
Mul iscale coupling: De i a ion
a smoo h and con inuous unc ion w(x), x2⌦C,TW: You de ine w on X o x which110
mono onically inc eases om ze o on @⌦DE ⌦C o one on @⌦FE ⌦C, as shown in Fig. 1a.111
The i ual wo k done in he DEM model WDE is weigh ed by 1 w(x)such ha he112
coupling weigh s on he FEM and DEM sides sum o uni y a any gi en loca ion. The113
coupled go e ning equa ions also need o be adap ed by he espec i e coupling weigh s, as114
demons a ed in Sec. 2.3.115
(a) (b)
Figu e 1: (a) 2D illus a ion o olume coupling be ween disc e e pa icle and ini e elemen domains ⌦FE
and ⌦DE, in he o e lapping olume ⌦C=⌦
FE ⌦DE =SNe
e=1 ⌦C
e, and (b) mapping o pa icle eloci y ↵
on o FEM nodes.
2.2. Homogeniza ion using coa se-g aining116
Be o e applying cons ain s on he a iables (e.g., displacemen o eloci y) a di↵e en 117
scales, a compu a ional homogeniza ion is needed o b ing hem on o he same scale, as118
illus a ed in Fig. 1b. The ac ha FEM and DEM gi e solu ions a he pa icle and119
con inuum scales, espec i ely, makes hei olume coupling non- i ial: he leng h scale120
(ei he mic oscopic o mac oscopic) on which he a iables should be cons ained is no well121
de ined. A no el app oach we ake he e is CG-en iched homogeniza ion, using smoo h ke nel122
unc ions whose leng h scale can be chosen independen o pa icle and ini e elemen sizes.123
In he ollowing, we will demons a e how a compu a ional homogeniza ion is en iched wi h124
CG and in which limi his gene alized o mula ion educes o he con en ional one.125
Le us now conside Nppa icles ha eside wi hin a ini e elemen in he coupling domain126
⌦C
e, as illus a ed in Fig. 2. Using coa se-g aining, we can de ine a homogenized eloci y ield127
6
a smoo h and con inuous unc ion w(x), x2⌦C,TW: You de ine w on X o x which110
mono onically inc eases om ze o on @⌦DE ⌦C o one on @⌦FE ⌦C, as shown in Fig. 1a.111
The i ual wo k done in he DEM model WDE is weigh ed by 1 w(x)such ha he112
coupling weigh s on he FEM and DEM sides sum o uni y a any gi en loca ion. The113
coupled go e ning equa ions also need o be adap ed by he espec i e coupling weigh s, as114
demons a ed in Sec. 2.3.115
(a) (b)
Figu e 1: (a) 2D illus a ion o olume coupling be ween disc e e pa icle and ini e elemen domains ⌦FE
and ⌦DE, in he o e lapping olume ⌦C=⌦
FE ⌦DE =SNe
e=1 ⌦C
e, and (b) mapping o pa icle eloci y ↵
on o FEM nodes.
2.2. Homogeniza ion using coa se-g aining116
Be o e applying cons ain s on he a iables (e.g., displacemen o eloci y) a di↵e en 117
scales, a compu a ional homogeniza ion is needed o b ing hem on o he same scale, as118
illus a ed in Fig. 1b. The ac ha FEM and DEM gi e solu ions a he pa icle and119
con inuum scales, espec i ely, makes hei olume coupling non- i ial: he leng h scale120
(ei he mic oscopic o mac oscopic) on which he a iables should be cons ained is no well121
de ined. A no el app oach we ake he e is CG-en iched homogeniza ion, using smoo h ke nel122
unc ions whose leng h scale can be chosen independen o pa icle and ini e elemen sizes.123
In he ollowing, we will demons a e how a compu a ional homogeniza ion is en iched wi h124
CG and in which limi his gene alized o mula ion educes o he con en ional one.125
Le us now conside Nppa icles ha eside wi hin a ini e elemen in he coupling domain126
⌦C
e, as illus a ed in Fig. 2. Using coa se-g aining, we can de ine a homogenized eloci y ield127
6
(a) Coupling o ce, c=ϵ( FE − DE), penalises eloci y di e ences.
(b) CG used o p oduce a smoo h eloci y ield, DE =1
ρPimi iϕ( − i).
22 / 39
Mul iscale coupling: De i a ion
a smoo h and con inuous unc ion w(x), x2⌦C,TW: You de ine w on X o x which110
mono onically inc eases om ze o on @⌦DE ⌦C o one on @⌦FE ⌦C, as shown in Fig. 1a.111
The i ual wo k done in he DEM model WDE is weigh ed by 1 w(x)such ha he112
coupling weigh s on he FEM and DEM sides sum o uni y a any gi en loca ion. The113
coupled go e ning equa ions also need o be adap ed by he espec i e coupling weigh s, as114
demons a ed in Sec. 2.3.115
(a) (b)
Figu e 1: (a) 2D illus a ion o olume coupling be ween disc e e pa icle and ini e elemen domains ⌦FE
and ⌦DE, in he o e lapping olume ⌦C=⌦
FE ⌦DE =SNe
e=1 ⌦C
e, and (b) mapping o pa icle eloci y ↵
on o FEM nodes.
2.2. Homogeniza ion using coa se-g aining116
Be o e applying cons ain s on he a iables (e.g., displacemen o eloci y) a di↵e en 117
scales, a compu a ional homogeniza ion is needed o b ing hem on o he same scale, as118
illus a ed in Fig. 1b. The ac ha FEM and DEM gi e solu ions a he pa icle and119
con inuum scales, espec i ely, makes hei olume coupling non- i ial: he leng h scale120
(ei he mic oscopic o mac oscopic) on which he a iables should be cons ained is no well121
de ined. A no el app oach we ake he e is CG-en iched homogeniza ion, using smoo h ke nel122
unc ions whose leng h scale can be chosen independen o pa icle and ini e elemen sizes.123
In he ollowing, we will demons a e how a compu a ional homogeniza ion is en iched wi h124
CG and in which limi his gene alized o mula ion educes o he con en ional one.125
Le us now conside Nppa icles ha eside wi hin a ini e elemen in he coupling domain126
⌦C
e, as illus a ed in Fig. 2. Using coa se-g aining, we can de ine a homogenized eloci y ield127
6
a smoo h and con inuous unc ion w(x), x2⌦C,TW: You de ine w on X o x which110
mono onically inc eases om ze o on @⌦DE ⌦C o one on @⌦FE ⌦C, as shown in Fig. 1a.111
The i ual wo k done in he DEM model WDE is weigh ed by 1 w(x)such ha he112
coupling weigh s on he FEM and DEM sides sum o uni y a any gi en loca ion. The113
coupled go e ning equa ions also need o be adap ed by he espec i e coupling weigh s, as114
demons a ed in Sec. 2.3.115
(a) (b)
Figu e 1: (a) 2D illus a ion o olume coupling be ween disc e e pa icle and ini e elemen domains ⌦FE
and ⌦DE, in he o e lapping olume ⌦C=⌦
FE ⌦DE =SNe
e=1 ⌦C
e, and (b) mapping o pa icle eloci y ↵
on o FEM nodes.
2.2. Homogeniza ion using coa se-g aining116
Be o e applying cons ain s on he a iables (e.g., displacemen o eloci y) a di↵e en 117
scales, a compu a ional homogeniza ion is needed o b ing hem on o he same scale, as118
illus a ed in Fig. 1b. The ac ha FEM and DEM gi e solu ions a he pa icle and119
con inuum scales, espec i ely, makes hei olume coupling non- i ial: he leng h scale120
(ei he mic oscopic o mac oscopic) on which he a iables should be cons ained is no well121
de ined. A no el app oach we ake he e is CG-en iched homogeniza ion, using smoo h ke nel122
unc ions whose leng h scale can be chosen independen o pa icle and ini e elemen sizes.123
In he ollowing, we will demons a e how a compu a ional homogeniza ion is en iched wi h124
CG and in which limi his gene alized o mula ion educes o he con en ional one.125
Le us now conside Nppa icles ha eside wi hin a ini e elemen in he coupling domain126
⌦C
e, as illus a ed in Fig. 2. Using coa se-g aining, we can de ine a homogenized eloci y ield127
6
(a) Coupling o ce, c=ϵ( FE − DE), penalises eloci y di e ences.
(b) CG used o p oduce a smoo h eloci y ield, DE =1
ρPimi iϕ( − i).
22 / 39
Mul iscale coupling: Wa e p opaga ion in elas ic beam
(a) To al mass (b) Sub-model linea momen um
(c) To al linea momen um (d) To al ene gy
Figu e 20: E olu ion o (a) mass, (b-c) momen um, and (d) kine ic, (e) po en ial and ( ) o al ene gies in
he coupled FEM-DEM model, using he con en ional and CG-en iched o mula ions (wi h a ious coa se-
g aining wid hs).
ins abili y and a con e ged solu ion canno be ob ained. This is because when he pa icle691
eloci y is homogenized o e oo many ini e elemen s, he di↵e ence be ween uDE and uFE
692
is excessi ely la ge, especially o hose elemen s which do no o e lap wi h he pa icles.693
As a esul , he coupling e ms ha a ise by penalizing uDE uFE lead o non-con e ging694
i e a ions in inding he solu ion o app oach uDE =uFE. As shown in Fig. 21, o a sha p695
signal, he CG wid h should no be la ge han wice he elemen size.696
6.2.3. F om a sha p impulse o con inuous wa e o ms697
Al hough he p e iously conside ed bounda y condi ion o he DEM model (i.e., ee698
end on he le in Fig. 18b) ensu es exac mass, momen um, and ene gy conse a ion, i 699
43
Coa se-g aining educes ene gy loss in o e lap egion. 23 / 39
In de elopmen : The mo-mechanical (Juan Al a ez)
Apply a simila coupling o he momechanical p oblems (elas ici y + hea ),
using he p inciples o i ual hea .
The numbe o elemen s o he FEM analysis is 4 o xand z,and50
o ydi ec ion. The cons i u i e ela ion o he beam is ep esen ed by he
gene alized Hooke’s law, and he Poisson e↵ec is no aken in o accoun .
Two di↵e en empe a u es T1=1.0CandT2Ca ese o he opand
bo om su ace o he domain, espec i ely. Fig. 13 shows he de lec ion o a
can ile e beam unde di↵e en ↵.
Figu e 13: The mal de lec ion esul s.
Fig. 13 shows di↵e en de lec ions when inc easing he he mal ine ia
coe icien (↵). The compa ison wi h he analy ical solu ion is p esen ed
in Fig. 14 whe e he de lec ion o he can ile e beam o di↵e en he mal
ine ia alues is shown.
The ag eemen be ween he analy ical solu ion and simula ions is consis-
en h oughou he ange o ↵=0.07.1⇥1031/C. This indica es ha
he con inuum analysis using iso opic ma e ial p ope ies may be eliable
o p edic ing he mal de lec ion.
42
The numbe o elemen s o he FEM analysis is 4 o xand z,and50
o ydi ec ion. The cons i u i e ela ion o he beam is ep esen ed by he
gene alized Hooke’s law, and he Poisson e↵ec is no aken in o accoun .
Two di↵e en empe a u es T1=1.0CandT2Ca ese o he opand
bo om su ace o he domain, espec i ely. Fig. 13 shows he de lec ion o a
can ile e beam unde di↵e en ↵.
Figu e 13: The mal de lec ion esul s.
Fig. 13 shows di↵e en de lec ions when inc easing he he mal ine ia
coe icien (↵). The compa ison wi h he analy ical solu ion is p esen ed
in Fig. 14 whe e he de lec ion o he can ile e beam o di↵e en he mal
ine ia alues is shown.
The ag eemen be ween he analy ical solu ion and simula ions is consis-
en h oughou he ange o ↵=0.07.1⇥1031/C. This indica es ha
he con inuum analysis using iso opic ma e ial p ope ies may be eliable
o p edic ing he mal de lec ion.
42
Bending o an elas ic beam due o hea .
24 / 39

Pa icle- luid coupling
•Fully esol ed: Accu a e and well-de ined, bu compu a ionally expensi e.
Valid o h < 1
8πdmin.
•Unde - esol ed: Fas , bu equi es closu e ela ions, modelling assump ions.
Valid o h > 3dmax.
25 / 39
Fully esol ed pa icle- luid coupling
26 / 39
Unde - esol ed pa icle- luid coupling
We apply a wo-phase low model (Ande son & Jackson):
•Each pa icle expe iences a d ag o ce,
d
p=β(u −up).
•The luid phase is go e ned by:
∂(ϵρ)
∂ +∇·(ϵρu ) = 0,
ϵDu
D =−ϵ∇p+µ∇2u +ϵg− d.
•Coupled ia coa se-g ained d ag o ce and po osi y (Rui & Heng):
d=X
p
d
pϕ( − p), ϵ =X
p
Vpϕ( − p)
27 / 39
Unde esol ed pa icle- luid coupling
Pa icles se ling in a lid-d i en ca i y 28 / 39
Open-sou ce coupling wi h
34 / 39

Coupling wi h p eCICE (wi h Da id Schneide )
•Allows coupling be ween mul iple codes and models ia con inuum ields.
•Takes ca e o coupling de ails (mapping, ime-s ep in e pola ion, e c)
•He e, we will use Me cu yDPM and openFoam (mul iphaseEule ):
Adap e
ϵ( ), d( )
u ( )
Con inuum
model
35 / 39
Coupling wi h p eCICE (wi h Da id Schneide )
•Allows coupling be ween mul iple codes and models ia con inuum ields.
•Takes ca e o coupling de ails (mapping, ime-s ep in e pola ion, e c)
•He e, we will use Me cu yDPM and openFoam (mul iphaseEule ):
Adap e
ϵ( ), d( )
u ( )
Con inuum
model
35 / 39
Valida ion: Oscilla ing lows
Rayleigh-Ko he lows a e s anda d CFD/FSI es cases o es sol e accu acy.
He e we use i o compa e mapping unc ions (nea es -neighbou s b ):
•One-di ec ional coupling ( luid →pa icles).
•No ine ia (pa icles ollow s eamlines).
•Oscilla ing low (pa icles e u n o o iginal loca ion).
36 / 39
Valida ion: Oscilla ing lows
•Obse a ion: b p oduces a smoo he eloci y ield, e e sible.
37 / 39
Valida ion: Sedimen ing pa icles
•Bidi ec ional coupling
•Resul : s eady-s a e p edic ed well
•Bu : dynamics sensi i e o mapping
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0 0.05 0.1 0.15 0.2 0.25
w = 5x10-3
w = 4x10-3
empi ical co ela ion
Veloci y
Time
p essu e eloci y magni ude
38 / 39

Valida ion: Sedimen ing pa icles
•Bidi ec ional coupling
•Resul : s eady-s a e p edic ed well
•Bu : dynamics sensi i e o mapping
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0 0.05 0.1 0.15 0.2 0.25
w = 5x10-3
w = 4x10-3
empi ical co ela ion
Veloci y
Time
p essu e eloci y magni ude
38 / 39
Coupling wi h p eCICE: Nex s eps
•Compa e wi h Moomph coupling.
•Valida e wi h expe imen s o o a ing d um case.
•Use coa se-g aining o ex ac coupled ields.
39 / 39
Goldhi sch, I. 2010 S ess, s ess asymme y and couple s ess: om disc e e
pa icles o con inuous ields. G anula Ma e 12 (3), 239–252.
Luding, S. 2008 In oduc ion o disc e e elemen me hods DEM: Basics o con ac
o ce models and how o pe o m he mic o-mac o ansi ion o con inuum
heo y. Eu o. J. o En i o. Ci . Eng. 12 (7-8), 785–826.
Roy, S., Schepe , B. J., Polman, H., Tho n on, A. R., Tunugun la,
D. R., Luding, S. & Weinha , T. 2019 Su ace loow p o les o d y and we
g anula ma e ials by pa icle .
Tho n on, A. R. & Weinha , T. e al. 2009-2019 Me cu ydpm.
h p://Me cu yDPM.o g/.
Tunugun la, D. R., Weinha , T. & Tho n on, A. R. 2017 Compa ing and
con as ing size-based pa icle seg ega ion models. Compu a ional Pa icle
Mechanics 4(4), 387–405.
Weinha , T., Ha kamp, R., Tho n on, A. R. & Luding, S. 2013
Coa se-g ained local and objec i e con inuum desc ip ion o 3d g anula lows
down an inclined su ace. Phys. Fluids 25 (070605).
39 / 39
Weinha , T., Tho n on, A. R., Luding, S. & Bokho e, O. 2012 F om
disc e e pa icles o con inuum ields nea a bounda y. G anula Ma e 14,
289–294.
39 / 39