3D coupling approach between discrete and continuum models for dynamic simulations (DEM–CNEM)

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To cite this version :

Mohamed JEBAHI, Jean-Luc CHARLES, Frédéric DAU, Lounès ILLOUL, Ivan IORDANOFF - 3D

coupling approach between discrete and continuum models for dynamic simulations

(DEM–CNEM) Computer Methods in Applied Mechanics and Engineering Vol. 255, p.196209

-2013

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simulations (DEM-CNEM) MohamedJEBAHI a ,Jean-lu CHARLES b ,Frederi DAU b ,Lounes ILLOUL ,Ivan IORDANOFF b a

Univ. Bordeaux,I2M,UMR5295,F-33400Talen e,Fran e b

ArtsetMetiers ParisTe h,I2M,UMR5295CNRSF-33400,Talen e,Fran e

ArtsetMetiersParisTe h,151,Boulevardde l'Hpital,F-75013 Paris,Fran e

Abstra t

The oupling between two dissimilar numeri al methods presents a major hallenge, espe iallyin ase of

dis rete- ontinuum oupling. The Arlequin approa h provides a exible framework and presents several

advantagesin omparisontoalternativeapproa hes. Manystudieshaveanalyzed,instati s,theingredients

of this approa h in1D ongurations underseveral parti ular onditions. Thepresent study extendsthe

Arlequin parameter studiesto in orporate a dynami behavior using3D models. Based on thesestudies,

a new 3D oupling method adapted for dynami simulations is developed. This method ouples two 3D

odes: DEM-based odeandCNEM-based ode. The3D ouplingmethodwasappliedtoseveralreferen e

dynami stests. Goodresultsareobtainedusingthismethod, omparedwiththeanalyti alandnumeri al

results ofboth DEM andCNEM.

Keywords: multis ale method, oupling method, dis reteelement, natural element,Arlequin approa h,

dynami simulation

1. Introdu tion

Thedis rete element method (DEM)[1 , 2℄ presents an alternative way to study physi al phenomena

requiring a very small s ale analysis or those whi h annot be easily treated by ontinuum me hani s,

su h as wear, fra ture and abrasionproblems. Inthe past de ades,an in reasing interest inthe dis rete

element method has led to the development of many interesting variations of this method. The most

re ent variation involves modeling the intera tion between parti les by ohesive beams [1 ℄. Thismethod

orre tly simulates the 3D linear elasti behavior of the ontinua. However, numeri al simulations are

verytime onsuming(CPU-wise). Furthermore,averygreatnumberofparti lesarerequiredtodis retize

small domains. This methoddoes not onsider large stru ture simulations. However, inmost situations,

theee ts thatmust be aptured byDEM are lo alizedina small portion ofthestudied domain. Thus,

theuseofaspe i multis alemethodto treatthephenomena at ea hs ale appears to be advantageous.

A hallengethatarisesinthemultis ale oupling approa histhatthehighfrequen yportionofwavesare

oftenspuriouslyree tedatthesmall/ oarses aleinterfa e. Thisphenomenonhasalreadybeenaddressed

using theniteelement modelwithdierent element sizes[3℄.

Theimportan eof thismultis aleapproa h hasattra tedmanyresear hers. Therefore,severalpapers

have been published on the subje t, and many oupling methods have been developed. These methods

an bedivided intwo lasses: edge to edgemethods and methods withoverlapping zones( alled overlap

methods). The rst lass [4, 5℄ is mainly applied to stati studies. Indeed, using this method, it is very

di ult to redu e spurious ree tions at the interfa e between models. Therefore, this lass will not be

treatedinthispaper. These ond lassseemstobemoreappli abletodynami studies,whi histhes ope

ofthe present work.

BenDhia [6, 7, 8℄, in a pioneer work, developed the Arlequin approa h asa general framework that

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with the mole ular dynami s su h that the two Hamiltonians are averaged in an overlapping zone. A

damping wasusedin the overlapping zoneto redu e the spuriousree tions at theinterfa e between the

two models. Nevertheless, the hoi e ofthedamping oe ient remains di ult.

Smirnovaetal. [11℄developeda ombinedmole ulardynami s(MD)andniteelementmethod(FEM)

model witha transition zonein whi h the FEM nodes oin idewith thepositions of theparti les inthe

MDregion. Theparti lesinthetransitionzoneintera t viatheintera tion potential withtheMDregion.

At the sametime, they experien ethenodalfor es dueto theFEMgrid.

Belyts hkoandXiao[4,12℄havedevelopeda oupling methodfor mole ulardynami s and ontinuum

me hani s models based on a bridging domain method. In this method, the two models are overlaid at

theinterfa e and onstrained withaLagrange multiplier modelintheoverlappingsubdomain.

Ja obetal. [13 ℄formulatedanatomisti - ontinuum ouplingmethodbasedonablendofthe ontinuum

stress andtheatomisti for e. In term ofequations, thismethod isvery similarto theArlequin method.

In an interesting work, Chamoin et al. [14 ℄ have analyzed the main spurious ee ts in the

atomi -to- ontinuum oupling approa hes andtheyproposed a orre tive method basedonthe omputation and

inje tion ofdead for es intheArlequinformulation to osetthese ee ts.

Aubertin et al. [15℄ applied the Arlequin approa h to ouple the extended nite element method

XFEMwiththemole ular dynami s MD to study dynami ra kpropagation.

Baumanetal. [16 ℄developeda3Dmultis alemethod,basedontheArlequinapproa h,betweenhighly

heterogeneous parti le modelsand nonlinear elasti ontinuum models.

Re ently,Combes ure etal. [17 ℄ formulated a 3D oupling method, applied for fasttransient

simula-tions, between the smoothed parti le hydrodynami s SPH and the nite element method. This oupling

methodis, also,basedon theArlequin approa h.

Formoredetails,areviewofthesemethods anbefoundin[18 ℄. A ommonfeatureofoverlap oupling

methods is that a weight fun tion is introdu ed to partition a ertain quantity in the overlapping zone.

Herein, theArlequin approa h [6 , 7,19℄is usedto develop a 3Dmultis alemethod adapted for dynami

simulations between the onstrained natural element method (CNEM) and the dis rete element method

(DEM). TheDEM version, whi h isusedinthis work,isthe most re ent versiondeveloped byAndré[1℄.

TheCNEMisamesh-freemethod,butitisvery losetotheniteelement method. The ouplingmethod

developed here an avoid spurious wave ree tions without any additional ltering or damping. Indeed,

thenes ale solutionisproje tedonto the oarses alesolutionintheoverlappingzoneatea htimestep.

Thus, it ltersthe high frequen ies oming fromthe ne s ale model(dis rete model),whi h aregreater

thanthe utofrequen yofthe oarses alemodel( ontinuummodel). Thispaperisorganizedasfollows:

in Se tion 2, the governing equations of both the DEM and CNEM models are given. Subsequently, we

des ribehowbothmodels are oupledusingthe Arlequinapproa hinthemostgeneral ase. InSe tion3,

several interesting previous studies on the Arlequin parameter are summarized: in luding mathemati al

studiesofBenDhiaetal. [7, 20℄,the studiesofBauman etal. [21 ℄ andthestati 1Dnumeri al studiesof

Guidaultetal. [22 ℄. After,thedierent Arlequinparametersarestudieddynami allyusing3Dmodels. In

Se tion 4,this new oupling method is validated for tensile- ompression, bending and torsional loadings

on beams. Se tion 5 presents the on lusionsand outlooks.

2. The problem statements

Adomain

Ω

is onsideredwithboundary

∂Ω = ∂Ω

u

_{+ ∂Ω}

T

su hthat displa ementsand tra tionsare

pres ribed on

∂Ω

u

and

∂Ω

T

,respe tively. Thisdomainisdividedintotwosubdomains,

Ω

C

and

Ω

d

,whi h are modeled using the ontinuum approa h and the dis rete approa h, respe tively. An isotropi linear

elasti behaviorand small deformation gradients are assumed for simpli ity. The governing equations of

both the ontinuum and the dis rete subdomains are re alled in Subse tions 2.1 and 2.2, while ignoring

the oupling onditions. These onditions will be introdu ed after detailing the oupling approa h in

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2.1. Continuum subdomain

Ω

C

As an isolated system, the governing equations in the ontinuum subdomain

Ω











div(σ) + ρf

= ρ¨

u

in

Ω

C

σ

= A : ε(u)

ε(u)

=

1 ₂

(∇u + ∇

t

u)

u

= u

d

on ∂Ω

u

_C

σ.n

= T

d

on ∂Ω

T

C

(1)

where

ρ

is the density,

u

is the ontinuum displa ement ve tor,

σ

is the Cau hy stress tensor,

∈ U

ad

su h that, given the initial onditions,

∀ δu ∈ U

ad,0

:

ˆ

∂Ω

T

C

δ ˙u · T

d

dΓ −

ˆ

Ω

C

ε(δ ˙u) : A : ε(u) dΩ +

ˆ

Ω

C

ρ δ ˙u · f dΩ =

ˆ

Ω

C

ρ δ ˙u · ¨

u

dΩ

(2)

with

δ ˙u

asatest fun tion andtheadmissible solution spa es,

U

ad

and

U

ad,0

,aredened asfollows:

U

ad

=

u = u(x, t) ∈ [H

1 _(Ω

C

)]

3 ; u = u

d

on ∂Ω

u

C

; ∀ t ∈ [0, T ]

U

ad,0

=

u = u(x, t) ∈ [H

1 _(Ω

C

)]

3 ; u = 0 on ∂Ω

u

C

; ∀ t ∈ [0, T ]

2.2. Dis rete subdomain

Ω

d

Inan isolated system of thedis rete domain

Ω

d

whi h is a set of spheri al parti les thatintera t via ohesive beams, the governing equations an be written as: for

i = 1..n

p

and

t ∈ [0, T ]

, given theinitial onditions,nd

(d

i

, θ

i

, f

int

/i

, c

int

/i

) ∈ R

3 × R

3

su hthat:

(

f

ext

_/i

+ f

int

_/i

= m

i

d

¨

i

c

ext

_/i

+ c

int

_/i

=

I

i

θ

¨

f

ext

/i

and

c

ext

/i

represent the total external for es and the total external torques applied on the

i

th

f

int

/i

and

c

int

/i

are the total internal for es andthetotal internal torquesapplied byotherparti lesvia the ohesive beams on the

i

th











f

int

_i

=

nnp

X

j=0

f

_ij

=

nnp

X

j=0

(E

µ

S

µ

∆l

_l

_µ

µ

x

−

6E

_l

µ

2 I

µ

((θ

jz

+ θ

iz

)y + (θ

jy

+ θ

iy

)z))

c

int

_i

=

nnp

X

j=0

c

_ij

=

P

nnp

j=0

(

G

µ

I

Oµ

l

µ

(θ

jx

− θ

ix

)x −

2E

µ

I

µ

l

2 µ

((θ

jy

+ 2 θ

iy

)y + (θ

jz

+ 2 θ

iz

)z))

• (O

i

, x, y, z)

is lo alframe asso iatedto thebeam onne ting

i

th

and

j

th

parti les.

• θ

i

(θ

ix

, θ

iy

, θ

iz

)

and

θ

j

(θ

jx

, θ

jy

, θ

jz

)

are the rotations of beam ross se tions expressed in the beam lo alframe.

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PSfragrepla ements

Ω

C

Ω

O

Ω

d

Figure1: Globaldomainde omposition

• l

µ

,

S

µ

,

I

Oµ

and

I

µ

arethebeamlength,beam rossse tionarea,polarmomentofinertiaandmoment of inertiaalong

y

and

z

.

• E

µ

and

Gµ

arethe beam Young and shearmodulus.

As in the ontinuum, the asso iated weak formulation an be dened as follows: nd

(d, θ, f , c) ∈

D

ad

× O

ad

× F

ad

× C

ad

su h that, given theinitial onditions,

∀ (δ ˙d, δ ˙θ) ∈ ˙

D

ad,0

_{× ˙}

_O

ad,0

:

n

p

X

i=1

f

ext

_/i

· δ ˙d

i

+

n

p

X

i=1

f

int

_/i

· δ ˙d

i

+

n

p

X

i=1

c

ext

_/i

· δ ˙θ

i

+

n

p

X

i=1

c

int

_/i

· δ ˙θ

i

=

n

p

X

i=1

m

i

d

¨

i

· δ ˙d

i

+

n

p

X

i=1

I

i

θ

¨

i

· δ ˙θ

i

(5) with:

D

ad

= {d = d

i

(t) i = [1..n

p

] ∀t ∈ [0, T ]}

O

ad

= {θ = θ

i

(t) i = [1..n

p

] ∀t ∈ [0, T ]}

F

ad

_{= {f = f}

int

/i

(t) i = [1..n

p

] ∀t ∈ [0, T ]}

C

ad

= {c = c

int

_/i

(t) i = [1..n

p

] ∀t ∈ [0, T ]}

n

p

: total numberof DEM parti les. 2.3. Couplingapproa h

As mentioned in the previous se tions, the oupling approa h used here is based on the Arlequin

approa h [6, 7,8℄. Thisapproa h onsistsof:

1. Asuperpositionofme hani al statesinthegiven subdomains

Ω

C

and

Ω

d

withanoverlappingzone

Ω

O

(Fig. 1).

2. A weak oupling (based on theweakformulation):

(a) Denition ofthegluing zone

Ω

G

:

Inthis study,thegluing zone

Ω

G

isthe same astheoverlappingzone

Ω

O

. Hereafter,the term overlapping zone will beusedto design theoverlappingzone orthegluing zone.

(b) Mediator spa e

M

:

To ensure the orre t dialogue between the models, the ontrol quantities in the overlapping

zonemust be hosen arefully. Here, the velo ity oupling, ina weak sense in

Ω

O

), is hosen. Fromanalgorithmi pointofview,thevelo ity ouplingiseasierthanthedispla ement oupling

(Remark2). The mediatorspa edenotedby

M

isdenedasthespa eof thevelo itiesdened in

Ω

O

.

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a) Discontinuous f unctions

b) Linear continuous f unctions

c) Continuous dif f erential f unctions

f (x)

_{f (x)}

¯

_{f (x)}

¯

f (x)

_{f (x)}

¯

Ω

C

Ω

C

Ω

O

Ω

d

Ω

O

Ω

d

Ω

C

Ω

O

Ω

d

1

0

Figure2: Examplesofweight fun tions

( ) Proje tion operator andjun tion model:

Theproje tionoperator

Π

proje tsthe ontinuumanddis retevelo itiesonthemediatorspa e. The jun tion model denes the linking onditions between thetwo models in theoverlapping

zone. To proje t the velo ities on

M

, an interpolation, whose shape fun tions will be dened later,is used. The jun tionmodelusedinthis workis the

H

1 _(Ω

O

)

s alarprodu tdened by:

< λ, q >

_H

1 _(Ω

O

)

=

ˆ

Ω

O

λ

· (Π ˙

u

− Π ˙

d) + l

2 ε(λ) : ε(Π ˙

u

− Π ˙

d) dΩ

(6)

where

(Π ˙

u

− Π ˙

d)

is the dieren e between the proje ted ontinuum and dis rete velo ities on

Ω

O

,

λ

is the Lagrange multiplier eld and

l

, the jun tion parameter, is an

H

1

oupling

parameter.This parameter has the unit of a length, it is added to ensure the homogeneity of

theintegral termsof the

H

1

oupling model. Inthiswork,

l

is onsideredavariableparameter and will be studied in Se tion 3. If

l = 0

, the

H

1 _(Ω

O

)

s alar produ t be omes equivalent to the

L

2 _(Ω

O

)

s alarprodu t(7)knownasthe Lagrange multipliermodel.

< λ, u >

_L

2 _(Ω

O

)

=

ˆ

Ω

O

λ

· (Π ˙

u

− Π ˙

(7),anintepolationisdenedontheDEMparti lesin

Ω

O

usingshapefun tions,whi hwillbe dened later.

3. Theenergy partition between the ontinuum anddis rete media intheoverlapping zone:

Asshown inFigure 1, the two models oexist in

Ω

O

. Therefore, the energies inthis zonemust be weighted, and,a kindof partitionof unityintermsofenergy isperformed. Three weight fun tions,

α(x)

,

β(x)

and

γ(x)

, are introdu ed for the internal energy, the kineti energy and the external work ofthe ontinuum subdomain,respe tively. Allof thefun tionsverifythefollowing:

f (x) : Ω → [0, 1]

x

→







1 in Ω

C

\Ω

O

[0, 1] in Ω

C

∩ Ω

O

0 otherwise

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In a omplementary manner, the internal energy, the kineti energy and the external work of the

dis rete subdomain are weighted by

α(x) = 1 − α(x)

¯

,

¯

β(x) = 1 − β(x)

and

γ(x) = 1 − γ(x)

¯

, respe tively. Figure2presentsexamples ofweight fun tions.

This oupling approa h is applied to ouple the ontinuum and thedis rete models dened on

Ω

C

and

Ω

d

,respe tively. By introdu ing theweight fun tions, in(2)and (5), and the oupling ondition (6),the global weighted weakformulationbe omes: nd

(u, d, θ, λ) ∈ U

ad

_{× D}

ad

_{× O}

ad

_{× M}

su hthat, giventhe

initial onditions,

∀ (δ ˙u, δ ˙d, δ ˙θ, δ ˙λ) ∈ U

ad,0

_{× D}

ad,0

_{× O}

ad,0

_{× M}

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´

Ω

C

β ρ δ ˙u · ¨

u

dΩ +

´

Ω

C

α ε(δ ˙u) : A : ε(u) dΩ −

´

∂Ω

T

C

γ δ ˙u · T

d

dΓ −

´

Ω

C

γ ρ δ ˙u · f dΩ

+

n

p

X

i=1

¯

β

i

m

i

d

¨

i

· δ ˙d

i

+

n

p

X

i=1

¯

β

i

I

i

θ

¨

i

· δ ˙θ

i

−

n

p

X

i=1

(¯

γ

i

f

ext

/i

+ ¯

α

i

f

int

/i

) · δ ˙d

i

−

n

p

X

i=1

(¯

γ

i

c

ext

_/i

+ ¯

α

i

c

int

_/i

) · δ ˙θ

i

+ δ

ˆ

Ω

O

λ

· (Π ˙u − Π ˙d) + l

2 ε(λ) : ε(Π ˙u − Π ˙d) dΩ

= 0

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2.4. Spatial dis retizationand integration issues

Intheprevious subse tions, the global weak formulation (9)is presentedin a ontinuous form. Now,

thespatial dis retization is introdu ed. In the literature, thereare many interesting ontinuum methods

used for solving partial dierential equations, su h as SPH [23℄, NEM [24 ℄ and FEM[25℄. Ea h method

is distinguished byits apability to spatially dis retizethe studied model. Among them,the onstrained

natural element method CNEM[26, 27,28 ℄,whi h isanextension of thenatural element method(NEM)

[29 ℄ to non- onvex domains, is hosen in this study. This method has pra ti ally all of the advantages

of the FEM approa h, and it ir umvents the major drawba ks related to the meshing. Indeed, using

the FEMapproa h, the approximation is dependent on the mesh quality. In ontrast, using the CNEM

approa h, the approximation is dependent only on the relative position of the nodes [30 ℄. Unlike the

othermesh-freeapproa hes: (i)the supportsof onstrained naturalneighbor (CNN)shape fun tionsused

in CNEM approa h are automati ally dened, (ii) the values of CNN shape fun tions asso iated with

internal nodesarenullonthe borderofthedomain. Thislastpropertyis parti ularlyinteresting be ause

itallowsadire timpositionoftheboundary onditions,exa tlyasintheniteelementsframework. Given

thebroadsimilarity between the CNEMand FEMapproa hes, theCNEM-DEM oupling havethe same

performan esastheFEM-DEM oupling withbetterappli abilityon omplex domainsand/orbehaviors.

Therefore,the ontinuumsubdomainin

Ω

C

isdis retizedwiththeCNEMapproa h. Consequently,

Ω

C

is approximated byaset ofnodesinwhi h onne tivityis not ne essary[26℄.

Toobtain a ontinuouseldfromthe dis retequantities dened attheDEM parti lepositionsin

Ω

O

, a onstrainednaturalneighbor (CNN)interpolationisintrodu ed in

Ω

d|Ω

O

. Thus,theparti lesasso iated withthis subdomain are also onsideredCNEM nodes. TheCNN interpolationis only applied in

Ω

d|Ω

O

, whi h is assumed to be far from the ne s ale ee ts. The mediator spa e is also dis retized with the

CNEMapproa h. Wedenoteby

M

h

C

,

M

h

d

and

M

h

O

thedis retizedspa esof

U

ad

,

D

ad

and

M

,respe tively. The asso iated dis retized subdomains are designed

Ω

h

C

,

Ω

h

d

and

Ω

h

O

,respe tively. The dis rete domain

Ω

d

is a set of parti les, thenit is naturally dis retized and

Ω

h

d

= Ω

d

. A ording to the ongurations of thedis retized spa es inthe overlapping zone,four ases an bedistinguished (Fig. 3).

Inthisstudyand ontrarytoprevious studieson ontinuum/dis rete oupling approa hes,no oin iden e

onditionsareimposedonthe oexistingdis retizedsubdomainsin

Ω

O

. Therefore,thefourth onguration is studied here (Fig. 3-d)as thegeneral onguration thatin ludes the three other ongurations. This

simplies theuseofthis method in3D omplexdomains. Indeed,inthis ase,it issu ient to dis retize

thesubdomainsindependentlyandmountthemasindi atedinFigure3-d. Infa t,usingthis onguration,

itisverydi ulttoprovemathemati allythe existen eanduniquenessofthesolution. Alsoand ontrary

to thethreeother ongurations,there arenonumeri al works,inliterature, studying this onguration.

d|Ω

O

,the

displa e-ment elds

u

and

d

and theLagrange multiplier unknowns

λ

areapproximated by:

h

O

6= Ω

h

d|Ω

O

Nodesofthe ontinuumsubdomain

Ω

h

C

Nodesoftheoverlappingsubdomain

Ω

h

O

Parti lesofthedis retesubdomain

Ω

h

d

Figure3: Thedierent ongurationsofthedis retizedsubdomains

d

h

(x) =

n

pO

X

i=1

N

_i

d

(x)d

i

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λ

h

(x) =

n

O

X

i=1

N

_i

O

Ω

h

d|Ω

O

.

u

i

isthe nodaldispla ements,

d

i

˙d

,

˙θ

and

λ

,it an be reformulated asfollows: nd

(u, d, θ, λ) ∈ U

ad

_{× D}

ad

_{× O}

ad

_{× M}

su h that, given

theinitial onditions,

∀ (δ ˙u, δ ˙d, δ ˙θ, δ ˙λ) ∈ U

ad,0

_{× D}

ad,0

_{× O}

ad,0

_{× M}

:

´

Ω

C

β ρ δ ˙u · ¨

u

dΩ −

´

∂Ω

T

C

γ δ ˙u · T dΓ +

´

Ω

C

α ε(δ ˙u) : A : ε(u) dΩ −

´

Ω

C

γ ρ δ ˙u · f dΩ

+

´

Ω

O

λ

· δΠ ˙u + l

2 _{ε(λ) : ε(δΠ ˙u) dΩ = 0}

(13)

n

p

X

i=1

¯

β

i

m

i

d

¨

i

· δ ˙d

i

−

n

p

X

i=1

(¯

γ

i

f

ext

/i

+ ¯

α

i

f

/i

int

) · δ ˙d

i

−

´

_Ω

_c

λ.δΠ ˙d + l

2 ε(λ) : ε(δΠ ˙d) dΩ = 0

(14)

n

p

X

i=1

¯

β

i

I

i

θ

¨

i

· δ ˙θ

i

−

n

p

X

i=1

(¯

γ

i

c

ext

/i

+ ¯

α

i

c

int

/i

) · δ ˙θ

i

= 0

(15)

´

Ω

O

δλ · (Π ˙u − Π ˙d) + l

2 _{ε(δλ) : ε(Π ˙u − Π ˙d) dΩ = 0}

(16)

The integralterms will be omputed numeri ally using an integration te hnique. Integrationbya Gauss

(9)

to ompute the ontinuum termsusing theVoronoï ells astheba kground of theintegration. However,

on erning the oupling terms, this te hnique annot be applied dire tly. Indeed, the integrands in lude

variables dened on dierent Voronoï diagrams. The issue here is how to hoose the ba kground of

the integration. In this work, the Voronoï ells asso iated with the mediator spa e

M

h

O

are hosen as ba kground of integration in the overlapping zone

Ω

O

. All of thevariables that are not dened on

M

h

O

areproje ted on thisspa e.

By repla ing

u

¨

n ¨do = f

int

α

+ f

ext

γ

+ {f

c

}

[I

β

]

n¨θo

=

c

int

α

+ c

ext

γ

(17) where

(m

β

)

ij

= ¯

β

i

δ

ij

· m

i

,

(I

β

)

ij

= ¯

β

i

δ

ij

I

i

,

(c

int

α

)

i

= ¯

α

i

c

int

_/i

,

(c

ext

γ

)

i

= ¯

γ

i

c

ext

_/i

,

(f

int

α

)

i

= ¯

α

i

f

int

_/i

,

(f

ext

_γ

)

i

= ¯

γ

i

f

ext

_/i

and

{f

c

_{} = [c}

d

] {λ} = ([c

L

2 d

] + l

2 [c

H

1 d

]) {λ}

representsthetotal oupling for e.

c

L

_d

2 ≈

n

O

X

K=1

V

O

_I

h

N

d

(x

I

)

i

T

N

O

_(x

I

) dΩ

And

c

H

1 d

≈

n

O

X

K=1

V

O

_I

h ˜

B

d

(x

I

)

i

T

h ˜

_B

O

_(x

I

)

i

Where

x

I

arethe oordinatesofthe

I

th

nodeof

Ω

h

O

,

V

O

I

, respe tively.

h ˜

B

d

i

and

h ˜

B

O

i

are the smoothed gradient matri es [31 , 32℄ asso iated with

Ω

h

d

and

Ω

h

_O

,respe tively.

•

CNEMequations

[M

β

] {¨

u} = −

F

int

α

+ F

ext

γ

− {F

c

}

(18) where

(M

β

)

ij

= δ

ij

β(x

i

) M

i

,

M

i

isthelumpedmassofthe

i

th

nodelo atedat

x

i

position,

F

int

α

=

[K

α

] {u}

,

[K

α

]

isthe weightedstinessmatrix and

{F

c

_{} = [C}

C

] {λ} = ([C

L

2 C

] + l

2 [C

H

1 C

]) {λ}

repre-sentsthetotal oupling for e.

C

_C

L

2 ≈

n

O

X

K=1

V

O

_I

N

C

_(x

I

)

T

N

O

_(x

I

) dΩ

And

C

H

1 C

≈

n

O

X

K=1

V

O

_I

h ˜

B

C

(x

I

)

i

T

h ˜

_B

O

_(x

I

)

i

Where

N

C

and

h ˜

B

C

i

are,respe tively,theinterpolationandsmoothedgradientmatri esasso iated

with

Ω

h

C

.

•

Interfa e equations: Equation16 leadsto:

[C

O

]

.

u − [c

o

]

n

.

d

o

= 0

where:

[C

O

] =

h

C

L

2 O

i

+l

2 h

_C

H

1 O

i

=

h

C

L

2 C

i

T

+ l

2 h

_C

H

1 C

i

T

and

[c

o

] =

h

c

L

2 o

i

+l

2 h

_c

H

1 o

i

=

h

c

L

2 d

i

T

+ l

2 h

_c

H

1 d

i

T

.

(10)

The numeri al time integration is based on an expli it integration s heme that is well adapted for

dynami omputations. Many expli it s hemes an be used, su h as the Runge-Kutta, position Verlet

and velo ity Verlet s hemes. A omparison between these s hemes an be found in [33℄. A ording to

this referen e [33 ℄, the velo ity Verlet s heme provides good results and is also easy to implement. For

these reasons, this s heme is usedin this paperto solve theglobal dynami problem. Thiss heme gives

an

O(h

3 ₎

approximation for both velo ities and displa ements. Thus, the velo ity oupling, usedin this

work,doesnot ae t the oupling approa h a ura y ompared to thedispla ement oupling.

2.5.1. DEM algorithm (DEM ode)

n

: Theinitial onditionsortheinterfa eresults.

•

Computationof

{d}

n+1

and

{θ}

n+1

:

{d}

_n+1

= {d}

_n

+ ∆t

n

d

.

o

n

+

∆t

2

2 n ¨do

_n

{θ}

_n+1

= {θ}

_n

+ ∆t

n

θ

.

o

n

+

∆t

2

2 n ¨θo

_n

(19)

•

Computationof

f

int

α

n+1

,

f

ext

γ

n+1

,

c

int

α

n+1

et

c

ext

γ

n+1

•

Computationof the predi tive lineara elerations

n ¨do

∗

n+1

(omittingthe oupling for es

{f

c

_}

from Equation17).

n ¨do

∗

n+1

= [m

β

]

−1

_(f

int

α

n+1

+

f

ext

γ

n+1

)

(20)

•

Computationof the angular a elerations

n ¨θo

n+1

:

n ¨θo

n+1

= [I

β

]

−1

_(c

int

α

n+1

+

c

ext

γ

n+1

)

(21)

•

Computationof the predi tive linearvelo ities

n

.

d

o

∗

n+1

:

n

.

d

o

∗

n+1

=

n

.

d

o

n

+

∆t

2 (n ¨do

n

+

n ¨do

∗

n+1

)

•

Computationof the angular velo ities

n

.

θ

o

n+1

:

n

.

θ

o

n+1

=

n

.

θ

o

n

+

∆t

2 (n ¨θo

_n

+

n ¨θo

_n+1

)

•

Transfer of the predi tive linearvelo ities and a elerations to theinterfa e:

n

.

d

o

∗

n+1

and

n ¨do

∗

n+1

2.5.2. CNEM algorithm (CNEM ode)

•

Initialization

{¨

u}

n

,

.

u

n

et

{u}

n

: The initial onditionsor theinterfa e results.

•

Computationof

{u}

n+1

:

{u}

_n+1

= {u}

_n

+ ∆t

.

u

n

+

∆t

2

2 {¨

u}

n

(22)

•

Computationofthe predi tivelineara elerations

{¨

u}

∗

n+1

: (omittingthe ouplingfor es

{F

c

_}

from Equation18).

{¨

u}

∗

_n+1

= [M

β

]

−1

(−

F

int

α

n+1

+

F

ext

γ

n+1

)

(23)

•

Computationof the predi tive linearvelo ities

.

u

∗

_n+1

:

.

u

∗

_n+1

=

.

u

_n

+

∆t

₂

({¨

u}

_n

+ {¨

u}

∗

_n+1

)

•

Transfer of the predi tive linearvelo ities and a elerations to theinterfa e:

.

u

∗

_n+1

and

{¨

u}

∗

n+1

(11)

•

Re overy of the predi tive linear velo ities from both the CNEM and DEM odes:

.

u

∗

_n+1

and

n

.

d

o

∗

n+1

•

Computationof

{λ}

n+1

.

u

n+1

=

.

u

∗

n+1

−

∆t

2 [M

β

]

−1

_{F

c

_}

n+1

n

.

d

o

n+1

=

n

.

d

o

∗

n

+

∆t

2 [m

β

]

−1

{f

c

}

_n+1

(24)

{F

c

}

_n+1

= [C

C

] {λ}

_n+1

{f

c

}

_n+1

= [c

d

] {λ}

n+1

(25)

[C

O

]

.

u

n+1

− [c

o

]

n

.

d

o

n+1

= 0

(26)

By introdu ingEquations 24 and25 into Equation26,the interfa esystemof equations an be

written as:

[A] {λ}

_n+1

= {b}

_n+1

(27)

where the oupling matrix

[A]

and

{b}

n+1

are dened,respe tively,as:

[A] =

∆t

2 ([C

O

] [M

β

]

−1

_[C

C

] + [c

o

] [m

β

]

−1

[c

d

])

(28)

{b}

_n+1

= [C

O

]

.

u

∗

n+1

− [c

o

]

n

.

d

o

∗

n+1

and

n ¨do

n+1

:

{¨

u}

_n+1

= {¨

u}

∗

_n+1

− [M

β

]

−1

{F

c

}

n+1

n ¨do

n+1

=

n ¨do

∗

n+1

+ [m

β

]

−1

{f

c

}

_n+1

(30)

•

Transfer of

{¨

u}

n+1

and

.

u

n+1

to theCNEM pro ess and

n ¨do

n+1

and

n

.

d

o

n+1

theDEM pro ess.

Remark1:. The system (27) is solved using the well-known

LU

de omposition method [34℄. Con erning thesystem (28), sin e the massmatri es ofboth ontinuum and dis rete models arediagonals, itis easy

to derive their inverse matri esand ompute the oupling matrix

A

.

Remark2:. From an algorithmi point of view,the velo ity oupling used inthis work is easier than the

displa ement oupling. This is be ause thedispla ement oupling requires, inaddition to thepredi tive

a elerations and velo ities, the omputation of the predi tive displa ements whi h must be sent to the

Interfa e odefor orre tion. Therefore,inthe aseofdispla ement oupling,additionalstepsarene essary

to omputeand orre tthepredi tivedispla ements. Whereas,inthe aseofvelo ity oupling,the orre t

(12)

of the parti les in the overlapping zone. These quantities are orre ted impli itly. Indeed, the internal

for es are omputed a ounting for parti le displa ements and rotations. These for es are later used to

ompute thenew displa ementswhi h are orre tedusing the oupling ondition(16).

Remark4:. To be able to ompute the predi tive a elerations (Eqs. 20, 21 and 23), the lumped mass

matri es must be invertible. Thus, the weight fun tions

β

and

¯

β

must be stri tly positive in

Ω







1 in Ω

C

\Ω

O

[ε, 1 − ε] in Ω

C

∩ Ω

O

0 otherwise

(31)

where

ε

isa small stri tlypositive realnumberto be hosen.

2.5.4. Implementation

TheDEM al ulusisa hievedusingtheGranOOworkben h(GranularObje tOriented). Granoowas

developed at theMe hani s institute of Bordeaux (I2M)by J.L. Charles etal. The ode provides C++

librariesthatimplement lassesusefultodes ribeandsolvedynami me hani al problemsusingDEMand

expli ittemporalintegration s hemes. TheCNEM al ulusisa hieved usingaCNEM-based ode,whi h

was developed at the PIMM laboratory by G. Cognal et al. It provides C++ libraries that interfa e

with Python modules. The oupling between DEM and CNEM, des ribed in the previous se tions, is

performedbyan interfa e written inthe Python language. This interfa e ommuni ates dire tlywith

theCNEM-based pro ess using Python lasses. The Inter Pro ess Communi ation (IPC) toolis usedto

ensure asyn hronized ommuni ation between theGranOO pro ess and theinterfa e pro ess.

3. Parametri study of the oupling parameters

Several workshavestudied mathemati ally theArlequin method forboth ontinuum- ontinuum oupling

[7 ,19,20℄and ontinuum-dis rete oupling[21℄. Themainresults on erningthewell-posednessofthe

ou-pling problemarere alledinthis paper. Theweight fun tion

α

mustbestri tly positive in

Ω

O

. Without this onditionthe oer ivityof theinternal energy annotbe veried. Another signi ant result

on ern-ing the oupling jun tion models is that for the dis retized problem, ontrary to the

H

1

oupling whi h

yields a well-posed problem, the

L

2

oupling model an lead to an ill- onditioned system of equations,

espe ially in the ase of very small mesh size. In this ontext, Bauman et al. [21℄ have studied another

oupling model, the

H

1

seminorm, inwhi hthe rst term ofthe

H

1

modelisremoved. Thismodel leads

to a well-posed problem, but it does not onstrain enough the ontinuum and dis rete displa ements in

the overlapping zone. Other works [22, 35 , 21℄ have studied numeri ally the ingredients of theArlequin

method using

1D

models. Guidault et al. [22, 35 ℄ noted that, for the

L

2

oupling model, the weight

fun tion

α

mustbe ontinuousattheboundaryofthegluingzone

∂Ω

O

. Indeed,theuseofadis ontinuous weight fun tion an ause undesirable free onditions at

∂Ω

O

.

Con erning the hoi e of the mediator spa e, Ben Dhia [7 , 20 ℄ mentioned thatin the ase of ontinuous

domains, itis onvenient to hoose

M = H

1 _(Ω

O

)

;however,itis verydi ultto hoosethenite approx-imation spa e

M

h

O

. To address this di ulty, several works[22 , 35,21 ℄ proposed a 1Dnumeri al study of

M

h

O

. The dierent ongurations thatwerestudied arepresentedina,band of Figure3. Thestati studiesof Guidaultetal. [22℄ showthat: (i)inthe aseofa nemultiplier spa e(Fig. 3-b), theresponse

of the stru ture do not depend on the weight fun tions and a lo king phenomenon takes pla e, i.e, the

(13)

E

µ

= 265 GP a

ν

µ

= 0.3

r

˜

µ

= 0.71

Table1: Themi ropropertiesofthe ohesivebeambondsintheDEMsubdomain

˜

r

µ

isanadimensional ohesivebeamradius,denedastheratiobetweenthebeamradiusandthemeanparti leradius;

E

µ

and

ν

µ

are themi roYoungmodulusandthePoissonratioofthebeams,respe tively.

multiplier spa e, the weight fun tions has an inuen e on the solutions su h that the larger the weight

fun tion onthene mesh,the smaller be omesthe maximumjumpbetween thetwo meshes.

This work proposes a 3D numeri al dynami study using the general onguration given in Figure 3-d.

It will be demonstrated that some of the results proven in stati using 1D models are not valid in 3D

O

,

•

Thedis retization ofthe approximated Lagrangemultipliers spa e

M

h

O

.

A 3D beam model is used for the dynami study (Fig. 4), in whi h the length and the diameter are

L = 20 mm

and

D = 2 mm

,respe tively. The modelis dividedinto two subdomainswithan overlapping zone. TheleftsubdomainismodeledbytheCNEMapproa husing626nodes(theasso iated hara teristi

length is about

l

c

= 0.47 mm

) and xedat the leftend (

x = 0

). Theright subdomain is modeledbythe DEM approa h using

20 000

spheri al parti les having

r

c

= 0.05 mm

as mean radius. Based on the hara teristi length of DEM and CNEM dis retization (

l

x = L

) is submitted to a tensile loading with a very steep slope (Fig. 5). As shown in lower viewgraph of Figure 5, the Fourier spe trum ontains

powerful high frequen y waves (greater than

f

CN EM

c

). The material of the beam is the sili a: Young's modulus

E = 72 GP a

, Poisson's ratio

ν = 0.17

and density

ρ = 2200 Kg/m

3

. The orresponding mi ro

propertiesof the ohesivebeambondsintheDEM approa h aregivenin[1℄ and presentedinTable1.

To ontrol thewave propagation in themodel, four he k points are pla ed along this beam (Fig. 4) as

follows:

•

0.1

0.2

0.3

0.4

0.5 T ime

(µs)

0

200

400

600

800 1000

1200

1400

F

o

r

ce

(N

)

(a) F orce vs. T ime

20

40

60

80

100 F requency

(M Hz))

50

55

60

65

70 P

o

w

e

r

(d

B

)

(b) F ourier spectrum

Figure5: Tensileloadingofstudymodelandtheasso ietedspe tralanalysis( omputedfromFFT)

U

mean

(mm)

f

0 (Hz)

f

1 (Hz)

f

2 (Hz)

Theory

0.087 71 757

215 272

358 787

DEM

0.083 72 408

217 246

362 072

CNEM

0.088 71 359

214 023

356 491

(15)

( omputedfromFFT)

Remark5:. Inthisstudy,forthesakeofsimpli ity,theweight fun tionsare hosenasfollows:

α = β = γ

.

is dened in (31)). The onditioning de reases with

l

and rea hes a minimum at a small

l = l

opt

. Beyond this value, the onditioning in reases exponentially as

l

in reases. This is true for any hoi es of

L

O

,theweight fun tionsand

M

h

O

. Be ause

l

de reasesthe onditioning,

H

1

oupling (6)for a

small valueof

l

,isbetter than

L

Ω

O

c

). Intheremainder of thisse tion we will usethe

H

1

oupling with

l = l

opt

.

3.2. Inuen e of the weight fun tions

Inthissubse tion,a nedis retization oftheapproximated multiplierspa e

M

h

O

is hosen, i.e,atthe same s ale as

M

h

d

. The widthoftheoverlapping zone

L

O

isxed at

2 mm

.

3.2.1. Constant weight fun tions

α

CN EM

= α

DEM

= 0.5

Themeandispla ementobtainedwiththe ouplingmethodis

0.081 mm

. Thisisinagreementwiththe referen emean displa ements(Tab. 2). However, the temporal urve (Fig. 8) presents several deviations

withregardtothereferen e urves. Figure9presentsthevelo itiesinthedierent he kpoints(Figure4)

for therstround tripof the wave propagation. It an beseen thatthemajor partof thehighfrequen y

waves (HFW) are ree ted without entering the overlapping zone. Indeed, the HFW initially aptured

intheDemChe kPoint didnot appearinOverlapDemChe kPoint or OverlapCnemChe kPoint. This

explainsthedeviationinthetemporaldispla ementea htimetheglobalwave rossestheoverlappingzone.

Thus, onstant weight fun tions are not a good hoi e for dynami simulations. Indeed, the proje tion

me hanism, whi h o urs in

Ω

O

, annot dampen the HFW, and an additional ltering is required. In ontrast,the stati studiesofGuidaultetal. [22 ℄showed that onstant weight fun tions anbe usedwith

(16)

Figure7: Conditioningof

A

withrespe tto

l

L

O

= 6 mm

, oarseMultiplierspa e, ontinuousweightfun tionswith

ε

= 0.005

Figure8:Thefree-enddispla ementsobtainedusingDEMandCNEMseparatelyandthe ouplingmethod,andtheasso iated

spe tralanalyses( omputedfromFFT)

(17)

L

O

= 2 mm

,neMultiplierspa e, onstantweight fun tions

α

CN EM

= 0.5

3.2.2. Constant weight fun tions

α

CN EM

6= 0.5

Thissub-subse tion analyzestheinuen e ofthe weight onstant onthewavepropagation. Two ases

are studied here; the rst ase uses

α

CN EM

= 0.3

(then,

α

DEM

= ¯

α

CN EM

= 0.7

), and the se ond ase uses

α

CN EM

= 0.8

(then,

α

DEM

= 0.2

). Theasso iated resultsare presentedinFigure 10.

= 0.8

. To provide an explanation for these results, the temporal velo ities at the he k pointsarepresentedinFigure11.

It anbeseenthatfor

α

CN EM

= 0.8

,aportionoftheprin iplewaveispositively ree tedattheinterfa e between the two models, or more pre isely, without entering the overlapping zone. Furthermore, only a

(18)

(19)

t

num

DEM −CN EM

=

magnitude of the transmitted wave

magnitude of the incident wave

=

9,90

22,01

= 0.44

and

r

num

DEM −CN EM

= 0.56

Byanalogywiththewavepropagationbetweenmediawithdierenta ousti impedan es,thetransmission

and ree tion oe ients antheoreti ally be dened as:

t

th

DEM −CN EM

=

α

DEM

2 α

+α

DEM

CN EM

and

r

th

DEM −CN EM

=

α

CN EM

DEM

+α

−α

CN EM

DEM

Then, it an be veried that

t

num

DEM −CN EM

α

CN EM

= 0.3

,thesame ree tion me ha-nism takespla e but witha negative oe ient. Indeed, the velo itymagnitude ofthe transmitted wave

(measured at CnemChe kPoint) is greater than the velo ity magnitude of the forward wave (initially

measuredat DemChe kPoint).

t

num

_Dem−Cnem

=

31.49 _21.86

= 1.44

r

num

_Dem−Cnem

= 1 −

31.49 _21.86

= −0.44

and

t

th

_Dem−Cnem

=

2×0.7

₁

= 1.4

t

num

_Dem−Cnem

= 1 − 1.4 = −0.4

Then,for the aseofa onstantweighting,

α

CN EM

= α

DEM

= 0.5

mustbeused. Otherwise,therewillbe aree tionof apartofthe prin ipalforward wave. Thisresultprovesthatthe1Dstati studiesavailable

in literature annot be used to perform dynami oupling. In ee t, Guidault et al. [22℄ noted that, in

stati s andusing a nemultiplier spa e, thesolutions do not depend on theweight fun tions.

3.2.3. Continuous weight fun tions

Asexplained in Remark 4,the weight fun tions must not vanish at the boundary of theoverlapping

zone, and a small value

ε

must be adopted rather than

0

at

∂Ω

O

. Prior to studying theinuen e of the ontinuous weight fun tions, the inuen e of

ε

is studied. Figure 12 presents the free-end displa ement using ontinuous weight fun tions for

ε = 0.05

,

ε = 0.005

and

ε = 0.0005

and the same onditions for

L

O

and

M

h

ε

0.05

0.005 0.0005

Cond[A(l

opt

)]

2.53e4

8.93e4

5.67e5

Table 3: Conditioningof

A

withrespe tto

ε

L

O

= 2 mm

,neMultiplierspa e, ontinuousweightfun tions

In the remainder of this paper,

ε = 0.05

will be hosen ea h time a ontinuous weight fun tion is used. Figure 14 shows the free-end displa ement for the ase of a ontinuous weight fun tion (Fig. 2-b). No

high frequen y waves (HFW) are ree ted at the interfa e between the two models. Using ontinuous

weight fun tions, the HFW enter the overlapping zone, and then, they are dampened by the proje tion

ontothe oarse spa e. Thelowersubplot inFigure14eviden es thatwitha nemultiplier spa e,a small

overlappingzoneissu ient to an elout alloftheHFW.AsshowninFigure13,theuseofa ontinuous

weight fun tion signi antly improves the results. However, a small deviation from the referen e results

still persists and be omes greater ea h time the wave travels ba k (CNEM-DEM dire tion). Be ause of

thevery ne dis retizationof theDEM subdomain,theweight of theparti lesin

Ω

O

de reases smoothly when approa hing the CNEM subdomain. Therefore, the forward wave orre tly rosses the interfa e

between the pure DEM (

Ω

d

\Ω

O

) and the overlapping zone

Ω

O

. Thus, byexamining the rst round-trip (Fig. 13),it isapparent thatno deviation fromthe referen esis notedwhen wave travels fromthe DEM

subdomaintotheCNEMsubdomain. IntheCNEMsubdomain,a oarsedis retizationisused. Thejump

betweentheweightsoftwoadja ent nodesisrelatively large. Thedis reteweightfun tionsoftheCNEM

subdomain aredis ontinuous stair asefun tionswithlarge jumps. Thus, thesameree tion me hanism,

observedpreviously,o urswhenthewavetravelsba k(CNEM-DEMdire tion). Toredu ethedeviation,

thewidthoftheoverlapping zonemustbe in reasedto redu etheslope oftheweight fun tions. Another

solution onsists of using ontinuous dierentiable weight fun tions (Fig. 2- ) to redu e the weighting

jump in the vi inity of the overlapping zone boundary

∂Ω

O

. Figures 19, 20 and 21 present the results using the twosolutions. Thewave orre tly rosses

Ω

O

withoutanydeviation.

Figure 13: The free-end displa ements obtained using DEM and CNEM separately and the oupling method, and the

asso iatedspe tralanalyses( omputedfromFFT)

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Figure14: Thelinearvelo ities(a)atthedierent he kpointsforthe aseof ontinuous

α

and(b)atthe"DemChe kPoint" forthe aseof ontinuousand onstant

α

L

O

= 2 mm

,neMultiplierspa e, ontinuousweightfun tions,

ε

= 0.05

3.3. Inuen e of the approximated multiplierspa e

M

h

O

In the previous subse tion, a ne multiplier spa e was used. In this ase, the velo ity in

Ω

d|Ω

O

is pra ti allylo kedatavalueequaltothevelo ityinthe

Ω

C|Ω

O

,asshownintheuppersubplotofFigure15. Indeed,thevelo ity urveatOverlapDemChe kPoint oin ides withthatatOverlapCnemChe kPoint.

The same lo king phenomenon is noted when the se ond onguration (Fig. 3-b) is used [22℄. Now, to

study theinuen e of

M

h

O

onthe results,a oarse multiplierspa e isused, i.e,at thesames ale as

M

h

C

. Asshown in thebottom subplot of Figure 15,equality of thevelo ities in

Ω

O

issatised only in a weak sense and not in ea h multiplier spa e node. This allows the ne model (DEM model) to orre tly a t

in

Ω

O

. However, in this ase a small overlapping zone is insu ient to orre tly transmit the prin ipal tensile waveand an el the highfrequen y waves.

3.4. Inuen e of the width of the overlapping zone

L

O

Ω

O

redu es theHFWree tion andallows abetter transferoftheforward wave.

3.5. How to hoose the oupling parameters in a general ase?

Ina general ase, there is not an obvious method to determine, ina single way, thevarious oupling

parameters to avoid wave reexion. This subse tion gives several re ommendations and trends to hoose

orre tly these parameters. The weight fun tions must be ontinuous. Indeed, with onstant weight

fun tions,thehighfrequen ywaves(HFW)areree tedwithoutenteringtheoverlappingzoneand annot

bedampened bytheproje tion me hanism. The hoi es ofthewidthoftheoverlappingzonedependson

the hara teristi dimension of the dis retization inthis zone. In the aseof ne dis retization, a narrow

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L

O

= 2 mm

, ontinuousweightfun tion,

ε

= 0.05

Approximated Multiplierspa e Coarse Multiplierspa e

Widthofthe overlappingzone

L

O

As largeaspossible

Weight fun tions Continuous,

ε = 0.05

Table4: Re ommendedArlequinparametersfordynami simulations

in

Ω

O

. On the ontrary, oarse dis retization requires a large

Ω

O

. To minimize the onditioning of the oupling matrix

A

,the

H

1

oupling with

l = l

opt

isre ommended.

l

opt

an be hosenasthe hara teristi length of theoverlapping zone dis retization

l

Ω

O

c

(

l

opt

≈ l

Ω

O

c

). Table 4 presents the onvenient Arlequin parameters to orre tlyperform the oupling.

4. Validation

The previous parametri -based study on tensile loading has allowed us to retain the onvenient

pa-rameters to perform a orre t oupling. In this se tion, the results of this study areused to validatethe

oupling between CNEM and DEM in a general 3D ase. Contrary to the tensile ase, in bending and

torsion, thedeformationsinthe rossse tions aresigni ant. To a ount for theseee ts, newgeometri

hara teristi s of the 3Dmodelare used:

L = 100 mm

and

D = 20 mm

(

L/D = 5

). The DEM method is applied only for the portion lo ated

20 mm

from the right end (the se tionlo ated at

x = L

) and the remainder ofthemodelismodeled usingtheCNEM method (Fig. 18).

ThefollowingArlequin's parameters are hosen:

L

O

= 10 mm

, ontinuousdierentiableweight fun tions,

ε = 0.05

,

l = l

opt

and oarse multiplier spa e.

Figures 19 and 20 present the temporal free-end displa ements with respe t to the x-axis and y-axis,

respe tively, using tensile and bending loading. The deviation from the referen e, as observed in the

previous simulations when the wave rosses the

Ω

O

, disappeared in the present results. Then, for this model,

L

O

= 10 mm

is su ient to orre tly transmit the wave between the dis rete and ontinuum models.