Numerical Modeling of granular materials with multi ... - [SMAI]

Numerical Modeling of granular materials with multi physics coupling.Application to the analysis of granular flow

Frederic Dubois1 and Mathieu Renouf2

1 Laboratoire de Mecanique et Genie CivilUniversite de Montpellier 2 - CNRS

[email protected]

2Laboratoire de Mecanique des Contacts et des StructuresINSA de Lyon - CNRS

[email protected]

F. Dubois & M. Renouf (CNRS) Divided Material Modeling 2008 1 / 89

1 Preamble

2 Modeling

3 Numerical Strategies

4 Technical Aspects

5 NSCD Extensions

6 Multi-Physics

7 Bibliography

8 Annexes:

DEM::Preamble

Co-workers:

Jean Jacques Moreau, Michel Jean (LMA)Fahrang Radjai

Gilles Saussine (SNCF), Emilien Azema (LCPC)Vincent Acary (INRIA), Yann Monerie (IRSN)

PhD support:

M. Renouf (2004, Alart-Dubois)G. Saussine (2004, Bohatier-Dubois)

F. Perales (2005,Monerie-Chrysochoos)B. Chetouane (2004, Bohatier-Vinches)

R. Perales (2007, Bohatier-Vinches)E. Azema (2007, Radjai-Saussine)A. Rafiee (2008, Bohatier-Vinches)V. Martin (Radjai-Dubois-Monerie)

DEM::Preamble

Many solid materials and structures are made of components in interaction:

Dry granular material (divided media)

I component: grain, cluster of grain, piece of grain, coarse grain,etc.I interaction: unilateral contact, elasticity or/and cohesion, friction, etc.

DEM::Preamble

Dry granular material (divided media)

I component: grain, cluster of grain, piece of grain, coarse grain,etc.I interaction: unilateral contact, elasticity or/and cohesion, friction, etc.

DEM::Preamble

Masonry structures (divided structure)

I component: brick, stone, mortar, mortar grain.I interaction: unilateral contact, friction, cohesion, etc.

DEM::Preamble

solid grains in a solid matrix (composites, etc) and/or in a fluid matrix (colloids, concrete,etc)

idem as before but a solid or fluid matrix should be concerned.

DEM::Preamble

solid grains in a solid matrix (composites, etc) and/or in a fluid matrix (colloids, concrete,etc).

various situations

“binary interaction” “ ?? ” “composite - continuous”

All the situations can’t be addressed easily by DEMs.

DEM::Preamble

Many solid materials and structures are essentially discontinuous or divided:

changing the scale a continuous media may become discontinuous

I component: metal grain, crystal in a poly-crystal, molecules, atoms;I interaction: grain bond, defects, Wan der Waals effects, Lenhard-Jones potentials, atomic bonds,

DEM::Preamble

Considering divided media it exists various space scales :

microscopic mesoscopic macroscopic

Behavior depends on :

composition: shape (angularity, elongation, etc), dispersity

state: dense-loose, confining pressure, dry-wet-composite

load: static-dynamic

Depending on the solicitation it may behave like: a solid, a liquid or a gas

DEM::Preamble

It exists numerous methods to model these kind of media and structures.

At the macroscopic scale the divided media is considered as continuous.It is possible to use various discretization :

classical methods (Finite Difference, Finite Element Method, Finite Volume Method, etc)

dedicated methods (Thin Layer Model, SPH, etc)

it needs an ad-hoc rheological model to represent the complex behavior of the media (elasticity,plasticity, visco-plasticity, fracture, etc).

Drawbacks: building a model with the relevant phenomenological parameters and identifying theparameters.

At the microscopic or mesoscopic scale, methods rely on a modeling of :

the behavior of each component

the behavior of the interactions of each component with its neighborhood

A lot of methods aim to model divided media (and/or structures) at the component andinteraction scale: Cellular Automata, SPH, Lattice Element methods, Discontinuous DeformationAnalysis (DDA), etc and Discrete Element Methods (DEM).

Advantages: simpler model, less parameters, easier to identifyDrawback: computational cost.

DEM::Preamble

One can also consider DEMs to model fictitious divided media as in:

coarse grain approach

domain decomposition approach

These methods may also be used for modeling mechanism or robotics

DEMs can be mixed with continuous methods to model, for example, the transition between acontinuous and a divided media :

fragmentation of a media

powder compaction to obtain ceramics

DEM::Preamble

DEM::Modeling

1 Preamble

2 ModelingSmooth Dynamic

Equations of motionInteraction treatmentLocal-Global mappingInteraction law

Non Smooth DynamicEquations of motionInteraction lawShock law

Conclusion

4 Technical Aspects

5 NSCD Extensions

6 Multi-Physics

7 Bibliography

8 Annexes:

DEM::Modeling::Smooth Dynamic::Equations of motion

Assuming a smooth evolution of the system allows to describe the motion of each mechanicalcomponent by a semi-discretized in space system:

M(q, t) ˙q = F(q, q, t) + P(t) + r, (1)

q ∈ Rn represents the vector of generalized degrees of freedom,

q ∈ Rn the generalized velocities,

r ∈ Rn the contact forces,

P(t) the external forces,

F(q, q, t) the internal force (deformable bodies) and the nonlinear inertia terms (centrifugaland gyroscopic),

M(q, t) : Rn 7→ Mn×n the inertia matrix.

Initial and boundary conditions must be added to fully describe the evolution of the system.

details

DEM::Modeling::Smooth Dynamic::Equations of motion

Considering a rigid component, a more suitable dynamical equation may be used introducing thev and ω, the translation and rotation velocities of the center of mass.

The Eq.(1) is replaced by the well-known Newton-Euler system of equations:Mv = P(t) + rIω = −ω ∧ (Iω) +MP(t) +Mr

P(t) and r represent respectively the resultant of external and contact forces,

MP(t) and Mr represent respectively the momentum due to external and contact forces,

M and I represents respectively the mass and the inertia matrices.

Note that for 2D components or 3D components with geometric isotropy, the vector ω ∧ (Iω) isequal to zero.

The choice of angle parameters is very important, especially in 3D (inertia frame mapping, Eulerangles, quaternion, etc.)

DEM::Modeling::Smooth dynamics::Interaction description

At any time of the evolution of the system one needs to define the interaction locus and anassociated local frame in order to describe the interaction behavior.

Ü implicit a priori.

it is assumed that one is able to define for eachpoint (C) of the candidate boundary its (unique)nearest point (A) on the antagonist boundary.It allows to define for each couple of points alocal frame (t, n, s): with n the normal vector ofthe antagonist boundary and (s, t) two vectorsof its tangential plane.

Only in simplest cases (rigid body with strictly convex boundary) the interaction locus may beconsidered as punctual.

Less trivial in usual cases :

not strictly convex, i.e. cubes, bricks, etc.

only locally convex, i.e. general polyhedron, triangulated surface

not convex at all.Ü It may be decomposed in not strictly convex shapes.

In these cases many choices

punctual contact with extended law (transmission of torque)Ü how to define the normal ? The interaction law depends on the objects shape ! etc.

multi-punctual contacts with classical interaction lawsÜ how many contact points ? normal choice ? It mays introduce local indetermination of

contact forces, etc.

continuous surfaced description as in mortar methodsÜ needs to perform integration on non-conforming triangulation, etc.

Less trivial in usual cases :

not strictly convex, i.e. cubes, bricks, etc.

only locally convex, i.e. general polyhedron, triangulated surface

not convex at all.Ü It may be decomposed in not strictly convex shapes.

In these cases many choices

punctual contact with extended law (transmission of torque)Ü how to define the normal ? The interaction law depends on the objects shape ! etc.

multi-punctual contacts with classical interaction lawsÜ how many contact points ? normal choice ? It mays introduce local indetermination of

contact forces, etc.

continuous surfaced description as in mortar methodsÜ needs to perform integration on non-conforming triangulation, etc.

Implicitly DEMs rely on the following hypotheses :

1 the deformation concerns only the contact point neighborhoodÜ components of the system may be considered as rigid;

2 the contact area is small behind the size of the component;Ü locus of interaction may be supposed as punctual;

3 interactions are binary (no effect of connected interaction by particle on their behavior);Ü interaction law depends only on related component;

At least, for every potential contact α, it will be determined:

Contact point coordinates;

The local frame (tα, nα, sα),;

The gap (gα), i.e. the algebraic distance between two bodies;

The contact relative velocity between the two bodies (Uα).

Contact description is a key point of DEMs.

DEM::Modeling::Smooth dynamic::Local-Global mapping

Two sets of unknowns:

global unknowns (or kinematic space unknowns) related to the bodies: center of inertia ormesh node displacement and velocity (q, q), resulting force and momentum (r), etc.

local unknowns (or contact space unknowns) related to interactions: gap (g), relativevelocities (U), forces (R), etc.

Related by kinematic relations :g = D(q)

U = H?(q)q = ∇qD(q)q

Lets consider two rigid bodies.

The mapping between inertia center and contact point velocities will be write as follow:

v(M) = v(G) + ω × l

where l represents the vector between the inertia center (G) and the contact point (M),Then it is possible to write the relative velocity between the two points (Mi ) (i bodies) and (Mj )(j bodies) :

UX ,Y ,Z = v(Mi )− v(Mj ) = v(Gi )− v(Gj ) + ωi × li − ωj × lj

which can be written in a “matricial” way :

UX ,Y ,Z =˙

1 li −1 lj¸2664

v(Gi )ωi

v(Gj )ωj

3775 = H?i,j (q)q

And expressed in a local frame as:

Ut,n,s =˙

U · t U · n U · s¸

35H?i,j (q)q = H?i,j (q)q

More generally, using kinematic relations, one can write for a given contact α:

Uα = H?α(q)q

Using duality consideration (equality of power expressed in terms of global or local unknowns),the local contact force may be mapped on the global unknowns :

rα = Hα(q)Rα,

where H?α(q) is the transpose of Hα(q).

In the following, operators mapping all the local and global unknowns are introduced :

H(q) : R = Rα → r =

ncXα=1

Hα(q)Rα

H?(q) : q→ U = Uα = H?α(q)q

Remark : even if H?α(q) and Hα(q) have good theoretical properties (surjectivity and injectivity),it is not necessary the case for H? and H. Loose of these properties is due to the introduction ofkinematic relation between contacts.

Uα = H?α(q)q

rα = Hα(q)Rα,

H(q) : R = Rα → r =

ncXα=1

Hα(q)Rα

H?(q) : q→ U = Uα = H?α(q)q

Uα = H?α(q)q

rα = Hα(q)Rα,

H(q) : R = Rα → r =

ncXα=1

Hα(q)Rα

H?(q) : q→ U = Uα = H?α(q)q

DEM::Modeling::Smooth dynamic::Interaction law

In DEMs, a large part of the physics of the problem is described through interaction laws.Two ways of thinking :

1 - interaction behavior represents onlywhat happens at the boundary scale:impenetrability, plastic deformation ofasperities, capillarity effects, friction,wear, etc.

2 - interaction behavior represents whathappens both at the bulk and boundaryscale: idem as 1 + bulk behavior, etc.

The two ways have different space scales which imply different time scales.When considering rigid bulk behavior, one is able to describe :

1 - rigid body motion.Ü “Large time scale”.

Recover plastic behavior of the media.

2 - motion due to wave propagation.Ü Fine time scale.

Recover elastic and plastic behavior ofthe media. No indeterminacy.

From a “mathematical” point of view :

1 - set value function, defined by animplicit law : h(R,U, g) == true

2 - function, defined by an explicit law :R = f (U, g)

DEM::Modeling::Smooth dynamics::Interaction law

explicit laws

Normal part (δ =< −g >+)

Hertz law : Rn = knδ32 = Kn(δ)δ

Hook law with viscous damping : Rn = max(0,Knδ + ηnmeff Un.d

reff d)

JKR cohesion law :Rn = knδ32 + ηn

√meff Un − γn

√δreff .

where Kn is the contact stiffness, ηn viscosity coefficient and γn contact cohesion. U representsthe relative velocity between particles and d(= dn) is defined as center-center vector, meff andreff represent respectively the effective mass and radius associated to the contact.

explicit laws

Hertz law : Rn = knδ32 = Kn(δ)δ

Hook law with viscous damping : Rn = max(0,Knδ + ηnmeff Un.d

reff d)

JKR cohesion law :Rn = knδ32 + ηn

√meff Un − γn

√δreff .

where Kn is the contact stiffness, ηn viscosity coefficient and γn contact cohesion. U representsthe relative velocity between particles and d(= dn) is defined as center-center vector, meff andreff represent respectively the effective mass and radius associated to the contact.

explicit laws

For Hertz law Kn =E√

reff δ

3(1−ν2).

Behavior parameter are “structure” dependant (i.e. geometry).

For a given pressure P, the stiffness level may be characterized by :

κ = (E

P)2/3 (Hertz) or κ =

Prd−2eff

(Hook)

So the elastic deflection δ ∝reff

Remarks :

rigid grains if κ→∞ . “Good” value are 10000.

Quasi-static problem: δ < g >+ ( < g >+ interstice between grains)

Flow problem: τ <g>+

reff γ(τ duration of contact)

explicit laws

Considering the oscillator made by two particles in contact (Hook law) :

meffd2δ

dt2+ ηn

dt+ Knδ = 0

critical damping ηcn = 2

pmeff Kn. αn =

pulsation ω =

meff(1− α2

n) and contact duration τ = π/ω

restitution coefficient en = exp[−παnp1− α2

Remarks :

ηn ≤ ηcn

For Hertz law, same results with a varying Kn

explicit laws

Tangential part

viscous law with Coulomb threshold:RTr

t = −ηt meff Ut

if ||RTrt || ≥ µ|rn| then Rt = µ|rn|

||RTrt ||

else Rt = RTrt

incremental elastic law with Coulomb threshold:a previous Rt is known, ∆Rt = Kt hUt and RTr

t = Rt + ∆Rt

||RTrt ||

else Rt = RTrt

with Kt = 2(1−ν)2−ν Kn.

Remarks :

compute ηct as before by taking meff = m1m2

m1+m2+ m1m2(

I2) and replacing Kn by Kt

ηt ≤ ηct

difficult to give a physical meaning to ηn and ηt

explicit laws

Tangential part

viscous law with Coulomb threshold:RTr

t = −ηt meff Ut

||RTrt ||

else Rt = RTrt

incremental elastic law with Coulomb threshold:a previous Rt is known, ∆Rt = Kt hUt and RTr

t = Rt + ∆Rt

||RTrt ||

else Rt = RTrt

with Kt = 2(1−ν)2−ν Kn.

Remarks :

compute ηct as before by taking meff = m1m2

m1+m2+ m1m2(

I2) and replacing Kn by Kt

ηt ≤ ηct

difficult to give a physical meaning to ηn and ηt

explicit laws

Explicit laws allow to substitute the displacement (or the velocity) to the force in the dynamicalequation.

If considering a simplified problem described by the Newton equation :

Mv = F(q, q, t) + P(t) + r

using U = H?(q)v , g = D(q), r = H(q)R and R = f (U, g) one can write :

Mv = P(t) + H(q)f (H?(q)v ,D(q))

which is a “classical” non linear problem only written in term of kinematic unknowns.

The kinematic unknowns of each component are dependent due to the interactions.

explicit laws

Explicit laws allow to substitute the displacement (or the velocity) to the force in the dynamicalequation.

If considering a simplified problem described by the Newton equation :

Mv = F(q, q, t) + P(t) + r

using U = H?(q)v , g = D(q), r = H(q)R and R = f (U, g) one can write :

Mv = P(t) + H(q)f (H?(q)v ,D(q))

which is a “classical” non linear problem only written in term of kinematic unknowns.

The kinematic unknowns of each component are dependent due to the interactions.

implicit laws

Frictional contact: Signorini-Coulomb

RN > 0 g > 0 RN · g = 0

‖ RT ‖6 µRN ,

‖ RT ‖< µRN ⇒ UT = 0‖ RT ‖= µRN ⇒ ∃α > 0,UT = −αRT

Remark : for dynamical problems, it is more natural to formulate the unilateral contact in term ofvelocities

Assuming g(t0) ≥ 0 then ∀t > t0

if g(t) ≤ 0 then UN ≥ 0, RN ≥ 0, UN RN = 0else RN = 0

implicit laws

RN > 0 g > 0 RN · g = 0

‖ RT ‖6 µRN ,

implicit laws

RN > 0 g > 0 RN · g = 0

‖ RT ‖6 µRN ,

implicit laws

It’s not possible to substitute reaction by kinematic unknowns, the problem remains implicit.

One can consider explicit law (smooth) as a regularization of implicit law (non smooth). In thiscase the parameters are not due to physical considerations but numerical ones.

An explicit law may be written as an implicit law.

implicit laws

It’s not possible to substitute reaction by kinematic unknowns, the problem remains implicit.

One can consider explicit law (smooth) as a regularization of implicit law (non smooth). In thiscase the parameters are not due to physical considerations but numerical ones.

An explicit law may be written as an implicit law.

DEM::Modeling::Smooth dynamic::Interaction

Lets consider the basic example of a single ball bouncing on a plan:

t*temps

distance vitesse relative

Question: Is the usual formalism adapted to describe a dynamical system with unilateral contact?

So let try to solve the (simplified) problem:

Mq = R

0 ≤ q ⊥ R ≥ 0

initial conditions : q(0) = 0, q(0) = −1

t*temps

Mq = R

0 ≤ q ⊥ R ≥ 0

t*temps

Mq = R

0 ≤ q ⊥ R ≥ 0

t*temps

Mq = R

0 ≤ q ⊥ R ≥ 0

Using an implicit Euler scheme, one obtains the following system:qi+1 = (qi+1 − qi )/hqi+1 = qi + hqi+1

Thus the solution is: q1 = 0, q1 = 0,R1 = Mq0/hqk = 0, qk = 0,Rk = 0

So if h→ 0 then R1 =∞ !

Two solutions are possible:

Using smooth interaction law as before but needs reasonably small time steps

Adapting the formalism to face the problem

Introducing a deformable body will not solve the problem.

DEM::Modeling::Non smooth dynamic::Equations of motion

The differential system must be modified to describe collisions and other non smooth phenomena.

The “spirit” of the approach is to consider a weaken form of the dynamical system, e.g. balanceof momentum :

M(q+ − q−) =

t−(F(q, q, s) + P(s))ds + p

where the impulse p will contain both the sum of

the contribution of smooth load over the time interval (R t+

t− Rdt),

the percussion, denoted P, at shock time (supposed instantaneous).

To describe q one uses the local bounded variation framework over all sub-intervals ofITps = [0,T ], i.e. lbv(ITps ,Rn)) and one introduces differential measures allowing thegeneralization of the equation of motion to non smooth phenomena (Moreau 1988).

Remark :

Event driven approaches will consider smooth motion, but will re-initialize velocity(acceleration, etc.) at eachnon smooth event using shock law.Only reasonable to compute a granular gas or a loose flow of granular material.

M(q+ − q−) =

t−(F(q, q, s) + P(s))ds + p

t− Rdt),

Remark :

M(q+ − q−) =

t−(F(q, q, s) + P(s))ds + p

t− Rdt),

Remark :

M(q+ − q−) =

t−(F(q, q, s) + P(s))ds + p

t− Rdt),

Remark :

More precisely the classical equation of motion is reformulated in terms of a differential measureequation:

Md q = F(t, q, q+)dt + P(t, q, q+)dt + dp

q(t) = q(t0) +

In the previous equation, d q is the differential measure of q (atomistic at the discontinuity andq dt on the continuous part), dt is the Lebesgue measure on R while dp is the differentialmeasure of contact forces.

The measure dp contains:

The contribution of smooth contact (diffuse contribution Rdt):

The contribution of local impulsion densities exerted by shocks Pδ (atomic contributions),

DEM::Modeling::Non smooth dynamic::Interaction law

Local - global mapping:U+ = H?(q)q+

dp = H(q)dP

Frictional contact:

if g(t) ≤ 0 then U+N ≥ 0, dPN ≥ 0, U+

N dPN = 0else dPN = 0

Coulomb friction (threshold law):

‖ dPT ‖6 µdPN ,

‖ dPT ‖< µdPN ⇒ U+

T = 0‖ dPT ‖= µdPN ⇒ ∃α > 0,U+

T = −αdPT

DEM::Modeling::Non smooth dynamic::Interaction law

Local - global mapping:U+ = H?(q)q+

dp = H(q)dP

Frictional contact:

if g(t) ≤ 0 then U+N ≥ 0, dPN ≥ 0, U+

N dPN = 0else dPN = 0

Coulomb friction (threshold law):

‖ dPT ‖6 µdPN ,

‖ dPT ‖< µdPN ⇒ U+

T = 0‖ dPT ‖= µdPN ⇒ ∃α > 0,U+

T = −αdPT

DEM::Modeling::Non smooth dynamic::Shock law

When shocks occur in a rigid body collection, the equation of motion and the interaction law arenot sufficient to describe properly all the physics of the problem.

It must take into account:

Local phenomena as inelastic behavior of materials at the interface depending on both thecontact geometry and the material behavior.

Global phenomena as the wave propagation in the body bulk, body geometry dependencyand boundary conditions.

More complex effects: long distance effects due to simultaneous impact

In the case of binary shocks, three kind of restitution can be used:

Newton restitution, which relates the velocity after (U+) to the velocity before (U−) impactU+ = −enU−

Poisson restitution, which relates restitution impulsion (Rr ) to the compression impulsion(Rc ) according to the decomposition of the shocks in a compression and restitution phase:Rr = epRc

Energy or Stronge restitution, which relates restitution energy to compression energy:

R trtc

RUdtR tct?

Impact law for the normal component is well understood, it is not the case of the tangentialcomponent. The choice of the restitution coefficient is a difficult task for complex structures.

In the case of dense granular material, the effect of impacts may be neglected and previous lawscan be used.

Binary impact law are not sufficient to model phenomena such as Newton cradle (multipleimpacts).

R trtc

RUdtR tct?

R trtc

RUdtR tct?

R trtc

RUdtR tct?

R trtc

RUdtR tct?

R trtc

RUdtR tct?

DEM::Modeling::Conclusion

Various modeling choices are possible for DEM depending on:

space scale: interaction Ü smooth, component Ü non-smooth

time scale: waves Ü smooth, rigid body motion with impact Ü non-smooth

the shape of the components Ü non-smooth easier but introduce indetermination.

DEM::Numerical Strategies

1 Preamble

2 Modeling

3 Numerical StrategiesIntroductionsmooth-DEM

GeneralityTime integratorTechnical aspects

Non Smooth Contact DynamicsTime steppingManaging interactions :Contact problem formulationSolversPractical view

4 Technical Aspects

5 NSCD Extensions

6 Multi-Physics

7 Bibliography

8 Annexes:

DEM::Numerical Strategies::Introduction

Depending on modeling choices numerical strategies are build to solve the evolution problem.They depend on :

time evolution strategy: time stepping or event driven

time integrator over a time step: explicit or implicit

implicit contact solver if necessary (Lemke, Gauss-Seidel, bi-potential, etc.)

technical aspects: contact detection, rotation integration, etc.

Over a time step [t, t + h[, three important tasks can be underlined:

The contact detection

The computation of contact forces, calledcontact problem

The motion of the different element of themedia.

Depending on modeling choices numerical strategies are build to solve the evolution problem.They depend on :

time evolution strategy: time stepping or event driven

time integrator over a time step: explicit or implicit

implicit contact solver if necessary (Lemke, Gauss-Seidel, bi-potential, etc.)

technical aspects: contact detection, rotation integration, etc.

Over a time step [t, t + h[, three important tasks can be underlined:

The contact detection

The computation of contact forces, calledcontact problem

The motion of the different element of themedia.

Pioneer DEMs (Cundall et al. 1979, Allen et al. 1987) consider explicit interaction model(smooth) and use explicit time integration scheme. This kind of methods refer to smooth-DEM.

The so-called Granular Element Method, proposed by Kishino (1988) consider also an explicitmodel for contact but is based on an implicit time integrator. The solving method is similar tothe penalization techniques used in Finite Element Method.

In a different way, Moreau (1988) developed the Contact Dynamics method for implicit nonsmooth interaction model. It uses an implicit time integrator. It needs a dedicated contact solver.Further works lead to the extension of the method to multi-contact simulations of collections ofdeformable bodies (Jean 1999) and the method becomes the so-called Non Smooth ContactDynamics method (NSCD). This kind of methods refer to NSCD.

Carpenter et al. (1991) proposed a method to solve implicit interaction model using explicit timeintegrator.

Thus the end-user has various possibilities to perform numerical modeling.

DEM::Numerical Strategies::smooth-DEM::Generality

Using explicit interaction law leads to solve :

Mv = F(q, q, t) + P(t) + r

with r = H(q)R, R = f (U, g) and U = H?(q)v , g = D(q).

Ü This is a non-linear problem.

Using an implicit time integrator, a non-linear solver (Newton-Raphson) will be necessary (seeGEM of Kishino).

Using an explicit time integrator leads to an uncoupled set of equations :

M(vn+1 − vn) = hF(qn, qn, tn) + hP(tn) + hrn

Different explicit time integrator can be used to integrate the motion of particle:

Gear integrator for Molecular Dynamics

Velocity verlet

Time-centered scheme for the Distinct Element Method (Cundall)

Mv = F(q, q, t) + P(t) + r

Velocity verlet

Mv = F(q, q, t) + P(t) + r

Velocity verlet

Mv = F(q, q, t) + P(t) + r

Velocity verlet

DEM::Numerical Strategies::smooth-DEM::Time integrator

Predictor-corrector Gear integrator

Prediction: Using classical Taylor expansion (b and c are first and second time derivative ofacceleration) : 8>>>>><>>>>>:

qp(t + h) = q(t) + h ˙q(t) + h2

2!q(t) + h3

3!b(t) + h4

4!c(t) + ...

qp(t + h) = q(t) + hq(t) + h2

2!b(t) + h3

3!c(t) + ...

qp(t + h) = q(t) + hb(t) + h2

2!c(t)...

bp(t + h) = b(t) + hc(t) + ...cp(t + h) = c(t) + ...

Perform detection in the predicted configuration, compute contact forces, compute theacceleration (qc (t + h) = M−1F + P + r(t + h)).

Correction: considering ∆q = qc (t + h)− qp(t + h) perform the correction8>>><>>>:qc (t + h) = qp(t + h) +c0∆qqc (t + h) = qp(t + h) +c1∆qqc (t + h) = qp(t + h) +c2∆qbc (t + h) = bp(t + h) +c3∆qcc (t + h) = cp(t + h) +c4∆q

The correction step may be repeated.

qp(t + h) = q(t) + h ˙q(t) + h2

2!q(t) + h3

3!b(t) + h4

4!c(t) + ...

qp(t + h) = q(t) + hq(t) + h2

2!b(t) + h3

3!c(t) + ...

qp(t + h) = q(t) + hb(t) + h2

2!c(t)...

qp(t + h) = q(t) + h ˙q(t) + h2

2!q(t) + h3

3!b(t) + h4

4!c(t) + ...

qp(t + h) = q(t) + hq(t) + h2

2!b(t) + h3

3!c(t) + ...

qp(t + h) = q(t) + hb(t) + h2

2!c(t)...

qp(t + h) = q(t) + h ˙q(t) + h2

2!q(t) + h3

3!b(t) + h4

4!c(t) + ...

qp(t + h) = q(t) + hq(t) + h2

2!b(t) + h3

3!c(t) + ...

qp(t + h) = q(t) + hb(t) + h2

2!c(t)...

qp(t + h) = q(t) + h ˙q(t) + h2

2!q(t) + h3

3!b(t) + h4

4!c(t) + ...

qp(t + h) = q(t) + hq(t) + h2

2!b(t) + h3

3!c(t) + ...

qp(t + h) = q(t) + hb(t) + h2

2!c(t)...

Velocity Verlet scheme

Its also predictor-corrector approach.

Prediction: q(t + h) = q(t) + hq(t) + 1

2h2q(t)

q(t + h/2) = q(t) + 12

hq(t).

Perform detection in the predicted configuration, compute contact forces, compute theacceleration knowing q(t + h) and q(t + h/2) (q(t + h) = M−1F + P + r(t + h)).Correction:

q(t + h) = q(t + h/2) +1

2hq(t + h)

Note that the classical Verlet scheme (leap-frog) is not usable for DEM.

2h2q(t)

q(t + h/2) = q(t) + 12

hq(t).

q(t + h) = q(t + h/2) +1

2hq(t + h)

2h2q(t)

q(t + h/2) = q(t) + 12

hq(t).

Perform detection in the predicted configuration, compute contact forces, compute theacceleration knowing q(t + h) and q(t + h/2) (q(t + h) = M−1F + P + r(t + h)).

Correction:

q(t + h) = q(t + h/2) +1

2hq(t + h)

2h2q(t)

q(t + h/2) = q(t) + 12

hq(t).

q(t + h) = q(t + h/2) +1

2hq(t + h)

2h2q(t)

q(t + h/2) = q(t) + 12

hq(t).

q(t + h) = q(t + h/2) +1

2hq(t + h)

Time-centered scheme

Knowing q(t) and q(t − h/2) one perform contact detection, computes contact forces and updateacceleration q(t).

Then one updates q(t + h/2) = q(t − h/2) + hq(t)q(t + h) = q(t) + hq(t + h/2)

It is not written as a predictor-corrector scheme.But it’s the same as velocity verlet.

DEM::Numerical Strategies::smooth-DEM::Technical aspects

Some cares must be taken to obtain numerical results that keep a mechanical sense.

The time step h depends explicitly on the mechanical parameters of the system

(h = 1N

qmeffKn

) and must be small enough to ensure numerical stability and describe with

accuracy the contact.

Dissipation must be added to stabilize the problem (i.e. over-shoots). For example nonviscous dissipation can be added to study granular flow :

Fd = −α r sign(q)

α is chosen depending on the studied problem: 0.7 for quasi-static problem, less for dynamicproblem, and 0.1 for wave propagation

Thus control parameter of the simulation are:

the time step discretization (N),

the local stiffness (normal Kn and tangential Kt ),

the viscous (η) or non viscous (α) dissipation.

Evaluation criteria related to the computation quality:

Respect of interaction law,

Control of numerical damping.

(h = 1N

qmeffKn

(h = 1N

qmeffKn

(h = 1N

qmeffKn

DEM::Numerical Strategies::NSCD::Time stepping

Time integration of the equation of motion leads to:

M(qi+1 − qi ) =

Z ti+1

(F(q, q, s) + P(s))ds + pi+1

= pfree + pi+1

qi+1 = qi +

Z ti+1

pi+1 =R ti+1

tidp represents the value of the total impulsion over the time step the unknown in

the following

pfree the integral of applied forces over the time step;

A θ-method (Crank-Nicholson) is used to evaluateR ti+1

ti(F(q, q, s) + P(s))ds and

R ti+1ti

pfree = h(1− θ)(F(qi , qi , ti ) + P(ti )) +

hθ(F(qi+1, qi+1, ti+1) + P(ti+1))

qi+1 = qi + h((1− θ)qi + θqi+1) = qm + hθqi+1

with qm = qi + h(1− θ)qi .

If θ ∈ [0.5, 1] the scheme is implicit and stable unconditionally.

If θ = 0.5 the scheme is conservative for smooth evolution problem,

If θ = 0 the scheme is explicit

The order of the time integrator is weak (1 or 2).

The time discretization is imposed arbitrarily. Its mainly driven by the precision of the contacttreatment.

If discontinuities occur they are treated simultaneously. No limitation on the number ofinteractions but the time order is lost.

A θ-method (Crank-Nicholson) is used to evaluateR ti+1

ti(F(q, q, s) + P(s))ds and

R ti+1ti

pfree = h(1− θ)(F(qi , qi , ti ) + P(ti )) +

hθ(F(qi+1, qi+1, ti+1) + P(ti+1))

qi+1 = qi + h((1− θ)qi + θqi+1) = qm + hθqi+1

with qm = qi + h(1− θ)qi .

If θ ∈ [0.5, 1] the scheme is implicit and stable unconditionally.

If θ = 0.5 the scheme is conservative for smooth evolution problem,

If θ = 0 the scheme is explicit

The order of the time integrator is weak (1 or 2).

The time discretization is imposed arbitrarily. Its mainly driven by the precision of the contacttreatment.

If discontinuities occur they are treated simultaneously. No limitation on the number ofinteractions but the time order is lost.

If now we use this formulation for the previous elementary contact problem, one obtains:

M(qi+1 − qi ) = p

If qi ≤ 0, 0 ≤ qi+1 ⊥ hR ≥ 0

Initial conditions : q(0) = 0, q(0) = −1

The implicit euler scheme gives: q1 = 0, q1 = 0, p1 = Mq0

qk = 0, qk = 0, pk = 0

The obtained solution is independent of the time step.

If q(0) = ε then q1 = 0 and p1 = Mq0 and qk = ε.The error made on the gap decreases with the size of the time step.

If now we use this formulation for the previous elementary contact problem, one obtains:

M(qi+1 − qi ) = p

If qi ≤ 0, 0 ≤ qi+1 ⊥ hR ≥ 0

Initial conditions : q(0) = 0, q(0) = −1

The implicit euler scheme gives: q1 = 0, q1 = 0, p1 = Mq0

qk = 0, qk = 0, pk = 0

The obtained solution is independent of the time step.

If q(0) = ε then q1 = 0 and p1 = Mq0 and qk = ε.The error made on the gap decreases with the size of the time step.

DEM::Numerical Strategies::NSCD::Interaction management

The two mappings H? and H depend of the solution.

They may be evaluated in a “mean” configuration : qk+1 = qk+1 + h(γ1qk + γ2qk+1)

If the time step is small enough (small sliding motions, etc) and the curvature of componentshapes is small the mapping H? and H can be considered as constant on a whole time interval.

Remarks:

Usually the contact problem is solved in a pseudo-explicit configuration :qm = qi + h(1− θ)qi .

The pseudo configuration of the next time step will be determined by the final velocity qi+1:qm+1 = qm + hqi+1. This is a Leap-Frog technique.

Remarks:

DEM::Numerical Strategies::NSCD::Contact problem formulation

The problem to solve is written in terms of :

discretized equations of motion for each body expressed with global unknowns :

Mqi+1 = pfree + pi+1

interaction laws expressed with local unknowns (contact α):

Contact(gα,Uαn ,Pαn ) = TRUE Friction(Uαt ,Pαt ) = TRUE

mappings (H and H?) to pass from local to global unknowns

q, q ← Equations of motion → p

H? ↓ ↑ H

U ← Interaction laws → P

Using basic algebraic transformations the equations of motion may be expressed in terms of localunknowns

U = Ufree + W P

where W = H?M−1H and Ufree = H?M−1pfree .

DEM::Numerical Strategies::NSCD::Contact problem formulation

The problem to solve is written in terms of :

discretized equations of motion for each body expressed with global unknowns :

Mqi+1 = pfree + pi+1

interaction laws expressed with local unknowns (contact α):

Contact(gα,Uαn ,Pαn ) = TRUE Friction(Uαt ,Pαt ) = TRUE

mappings (H and H?) to pass from local to global unknowns

q, q ← Equations of motion → p

H? ↓ ↑ H

U ← Interaction laws → P

Using basic algebraic transformations the equations of motion may be expressed in terms of localunknowns

U = Ufree + W P

where W = H?M−1H and Ufree = H?M−1pfree .

DEM::Numerical Strategies::NSCD::Solvers

The classical NSCD approach rely on a Non Linear Gauss Seidel (NLGS) algorithm.

Considering, one by one, the local systems to solve for each contact α,

Uα = Uα,free + WααPα +Xβ 6=α

WαβPβ

Law(gα,Uα,nUα,t ,Pα,n,Pα,t ) = TRUE

we freeze the contributions of other contact (β 6= α) taking updated values if β < α or old valuesif β > α.

we solve the α block of equations using:

An explicit uncoupled resolution if Wαα is diagonalA coupled (n, t) graph intersectionA pseudo-potential approach (bi-potential)LCP like local solver

An explicit resolution if Wαα is diagonalA pseudo-potential approachA Generalized Newton algorithmLCP like local solver

We repeat the process until convergence

WαβPβ

A parallel treatment of the NLGS is possible

quasi NLGS may be derived, rewriting the system :

Uα = Uα,free + Wαα(Pα − Pα,esti ) +Xβ

WαβPβ

Law(gα,Uα,t ,Pα,n,Pα,t ) = TRUE

where Pα,esti is the impulsion computed at the previous Gauss-Seidel iteration.

Noting that when the algorithm goes close to the solution, Pα − Pα,esti → 0 one may derive aquasi NLGS replacing the original Wαα by an arbitrary one.

Various alternatives are possible for the contact solver :

Conjugate Projected Gradient Algorithm

Lemke (for small collection of rigid bodies)

Other possibilities, see Siconos NSSPack

WαβPβ

DEM::Numerical Strategies::NSCD::Scheme

Resolution Scheme:

Iteration matrix computation (M)Time loop

Free velocity computation (qfree )Temporary configuration computation(qm)Contact detectionContact problem resolutionVelocity correction (qi+1)Update of the kinematics and the configuration

Halt criteria

DEM::Numerical Strategies::NSCD::Practical view

Numerical model parameters are:

The θ value

The time step

The convergence norm of the Gauss-Seidel algorithm

The direction of reading the contact set

Quality of the computation lays on:

The respect of interaction laws

The number of iteration performed

The free evolution of bodies

DEM::NSCD Extensions

1 Preamble

2 Modeling

4 Technical AspectsContact detectionRotation

5 NSCD Extensions

6 Multi-Physics

7 Bibliography

8 Annexes:

DEM::Technical Aspects::Contact detection

Contact detection becomes quickly a complex problem, using a large part of the CPU time.This is directly related to the number and to the geometry of the different elements.

A first way to simplify and optimize the process is to perform:

a neighborhood construction based on the sort of closed bodies (surrounding box method,tessellation,etc.)

a rough detection based on the research of proximal objects (oct-tree, shadow-overlap,separator plan, etc.)

an accurate detection based on the research of proximal points (intersection, projection,etc.)

DEM::Technical Aspects::Rotation

A relevant computation of “rotation” is necessary for

interaction law: friction, cohesion, rolling friction, etc.

numerical precision: respect objectivity

Usually geometry is represented in a moving frame (R).

Integrating Euler equation : ωf − ωi = hI−1MÜ update the locate frame through the transformation matrix Rf = PRi

Various possibility :

Euler anglesω = ψe1 + φe2 + ζe3R f

i ψds Ü ∆Ω1,R f

i φds Ü ∆Ω2 andR f

i ηds Ü ∆Ω3

P = ∆Ω3 ·∆Ω2 ·∆Ω1

drawback: indeterminacy

quaternion

transformation matrix :I linearization : for each frame vector vk = ω ⊗ ek then e f

k = e ik + hvk

drawback: lose orthogonality and normalityI incremental objective integrator (Hughes, 1980), (Geradin et al.,)

DEM::NSCD Extensions

1 Preamble

2 Modeling

4 Technical Aspects

5 NSCD ExtensionsDeformable bodiesInteraction lawsExplicit Integration

6 Multi-Physics

7 Bibliography

8 Annexes:

DEM::NSCD extensions::Deformable bodies

When deformable bodies are used, the local dynamic system may be written in the same way

U = Ufree + W P

rigid: M = Mpfree = h(1− θ)(Fi + Pi ) + hθ(Ff + Pf )

linear: M = M + hθC + h2θ2Kpfree =

ˆM− h(1− θ)C− h2θ(1− θ)K

˜ui − hKqi + h [θPf + (1− θ)Pi ]

non-linear: Mk = M + hθCk + h2θ2Kk

pfree = Mk ukf + M(ui − uk

f ) + h[(1− θ)(Fi + Pi ) + θ(Fkf + Pf ))]

Remark :

due to “energy conservation” one may replace h[(1− θ)Fi + θFkf ] by more suitable expression

(Gonzalez, 2000)

DEM::NSCD extensions::Interaction laws

Different interaction laws may be used:

Newton shock law using the variable un = un + eu−n and ut = ut

uα = uα,free + max(euiα, 0)T + Wααhrα +

Xβ 6=α

Wαβhrβ

Law(uα, rα) = TRUE

Gap Signorini law using the variable un = g/h = g0/h + un, ut = ut

uα = uα,free + max(g0/h, 0)T + Wααhrα +Xβ 6=α

Wαβhrβ

Elastic law using the variable un = g0/h + un + hrn/h2kn

Xβ 6=α

Wαβhrβ

Xβ 6=α

Wαβhrβ

DEM::NSCD Extensions::Explicit Integration

In some cases it is necessary to decrease the time step (sharp body shape, impact with non linearbulk behavior, wave propagation, etc). It leads to reconsider the use of implicit time integrator.Carpenter et al. (1991) proposed a method to respect implicit law (impenetrability) when usingan explicit time integrator.

The key point of the method is to apply the geometrical constraint in the actual (and unknown)configuration.Considering the time increment [t, t + h] it leads to

Mqi = Fi (qi , qi ) + Pi + Hi+1ri

H∗i+1qi+1 ≥ 0, (1)

where Fi (qi , qi ) represents the internal forces. As in the previous section, quantities indexed by i(resp. i + 1) refer to time t (resp. t + h).

In some cases it is necessary to decrease the time step (sharp body shape, impact with non linearbulk behavior, wave propagation, etc). It leads to reconsider the use of implicit time integrator.Carpenter et al. (1991) proposed a method to respect implicit law (impenetrability) when usingan explicit time integrator.

The key point of the method is to apply the geometrical constraint in the actual (and unknown)configuration.Considering the time increment [t, t + h] it leads to

Mqi = Fi (qi , qi ) + Pi + Hi+1ri

H∗i+1qi+1 ≥ 0, (1)

where Fi (qi , qi ) represents the internal forces. As in the previous section, quantities indexed by i(resp. i + 1) refer to time t (resp. t + h).

In this approach, a second order scheme is taken into account, using the first and second timederivatives of the configuration parameter that appears to be the primary unknown of theproblem as contact forces. Thus acceleration and velocity are related to configuration parameterusing, for example, a β method,8><>:

1 + 2βqi−1 + h(1− β)qi−1 +

h(qi+1 − qi )

h2(qi+1 − qi − hqi )

where β (∈ [0.5, 1]) represents a numerical damping parameter.

To take into account the displacement constraints in system (1), the displacement qi+1 isdecomposed as

qi+1 = qfreei+1 + qc

where qfreei+1 represents the prediction of the displacement without contact forces and qc

i+1 thecorrection due to the contact forces only.

In the case of the central different method (β = 0.5), qfreei+1 is defined by

qfreei+1 = h2M−1Pi − Fi (qi , qi )+ 2qi − qi−1 (3)

To solve the frictional contact problem and to determine the solution of the equation of motion,the couple of unknown (ri , q

ci+1) solution of the following system must be found

h2H∗i+1M−1Hi+1ri = H∗i+1qfreei+1

qci+1 = h2M−1Hi+1ri

To take into account the displacement constraints in system (1), the displacement qi+1 isdecomposed as

qi+1 = qfreei+1 + qc

where qfreei+1 represents the prediction of the displacement without contact forces and qc

i+1 thecorrection due to the contact forces only.

In the case of the central different method (β = 0.5), qfreei+1 is defined by

qfreei+1 = h2M−1Pi − Fi (qi , qi )+ 2qi − qi−1 (3)

To solve the frictional contact problem and to determine the solution of the equation of motion,the couple of unknown (ri , q

ci+1) solution of the following system must be found

h2H∗i+1M−1Hi+1ri = H∗i+1qfreei+1

qci+1 = h2M−1Hi+1ri

DEM::Multi-Physics

1 Preamble

2 Modeling

4 Technical Aspects

5 NSCD Extensions

6 Multi-PhysicsGeneralityFluidThermalElectricity

7 Bibliography

8 Annexes:

DEM::Multi-Physics::Generality

It appears more and more works related to the multi-physical extension of classical DEM, theclassical feature refers to the mechanical one:

fluid effect (...)

thermal effect (Vargas et al., 2001) (Luding et al., 2005) ...

electrical (Renouf et al., 2007)

physico-chemical

Coupling between mechanics and an added physical model could be performed using variousapproaches:

Serial treatment: the results of the added physical model is used to compute the mechanicalmodel without “retroaction”.

Ü thermal dilatation effects in a granular assembly

Staggered treatment: The new problem is computed using mechanical data in input andoutput are used to update some mechanical variables.

Ü thermal conduction with thermal resistance at contact depending on pressure andthermal dilatation effects in a granular assembly

Monolithic treatment: The problem is solved as a full multi-physical problem where themechanics and the new physics are computed simultaneously.

DEM::Multi-Physics::Generality

It appears more and more works related to the multi-physical extension of classical DEM, theclassical feature refers to the mechanical one:

fluid effect (...)

thermal effect (Vargas et al., 2001) (Luding et al., 2005) ...

electrical (Renouf et al., 2007)

physico-chemical

Coupling between mechanics and an added physical model could be performed using variousapproaches:

Serial treatment: the results of the added physical model is used to compute the mechanicalmodel without “retroaction”.

Ü thermal dilatation effects in a granular assembly

Staggered treatment: The new problem is computed using mechanical data in input andoutput are used to update some mechanical variables.

Ü thermal conduction with thermal resistance at contact depending on pressure andthermal dilatation effects in a granular assembly

Monolithic treatment: The problem is solved as a full multi-physical problem where themechanics and the new physics are computed simultaneously.

DEM::Multi-Physics::Fluid

Modeling gaz grain mixture (see MacNamara et al., 2000).

granular material with drag force due to fluidpressure (micro)

fluid mechanics in an evolutive porous material(macro)

Notations:I P: pressure.I η: fluide viscosity.

I φ = 1− V G

V : granular material porosity.I κ: permeability of the granular material.I −→u : mean velocity of the grains.

Hypotheses:I perfect gas.I neglect fluid inertia.I low Reynolds number (Darcy law).I spherical grains (radius r)

Carman-Kozeny : κ (φ) =`

r 2/45´φ3/

`1− φ2

Modeling gaz grain mixture (see MacNamara et al., 2000).

granular material with drag force due to fluidpressure (micro)

fluid mechanics in an evolutive porous material(macro)

Notations:I P: pressure.I η: fluide viscosity.

I φ = 1− V G

V : granular material porosity.I κ: permeability of the granular material.I −→u : mean velocity of the grains.

Hypotheses:I perfect gas.I neglect fluid inertia.I low Reynolds number (Darcy law).I spherical grains (radius r)

Carman-Kozeny : κ (φ) =`

r 2/45´φ3/

`1− φ2

Fluid pressure evolution

Mass conservation of the fluid and grains :

„∂P

∂t+−→u ·

−→∇P

«=−→∇ ·

„Pκ(φ)

−→∇P

«− P−→∇ · −→u

convection term: drag of the fluid due to the moving granular material.

diffusion term: fluid diffusion in the porous granular material.

Biot term: variation of the porous pressure due to evolution of the pore size.

0B@∂P

∂t+ −→u ·

−→∇P| z

convection term

1CA =−→∇ ·

„Pκ(φ)

−→∇P

«− P−→∇ · −→u

0B@∂P

∂t+ −→u ·

−→∇P| z

convection term

1CA =−→∇ ·

„Pκ(φ)

−→∇P

diffusion term

−P−→∇ · −→u

0B@∂P

∂t+ −→u ·

−→∇P| z

convection term

1CA =−→∇ ·

„Pκ(φ)

−→∇P

diffusion term

− P−→∇ · −→u| z

Biot term

Hydrodynamic force

Notations:I V G : grain volume.I−→∇P : pressure gradient at the center of the grain.

Force on the grain due to the fluid pressure gradient :

the force increases proportionally to the fluid volume around the grain.

Hydrodynamic force

−→F h = −

1− φ−→∇P

Hydrodynamic force

−→F h = −

1− φ−→∇P

Numerical strategy

coupling is solved with fixed point method on the grain velocity.

for one iteration :

Numerical strategy

coupling is solved with fixed point method on the grain velocity.

for one iteration :

`−→x G ,−→v G´ interpolation // `φ,−→u ´

fluid resolution

−→F h

mecanichal resolution

Plocalization

Numerical strategy

interpolation: interpolation, aux nœuds du maillage utilise pour le calcul de la pression, de laporosite locale : φloc = 1− V G/Vref , definie au centre des grains.

fluid resolution: compute pressure and pressure gradient on the mesh node :I “directional” operator splitting,I centered finite difference (space),I Crank-Nicholson (time),I Newton-Raphson (Non linearity)

localization: compute hydrodynamic force by interpolating (at the grain center),

1/(1− φ)−→∇P known at the mesh load.

mechanical resolution: compute velocity and position with a DEM code

Numerical strategy

Exemple: Sedimentation

DEM::Multi-Physics::Thermal

Dilatation effect Resolution Scheme:

Iteration matrix computation (M)Time loop

Compute temperatureFree velocity computation (qfree )Temporary configuration computation(qm)Contact detection (A)Compute thermal velocity: Uth = dth/h (B)Contact problem resolutionVelocity correction (qi+1)Update of the kinematics and the configuration (C)Update shapes (D)

Halt criteria

DEM::Multi-Physics::Electricity

As granular assemblies are considered, the formulation of the electrical problem relies on ananalogy between an electrical and a contact network. Each particle of the granular assembly isconsidered as a node of the electrical network while each contact is considered as a branch of it.The formulation of the electrical problem relies on an analogy between an electrical and thecontact network of a granular material.

An linear electrical network is a set of linear dipoles, linked by conductor with weak resistivity.This network is composed of nc branches (contacts) connected by nb nodes (particles) andrealizing nm meshes. Thus, a node is a junction of several conductors, a branch a part of thenetwork between two nodes and a mesh a closed run, composed of branch using a node only once.

To determine the global resistivity/conductivity of the sample, two aspect must be considered:the local conductivity of each branch and the relation between each branch of the network.

DEM::Multi-Physics::Electricity::Local formulation

To determine the conductivity of a contact between to particle, both the conductivity of particlesand the mechanical conductivity (related to the contact force) must be determined. If the contactindexed by α involved particles i and j , their conductivity will be noted respectively Ci and Cj .The mechanical conductivity will be noted Cm.

Then, the contact conductivity can be express asa function of Ci , Cj and Cm :

Cα =Ci Cj

Ci + Cj +Ci Cj

The equation (5) suppose that Ci and Cj as wellas Cm are not equal to zero. If one of the con-ductivity is equal to zero (insulating particle ornon contact) then Cα = 0.

Remark : it is possible to take into account more complex local phenomena (oxidation,breakdown,...). These ones will not be evoked here.

DEM::Multi-Physics::Electricity::Local formulation

To determine the conductivity of a contact between to particle, both the conductivity of particlesand the mechanical conductivity (related to the contact force) must be determined. If the contactindexed by α involved particles i and j , their conductivity will be noted respectively Ci and Cj .The mechanical conductivity will be noted Cm.

Then, the contact conductivity can be express asa function of Ci , Cj and Cm :

Cα =Ci Cj

Ci + Cj +Ci Cj

The equation (5) suppose that Ci and Cj as wellas Cm are not equal to zero. If one of the con-ductivity is equal to zero (insulating particle ornon contact) then Cα = 0.

Remark : it is possible to take into account more complex local phenomena (oxidation,breakdown,...). These ones will not be evoked here.

DEM::Multi-Physics::Electricity::Global formulation

The global formulation of the electrical problem refers to the first Kirchhoff law and to the Ohmlaw.The first Kirchhoff law can be given under a matricial equation as:

NI = I0, (6)

where N ∈ Rnb×nc , I ∈ Rnb and I0 ∈ Rnb . The matrix N denotes the incidence matrix between thenodes of the network. Its role is equivalent to the one of matrix H defined in the mechanicalsection. The vector I0 is the intensity vector where each component Iα is linked to one of thenetwork branches.

Thus writing the Ohm law for the set of contact, one obtains:

I = CNT V. (7)

Finally, using Equation (7) in Equation (6) one obtains:

NCNT V = −I0. (8)

where the electrical potential V, expressed in each node of the network, become the primaryunknown of the problem. The matrix G is equal to the product NCNT and is, as the W matrix ,symmetric semi positive defined.

DEM::Multi-Physics::Electricity::Global formulation

The global formulation of the electrical problem refers to the first Kirchhoff law and to the Ohmlaw.The first Kirchhoff law can be given under a matricial equation as:

NI = I0, (6)

where N ∈ Rnb×nc , I ∈ Rnb and I0 ∈ Rnb . The matrix N denotes the incidence matrix between thenodes of the network. Its role is equivalent to the one of matrix H defined in the mechanicalsection. The vector I0 is the intensity vector where each component Iα is linked to one of thenetwork branches.Thus writing the Ohm law for the set of contact, one obtains:

I = CNT V. (7)

Finally, using Equation (7) in Equation (6) one obtains:

NCNT V = −I0. (8)

where the electrical potential V, expressed in each node of the network, become the primaryunknown of the problem. The matrix G is equal to the product NCNT and is, as the W matrix ,symmetric semi positive defined.

DEM::Multi-Physics::Electricity::Resolution

The resolution of the electrical problem is realized using the iterative Gauss-Seidel algorithm, usedto solve the contact problem. On the same level as the mechanical problem, the electricalproblem G have a multiplicity of solution and thus is strongly dependant of initial condition andthe direction of reading of the contact loop.

Gauss-Seidel algorithm

Step 0: Initialization with V0 non negative.Step 1: Iteration k

For all i in 1, ..., nbCompute Vi = I 0

i −P

j<i Gij Vkj −

Pj>i Gij V

k−1j

V ki = 1

Step 3: If error criterion is satisfied then Vk is solutionElse one iterates k and go to Step 1.

The error criteria used is a simple relative maximal variation criterion applied on the electricalpotential.

DEM::MP::Electrique::Resolution

The construction of the electrical-mechanical problem is easy. For that, the different part of theelectrical problem are introduced in the mechanical scheme after the contact resolution andbefore the correction of the position.2666666666666666664

i = i + 1 (time loop)q(i + 1) predictionContact detectionq(i)free computation266664

k = k + 1 (NSGS iteration)24 α = α+ 1 (contact loop)bkαcomputation

Contact problem resolution :(vk+1α , rk+1

α ) solution

35convergence test

Electrical problem formulationElectrical problem resolutionq(i + 1) correction

Then, the electrical problem construction is fully dependant of the mechanical problem. In thisscheme, the electrical problem have any influence on the mechanical problem.

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Concurrent multiple impacts modeling - Case-study of a 3-ball chain,in Second MIT conference on computational Fluid and Solid Mechanics (K. J. Bathe Eds.), 2003, p. 1842-1847.

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A mixed formulation for frictional contact problems prone to Newton like solution methodin Comput. Methods Appl. Mech. Engrg. (1991), vol. 92(3), p. 353-375.

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Computer simulation of liquidsOxford University Press, 1987.

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Nonlinear Finite Elements for Continua and StructuresJohn Wiley & Sons, 2000.

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Lagrange Constraints for transient finite element surface contact,in Int. J. Numer. Methods Engrg., 1991, vol. 32, p. 103-128.

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A computer model for simulating progressive large scale movements of blocky rock systemsin Proceedings of the symposium of the international society of rock mechanics, 1971, vol. 1, p. 132-150, Nancy, France.

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A discrete numerical model for granular assembliesGeotechnique, 1979, vol. 29, n 1, p. 47-65.

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Numerical simulation of faults and shear zonesin Geophys. J. Int., 1994, vol. 116, p. 46-52.

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LMGC90 une plateforme de developpement dediee a la modelisation des problemes d’interactiondans les Actes du sixieme colloque national en calcul des structures, vol. 1, p. 111-118, 21-25 Mai 2003, Giens (VAR), France.

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On dense granular flowsin Euro. Phys. J. E, 2004, vol., p.

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Solid Third Body Analysis Using a Discrete Approach: Influence of Adhesion and Particle Size on the Macroscopic Behavior of the Contactin ASME J. Tribol., 2002, vol. 124, p. 530-538.

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The Non Smooth Contact Dynamics Methodin Compt. Methods Appl. Math. Engrg., 1999, vol. 177, p. 235-257.

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Non-smooth contact dynamics approach of cohesive materialsin Phil. Trans. R. Soc. Lond. A, 2001, vol. 359, p. 2497-2518.

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A Discrete Model For Long Time Sinteringin Journal of the Mechanics and Physics of Solids, 2005, vol. 53(2), pp. 455-491.

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Comparisons and coupling of algorithms for collisions, contact and friction in rigid multibody simulationsin III European Conference on Computational Mechanics Solids Proceedings (C.A. Mota Soares et.al. eds.), Lisbon, Portugal, 5-8 June 2006.

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Annexes::Deriving the Lagrangian system

Lets consider the smooth equations of motion of a collection of N continuum media.The continuum medium is identified at time t ∈ [0,T ] by its volume in Rd .The integer d = 1, 2, 3 denotes the space dimension, of interior

Ω(t) ⊂ Rd , i ∈ 1, . . . ,N

and boundary ∂Ω(t).

If Ω is a deformable continuum medium, the equations of motion are introduced through theprinciple of virtual powers in a finite strain Lagrangian setting permitting a space-discretizationbased on a conventional finite element method.If Ω is assumed to be a rigid body, the equation of motion will be described by a finite set ofcoordinates.

In both cases, possibly after a space-discretization, the equations of motion will be formulatedand treated in a single finite-dimensional framework.

A material particle is described by its position, X in a reference frame at t = 0 and by its currentposition x = ϕ(X , t) at time t.

For a Lagrangian description, we also assume we know at least formally the function X = ψ(x , t).The displacement is defined by u(x , t) = x − X = x − ψ(x , t) and the velocity and theacceleration are denoted by u and u.

Most of the Lagrangian variables expressed in terms of X are denoted by capitals letters, forinstance, U(X , t) for the displacement, and denoted by lower case for the associated Eulerianvariables, in this case u(x , t).This convention can be summarized by u(x , t) = u(ϕ(X , t), t) = U(X , t).

Starting from the equation of motion in Eulerian coordinates,

divσ(x , t) + ρ(x , t)b(x , t) = ρ(x , t)u(x , t), ∀x ∈ Ω(t),

where σ(x , t) is the Cauchy stress tensor and b(x , t) is the density of body forces, the principle ofvirtual power states thatZ

Ω(t)(u(x , t)− b(x , t))v(x , t) dm(x , t) =

ZΩ(t)

divσ(x , t)v(x , t) dω(x , t), (9)

for all virtual velocities denoted by v(x , t). The measure dω(x , t) denotes the Lebesgue measurein Rd at x and the measure dm(x , t) = ρ(x , t) dω(x , t) is the mass measure.With the help of the Green formulas, the principle of virtual power is usually reformulated asZ

Ω(t)(u(x , t)− b(x , t))v(x , t) dm(x , t) = −

ZΩ(t)

σ(x , t) : ∇v(x , t) dω(x , t)

Z∂ΩF (t)

t(x , t)v(x , t) ds(x , t) +

c (t)r(x , t)v(x , t) ds(x , t)

where A : B = Aij Bij is the double contracted tensor product, and t(x , t) = σ(x , t).n(x , t) is the

applied forces on the boundary of outward normal n and r(x , t) the reaction forces due to thecohesive interface. The measure ds(x , t) is the Lebesgue measure at x ∈ ∂Ω.

The symmetry of the Cauchy stress tensor in absence of density of momentum allows one tointroduce the symmetric deformation rate tensor,

D(x , t) =1

2(∇T v(x , t) +∇v(x , t)) (11)

leading to the standard expression of the virtual power of the internal forces of cohesion

Pint = −Z

Ω(t)σ(x , t) : ∇v(x , t) dω(x , t) = −

ZΩ(t)

σ(x , t) : D(x , t) dω(x , t) (12)

In order to formulate the principle of virtual power in a total Lagrangian framework, the secondPiola–Kirchhoff tensor,

S(X , t) = F−1 det(F )σT F−T

is introduced, where F =∂x

∂X=∂ϕ(X , t)

∂Xis the deformation gradient.

The virtual power of the internal forces is then rewritten, as

Pint = −Z

Ω(t)σ(x , t) : ∇v(x , t) dω(x , t) = −

ZΩ(0)

S(X , t) : L(X , t) dΩ(X , 0) (13)

where L = F .

Finally, the principle of virtual power in a total finite strain Lagrangian framework in terms of theconvected Lagrangian variable X , isZ

Ω(0)(U(X , t)− B(X , t))V (X , t) dM(X , 0) = −

ZΩ(0)

S(X , t) : L(X , t) dΩ(X , 0)

Z∂Ω(0)

T (X , t)V (X , t) dS(X , t) +

c (0)R(X , t)V (X , t) dS(X , t)

where the applied forces laws on the boundary satisfies T (X , t) = S(X , t).F T (X , t).N(X , t) andR(X , t) = S(X , t).F T (X , t).N(X , t). It is noteworthy that the interactions between bodies thatare taken into account in this model are only given by the forces through the interface withunilateral contact and friction.

Annexes:: Deriving the Lagrangian system

For the constitutive material laws, a large panel of models can be taken into account in thisframework and in the numerical applications. The formulation of the constitutive laws is based onthe standard thermodynamics of irreversible processes or based on a variational formulation ofincremental stress-strain relation deriving from a pseudo elastic potential.

If the bulk response of the material is linear elastic

S(X , t) = K(X ,T ) : E(X , t), (15)

where E is the Green-Lagrange strain tensor,

E(X , t) =1

2(F T (X , t)F (X , t)− I (X , t)), (16)

the tensor I is the identity tensor and K(X ,T ) is the fourth order tensor of elastic properties.back

The finite element discretization is conventional and is based on this principle of virtual power inthis total Lagrangian framework. Choosing some isoparametric element leads to the followingapproximation

U(X , t) =X

Nh(X , t)Uh(t), u(X , t) =X

Nh(X , t)Uh(t), U(X , t) =X

Nh(X , t)Uh(t),

(17)where Nh are the shape functions and Uh the finite set of displacement at nodes. Substitutingthis approximation into the principle of virtual power and simplifying with respect to the virtualfield yields a space-discretized equation of motion of the form

M(Uh)Uh + F (t,Uh, Uh) = R, (18)

where M(Uh) is the consistent or lumped mass matrix, the vector F (t,Uh, Uh) collects theinternal and external discretized forces and R are the discretized forces due to the interaction.

In rigid body mechanics, it is assumed that the power of the cohesion internal forces vanishes fora rigid motion given by the following set of virtual velocity field,

V = v(x , t) = vO (t) + ω(t)× (x − xO ), ∀x ∈ Ω(t), (19)

where O is a geometrical point fixed with respect to the body, xO is the position of this pointvO (t) is its velocity, and ω(t) the angular velocity of the body at O. This assumption yieldsZ

Ω(t)(u(x , t)− b(x , t))v(x , t) dm(x , t) =

Z∂Ω(t)

t(x , t)v(x , t) ds(x , t), ∀v(x , t) ∈ V (20)

The equation of motion can be derived choosing a particular virtual velocity as:8>>>>>>>><>>>>>>>>:

ZΩ(t)

u(x , t) dm(x , t) =

ZΩ(t)

b(x , t) dm(x , t) +

Z∂Ω(t)

t(x , t) ds(x , t),

ZΩ(t)

(x − xO )× u(x , t) dm(x , t) =

ZΩ(t)

(x − xO )× g(x , t) dm(x , t)

Z∂Ω(t)

(x − xO )× t(x , t) ds(x , t).

Various descriptions of the equations of motion of a rigid body can be deduced from the principleof virtual power choosing particular kinematics. Without going into further details, theNewton-Euler formulation can be chosen to write the kinematics with respect to the center ofmass Gi of the body Ω in Eulerian coordinates:(

u(x , t) = vGi(t) + ωi (t)× (x − xGi

) + ωi (t)× (ωi (t)× (x − xGi)).

Substituting (22) into the equations of motion (21) yields the well-known Newton-Eulerequations, »

M 00 I

(t)ωi (t)

ωi (t)× Iωi (t)

»fext (t)

mext (t)

–, (23)

ZΩ(t)

dm(x , t) =

ZΩ(0)

dM(X , 0),

ZΩ(t)

(x − xGi)T (x − xGi

) dm(x , t),=

ZΩ(0)

(X − XGi)T (X − XGi

) dM(X , 0),

fext (t) =

ZΩ(t)

b(x , t) dm(x , t) +

Z∂Ω(t)

t(x , t) ds(x , t),

mext (t) =

ZΩ(t)

(x − xGi)× b(x , t) dm(x , t) +

Z∂Ω(t)

(x − xGi)× t(x , t) ds(x , t).

Usually, a second order form of the dynamics is obtained with the help of the followingparametrization of the vector ωi ,

ωi (t) = Di (Ψ, t)Ψi (t), (25)

where D(Ψ, t) is supposed to be a diffeomorphism. For a collection of N rigid bodies, a usual wayis to introduce a set a generalized coordinates z such that

z =hˆ

xGi, Ψi

˜i∈1...N

iT(26)

assuming that the positions and the orientations of the bodies are uniquely determined by z.With this variable, after some algebraic manipulations, the equations of motion can be written as:

M(z)z + F (t, z, z) = r (27)

It is noteworthy that this formulation allows us to add some internal forces between bodiesexpressed in terms of the generalized coordinates z.

Clearly, the equations of motion can also be obtained straightforwardly thanks to the Lagrangianformalism postulating the existence of the Lagrangian of the system,

L(z, z) = T (z, z)− V (z),

composed of the kinetic energy,

T (z, z) =1

2zT M(z)z

and the potential energy of the system, V (z). The Lagrange’s equations can be written as

„∂L(z, z)

«−∂L(z, z)

∂zi= Qi (z, t), i ∈ 1 . . . n, (28)

where the vector Q(z, t),∈ Rn denotes the set of generalized forces.back

With some standard algebraic manipulations, the Lagrange equations (28) can be put in a moreusual form:

M(z)dz

dt+ N(z, z) = Q(z, t)−∇z V (z) (29)

where the vector N(z, z) =

∂Mik

∂zl+∂Mil

∂zk−∂Mkl

∂zi, i = 1 . . . n

collects the nonlinear

inertial terms i.e., the gyroscopic accelerations.If we allow one to introduce non linear interactions between bodies of the system and externalapplied forces which do not derived from a potential, we will use the following more general formfor the equation of motion:

M(z)z + F (t, z, z) = 0 (30)

where F : Rn × Rn × R→ Rn collects the internal non linear interactions between bodies whichare not necessarily derived from a potential and the external applied loads.

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