granular as a continuum - project.inria.fr · figure 5: rescaled granular proﬁles: comparison...

Granular as a continuum

Pierre-Yves Lagrée

Gilles Daviet

Graphyz — October 24, 2019 — Montbonnot, France

Granular as a continuum

Lagrée Pierre-Yves

Institut Jean Le Rond ∂’Alembert, Paris

Graphyz — October 24, 2019 — Montbonnot, France

Lydie Staron, Stéphane Popinet, Stéphanie Deboeuf (∂’Alembert)

Pascale Ausillous (IUSTI), Pierre Ruyer (IRSN)

Guillaume Saingier (∂’Alembert-SVI),

Yixian Zhou, Zhenhai Zou (IUSTI & IRSN)

Sylvain Viroulet (IPGP-IMFT)

motivation, facts

• sand, granulates: 6 103 kg/french/year

• 2nd most used "fluid" after water >> petroleum

(water 1.0, granulates 0.1, petroleum 0.025)

• corn: 450 kg/french/year,

• Food, Medicines

• Environmental flows (avalanches, mud flow….)

Jean Le Rond ∂'Alembert1717 1783

"useful" for industry and real live

PYL

motivation, facts

PYL

Beetroots spoil tip (“terril”) silos

North of France

motivation, facts

motivation, facts

PYLVal André le Pissot

PYLRio de Janeiro, Ipanema

motivation, facts

pyl

Paris, Univ. Pierre & Marie Curie, Sorbonne University, Jussieu

motivation, facts

IPGP

http://books.google.fr/books?id=HY6Z5od4-E4C&pg=PA49&dq=granular+flow&hl=fr&ei=lamtTaa_NYyVOoToldcL&sa=X&oi=book_result&ct=result&resnum=10&ved=0CFkQ6AEwCTgK#v=onepage&q&f=true

40km8km

http://www.cieletespace.fr/image-du-jour/5126_la-saison-des-avalanches-sur-mars

Mars

motivation, facts

NASA’S Mars Reconnaissance Orbiter (MRO)

-

re-not

v--

er,--

t

e-

g

g g

(c)(a) (b)

(d) (e) (f)

Fig. 1. The six configurations of granular flows: (a) planeshear, (b) annular shear, (c) vertical-chute flows, (d) inclinedplane, (e) heap flow, (f) rotating drum.

• Solid - Liquid - Gas

• Looking for a continuum description for liquid phase

• Many experiments in simple configurations: shear/ inclined plane, with model material (glass beads, sand…)

• Simulations with discrete elements (disks, polygona, spheres)

DOI 10.1140/epje/i2003-10153-0

Eur. Phys. J. E 14, 341–365 (2004)

On dense granular flows

GDR MiDia

THE EUROPEAN

PHYSICAL JOURNAL E


-

re-not

v--

er,--

t

e-

g

g g

(c)(a) (b)

(d) (e) (f)

Fig. 1. The six configurations of granular flows: (a) planeshear, (b) annular shear, (c) vertical-chute flows, (d) inclinedplane, (e) heap flow, (f) rotating drum.

• Solid - Liquid - Gas

• Looking for a continuum description for liquid phase

• Many experiments in simple configurations: shear/ inclined plane, with model material (glass beads, sand…)

• Simulations with discrete elements (disks, polygona, spheres)

DOI 10.1140/epje/i2003-10153-0

Eur. Phys. J. E 14, 341–365 (2004)

On dense granular flows

GDR MiDia

THE EUROPEAN

PHYSICAL JOURNAL E


Utopian research paper with collective author name

Pouliquen 99

Pouliquen Forterre JSM 06

Da Cruz 04-05

GDR MiDi 04

falling time

displacement time I =

d∂u

∂y

P/ρ

u(y)

Coulomb friction law

by grain dynamics

I2

non dimensional number: «Froude»

local «Inertial Number» (Da Cruz 04-05)

(Savage Number 89 / Ancey 00 )

The µ(I)-rheology

«Drucker-Prager» plastic flow

«Inertial Number»

τP

Da Cruz PRE 05

τ

τ = μ(I)P

Charles-Augustin Coulomb1736-1806

µ(I) = µ1 +µ2 − µ1

I0/I + 1µs

µ2

(c)

quasi-static

kinetic regime

I

µ(Ι)

µ1 0.32 (µ2 µ1) 0.23 I0 0.3 µ1

Lacaze Kerswell 09

I =

d∂u

∂y

P/ρ

µ(I)

I

Coulomb friction law

by grain dynamics

Jop Forterre Pouliquen 2005

«Inertial Number»

τP

τ = μ(I)P«Drucker-Prager» plastic flow

Charles-Augustin Coulomb1736-1806

u(y)

The µ(I)-rheology


• Introduction

• presentation of µ(I) rheology

• Simplification with shallow water (depth average) 1D continuum model

• use of contact dynamics (simulation all grains, discrete, here 2D)

• implementation of µ(I) in a continuum Navier Stokes 2D code

• applications to archetypal flows: collapse of columns - hourglass

Outline

y

−→g

H grains

Saint-Venant Savage Hutter

θ

Dominant terms in the shallow layer / thin layer (boundary layer equations)

∂u

∂x+

∂v

∂y= 0

ρ(∂u

∂t+

∂u2

∂x+

∂uv

∂y) = −ρg tan θ −

∂p

∂x+

∂τxy

∂y+

∂τxx

∂x

ρ(∂v

∂t+

∂uv

∂x+

∂v2

∂y) = −ρg −

∂p

∂y+

∂τxy

∂x+

∂τyy

∂y

Write thin layer (boundary layer) and shallow layer equations, test the most simple implementation of µ(I) in shallow water

Shallow water/ depth averaged

Adhémar Barré de Saint-Venant1797-1886

Mass and Momentum conservation

y

−→g

H grains

θ

Dominant terms in the shallow layer / thin layer (boundary layer equations)

∂u

∂x+

∂v

∂y= 0

ρ(∂u

∂t+

∂u2

∂x+

∂uv

∂y) = −ρg tan θ −

∂p

∂x+

∂τ

∂y

0 = −ρg −∂p

∂yτ = µ(I)p

Saint-Venant Savage Hutter Shallow water/ depth averaged


Write thin layer (boundary layer) and shallow layer equations, test the most simple implementation of µ(I) in shallow water


y

α

−→g

H grains

µ(I) = µ0 +∆µ

I0/I + 1,

Q = hu

Closure (here use Bagnold profile, next slide), if flat profile α=1, if half Poiseuille α=6/5


θ

∂h

∂t+

∂

∂x(hu) = 0

∂

∂t(hu) + α

∂

∂x(hu2) = hg(tan θ −

∂h

∂x− µ(I))

α =1

h

R h

0u2(z)dz

1

h

R h

0u(z)dz

2, I =

5

2

udg

h√

gh,

Saint-Venant Savage Hutter Shallow water/ depth averaged

Integrate across the layer

α =5

4


α =1

h

R h

0u2(z)dz

1

h

R h

0u(z)dz

2,

kind of Nusselt solution

Bagnold 1954

«Bagnold» avalanche

Ralph Bagnold1896-1990

• explored Lybian desert in 30’

• low pressure in tires when driving on sand

• waffle-boards (tôle de désensablement)

• commando in desert WW2 (Long Range Desert Group)

• field observations (The Physics of Blown Sand & Desert Dunes, 1941)

y

u(y)α

−→g

H grains

µ(I) = tanα

u =2

3Iα

r

gd cos α

H3

d3

1−

1−y

H

3/2

,

v = 0, p = ρgH

1−y

H

cos α.

Ralph Bagnold1896-1990

«Bagnold» avalanche

An analytical solution of the µ(I) rheology for an infinite layer on an inclined planeanalog of Nusselt flow (or half-Poiseuille) in Newtonian fluids

Bagnold velocity profile

α =5

4α =

1

h

R h

0u2(z)dz

1

h

R h

0u(z)dz

2,

Experimental set up:

Pouliquen 99 & 99

Front in Savage Hutter St Venant

With Stéphanie Deboeuf and Guillaume Saingier

0 100 200 300 400 500 6000

5

10

15

x (mm)

h (

mm

)

t = t0

t = t0 + 0.667 s

t = t0 + 1.333 s

t = t0 + 2.000 s

spato−temporal profile

0 200 400 6000

2

4

6

h (

mm

)

x − u0.t (mm)

(b)

(a)

front moving at

constant velocity u0


glass bead 400µm

X(H) = X0−

1

3(−1 + d)(3H(d−1)−2

√

3 tan−1(1 + 2

√

H√

3)−2 log(1−

√

H)+

3d(1− α)Fr2 log(H) + log(1 +√

H +H)− 2(1− α)Fr2 log(1−H3/2)),

analytical implicit solution:

superposed on Pouliquen 99 (who supposed =1)α

ξ = x− u0t

In a moving framework

X =ξ(tan θ − µ0)

h∞

, H =h

h∞

, d =µ0 − tan θ

∆µ,

((α− 1)Fr2h∞

h+ 1)

dh

dξ= tan θ − µ(I).


Figure 5: Rescaled granular profiles: comparison between experiments and analyticalpredictions for different inclinations and different thicknesses h∞. Analytical solutions

(colored lines) are calculated by using the thickness h∞ and the front velocity u0 measuredfor each experimental front (colored circles) with a shape factor α = 5/4. The analytical

solution evaluated for α = 1 is plotted in black line.

with our inclined plane

importance of the up to now neglected inertial effects

((α− 1)Fr2h∞

h+ 1)

dh

dξ= tan θ − µ(I).


• implementing the µ(I) friction law in Shallow Water (SVSH)

- friction is only at the bottom

- pay attention to the shape factor: should be α=5/4

- but: α=1 in the SVSH for Galilean invariance and entropy

• Widely used in geophysics with α=1

• 1 D model (like rods or shell in elasticity)

Go now to grains (contact dynamics) vs full Navier Stokes


t

• Direct simulation of movement of thousands of grains

Newton’s law

m(−→

U+−

−→

U−) =

−→

F δt

fn

un

ft

µfn

−µfn

ut

take the form of an equality between the change of momentum and the average impulse during δt.

written for each grain at the contactun, ut

Contact Dynamics 1988

!

u,! né! en! 1923! à! Blaye! (Gironde),! fut! agrégé! enJean-Jacques Moreau 1923-2014

More details in Gilles talk

With Lydie Staron

first order fluids (linear Stokesian or linear Reiner Rivlin)

σij = f(Dij)

D2 =

p

DijDij

Dij =ui,j + uj,i

2

With strain rate tensor

second invariant of strain rate tensor

σij = −pδij + 2η(D2)Dij

simple form γ =∂u

∂y

Non newtonian flows:

local constitutive law

(Stokesian or Reiner Rivlin)

General formulation

tensorial formulation

τ =

η(∂u

∂y)

∂u

∂y

τij = 2(η(D2))Dij

σij = f(Dij)

D2 =

p

DijDij

Dij =ui,j + uj,i

2

With strain rate tensor

second invariant of strain rate tensor

τ =

µp

∂u

∂y

!

∂u

∂y

classic formulation

practical formulation

τ = µp

I = d√

2D2/p

(|p|/ρ).


τ =

η(∂u

∂y)

∂u

∂y

tensorial formulation

η =

µ(I)√

2D2

p

µ(I) = µ1 +µ2 − µ1

I0/I + 1µ1 0.32 (µ2 µ1) 0.23 I0 0.3

τij = 2(η(D2))Dij

General formulation

- the «min» limits viscosity to a large constant value - always flow, even slowly

construction of a viscosity based on the D2 invariant and redefinition of I

Boundary Conditions: no slip and p=0 at the interface for µ(I)

D2 =

p

DijDijDij =

ui,j + uj,i

2


I = d√

2D2/p

(|p|/ρ).η = min(ηmax,max (η(D2) , 0))

Implementation in Basilisk flow solver?

η =

µ(I)√

2D2

p

- the «min» limits viscosity to a large constant value - always flow, even slowly



D2 =

p

DijDijDij =

ui,j + uj,i

2


I = d√

2D2/p



η =

µ(I)√

2D2

p

see Gilles talk for a better model

D2 =

p

DijDijDij =

ui,j + uj,i

2

· u = 0, ρ

∂u

∂t+ u ·u

= p + · (2ηD) + ρg,

∂c

∂t+ r · (cu) = 0, ρ = cρ1 + (1 c)ρ2, η = cη1 + (1 c)η2


The granular fluid is covered by a passive light fluid (it allows for a zero pressure boundary condition at the surface, bypassing

an up to now difficulty which was to impose this condition on a unknown moving boundary).



I = d√

2D2/p



η =

µ(I)√

2D2

p

Volume Of Fluids, projection method, finite volumes

−→g

fluidsmall density

small viscosity

The granular fluid is covered by a passive light fluid (it allows for a zero pressure boundary condition at the surface, bypassing

an up to now difficulty which was to impose this condition on a unknown moving boundary).

∂c

∂t+ r · (cu) = 0, ρ = cρ1 + (1 c)ρ2, η = cη1 + (1 c)η2

neumann and p=0

no slipno slip

no slip

neumann and p=0



granular fluid

· u = 0, ρ

∂u

∂t+ u ·u

= p + · (2ηD) + ρg,

Volume Of Fluids, projection method, finite volumes

Implementations of NS-µ(I)- Lagrée Staron Popinet 2011

- Mangeney, Ionescu, Bouchut, Lusso 2016

- Krabbenhoft 2014

- Dunatunga & Kamrin 2015

- Barker & Gray 2015

- Daviet & Bertails-Descoubes 2016

Implementations of Bingham

- Liu, Balmforth, Hormozi, Hewitt, 2016,

- Dufour and Pijaudier-Cabotz 2005

- Vinay Wachs, Agassant 2005

- Vola, Babik, Latché 2004

τ =

τ0∂u

∂y

+ η

!

∂u

∂y

τ =

µp

∂u

∂y

!

∂u

∂y

Granular Column Collapse

L0

L∞

H∞

H0

t = 0

t = ∞

aspect ratio a = H0/R0 = H0/L0

The sand pit problem: quickly remove the bucket of sand

Collapse of columns simulation Basilisk µ(I)

http://basilisk.fr/sandbox/M1EMN/Exemples/granular_sandglass.c

reproduce Lagrée Staron Popinet 2011

Collapse of columns

DCM vs Navier Stokes µ(I)

at the tip, a=6.6 t=1.33 2 2.66

Collapse of columns

DCM vs Navier Stokes µ(I)

0.5

1

2

4

8

16

32

64

0.25 0.5 1 2 4 8 16 32 64

a

1.85*x3.5*x**(2./3.)

9 levels10 levels11 levels12 levels

Collapse of columns simulation µ(I)

We recover the experimental scaling [Lajeunesse et al. 04] and [Staron et al. 05]. Differences in the prefactors are due to the difficulties to obtain the run out length (discrete simulation shows that the tip is very gazeous, it can no longer explained by a continuum description).

Normalised final deposit extent as a function of aspect ratio a. W e l l - d e fi n e d p o w e r l a w dependencies with exponents of 1 and 2/3 respectively.

$L

Li

! !a a " 3,

a2/3 a # 3."

simul. 2D

exp. 2D /axi

a

a2/3

a

a2/3

Collapse of columns

Solids are with friction at the wall

implement solid friction at the wall

instead of no slip

τ = µsp

Neumann condition, instead of no slip

under work

with Sylvain Viroulet, Anne Mangeney IPGP

http://basilisk.fr/sandbox/M1EMN/Exemples/granular_column_muw.c

∂u

∂y|0 =

µsp

η

Results identical, good fit with experiment,

better than Ionescu 2015

• Problem:

Simulate the hour glass with discrete and continuum theories

• try to recover the well know experimental result: Beverloo 1961 Hagen 1852 law from discrete and continuum simulations

Gotthilf Hagen 1797-1884

• Flow in a Hourglass Discharge from Hoppers

simulation DCM

simulation Navier Stokes µ(I)


no influence of the hight nor the widthinfluence of D, d and , so by dimensional analysis:

• Hagen 1852 Beverloo 1961 constant discharge law

mass flow rate

ρ

d

Du ∼

gD

Gotthilf Hagen 1797-1884

Q3D ∼ ρ

p

gDD2

Q2D ∼ ρ

p

gDD

π-theorem

Q2

3— —

Q—

L—

L—

Width of the aperture

discrete vs continuum 2D

Beverloo (1961)Hagen (1852)


Q2D ∼ ρ

p

gD3

Width of the aperture

continuum 3D

Beverloo (1961)Hagen (1852)


0 8 16 24 32

D

0

1000

2000

3000

4000

Q

Q = 0.87D5/2

Q3D ∼ ρ

p

gD5

comparing Torricelli

0 2 4 6 8 10 12 14 16 180.0

0.3

0.6

0.9µ(I) plastic flow

y = 0.9 - 0.053 t

Newtonian flowTorricelli's flow

time

Volu

me

viscosity of the Newtonian flow extrapolated from the µ(I) near the orifice

Evangelista Torricelli 1608 1647


discrete vs continuum (at same rate)Staron Lagrée Popinet 2014

Staron Lagrée Popinet 2014 discrete vs continuum (at same rate)

r · u = 0, ρ

∂u

∂t+ u ·ru

= rp+r · (2ηD) + ρg 2µw

p

W

u

|u|

r · u = 0, ρ

∂u

∂t+ u ·ru

= rp+r · (2ηD) + ρg 2µw

p

W

u

|u|Z· dz

u = (u, v)

u = (1

W

Z W/2

−W/2

udz,1

W

Z W/2

−W/2

vdz, w = 0)

redefinition of velocity

u = (u, v, w)

3D as Hele-Shaw approximation

The 3D equations are averaged across the cell of thickness W

−µwpu

|u|

−µwpu

|u|

u

W

suppose an almost transverse flat profile:

non linear closure coefficient is one

extra wall friction source term

With Pascale Aussillous Pierre Ruyer and Yixian Zhou

2D width averaged Equations

Q = ρW 2p

gW F(D/W )rescaling with:

(D/W ) 1 (D/W ) 1small thicknesslarge thickness

plot as function of D/W

(D/W ) < 1, Q ∼ D3/2W (D/W ) > 1, Q ∼ W 3/2D

NS Hele-Shaw µ(I) vs Experiments

Exp. NS HS µ(I)

http://basilisk.fr/sandbox/M1EMN/Exemples/granular_sandglass_muw.c

Hagen friction dominated

Hagen friction dominated

0 10 200

0.2

0.4

0.6

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Streamlines

Granular

Fluid

r · up= 0

r · uf= 0

Coupling unsteady Darcy Forchheimer (porous)

With granular flow

ρp

∂up

∂t+ up

·rup

= rpp +r · (2ηD)rp

f + ρg

Rf ∂uf

∂t= rpf Bl(u

f u

p)Bi|(uf u

p)|(uf u

p)

The flow rate is increased by the constant pressure drop

axi

With Pascale Aussillous Pierre Ruyer and Zhenhai Zou

Coupling granular with air: Darcy Forchheimer

Using simplified Jackson 00 two fluids equations model

Obvious societal problem

Experiment/ simulation/ modelisation

Granular µ(I) rheology shows agreement with experiments

at least qualitatively

Method conserving exactly mass, fast code.

Problems here: - it always flows (regularisation at small shear) - no void formation (constant density) - no steady flow - no solid - no "h stop" - instabilities (ill posed)

Next: - non locality, - coupling with air/water

Conclusion perspectives

Obvious societal problem

Experiment/ simulation/ modelisation

Granular µ(I) rheology shows agreement with experiments

at least qualitatively

Method conserving exactly mass, fast code.

Problems here: - it always flows (regularisation at small shear) - no void formation (constant density) - no steady flow - no solid - no "h stop" - instabilities (ill posed)

Next: - non locality, - coupling with air/water

Conclusion perspectives

see Gilles talk

Lydie Staron

Stéphanie Deboeuf

Pascale Ausillous

Pierre Ruyer

Anaïs Abramian

Yixian Zhou

Zhenhai Zou

Luke Fullard

Sylvain Viroulet

Guillaume Saingier

Thomas Aubry

& Stéphane Popinet

Special Thanks to

∂’Alembert par Félix LECOMTE. Avant 1786, Le Louvre Lens


+ Gilles & Florence

Granular as continuum(for Computer Graphics)

Gilles Daviet

with Florence Bertails-Decoubes

Graphyz — October , — Montbonnot, France

What we want to capture

To what extent do we need physical realism?

I We are not making predictions

• Quantitative accuracy not required• Does not need to be conservative

To what extent do we need physical realism?

I We are not making predictions

• Quantitative accuracy not required• Does not need to be conservative

I Physics-based models are worth it:

• Parameters can be measured (or read from tables)• Model validation settles debates about plausibility• =⇒ reduces time to get look approved• Provable energy conservation/dissipation properties

Numerical simulation

How to simulate granular materials numerically?

Discrete Element Modeling

Simulate each constituant individually, and the interactions

between them

I Controllabilty

I Computational cost

Numerical simulation

How to simulate granular materials numerically?

Continuum approach

Considers the “averaged” behavior of many constituants

(zoom-out).

I E.g. Navier-Stokes for Newtonian uids

I Cost no longer depends on the system’s size

I Inhomogeneities must be relatively small

I Macroscopic model has to be derived

Outline

Dense, dry granular materials in D

Continuous model

Results

D Simulations

Granular continuum model

Time discretization

Spatial discretization: the Material Point Method

Results

Concluding remarks

DP (µ) rheology

σtot = ηε|z

Newtonian part

+ τ pI| z

Contact stress

DP (µ) rheology

σtot = ηε|z

Newtonian part

+ τ pI| z

Contact stress



8

<

:

τ = µpDev ε



Dev ε

τ

Normal stress p

?

Observation

Pressure and stress elds in a granular medium around an

intruder moving left to right

Seguin et al.

DP (µ) rheology

σtot = ηε|z

Newtonian part

+ τ pI| z

Contact stress


8

<

:

τ = µpDev ε



Dev ε

τ

Normal stress p

r · u ? p r · u

p

Analogy with discrete Signorini-Coulomb law

Let λ := pI τ (opposite of contact stress);

Dev ε

|Devλ| = µTrλ

Tr ε

Trλ



Dev ε

|Devλ| = µTrλ

Tr ε

Trλ

(

λ f (contact force)

ε u (relative velocity)



Dev ε

|Devλ| = µTrλ

Tr ε

Trλ

With χ orthonormal mapping for the norms k · k and | · |:

(λ; ε) 2 DP

r

d

µ

!

() (χ(λ);χ(Dev ε)) 2 C (µ)

=) we can leverage the same numerical solvers

Creeping ow

Dimensionless equations

I Characteristic velocity and pressure U :=p

gL etP := ρgL.

• u := Uu and λ := Pλ

I Dimensionless number Re := ρULη

Rer · [D (u)] +r · λ = eg on Ω

u = uD on BD

ru.nΩ = on BN

Flow around obstacle

.

.

.

.

.

|u|

-. -. .

.

.

.

.

.

p

-. -. .

Silo

.

.

.

.

.

|u|

-. -. -. -. -. .

.

.

.

.

.

p

-. -. -. -. -. .

Validation: Beverloo scaling(Using timestepping scheme for momentum advection)

. .

Outlet diameter L

Dischargerate

Qµ = µ = . µ = . µ(I)

Outline


Continuous model

Results

D Simulations


Time discretization


Results

Concluding remarks

Distinct regimes

φ(x, t) volume fraction eld: fraction of space occupied by the

grains.

φ < φmax : Gaseous regime, energy dissipation through

random collisions

Distinct regimes


grains.

φ = φmax : Frictional regime, pressure-dependent yield stress.

Distinct regimes


grains.


I Below the yield stress: solid regime

Distinct regimes


grains.


I Below the yield stress: solid regime

I At the yield stress: liquid regime

Drucker–Prager viscoplastic rheology

σtot = ηε|z

Newtonian part

+ τ pI,| z

Contact stress

Conservation equations

Conservation of mass

Dφ

Dt+ φr · [u] =

Conservation of momentum

ρφDu

Dtr ·

2

64φ (ηε+ τ pI)

| z

σtot

3

75 = ρφg

Time discretization

Semi-implicit integration

For each timestep ∆t

(i) Solve momentum balance using the current volumefraction eld φ(t) so that the rheology constraintshold at the end of the timestep

• Get u(t+∆t), p(t+∆t), τ (t+∆t)

Time discretization

Semi-implicit integration

For each timestep ∆t

(i) Solve momentum balance using the current volumefraction eld φ(t) so that the rheology constraintshold at the end of the timestep

• Get u(t+∆t), p(t+∆t), τ (t+∆t)

(ii) Solve the mass conservation equation using thenewly computed velocity eld u(t+∆t) to getφ(t+∆t)

• Hybrid method: move particles• We use APIC: Ane Particle-in-Cell [Jiang et al. ]

Spatial discretization

We must restrict ourselves to a limited number of degrees of

freedom for:

I Scalar volume fraction eld

I Vector velocity eld

I Symmetric tensor stress and strain eld


Particle-based

methods

(e.g. SPH)[Alduán & Otaduy

]

Mesh-based

methods

(e.g. FEM)[Ionescu et al. ]


Hybrid methods: Particles for material state + mesh for

velocities

[Zhu and

Bridson

]

[Narain et al.

]


Hybrid methods: Particles for material state + mesh for

velocities

[Zhu and

Bridson

]

[Narain et al.

]

[Klar et al. ]

We use the Material Point Method (MPM)

MPM for sand

I Computational mechanics: [Wieckowski ],

[Dunatunga et al. ], . . .

I Graphics: [Klar et al. ], [Tampubolon et al. ], [Hu

et al. ], [Gao et al. ], [Yue et al. ], [Gao et al.

], . . .

I See next session

MPM for sand

I Computational mechanics: [Wieckowski ],

[Dunatunga et al. ], . . .

I Graphics: [Klar et al. ], [Tampubolon et al. ], [Hu

et al. ], [Gao et al. ], [Yue et al. ], [Gao et al.

], . . .

I See next session

Peculiarities of our approach:

I Purely inelastic

I Implicit regime switching

MPM: Principle

Volume fraction eld φ discretized as a sum of Dirac point

masses:

φ(x) =X

p

Vp|z

Particle volume

δ(x− xp|z

Particle position

)

Integration over the simulation domain Ω:

Z

Ω

φv =X

p

Vpv(xp)

MPM: Principle

Volume fraction eld φ discretized as a sum of Dirac point

masses:

φ(x) =X

p

Vp|z

Particle volume

δ(x− xp|z

Particle position

)

Integration over the simulation domain Ω:

Z

Ω

φv =X

p

Vpv(xp)

Interpretation: xp ∼ quadrature points and Vp corresponding

weights. Not grains!

MPM: Application

Weak momentum balance

ρ

∆t

φur · [φ (ηD (u) λ)] = ρφf

MPM: Application


ρ

∆t


FEM: multiplying by a test function v and integrating over Ω +

Green formula:Z

Ω

ρ

∆t

φu.v+

Z

Ω

φ (ηD (u) λ) : D (v) =

Z

Ω

ρφf.v 8v

MPM: Application


ρ

∆t


FEM: multiplying by a test function v and integrating over Ω +

Green formula:Z

Ω

ρ

∆t

φu.v+

Z

Ω

φ (ηD (u) λ) : D (v) =

Z

Ω

ρφf.v 8v

MPM: φ(x) =P

p Vpδ(x xp)

P

p Vp

ρ

∆tu.v+ ηD (u) : D (v)

(xp)P

p Vp(λ : D (v))(xp)

= ρX

p

Vp(f.v)(xp) 8v

Basis functions

We still need to discretize u (velocity, vector) and λ and τ

(stress / strain, tensors) using a nite number of degrees of

freedom (grid nodes).

u(x) =X

i

Nv

i(x)ui, ui = u(yi), (yi)degrees of freedom

yi yi+yi− yi yi+yi− yi−

yi+

yi yi+yi−

Basis functions

We still need to discretize u (velocity, vector) and λ and τ

(stress / strain, tensors) using a nite number of degrees of

freedom (grid nodes).

u(x) =X

i

Nv

i(x)ui, ui = u(yi), (yi)degrees of freedom

yi yi+yi− yi yi+yi− yi−

yi+

yi yi+yi−

I Aects well-posedness of the numerical system,

spatial convergence and computational performance

I May create visual artifacts

Discrete System

Concatenating all unknown components and writing

constraints at stress quadrature points leads to

Au = f + B>λ (Momentum balance)

γ = Bu+ k (Strain from velocity)

(γi,λi) ∈ DP (µ) ∀i = . . .n (Strain–stress relationship)

I Similar to discrete contact mechanics with Coulombfriction

• . . . in dimension

• . . .A− dense for non-zero Newtonian viscosity

I In practice: Matrix-free Gauss–Seidel solver with

Fischer-Burmeister local solver ( NSCD)

Test case: collapse of a granular column

Silo

Constant discharge rate

.

t

V

V

µ =

µ = .

µ = .

Rigid-body coupling

Outline


Continuous model

Results

D Simulations


Time discretization


Results

Concluding remarks

Conclusion

I Qualitatively retrieves granular features

I Stable simulations at reasonable cost

• Avoids having to simulate elasticity timescale

Perspectives

I Explore other shape functions

• Volume preservation• Visual artifacts in degenerate cases

I Interactions with surrounding uid

Thank you for your attention

Many thanks for the insightful discussions:

I Florence Bertails-Descoubes

I Pierre-Yves Lagrée

I Pierre Saramito

This work has been partially supported by the LabEx PERSYVAL-Lab

(ANR--LABX--) funded by the French program Investissement

d’avenir

Time for questions?

comparisons

-2D

-RNSP Multilayer/

-1D (integral)

compared experiments, discrete and continuum simulationshttp://basilisk.fr/sandbox/M1EMN/Exemples/column_SCC.c

http://basilisk.fr/sandbox/M1EMN/Exemples/granular_sandglass_muw.chttp://basilisk.fr/sandbox/M1EMN/Exemples/granular_sandglass.chttp://basilisk.fr/sandbox/M1EMN/Exemples/granular_column_muw.c

the model for the pertinent level of simplification

http://basilisk.fr/sandbox/M1EMN/Exemples/

viscous_collapse_ML.chttp://basilisk.fr/sandbox/M1EMN/Exemples/

bingham_collapse_ML.c

http://basilisk.fr/sandbox/M1EMN/Exemples/bingham_collapse_noSV.c

Conclusion: a simple class of non newtonian

flows solved with Basilisk

+ shallow water Savage Hutter on the webhttp://basilisk.fr/sandbox/M1EMN/Exemples/front_poul_ed.c

granular as a continuum - project.inria.fr · figure 5: rescaled granular proﬁles: comparison...

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