particle simulations benjamin glasser. particle simulations 2 overview physics of a collision...

Particle Simulations


Benjamin Glasser

Particle Simulations 2

Overview

• Physics of a collision– Experimental perspective

• Instantaneous collisions• Sustained contacts

• Particle Simulations– Hard particle models

• Event-driven

– Soft particle models• Time stepping

• Continuum models

http://www.ux1.eiu.edu/~cfadd/1150/07Mom/Images/NewtPend.gif

http://www.ux1.eiu.edu/~cfadd/1150/07Mom/Images/boing.gif


Why?

• To better understand, control, and optimize– Fluidization processes

• Fluid bed reactors• Catalyst manufacture

– Solids handling operations• Powder mixing• Hopper flow

– Geophysical flows• Avalanches• Mudslides

– Geophysical formations• Sand dunes• Martian topography

http://www.rrdc.com/images/ph_peru_rockslide_lrg.jpg

http://photojournal.jpl.nasa.gov/jpegMod/PIA02405_modest.jpg


History

• Leonardo da Vinci (1452-1519) – first to device a simple and convincing experiment demonstrating dry friction.

• Charles de Coulomb (1736-1806) – Coulomb laws of dry friction between solids – would be extended to granular

materials.• Michael Faraday (1791-1872)

– examined how vibrations affect sand piles.• William Rankine (1820-1872)

– examined friction in granular materials.• H. Jannsen (1880’s)

– model of stresses in silos (granular material in a cylinder)• Lord Rayleigh (1842-1919)

– further work on stresses in containers• Osborne Reynolds (1842-1912)

– dilatency – expansion of material during shear• Ralph Bagnold (1950’s)

– sand dunes, role of particle-particle interactions vs. fluid-particle interactions


Benefits

• Ability to see “inside” granular flows• Relatively cheap• Permit theoretical investigations• Investigate transitions between fluid-like and solid-like behavior• Safer to run computer simulations• Validate granular experiments• Trace every particle

– Part of a force chain?• Versatility to be used for similar systems• Quick answers

– Industry pleasing• Manipulate parameters

– Coefficient of restitution– Coefficient of friction

Poschel and Scwager. Computational Granular Dynamics. Springer, New York, 2005.

http://www.nature.com/nature/journal/v435/n7045/images/4351041a-f1.2.jpg


A Simple Scenario

Two particles approach one another with known initial velocity in a frontal (normal) impact v1

m1 m2

v2

Before:

W. Goldsmith, Impact: The Theory of Physical Behavior of Colliding Solids, 1960

m1

u1

m2

u2

After:

22112211 umumvmvm

2 unknowns (u1 and u2)

Conservation of momentum!

Require another equation


A Simple Scenario

If kinetic energy were conserved (elastic spheres):

222

211

222

211 2

1

2

1

2

1

2

1umumvmvm

221

21

21

211

2v

mm

mv

mm

mmu

221

121

21

12 v

mm

mmv

mm

mu

and

Then:

However, energy is not conserved

inelastic collisions


Inelastic Collisions

• Initial velocity is v, rebound velocity is –v• Unique to granular materials

• Why?– Permanent deformation

• Microcracks• Deformed surface

– Acoustic Waves• Dissipated through heat http://

www.mathworks.com/access/helpdesk/help/techdoc/math/ballode.gif


Coefficient of Normal Restitution

• ε is the coefficient of normal restitution•

• Ratio of pre-collisional to post-collisional velocities

• Change in Kinetic Energy

• Function of approaching velocity

• Common values:

1

21

21

vv

uu

221

212

1vvmEkin

Glass spheres 100 cm/s 0.95 - 9

Steel spheres 100 cm/s 0.91 - 0.95

Brass spheres 100 cm/s 0.9

Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.


Matrix Equations

• Reference: System center of gravity– Particle - Particle

– Particle – Wall (infinite mass, rigid body)

2

1

2

1

2

1

2

12

1

2

1

v

v

u

u

1

0

1

0

1

01

v

v

u

u

u0: velocity of wall

u1: particle velocity



Normal Collision - No Friction

Only translational motion x and y are components of linear

momentum• ux = -vx

•

• uy = vy

• 1 = 0

Xvum xx

x

y


Spin and Friction

Spherical, Spinning Ball - Vertical Wall Precollisional velocities

• vx

• vy

• 0

Compute:• ux

• uy

• 1

x

y

x

y


Normal Collision - Friction

Case I: Gliding velocity remains positive

and non-zero Xvum xx

Xvum yy

Xama 012

5

2

xx vu

: coefficient of friction between the ball and the wall

Can distinguish between two cases based on

x

y

x

y

v

av

01

2

7

True when:

Coulomb

Rotation

4 Equations, 4 Unknowns (X, ux, uy, 1)

a: radius



Normal Collision - Friction

Case II: Gliding velocity drops to 0 during the

collision

aYma 012

5

2

Xvum xx

Yvum yy

xx vu 01 au y

x

y

x

y

v

av

01

2

7

True when:

Rotation

Pure rotation


Contribution from spin

Non-Frontal Collision with Friction

• Two particles– Diameter d1 and d2

– Mass m1 and m2

– Unit vector normal to contact•

– Relative velocity at point of contact•

21

21

rr

rrn

ndd

vvvc

2

21

121 22

Relative linear velocity

The magnitude of the relative velocity |vc| increases when the individual velocities point in opposite directions and when the rotations are in the same direction

12

v2

v1

n

1

2


Tangential Velocity - Rotational Motion

Normal component of vc

• Tangential component of vc

• Tangential unit vector

•

Angle of impact• From normal vector n to

relative velocity vc

•

nnvv cnc

ncc

tc vvv

tc

tc

v

vt

Normal collision: =

Glancing collision: = /2

vc

n

vc,, n

12

v2

v1

vc

n

2


Momentum Change

• Linear change of momentum from a collision

• Tangential contribution to angular momentum (torque)– Normal component of P has no contribution

for i=1,2

– Ii: moment of inertia of particle i

– The change in angular momentum is the same for both particles

222111 vumvumP

iii

i

d

IPn '2


Outcome of the CollisionMaking use of the equations, one can compute the outcome of the collision

Linear Velocity

PnI

d

1

11

'1 2

PnI

d

2

22

'2 2

Angular Velocity

111 m

Pvu

222 m

Pvu

nc

nc vu

ndd

vvvc

2

21

121 22

Where:

nc

n vmP 112

21

2112 mm

mmm

cos112 tvmP ct

21

21

rr

rrn

nt PPP The translation velocities

before and after the

collision are still related by

this


Rolling

• The previous equations describe all collisions on the basis of Coulomb’s Law () and and nothing else

• Ignored an important physical mechanism: rolling– Heuristic model to agree with experiments

– Fire two spheres together with initial spin and examine the outcomes

– Coefficient of tangential restitution, • Equal to the smallest of two values:

• 0 – Rolling; 1 – Sliding/Gliding

1 2


Critical Angle – 0

Applies for this form of the momentum equation

tc

nc vmvmP 1

7

21 1212

0

0 1

1

2

7tan

Applies for this form of the momentum equation

For < 0:Sliding/Gliding regime

1,10

cot12

711

S.F. Forrester et al., Phys. Fluids, 6, p.1108 (1994)

Small values of correspond to dry friction

(previous result)

For > 0:Rolling regime


Sustained ContactsWhat about particles that are not completely rigid?

Spheres can deform Can have interpenetration of the spheres, resulting in long contact times

Consider 2 identical spheres– Mass, M– Radius, R relative velocity v

Fan and Zhu, Principles of Gas-Solid Flow, 1998

Hertz’s elastic energy

2

5

2

1 kEe

where

RE

k2115

24

E – Young’s modulus – Poison’s ratio

R R

Stored by each sphere during contact

Ratio of transverse strain to longitudinal strain



Sustained Contact Quantities

• Upon impact the kinetic energy is converted into a reduced kinetic energy and stored elastic energy– Energy balance:

2

2

52

dt

dmkmv

• Velocity drops to zero when the two spheres have overlapped by a distance 0

5

45

2

0 vk

m

• The entire duration of the collision (up to 0 and recoil)

0

0 2

52

2

mk

v

d 5

1

2

2

94.2

vk

m

0dt

d

Important: only depends weakly on v – exponent is 1/5. Thus the duration of impact is only a weak function of the initial relative velocity


Example Calculation

Consider aluminum beads ( = 2.70 g/cc) 1.5 mm in diameter moving towards one another at 5 cm/s.

E = 6 x 1011 dynes/cm2

= 0.3What is the duration of contact?

D=1.5 mm

Answer:

k = 7 x 1010 dynes/cm3/2RE

k2115

24

3

3

4Rm m = 4.77 x 10-3 g

5

1

2

2

94.2

vk

m = 1.15 x 10-5 s


Issues

• Exceed elasticity limit– Plastic, not elastic deformations– Model can be adjusted to handle this

• Energy dissipation– Sound waves– Heat

• Spin complications– Can handle similar to rigid spheres



• Follow trajectories of individual particles– Incorporate statics and dynamics

• Methods– Particle dynamics

• Hard Particles• Soft Particles

– Cellular automata• Motion evolves according to simple rules based on lattice

sites– Monte-Carlo

• Analogous to molecules but change probabilities to match particles

• Assumption of molecular chaos

Baxter and Behringer (1990) Cellular Automata of Granular Flows. Phys. Review A., 42, 1017-1020

Rosato et al. (1987) Monte Carlo simulation of particulate matter segregation. Powder Technology, 49, 59-69


Source: Thorsten Poschel, Thomas Schwager, Computational Granular Dynamics, Springer, 2005.


Boundary Conditions

• Wall constructed from individual particles• Containers do not follow Newton’s

equation of motion– Predetermined path as a function of time

• Vibrated bed• Moving plate• Rotating vessel

• Periodic boundaries– Can mimic infinitely-wide regions


Initial Conditions

• Depending on the algorithm (predictor-corrector), you may need to define higher order derivatives

• Most long-term behavior is independent of initial conditions

• Often, random (non-overlapping) positions and velocities are assigned


Hard Particles

• Event Driven (ED)• Strictly binary collisions• No integration of the equations of motion

– More efficient

• Without gravity – straight line paths• With gravity – parabolas• Time to collision is identical

With gravityWithout gravity

Collision at (xc, yc + 1/2gt2)Collision at (xc, yc)

v10 v2

0


di

Particle Motion

• Consider a particle– Position vector xi

– Velocity vector vi

– Initial position (t=0) • xi = xi

0

• vi = vi0

– Undeterred position at time t200

2

1attvxx iii

x

yxi

0 = (x, y)vi

0 = (vx, vy)


Collision Prediction

Two particles collide if:

i

jdj/2

di/2

22ji

ji

ddxx

Insert the equation of motion:(same force on each particle, so the accelerations are

equal as well and cancel)

22 0000 ji

jiji

ddtvvxx

2

22000000200

22 2

ji

jijijiji

ddtvvtvvxxxx

Expand : Real root > 0 collision


Scheduling• Can schedule the events for N particles in a box

– 4 walls of a box 4N events– N particles N(N-1) events

• An example stack from t=0

• March to t = 0.1, execute the collision, then recalculate the stack– The whole stack, or– Just events with particles 5 or 7 (much faster)

time wall event particle event type

0.1 no yes particle 5 and 7

0.25 yes no particle 3 and wall 1

0.33 no yes particle 2 and 5

…


Gridding

Predicting collisions between all particles wastes time

• Black arrows will rarely collide

Divide region into cells• Search within cells for collisions• Include cell crossings as events• Track the cell location of each

particle


Inelastic collapse

Left and right particle collide with the middle particles alternately until the motion of all three particles is zero

Only for a constant coefficient of restitution– Not valid for real materials– Experiments suggest is a function of v

Poschel and Scwager. Computational Granular Dynamics. Springer, New York, 2005.


2D - Couette Flow

U

-U


3D - Couette Flow


Soft Particles

• Force-based• Time stepping• Small overlap allowed• Useful for

– Statics– Dense quasi-static flows

• To follow particles:

F is the normal force on particles proportional to amount of overlap

m

Ftvv ii '

''iii tvxx


Advanced Algorithms

• Verlet Algorithm– Position from acceleration

42)()()(2 tOttattrtrttr

?

program termination

data output

initialization

predictor

force computation

corrector

Poschel (2005) Computational Granular Dynamics. Springer, New York.

• Predictor – Corrector (left)– Predict future acceleration using

previous position time derivatives– Acceleration from force– Adjust the predicted value

• Acceleration from force

maF

– Back out velocity

t

ttrttrtv

2

)(


Contact Models

• Spring and Dashpot

0 jijiij rrRR

nn

nij kF

– may be related to kn and n

• Mutual compression of particles i and j

otherwise 0

0 if ijt

ijn

ijij

FFF



Force Calculation

• In addition to Fn

– Gravity– Wall forces– Interstitial fluid– Spin– Cohesion

Total of all is the resultant force on the particle, F

Fw

Fg

F2

F3


More Contact Models

Hertz2/3n

nij kF

2/12/3nn

nij kF

)(unloading 0 ,

(loading) 0 ,

02

1

k

kF n

ij

Kuwabara and Kono

Walton and Braun (right)

Can also include friction in the mechanism



Discrete ModelsShape Issues:

Spheres

Collections of spheres

Arbitrary surfaces

Simplest

Most Complex

Needles

Flakes


Results: Constant inter-particle force Model

• Optimized Condition: 20,000 particles, 2mm diameter, in an axially smaller rotating drum of 9 cm radius and 1 cm length are considered. The sidewalls are made frictionless to avoid end wall effects.

• Inter-particle or particle-wall cohesive force is varied such that K=45-75

Fast-Flo Lactose

Size: 100 micron

RPM = 7 K = 45 ; RPM = 20

sp = 0.8 ; dp = 0.1 ;

sw = 0.5 dw = 0.5 ;

• Avalanches start appearing at K = 30 and become bigger at K =45.

• Distinct angles of repose are visible at top and bottom of “cascade” layer.

Credit: F. Muzzio


Similarities:Increase in size of avalanches.Mixture of splashing and timid bulldozing.

Avicel-101; Size : 50 m; RPM = 7

Results: Constant inter-particle force Model

Dynamic friction within the particles and the cohesion are increased to simulate the flow of more cohesive material. Wall friction is increased.

Reg. Lactose; Size : 60 m; RPM = 7

K = 75 ; RPM = 20

sp = 0.8 ; dp = 0.6 ;

sw = 0.8 ; dw = 0.8

K = 60 ; RPM = 20

sp = 0.8 ; dp = 0.6 ;

sw = 0.8 ; dw = 0.8

Similarities:Mixture of chugging and bulldozing. Periodic AvalanchesBigger Avalanches

Credit: F. Muzzio


Uniform Binary SystemComparison of model and experiment

• Optimized Condition: 10,000 red and 10,000 green particles of same size (radius: 1mm) are loaded side by side along the axis of the drum. The drum of radius 9 cm and length of 1 cm is considered. The sidewalls are made frictionless to avoid wall effects.

• Inter-particle or particle-wall cohesive force is varied such that K=0 – 120Glass beads (40 m)

RPM=10

K=0

RPM = 20

•No avalanches. •Mixes well in 3-4 revolutions

•Avalanches appearing•Slower mixing

K=60

RPM = 20

Colored Avicel (50 m)

RPM=10

Credit: F. Muzzio


Non-uniform Binary System

Experiment: Blue (30 mm) and Red glass beads (50 mm) of equal mass are axially loaded side by side in a drum of radius = 7.5 cm and length 30 cm.

Simulation: 8000 blue particles (1mm) and 2370 red particles(1.5mm) of same density are loaded side by side along the axis of the drum. Red and blue particles of are of the same total mass.

Inter-particle Force ModelFRR = KRR WB (red-red pair)FRB = KRB WB (red-blue pair)FBB = KBB WB (blue-blue pair)WB is the weight of a blue particle.

Non-cohesive binary mixture : Axial Size Segregation is evident in both the simulation and experiments.

RPM =12 RPM =20

(K.M.Hill et.al Phys. Rev. E.,49,1994).Credit: F. Muzzio


Example –A Particle/Wall Contact

2D Disk - Flat, Vertical Wall• R = 0.5 mm

• kn = 10 N mm-3/2

• t = 0.25 s• m = 0.148 kg

• vp0 = (vx

0, vy0) = (1 mm s-1, 0 mm s-1)

• xp0 = (xx

0, xy0) = (1.2 mm, 1 mm)

• xw = 0 (line from origin to +∞)

xy

xy


Results

2/3nn

ij kF

0 jijiij rrRR

maF

Force

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

Overlap

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Overlap Rate

-1.2-1

-0.8-0.6-0.4-0.2

00.20.40.6

0 0.5 1 1.5 2

Time (s)

Collision!!

tavvii 0

tvxx ii i 0

Position

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Velocity

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Acceleration

00.5

11.5

22.5

33.5

44.5

0 0.5 1 1.5 2

Time (s)

Wall


Hard vs. Soft

• When to use hard (event) instead of soft (force)– Average collision duration <<

Time between collisions• Granular gases cosmic dust

clouds

– Unknown interaction force• Non-linear materials• Complicated particle shapes• Can experimentally determine pre- and post-collisional

velocities



Drawbacks

• Computationally expensive– t << to calculate forces– 20,000 particles

• Real time of seconds to minutes

• Dynamic issues– Strain hardening– Contact erosion over time


Continuum

• Discrete particles replaced (averaged out) with continuous medium

• Quantities such as velocity and density are assumed to be smooth functions of position and time

• Volume element (dv) contains multiple particles• Time (dt) should be large compared to time

required for a particle to cross dv

Truesdall, C. and Muncaster, R.G. (1980) – Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatonic Gas. Academic Press, pp. xvi.


Continuum

Stay in this region where the average quantities are equal to the bulk

10-10 10-6 10-5 10-4 10-3 10-2 10-110-710-810-9

= mass/volume

l (in meters)


Continuum Simulation for Granular Flows

Consider a mass balance over a stationary volume element of size ΔxΔyΔz. Rate of mass accumulation = rate of mass in – rate of mass out

]|)(|)[(

]|)(|)[(

]|)(|)[(

zzzszzs

yyysyys

xxxsxxss

vuvuyx

vuvuzx

vuvuzyt

vzyx

Δx Δy

ΔzGranular Flow

Dividing by ΔxΔyΔz and taking limits, we obtain

z

vu

y

vu

x

vu

t

v zsysxss

or )( vut

vs

We can extend this approach to calculate momentum and energy balance. Particle Dynamic Simulations – Limited by computational resources Continuum Simulations with physically realistic closures (Use a set of equations for design, control and optimization) More efficient design, scaling & control through Continuum modeling

Average over a small region in a granular flow.


Hydrodynamic Model

Boundary Conditions (Johnson and Jackson, 1987)

Force Balance

Energy Balance

0)(

uvt

v

gvvut

uv sss

][

Momentum Balance

Pseudo-Thermal Energy Balance

uquTv

t

vTcss

s

:)2

3(

)23

( ''

3

1vvm

Mass Balance

0))/(1(6

||3'

|| 3/1maxmax

2/1

vvv

uvT

u

nusls

sl

ssl

0])/(1[

)3()]1(

4

1[

3/1maxmax

2/12

sslws u

vvv

TveTqn

u solids velocityv solids fractionT granular temperature


Computational Approaches

Steady State Simulations

time

1 Steady state solution is augmented by a small perturbation.The small perturbation has a periodic form.

s = - i determines the rate of growth or decay of the perturbation waves.

)exp()exp()(1̂1 xiksty x

Linear Stability Analysis

Transient Integration on Linear Instabilities

Direct Integration

Bifurcation Analysis

Bubble formation in a gas-particle system.

( Anderson et al. 1995)


Steady State Solutions for Couette Flows

(Nott et al. 1999)

The structure of fully developed solutions (a) Particle volume fraction, (b) pseudo-thermal temperature, (c) axial velocity.

Broken lines: wall is a source of energy.Solid lines: wall is a sink of energy.

T*


Linear Stability for Couette Flows

(Alam and Nott, 1998)

Eigenfunctions Associated with Solids Fractions

Instability dominated by symmetric patterns

Instability dominated by anti-symmetric patterns


Transient Integration in Couette Flows

(Wang and Tong 1998)


Couette Flow with Binary Particles (Steady State)

A: large/heavy particles

B: small/light particles

(R=dA/dB M=mA/mB)

y

x

u0

-u0

Non-uniform Solids DistributionSpecies SegregationNon-linear Velocity Distribution

(Equal Density: R=2, ep=0.9 Mean solids fraction=0.1)


Steady State Solutions for Channel Flows

(Wang et al. 1997)

u* dimensionless velocityT* dimensionless granular temperaturev solids fraction

Solid line: wall is a sink of energy.Broken line: wall is a source of energy.


Linear Stability for Channel Flows

Instability dominated by symmetric patterns

Instability dominated by anti-symmetric patterns

(Wang et al. 1997)

Eigenfunctions Associated with Solids Fractions


Transient Integration for symmetric patterns

Transient Integration for anti-symmetric patterns

(Wang and Tong 2001)

Solids Fraction Distribution in the Channel

Transient Integration in Channel Flows


Channel Flow with Binary Particles (Steady State)

A: large/heavy particles

B: small/light particles

(R=dA/dB M=mA/mB)

y

x

(Equal Density: R=2, ep=0.9 Mean solids fraction=0.1)

particle simulations benjamin glasser. particle simulations 2 overview physics of a collision...

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