event driven programming - eth z...event driven programming 1 as opposed to „batch“ programming...

Event driven programming

1

As opposed to „batch“ programming

the flow of the programme is not

determined by loops but by events

and therefore has branching points

and conditional logic.

Paradigma in Computer Science and

important for GUI and Java.

Event driven Molecular Dynamics

B.J. Alder and T.E. Wainwright (1957)

Consider rigid objects of finite volume (e.g. billiard

balls) which normally one would like to describe

by a hard core potential which in classical MD

cannot be handled because of the infinite forces.

In „event driven“ simulations the collisions between

particles are considered as instantaneous events

and between them particles do not interact.

Event driven MD

Event driven MD

In this method no forces are calculated. Only

binary collisions are considered, i.e. interactions

between three or more particles are neglected.

Between two collision events the particles follow

ballistic trajectories. One needs to calculate the

time tc between two collisions and then obtain

the velocities of the two particles after the collision

from the velocities of the particles before the

collision from a look-up table.

The collision event (2d)

ij i jv v v

ij i jr r r

Consider the collision of two rigid disks i and j:

The „collision angle“ θ is the angle between

and relative

velocity

t

Collision time tc

0

( )

( )

ij ij i j

ij ij ij i j

r t R R

r t v t R R

2 22 2 0 02 0ij ij ij ij ij ij i jv t r v t r R R

min ( )c ij ijt t

Calculate for each pair of

particles (i, j) the time tij when

the next collision will occur.

The time tc when the next collision occurs in the

system is then the minimum over all pairs (i, j):

occuring for pair (i*, j*).

t0 is the time at which the last collision occurred and we set rij0 rij(t0) .

Propagation step

Due to the global minimization the algorithm

is not easily parallelizable nor vectorizable.

Once tc has been determined

one moves particles by:

! !i i i i c i i cr r v t t i and

Then the collision between the pair (i*, j*) occurs.

Propagation step

Knowing position and angle of a particle i

at a time t*, since which it had no collision, one

can determine its position at a later time as:

2* * *

* *

1

2

i i i z

i i i

r t r t v t t g e t t

t t t t i

One can also add simple accelerations, like gravity g.

Identifying an event

Unfortunately the loop to

calculate tc is of order N 2 when

simply checking all pairs.

Tricks due to Lubachevsky (1992)

allow to reduce this to order N log(N).

The idea behind is based on lists of events

and binary stacks.

Boris D. Lubachevsky

Lubachevsky trick

Store additionally to the particle position and

velocity for each particle the last event and the next

event, i.e. keeping track of the time of the event and

the partner particle involved in the event.

In practice, this can be implemented in 6

arrays of dimension N (number of particles in the

system). Alternatively, one creates a list of pointers

pointing to a data structure for each particle

consisting of 6 variables.

Note: Same time tij may occur twice in t (i) (i = 1..N).

From this we get for the next collision in the system:

Lubachevsky trickStoring the last event is needed as particles are only

updated after being involved in an event. For each

particle i the time t (i) is the minimal time of all

possible collisions involving this particle:( ) min ( )i

j ijt t

( )min ( )ic it t

Straightforward strategy would require order O(N).

Lubachevsky trick Organize the t (i) in increasing order implicitely in a stack:

• The vector part[m] points to particle i which is at the position m in the stack. (Sometimes also a vector pos[i] is stored giving position m of particle i in the stack.)

• This constitutes an implicit ordering of the collision times t (i) , where m = 1 points to the smallest time.

• part[1] is particle with minimal collision time:tc= t (part[1]) .

• After the event for both particles all 6 entries (event times, new partners, positions and velocities) have to be updated. Additionally, the vector part[m] has to be re-ordered.

Lubachevsky trick

Straightforward calculation of the new collision

times would be of order O(N) when checking for

both particles with all other particles. Collision

lists for each particle or division into „sectors“

(cells) reduce the order to O(1) per event. These sector boundaries have to be treated similaras obstacles, i.e. when particles cross sector boundaries a „collision event“ happens.

Lubachevsky trick

Re-ordering the heap after each event

is of order O(log N) when using,

e.g. binary trees for sorting

For sorting so called „tournament

trees“ can be used (M. Marin, 1993). For sorting algorithmssee also D. Knuth, „The Art of Computer Programming“.

Tournament tree

Example of tournament tree for 4 particles:

before collision after collision

Depth of the tree (log2N) determines maximum

number of comparisons for re-ordering.

Scaling with sizePerformance benchmark for fixed number of events

~ log (N)

Typically: number of events proportional to N

→ order of method N log(N).

Propagation step

Note: using the event lists,

once tc has been determined,

one moves only particles i*, j*

involved in the collision by:

! ! * *i ( , )i i i c i i cr r v t t i i j and

Then the collision between the pair (i*, j*)

occurs according to the collision rule.

Collision with perfect slip

, after before after beforen ni i j j

i j

p pv v v v

m m

Use energy and momentum conservation at collision.

From momentum conservation we obtain:

Insert into energy conservation:

2 2 2 21 1 1 1

2 2 2 2before before after after

i i j j i i j jm v m v m v m v


222 2

22 2

1 1 1 1

2 2 2 2

1 1

2 2

2 2

before before before beforen ni i j j i i i j

i j

before before before beforen n ni i i j j j

i i j

p pm v m v m v m v

m m

p p p pm v v m v v

m m m

2

22

1 1

2 2 0 2 2

2 2

n

j

before beforen n n ni i j j

i i j j

before beforeni i j j

i

m

p p p pm v m v

m m m m

p pm v m v

m

2 2

n nn

j i j

p p

m m m


i jeff

i j

m mm

m m

2 before beforen eff i jp m v v n n

Define „effective mass“:

Use the fact that the momentum change must be

parallel to the normal direction n and therefore

multiply this equation on both sides with n.


for collision between particles i and j:

2

2

jafter before before beforei i i j

i j

after before before beforeij j i j

i j

mv v v v n n

m m

mv v v v n n

m m

after before before beforei i i j

after before before beforej j i j

v v v v n n

v v v v n n

for mi =mj


for collision between particles i and j

make look-up table

and mi =mj :

after before ni i ij

after before nj j ij

v v v n

v v v n

Event driven simulation

perfect slip

Collision with rotation

' '

' '

' '

i i i i

j j j j

i j i j

I Rm v v n

I Rm v v n

v v v v

' ' 'i i j j i i

Rmv v n

I

i iij

d dvI r f mr

dt dt

Eq. of motion for rotation:

Consider collision

between spheres i

and j having the same

radius R, moment of

inertia I and mass m.

Collision with general slip

nij ij

tij ij i j i j

u u n n

u u n v v R n

This means a condition for the relative velocity u

between the particles at their contact point.

't tij t iju e u

1te energy conserving:

general slip condition:

1 perfect slip

0 no slipt

t

e

e


1 '

' ' ' '

' ' ' '

t t tt ij ij ij

i j i j i j i j

i i j j i i j j

e u u u

v v R n v v R n

v v v v R n

tij ij i j i ju u n v v R n

't tij t iju e u

Consider only the tangential component:


1 ' ' ' 'tt ij i i j j i i j je u v v v v R n

' ' ' 'i i j j i i j j

Rm Rmv v n v v n

I I

2

1 2 ' 2 '

1' ,

2 1

1analogously: '

2 1

tt ij i i i i

tt ijt t

i i

tt ijt t

j j

e u v v q v v n

e u mRv v q

q I

e uv v

q


1 ' ' ' 'tt ij i i j j i i j je u v v v v R n

' ' ' 'i i j j i i j j

Rm Rmv v n v v n

I I

1

2

1

1

1 2 ' 2 '

1' ,

2 1

1analogously: '

2 1

tt ij i i i i

tt ij

i i

tt ij

j j

e u R q n

e u n mRq

IR q

e u n

R q


2eff

eff

m Rq

I

1

1

1'

2 1

1'

2 1

1'

2 1

1'

2 1

tt ijn

i i ij

tt ijn

j j ij

tt ij

i i

tt ij

j j

e uv v u

q

e uv v u

q

e u n

R q

e u n

R q

make look-up table

Inelastic collisions

Even collisions between billiard balls are inelastic.

Energy is dissipated through vibrations (sound)

and eventually also small plastic deformation or

heat production. Dissipation is quantified through

the material dependent „restitution coefficient“ r

which is the fraction of not dissipated energy after

a collision. One has r = 1 for elastic collisions and

r = 0 in case of perfect plasticity.


2afterafter finaln

n before initial beforen

vE hr r

E h v

The restitution coefficient r can be measured by

letting the particle fall from a height hinitial on a plate

of same material and measuring the rebounce

height hfinal:

, after aftern t

n n t tbefore beforen t

v ve r e r

v v

One also defines normal and tangential coefficients:


after after before beforej i n j iv v n e v v n

steel 0.93

aluminum 0.8

plastic 0.6

examples for en:

Inelastic normal collision

nij ij i ju u n n v v n n

The normal component of the relative velocity u

between the particles at their contact point:

'n nij n iju e u

1 ' ' '

' ' 2 ' 2 '

n n nn ij ij ij i j i j

i i j j i i j j

e u u u v v n n v v n n

v v v v n n v v n n v v n n

dissipation through normal coefficient of restitution en :

' 'i j i jv v v v

using momentum conservation:

Inelastic normal collision

1 2 ' 2 'nn ij i i j je u v v n n v v n n

1 1' , '

2 2n nn n

i i ij j j ij

e ev v u v v u

For perfect slip we get:

(1 ) before beforen eff n i jp m e v v n n

Inelastic collision with general slip

2eff

eff

m Rq

I

1

1

11'

2 2 1

11'

2 2 1

1'

2 1

1'

2 1

tt ijnn

i i ij

tt ijnn

j j ij

tt ij

i i

tt ij

j j

e uev v u

q

e uev v u

q

e u n

R q

e u n

R q

make look-up table

Inelastic collapse

Physical effectis the formation of clusters in a granular gas.

Inelastic collapse

If an inelastic sphere jumps on a plate it will perform in a finite time ttot an infinity of collisions.

Finite time singularity

1 1

2 2 12 =2 1

1

initial initialj

tot jj j

h ht t r

g g r

S. McNamara and W.R. Young (1994)

Finite time singularity

7 4 3r <The effect occurs for three particles if

→ ←↔

min

ln(1 )

1

rn

r

Minimum number of particles for

which the effect occurs if r 1:

PROBLEM FOR LARGE DENSITIES

S. McNamara and W.R. Young, Phys. Fluids A, 4, 496 (1992)

TC-model for ED

Real collisions are not instantanous,

but they have a certain duration.

Solution in ED: use coefficient of restitution

dependent on time elapsed since last event

(collision with another particle).

With this one can overcome finite time singularity.

S. Luding and S. McNamara (1998)

Stefan Luding

Sean McNamara

TC-model for ED

Simplest implementation: set coefficient of

restitution to one (perfectly elastic) if time t (i)

since last collision for both partners

is less than tcontact :

( ) ( )( , ) contact contactif or

1 otherwise

i ji j r t t t t

r

More complex dependency on t (i) can be used,

adjusted to the behavior of specific materials.

TC-model for EDCompare dissipation affected by tcontact:

soft particles hard particles (TC-model)

contactt contactt

TC-model for ED

During the contact between two particles

additional collisions with other particles

can occur (non binary collisions)

and then dissipation is smaller.

Also for soft particles it is known

that they dissipate less energy

when they are in contact with

other particles.

TC-model for ED

Sand swimming in a dense granular medium

by T. Shimada et al (2009)

Inelastic collisions with MD

efff kr r m r i j i jr R R x x

( ) sin( )before

tvr t t e

2 2 20 0 , ,

2eff eff

k

m m

Consider a one dimensional damped oscillator:

with

The solution is:

with

One can also introduce inelasticity in MD

with potentials and finite time steps Δt.

viscous damping

Inelastic collisions with MD

2

4

22 2

2 2

( )

( )

4 ln

2 lnln

effm k

n

eff n

effn

n

Tr t Te

r t

m k e

m ke

e

e e

The collision time T lasts a half period T=π/ω.

A viscous damping γ uniquely describes

the dissipation by a restitution coefficient en.

2 20

20

2

eff

eff

k

m

m

Granular interaction

Straightforward implementation

of damped harmonic oscillator

(„spring-dashpot“)

leads to artificial attraction.

Solution:

Allow for repulsive forces only.

Plastic deformation

elasticplastic

Plasticity involves a remnant deformation which corresponds effectively to a change in the radius.

Plastic deformation

O.R. Walton and L.Braun (1986)

force

displacement

plastic deformation

Stiffness largerwhen moving apart

Plastic deformation

1

hys2 0

for loading

for unloading

kf

k

1 22ij ij

c

m mt

k k

asymmetric collision time

green area is dissipated energy

max max

max max 0

1

2

1

2

1

2

before

after

after

before

E f

E f

kEr

E k

max max1 2

max max 0

, f f

k k

Calculate k1 and k2 from r and tc .

Coulomb friction

inclined chute

tant nF F

µ is frictioncoefficient

Coulomb friction

if

if

0

0

t s n t

t d n t

f f v

f f v

•

•

μμs

μd

vt

s d experimentally:

μd ≈ 0.1 – 0.5

static case

dynamic case

= friction coefficientstatic dynamic

Coulomb friction with ED

for

for

1 1

1 1

before beforet eff t i j s

before beforet d eff n i j s

p m e v v t v

p m e v v n v

The tangential restitution coefficient et ≈ ½en.

Tangential momentum transfer

due to dynamic Coulomb friction:vt

S. Luding: http://www2.msm.ctw.utwente.n/sluding/PAPERS/coll2p.pdf

Coulomb friction with MD

min , ( )

min , ( )

t t d n t

t s s n t

f v f sign v

f k f sign v

s s nk f

P.A. Cundall and O.D.L. Strack (1979)

DEM = Discrete Element Method

dynamic

static

When then spring is removed.

Put spring with spring constant ks .

Peter Cundall

Applications of DEM

P.A. Cundall: Founder of Itasca Consulting group

→ PFC3D (DEM-modelling tool)

Applications of DEM



Applications of DEM



Elliptic particles (2D)

Overlap algorithm for two ellipses:2 2 2 2

1 2 1 2

1, 1a a b bx x y y x x y y

a a b b

1a2a

1b

2bar

br

Orthonormal vectors

along major axes:

1 2 1 2, and ,u u v v

2 2,T Tk k k k k k

k k

A a u u B b v v

Rotated ellipses defined

by matrices A, B:↔ ↔

J. W. Perram andM. S. Wertheim

(1985)

A B


( ) ( ) ( )TA a aG r r r A r r

where

Ellipse A can be described by the function:

1 if is inside ellipse

( ) 1 if is on ellipse

1 if is outside ellipse A

r A

G r r A

r A

Find the global minimum over all r for fixed λ by


1

2 ( ) 2 1 ( ) 0

( ) 1 ( ) 1

( ) 1 1

m a m b

m m a b

m a b

A r r B r r

Ar Br Ar Br

r A B Ar Br

( , ) ( ) (1 ) ( )A BG r G r G r

Define for λ [0,1] the function :

( , ) 0r mG r

( ) ( ) ( )TA a aG r r r A r r


( ) 1 ( ) 1m m a bAr Br Ar Br

ab b ar r r

( ) 1 ( ) 1 1m m a ab aAr Br Ar Br Br

1 1 1

1 ( ) 1

1 ( ) 1

m a ab

m a ab

A B r r Br

B A r r A r

define:


This minimum rm(λ) describes a path from the center

of B to the center of A parametrized by λ [0,1]

11 1 1

11 1 1

( ) (1 ) (1 )

( ) (1 )

m a ab

m b ab

r r A A B r

r r B A B r

and analogously :

1 1 11 ( ) 1m a abB A r r A r


11 1 1

11 1 1

( ) (1 ) (1 )

( ) (1 )

m a ab

m b ab

r r A A B r

r r B A B r

( , ) ( ) ( ) (1 )( ) ( )T Ta a b bG r r r A r r r r B r r

2 22 1 1 1 2 1 1 1

1 1

( ), ( ) ( ) (1 )( ) ( )

1 (1 ) 1 (1 )

1 (1 ) (1 )

( ) T Tm m a m a m b m b

T Tab ab ab ab

Tab

G r r r A r r r r B r r

r A A B r r B A B r

r A B A

S

21 1

abB r

Inserting the minimum rm(λ) into G (r, λ) defines

the overlap function S(λ) :


One gets for the overlap function S(λ)

(non-negative for λ [0,1]):

11 1( ) (1 ) (1 ) ,T

ab ab ab b aS r A B r r r r

11 1 2( , ) (1 ) (1 ) ,T

m m r m m r ab ab rF A B e A B e r r re

Maximize S(λ) for λ [0,1] → λm , defines contact index:

overlap if F(A,B) < 1, tangent if F(A,B) = 1, no contact if F(A,B) > 1


Set F(A,B) to unity to find distance r* at which

the particles would just touch:

*

2r abr e r

1/ 21* 1 1(1 ) (1 )Tm m r m m rr e A B e

→ overlap vector:

Obtain contact point:

*pc pcr r


example:

Influence of elongation

on packing fraction

for elliptic particles

N. Berntsen, 2001



Superellipsoids (3D)G.W. Delaney, P.W. Cleary (2009), F. Alonso-Marroquín

Generalized ellipsoid:

a,b,c: major axes

shape parameter m

(m = 2: ellipsoid)

here: a = b = 1, c = α

→ elongated particles,

platelets, etc.

1m m m

x y z

a b c

Superellipsoids (3D)

G.W. Delaney,

P.W. Cleary (2009):

1125 particles to

study shape influence

on packing fraction.

Advantage:

good control of shape

by two parameters

α and m

ijF

ijF

jv

jv

1 ( )i i ix t x t v t t

1i ii

i

v t v t tm

F

1i i it t t t

1i ii

t t tI

iT

min ,ij

n t nij ij

Yn v n v F t

l

ijF

A

ij

m g

i ijF F i ijj

TT

MD for polygons

A = overlap area

i j

two dimensions

Polygonal particles

Area detected by using

simple geometry, e.g.

area S1AiS2Aj can be

decomposed in triangles.

Many different types of

contacts, when consider-

ing polygonal particles.

Some types have to be

treated separately.

Polygonal particles

Edge-edge contact: cut area has to be added

discontinuities can appear when particles move→ change in particle area is added,such that total area changes continuously.

Torque at contact

Particles can rotate against each other:

Torque acting against this rotation also depends on

size and shape of the contact area.

Polyhedra (3d)

A.V. Potapov, C. S. Campbell (1996):

Huge number of different contact types

approximately

12000

Delauney tetrahedra

Polyhedra (3d)

A.V. Potapov, C. S. Campbell (1996)

Fragmentation by impact

SpheropolygonsFernando Alonso-Marroquin, EPL, 83, 14001 (2008)

Minkowski sum

, ,

( , ) ( , )

,

if 0 ...

0 if 0

ij i j j ii j i j

i j V E

F F d V E F d V E

d V E r r x x

y y

y

Spheropolygons

polygon of 62 edges 726 disks

Spheropolygons

Dilation Technique

Mathematical Morphology (1964)

Jean Serra

simulating snow with cylindrical ice particles (=dilated lines) :

Dilation Technique

2 2 2

2 2 2

2 2 2

2 2 2

1

1

x y z

a b c

x y z

a R b R c R

example ellipsoids

2 2 1 1d p r p r

apply elastic band algorithm on:

Force at the end of d is the tangential component of a spring along d (by substracting the normal component). → d converges towards becoming the shortest distance.

Rocks fallingon a pileusing the dilatation technique of Serra (1986)

M.A. Hopkins

DEM simulation of heap

90

Constraint DynamicsConsider rigid objects of finite volume (e.g. billiard balls) which normally one would like to describe by a hard core potential which in classical MD cannot be handled because of the infinite forces.

But we would like to know the forces so that also event driven simulations are not suited.

Idea: Interactions fully determined by constraints!

Contact dynamics is typically used for granulates, rocks, machine parts, etc.

91

Contact Dynamics (CD)

Contact Dynamics treats rigid particles and is good

for long-lasting contacts as they occur in dense

packings. At contacts one imposes constraint forces

that prevent the penetration of particles.

Advantage: Δt not related to duration of collision

P. Lötstedt (1982)

J.J. Moreau and M. Jean (1992)

Jean Jacques Moreau

Per Lötstedt

92

Signorini-Graph: perfect volume exclusion (perfectly rigid particles)

Coulomb-Graph: friction between particles with relative tangential velocity vs

Contact Dynamics

Non-smooth dynamics n

n

n

s

ss

93

First: no friction between particles with relative tangential velocity vs

considered (added later).

Thus: only normal forces at contacts!

Contact Dynamics

n

94

Contact DynamicsCollisions of rigid bodies give rise to discontinuous velocities, thus higher order schemes are not beneficial.

Use implicit Euler integration:

1( ) ( ) ( )

( ) ( ) ( )

i i ii

i i i

v t t v t F t t tm

r t t r t v t t t

exti iF R

external contact

95

2extF1

extF

Example: two particles (1D)Matrix H transforms contact forces (loc) into particle forces, HT particle velocities into relative velocities (loc).

1 12 1

2 2

1

2

( 1,1)

1

1

locn

locloc locnn nloc

n

v vv v v

v v

R RR R

R R

TH

H

1 2

1 2m m m

1v2v

96

Example: two particles (1D)

Newton‘s equation in vectorial form:

1 1 1

2 2 2

1 ext

ext

v R Fd

v Rdt m F

Use for contact quantities (transformation by H):

12 1

2

11 1 1( 1,1)

1

extlocloc loc ext extnn next

eff

FdvR R F F

dt m m mF

effective mass

2 1

1 ext extF Fm

/ 2effm m

acceleration without contact forces

97


The equation has two unknowns:

Using the implicit Euler integration leads to relation:

2 1

1 1 ext ext

eff

locn l

n

lc

ocn ov t t

R t F Ft m m

vt

t

Use the constraint (Signorini) as condition

to calculate the contact force Rnloc.

2 1

1 1locl ext ex

e fn

f

ocn tdF F

d mR

t m

v

98


Split solution of this equation in two steps:

, ( )( )( )

locloc n

loc freen

en ff

v t v t tm

t

tR t t

,2 1

1( ) ( )loc free loc ext ext

n nv t t v t F F tm

where

2 1

1 1locn ext ext

eff

locn loc

n

v t tR t t

v tF F

t m m

99

illustrates solutions for persisting contactand for open contact.


Graphical interpretation:

Signorini condition at contact (d = 0) can be re-formulated in terms of the local velocity.

Intersect with

locnR

locnv

, ( )( )( )

locloc n

loc freen

en ff

v t v t tm

t

tR t t

100


No overlap (after Δt):

Graph is shifted by –d/Δt to account for non-zero distance.

Now intersection works for all contact cases:

• no contact

• closing contact

• persisting contact

• opening contact

locnR

locnv

,loc freenvd

t

effm

t

( ) ( ) ( ) 0locnd t t d t v t t t

→ simple implementation

101

i.e. no contact or closing contact.

Otherwise a contact persists or forms during the time step, i.e. the gap d closes. In that case impose:

and thus, the contact force can be calculated:

Example: two particles

,( ) ( ) ( ) 0loc freend t t d t v t t t

( )( ) ( ) 0 ( )loc loc

n n

d td t v t t t v t t

t

Check what would happen without the contact force.

0locnR

,( ) /( )

( )loc freen

efo

fl cn

d t t v t tR m

tt t

If then

102

Dissipation

Note: Collisions between grains are usually inelastic, i.e. energy is dissipated through vibrations (sound) and eventually also small plastic deformation or heat production.

Here we assumed the simplest case of perfect plasticity, where the full energy is dissipated.

103

Coulomb friction

if

if

0

0

s s n s

s d n s

f f v

f f v

•

•

μμs

μd

vs

s d experimentally:

μd ≈ 0.1 – 0.5

static case

dynamic case (dissipation)

= friction coefficientstatic dynamic

104

Coulomb-Graph: Friction between particles with relative tangential velocity vs ,simplification

Additional unknown, solution as for normal force:Intersect with linear relation of and obtained from Newton‘s equation of motion.

Note: for spheres not coupled to normal direction!

Contact Dynamics

n

n

s

ss

loctR

loctR

loctv

s d

105

Contact DynamicsOne frictionless contact in 3D, spherical particles:

Angular velocities and torques are not needed.

Particles velocities and forces are:,

1,2 1,2 1,2,

1,2 1,2 1,2 1,2 1,2 1,2,

1,2 1,2 1,2

, ,

x x x ext

y y ext y ext

z z z ext

v R F

v v R R F F

v R F

(R1,2 are forces on the particles due to contact forces)

At contact: only normal components Rnloc and vn

loc

have to be considered.

106

Simple transformation:

Note: compressive contact force is positive

Contact Dynamics

One frictionless contact in 3D, spherical particles:

normal vector:

2 1 1 2( ), ,loc loc locn n nv n v v R nR R nR

1 2

1v

2vn

x

y

z

n

n n

n

107

Simple geometrical transformation:

For friction one would have to add angular velocities and torques.

Contact Dynamics

2 1 1 2( ), ,loc loc locn n nv n v v R nR R nR

1 1

2 2

,loc T locn n

v Rv H HR

v R

,, , , , ,Tx y z x y zH n n n n n n

→ Matrices H and HT:

108

Contact Dynamics

General case:

Consider N particles with generalised coordinates q and forces R, Fext .

11 1

1 1 0

, ,

0

ext

ext

extN N N

NN

Rv F

T

q R F

v R F

T

In 2D one has 3N components (2 translational and 1 rotational per particle) ,

in 3D one has 6N components (3 translational and

3 rotational per particle) .

109

Contact DynamicsContact network:

Be c the number of contacts. Define generalised relative velocities u and contact forces Rloc :

1 1

,

loc loc

loc

loc locc c

v R

u R

v R

2D: 2c components (1 normal and 1 tangential per contact)

3D: 3c components (1 normal and 2 tangential per contact)

Here we do not consider contact torques.

110

Contact Dynamics

,T locu H q R HR

The transformation H is given by a

2c×3N matrix in 2D and a 3c×6N matrix in 3D.

( ) ( ) extM q t R t F Newton‘s equation of motion:

1 0 0 0

0 0

0 0 0

i

i i

N i

m

M m

I

with in 2D

Define the diagonal mass matrix M:

111

Contact Dynamics

,T locu H q R HR

( ) ( ) extM q t R t F

1 1

1 1

( )

T

T ext

T T exloc t

H q

H M R t M F

H M H H M F

u

R

1 1 ( ) ( ) extq t M R t M F

112

1 1TeffM H M H

Contact Dynamics relation between the contact quantities:

1 1locT T extH M H H Mu R F

effective inverse mass matrix

inverse mass matrix

(( ) )( )

free

efl c

fo u tu t t

R tt

tt

M

implicit Euler integration:

113

Contact Dynamics

Perform direct inversion of local inverse mass matrix.

Then implement the constraints by checking if contacts close and determine u(t+Δt) .

This is difficult because it is not unique.

iterative solution(neighboring contacts as external forces)

114

Contact Dynamicswithout friction with friction

More examples:http://alert.epfl.ch/Archive/ALERT2008/school08/Radjai&Dubois/slides_alert2008_public.mov

many applications: granular materials, masonry, ...

115

CD: more contact laws

rolling friction:Add additional contact torque. more unknowns(1 in 2D, 3 in 3D) more constraints

ωr

r,eff

T loc

r,eff

FN FC+( )

FN FC+( )

−μ

μ

cohesion:Add to Signorini conditionan attractive force within a range.

Many variations of contact laws are possible.

116

CD: more contact laws

Collapsing soil/granular structure:Gravity leads to collapse, cohesive force stabilizes structures.

Two extremes:

• very fast deposition

• very slow depositon

117

Non-spherical objects

Up to now only spherical particles. Non-spherical objects can be composed of spherical objects, e.g. by additional constraints between particles.

For non spherical rigid objects of finite volume (e.g. polyhedra) this can be done approximately.

Contact Dynamics

Sandstorm

Particles in Fluids

• Sedimentation

• Fluidized beds

• Size segregation under shear

• Hydraulic transport

• Filtering

• Saltation

• Rheology of suspensions (e.g. blood)

• Sandstorms

• Oceanic pollution and planktion

• Volcanic explosions....

Fluidized Bed

Incompressible Navier-Stokes equation:

Equation of motion of fluid

1( )

uu u p u

t

0u

and are velocity and pressure field of the fluid, and its density and dynamic viscosity.

( )u x

( )p x

0

( ) u const

t

Reynolds number Re

Reuh

u is characteristc velocityh is characteristic length

µ is dynamic viscosity

Re << 1 is the Stokes limit (laminar flow)

Re >> 1 is turbulent limit (Euler equation)

Solvers for NS equation

• Penalty method with MAC

• Finite Volume Method (FLUENT)

• Turbulent case: k-ε model, DNS, LES or spectral method

• Lattice Boltzmann Method

• Discrete methods: DPD, SPH, SRD, DSMC..

CFD = Computational Fluid Dynamics

One particle in fluid

0vu

fluid

e.g. pull sphere through fluid

0v

Γ

no-slip condition:

create shear in fluid : exchange momentum

movingboundary condition

Drag force

AdFD

jiij ij

j i

uup

x x

drag force

stress tensor

η = µ is dynamic viscosity

(Bernoulli‘s principle)

Homogeneous flow

Re << 1 Stokes law:

FD = 6π η R u(exact for Re = 0)

R

u

Re >> 1 Newton‘s law: FD = 0.22π R2 u2

general drag law:

CD is the drag coefficient

22Re

8

D DF C

Drag

no dragStokesNewton

Compare Sphere and Cylinder

Drag coefficient CD

0.687241 0.15Re Re 1000

Re0.44 Re 1000

p pD p

p

C

3 22 3

321 , Re

Ref av

D pp

DC

v u

Examples for empirical relations for drag coefficient

Gidaspow, 1994:

Chen, 1997:

Drag

Inhomogeneous flow

In velocity or pressure gradients: Lift forcesare perpendicular to the direction of the external flow,

important for wings of airplanes.

lift force: CL is lift coefficient21

2L LF C Au

Torque on particle

CMT r dA

vorticity

Magnus effect

Gustav Magnus

important for soccer22MF R u

when particle rotates

for cylinder:

Many particles in fluids

22

Re8 DD CF

•The particles are dragged bythe fluid with a force:

simulating particles moving in a sheared fluid

This image cannot currently be displayed.

•The fluid mediates an effectiveforce between the particles ,called

•„hydrodynamic interaction“.

1 2 1 2

1 2

r r R R

R R

1, Re 1

Lubrication

Sabri Ergun, 1952

Drag at high volume fraction

Re << 1

FD = 6π K η R u

4.7

3

1 , 0.225 , 0.2 3 1

K

Stokes limit

Stokesian Dynamics (Brady and Bossis, 1985)

John F. Brady

Georges Bossis

3

1( )

8S r r

G rr r

Stokeslet = Greens function of the Stokes equation

1up u

t

Re = 0

Stokes equation

Stokes limit

Hydrodynamic interaction between the particles:

( )i ij i j jj i

u M r r u

mobility-matrix

for Re = 0 mobility matrix exact

invert a full matrix only a few thousand particles

1

( ) i

N S

ii

u x G n d

General solution

for N particlesthrough convolution:

ijk uup

x x

Calculation of stress tensor on a grid

Calculate pressure and velocity gradientthrough interpolation.

ijk

ijkf n

Calculate drag forceat surface element ijk

Re < 100

Numerical technique

Calculate stress tensor directly by evaluating the gradients of the velocity field

through interpolation on the numerical grid,e.g. using Chebychev polynomials:

0

N

n nn

f x c T x

cos arccosh cosh arccosnT x n x n x

0

1

22

1 1

1

2 1

2n n n

T x

T x x

T x x

T x xT x T x

1 1( ) walls

uu u p u f

t

1 21

1 1

( )

k kk k k kk

u up u u u f

t

Momentum loss of fluid

Fluid goes through particle but loses momentum.Momentum should not be substracted at center ofmass of the particle but at each surface element.

Impermeable particles

commercial discrete volume solver on an adaptive triangulated mesh

Constantly remeshing ,is computationally very costly.

Numerical technique for 0 < Re < 200

Method of A.L. Fogelson and C.S. Peskin:Advect markers that were placed in the particle and then put springs between

their new an their old position.These springs then pull the particle.

Sedimentation

• Sedimentation ist the descent ofparticles in a fluid due to the action ofgravity.

• The interaction between the particlesand the fluid is given by the conditionthat the velocity of the fluid on theentire surface of each particle is equalto the velocity of this particle.

• Measure settling velocity, i.e. velocityof the upper front.

• If particles are of different speciesthen one has several fronts.

• Open question: size dependence ofthe density fluctuations.

Sedimentation

comparing experiment and simulation

Glass beadsdescendingin silicon oil

using penaltymethod withMAC grid

Sedimentation velocity

1954

settling velocity vS = v0 (1-Φ)5

Φ = volume fraction of particles

Sedimentation of platelets

Oblate ellipsoids descendin a fluid under the actionof gravity.

This has applications inbiology (blood), industry(paint) and geology (clay).

Draft, Kiss and Tumble

draft kiss tumble

Draft, Kiss and Tumble

vertical velocity horizontal velocity

Lattice Boltzmann Method

Sauro Succi : The Lattice Boltzmann Equation for Fluid Dynamics and Beyond

(Oxford Univ. Press, 2001)

where the equilibrium distribution is defined as:

Define on each site x of a lattice on each outgoing bond ia velocity distribution function f(x,vi,t) which is updated as:

weights wi in equilibrium distribution:

two dimensions (D2Q9)

three dimensions (D3Q19)

Lattice Boltzmann Method

Tony Ladd1994

Suspensions of solid particles with LBM

Apply no-slip condition, i.e. elastic bounce back for each individual streamimg direction that crosses boundary.

Sedimentation

Settling of four spherocylinders

Shear flow

A.J.C. Ladd, J.Fluid Mech., 1994

using Lattice

BoltzmannMethod

Shear cell

Shear flow

Velocity distributions under shear

exponential tailsJens Harting 2006

using LBMtechnique ofT. Ladd

different number of particles N

average over1010 velocities

Stochastic Rotation Dynamics (SRD)

• introduction of representative fluid particles

• collective interaction by rotation of local particle velocities

• very simple dynamics, that recovers hydrodynamics correctly

• Brownian motion is intrinsic

A. Malevanets and R. Kapral (1999)

or Multi-particle Collision Dynamics (MPC)

Sedimentationusing SRD

Péclet number :Pe = Lu/D = 5

by J. Padding

• Assume spherical particles

• Use molecular dynamics for the simulation

• DLVO potentials describe the dominant

particle-particle interaction:

- screened Coulomb potential

(ions / counterions), repulsive

- Van-der-Waals-attraction for

short distances

• Hertz force for overlapping particles

• Lubrication force ~

Simulating clay

Al2O3 interaction potentials

Shear flow

Shear viscosity

Viscosity at small vs

Martin Hecht, 2006

Density correlation

Martin Hecht, 2006

Direct Simulation Monte Carlo (DSMC)

G.A. Bird (1963)

Good for large Knudsen numbers,i.e. mean free path is greater thana representative physical length scale.Popular in aerospace engineering.

G.A. Bird, Molecular Gas Dynamics and Direct Simulation of Gas Flows, Clarendon, Oxford (1994) F. Alexander and A. Garcia, Computers in Physics, 11 588 (1997)

Divide the system into cells and generate particles in each cell according to desired density, fluid velocity, and temperature.For fluid velocity and temperature, assign to each particle a velocity from Maxwell-Boltzmann distribution.

Molecules and “Simulators”

Physical Molecules DSMC Simulators

In DSMC the number of simulation particles (“simulators”) is typically a small fraction of the number of physical molecules.Each simulator represents Nef physical molecules.

Nef = 2

Accuracy of DSMC goes as 1/N; for traditional DSMC about 20 particles per collision cell is the rule-of-thumb.

DSMC Collisions

• Sort particles into spatial collision cells

• Loop over collision cells– Compute collision

frequency in a cell– Select random collision

partners within cell– Process each collision

Collision pair with large relative velocity are more likelyto collide but they do not have to be on a collision trajectory.

Selectedcollisionpartners

Thermal Walls

A more realistic treatment of a material surface is a thermal wall, which resets the velocity of a particle as a biased-Maxwellian distribution.

x

y

z

uw

Walls can also be part-thermal, part-specular.

DSMC vs ED

Dissipative Particle Dynamics (DPD)

P.J. Hoogerbrugge and J.M.V.A. Koelman (1992)

Make MD of soft particles with forces:

with conservative force:

random force:

and dissipative force:

C R Di ij ij ij

i j

F f f f

1 1

0 1

ij ij ij ijCij

ij

a r r rf

r

( ) R Rij ij ij ijf w r r

( ) D Dij ij ij ij ijf w r v r r

unit vector indirection connectingthe centers of mass


( ) R Rij ij ij ijf w r r ( ) D D

ij ij ij ij ijf w r v r r

Both forces conserve momentum. θij is a Gaussian distributed random number.

22 1 1

( ) ( ) 0 1

ij ijD Rij ij

ij

r rw r w r

r

The weight functions wR(rij) and wD(rij) must be chosen such as to fulfill thermal equilibrium . One possibility is:

P. Español and P. Warren (1995)

random force dissipative force


Dense suspension of ellipsoids to simulate the rheology of high performance concrete using more than one million fluid particles (NIST)

Hybrid simulation

A. Donev

Flow through a porous medium

two-dimensionalrealization by

placing randomlydisks that do notoverlap (RSA)

important foroil recovery,filtration andfluidized beds

Stokes number:

uinertial impaction

direct interception

diffusion

Massive tracerparticles of diameter dp,

velocity upand density p

are released.

Filtration

δ D

x

y

u

St : = D and St 0: = 0

= capture efficiency

Filtration: Trajectories of particles

St = 2.0610-4

St = 8.1210-3

Stokes number:

Non-Newtonian fluids

11 2 , nA Cross fluid:

Bingham fluid:

0 00

0 0 0

0 0

for

for BA

Example: styling gel, i.e.vinyl acetate/vinylpyrrolidone copolymer (PVP/PA)

Example: tooth paste

Non-Newtonian flow through filter

Cross fluid

porosity ε = 0.7

Particles in Turbulent Fluids

• Saltation

• Aerosols

• Fluidized beds

• Local plankton concentration

• Pneumatic transport

• Turbulent mixing

Aeolian Saltation

The Mechanism of Saltation

h

Windh

• Grains are drawn from the ground and accelerated by the wind. With moreenergy they impact again against the surface and eject a splash of newparticles. In this way more and more grains saltate until saturation is reacheddue to momentum conservation.

Ralph A. Bagnold

The turbulent air flow

*

0

lnx

u zu z

z

logarithmic velocity profile of the horizontalcomponent of the velocity as function of height y:

0.4 is the von Kármán constant

z0 is the roughness lengthLudwig Prandtl, 1924

wind strength

Sketch of the system

• Logarithmic wind profile above the sand bed starts at height , which must be determined

0 0*

0

( ) lnx

z z huu z

z

0h

2 ( )( )

8p

p p

dF C u v u v

D D

DEM model of bed in moving fluid

The fluid exerts on the particles the drag force:

i i D collm x F F

3 22 332

1 , ReRe

f avD

DC

v u

coll n ij n ij ij

i jij

i j

ij i j i j

k c

R R

F n

x xn

x x

x x

Dissipative collisions between particles

kn elastic spring constant, cn damping coefficient

2exp

4n

n

eff n n

ce

m k c

The restitution coefficient en

is given by cn through:

Including Feedback

• The reaction-drag forces exerted by the particles on thewind slow down the wind.

• The grain-stress generated by the accelerated particles iscalculated through (A = area of sand bed, f = force density)

• The grain-stress modifies the shear velocity in the saltationlayer by

• With this one determines the wind profile using

:

( ) ( ') 'i

ig

i z zz

Fz f z dz

A

* 2*

( )( ) : 1 g

a

zu z u

u

( )u zdu

dz z

Apply drag forces on particles

( )jtV z

1( ) ( )j jt tv z v z 0

1( ) : ( )jt tv z v z

Wind Profile

Calculate drag forces

Calculate grain stress

Integrate

( )jtV z

[ ( )]i tF V z

:

( )i

ig

i z z

Fz

A

( )u zdv

dz z

* 2

*

( )( ) : 1 g

a

zu z u

u

: 1j j

Feedback Algorithm

u* = 0.5m/s u* = 1.0m/s u* = 1.5m/s

Aeolian Saltation

Role of collisions in preferential concentation

Pico-Phytoplankton distribution

Particles in Turbulent Flow

Particle motion

0.687

3 ,

8

2421 0.15Re Re 1000

, ReRe0.44 Re 1000

,

fDi i drag coll drag i i i

i i

i ip pD pp

fp

i jcoll n ij n ij ij ij

i

Cm m

R

RC

k c

x F F F u v u v

u v

x xF n n

x

j

ij i j i jR R

x

x x

2exp

4n

n

eff n n

ce

m k c

K. Alvelius, Phys. of Fluids 11, 1880 (1999)

The random force g (x, t) should generate an isotropicand homogeneous turbulent velocity field.To be divergence free its Fourier transform must fulfill:

, 0t k g k

1 2, , ,ran rant A t B t g k k e k k e k

Turbulent forcing

1 2 2

1

0

y

x

x y

ke k

k k

2 2 22 2

1 -

x z

y z

x yx y

k ke k k

k k k k k

Turbulent forcing

12exp sin

2ran

S kA i

k

22exp cos

2ran

S kB i

k

21exp 1 , 2 1

0 otherwisef fA c k k k kS k

with

121exp

b

a

kin

fk

PA c k k dk

t

and

1

2in i iP g t g t

where the powerinput is given by:

kf corresponds to the length scale of energyinjection

Preferential concentration

2d slice of a 3d system

Particle mixing in turbulent channel flow

• Consider a channel through which particles are dragged by a turbulent flow.

• Do not calculate the whole fluid field, but rather use a stochastic approach to model the influence of the turbulent velocity field on the particle movement.

• The particle density is constant throughout the system.• Concentrate on a small region in the center of the channel, which means

we ignore the effects of the walls.• Study the mixing of two types of spherical particles.


Fluid velocity inside the channel: )()( tuutu t

mean fluid velocity intrinsic fluctuations

Use the empirical drag law to couple the particle movement to the fluid velocity )(tu

Intrinsic velocity fluctuations

The mean fluid velocity is kept constant:

The fluctuating part is determined by using the stochastic model introduced by A. M. Reynolds (Phys. Rev. Lett. 91 (2003), 084503) that reproduces well the experimentally observed distributions and autocorrelations of velocities and accelerations of tracer particles in fully developed turbulence.

The model calculates a time series for the velocity of a tracer particle. For every “real” particle we generate a tracer particle and evolve it in time. The velocity of the tracer particle then gives us a stream line and the “real” particle is dragged into the direction of this line.

.constu

)(tut

1 2 11

1 1 1 1 1

2 1 1 1 12

2

2 ,

t

t

a

t t La

u L L

t t t t

d T dt T dd

da T t a dt T tdt

T t T t ddu a dt dx u dt

ln

Stochastic differential equations for and each component of the acceleration at .

Intrinsic velocity fluctuations

Log of local dissipation rateA.M. Reynolds (2003)


QM in a nutshell I

2*

( )

1

r

r r r r

r dr

wave function

QM deals with wave functions. Their absolute value is related to the probability density of finding a particle at a given position.

Probablitiy of finding a particle at position r.normalization

226

QM in a nutshell II

2

2

2

H E

H V rm

The wave functions are the eigenfunctions of the Schrödinger equation (SG). Their eigenvalues correspond to their energies E.

Hamiltonian with kinetic energy and potential V(r)

Time independent SG

The ground state is the eigenfunction corresponding to the smallest eigenvalue. This is the wave function of interest.

227

QM in a nutshell III

For more than one particle the wave function depends on all particle positions.

The density has now the meaning of the probability of finding the particles at the given positions.

*

1 1 1

*

1 1 1

, , , , , ,

, , , , 1

n n n

n n n

r r r r r r

r r r r dr dr

normalization228

For more than one particle the wave function has a symmetry: „Bosons“ are symmetric and „Fermions“ are antisymmetric when exchanging two particle positions.

1 1

1 1

,..., ,..., ,..., ,..., ,..., ,...,

,..., ,..., ,..., ,..., ,..., ,...,

i j n j i n

i j n j i n

r r r r r r r r

r r r r r r r r

Symmetric (ex: photons, like to be in the same state):

Antisymmetric (ex: electrons, don‘t like to be in the same state):

229

QM in a nutshell IV

QM in a nutshell V

QM particles own a special character: their spin.

Electrons can have two kind of spins (up or down).

Since electrons are fermions every state can be occupied only once.

Different spins count as different states.

Therefore every eigenfunction can be occupied with an up- and a down-spin electron.

230

Many particles

For many-particle wave functions often single body wave functions are glued together. To keep the antisymmetry the Slater determinant is used:

1 1 1

1

1

1, ,

!

N

N

N N N

r r

r rN

r r

Calculation of this determinant is a computationally expensive job.

231

Note: Using Slater determinants of single body wave functions is already an approximation. The differences are called correlation effects.

Electrons vs ionsQM is important for particles with small masses:

22

2T

m

For large particle masses

as ions (atomic cores) the QM is not anymore important.

Electrons need to be treated with QM, ions not.As the ion mass is 3 orders of magnitude larger than the electron mass, the motion can therefore be separated (Born-Oppenheimer).

=> It makes sense to calculate the electron distribution for fixed ions.

232

Born-Oppenheimer Approximation

The Born-Oppenheimer Approximation implies thatthe time step for the motion of ions has to be smallenough, such that when ions move, the electrons donot skip any energy level transition.

1( ) ( ) ( ) ( )i i i iR R R R R

R u t

Small time step

BO Approximation Hydrogen Molecule

Otherwise the dynamics is not correct as for example in the oscillationof the hydrogen molecule.

Large time steps

235

Hohenberg-Kohn Theorem

(1964)

Pierre Hohenberg Walter Kohn . (Nobel prize 1998)

If the ground state is not degenerated, all information of this ground state is present in its density distribution. All the observables can be calculated by only using the density distribution. The knowledge of the wave functions is not needed.

E E

Kohn – Sham equation

(1965)

Lu Jeu Sham

Proposing non-interacting single particles moving in a potential, only depending on the density distribution. Many-particle effects are handled using correction terms.

Walter Kohn

236

Kohn – Sham energy

2* 2

*

*

( ) ( )2

1 ( ) ( )( ) ( ) '

2 '

( ) ( )

( ) ( )

KS i i i effi

eff extern xc

i i ii

i j ij

E f r r dr Vm

r rV r V r dr dr dr E

r r

r f r r d r

r r d r

237

occupation number

Instead of Slater determinant use single particle wave functions.

orthonormality condition

kinetic energy

xc x cE E E

correction for many particle effects

exchange

correlation

22

*

( ) ( ) ( )2

1 ( ) ( ')( ) ( ) ( ) '

2 '

( ) ( )

eff i i i

eff extern xc

i j ij

V r r rm

r rV r r V r dr dr E

r r

r r dr

Kohn – Sham equation2

( ) i ii

r f

xE

Since no Slater determinant is used, the antisymmetry is missing. It is considered by adding a correction term in .

potential from ions

238

orthonormality

xcE

Exchange-Correlation Energy

239

1 21 1 2

1 2

( , )1[ ] ( )

2xc

D r rE dV r dV

r r

Exchange-correlation hole density: Probability offinding an electron at r1, knowing that there is anelectron at position r2.

Until here, density functional theory is formallyexact to describe a many body quantum system.Since computing the exchange-correlation holedensity is too complicated, there areapproximations.

Kohn – Sham orbitals

Unlike for real many-particle orbitals theeigenvalues of the single particle orbitalshave no physical meaning. The sum ofthem is not the total energy!

The only physical meaning of the singleparticle wave functions is that they yieldthe correct density.

240

Local density approximation (LDA)

How to get the correction term?

The free electron gas can be calculated analytically using Slater determinants.

The LDA approximates the Kohn-Sham correction energy locally with the correction energy of the free electron gas with the same density.

1

3 433 3

( ) ( )4

LDAx xE r dr r dr

Using the free electron gas

241


As single body wave functions are used, also a correction to consider the correlation effects has to be included. This correction is called . And it is also treated using LDA assuming that the electrons are homogeneously distributed.

No analytic expression for is known.

242

( )LDAc cE r d r

cE

c


LDA has been applied for calculations of band structures and the calculation of the total energy in solid-state physics.

In quantum chemistry it has been much less popular since chemical bonds need more accurate results.

=> A better approximation is needed.

243

Generalized gradient approx. I

It is possible to improve the LDA by adding a dependence on the gradient of the density. This approach is called General Gradient Approximation (GGA).

GGA improves the resulting energies.

( ), ( )GGAx xE r r dr

More and more derivatives could be added, but results do not improve much.

244

and similar for the correlation correction

245

Generalized gradient approximation II

The most used GGA is the one of Axel Becke (his paper has been cited more than 17000 times) .

Axel Becke

(1988)

Several proposals exist for such an approximation:

2

43

LGC LDAX XE E d r

2

43

1

4 3

1 6 sinh

with , 0.0042

Becke LDAX X

xE E dr

x x

x

Frank Herman

(1969)

Generalized gradient approx. III

GGA gives much better results than LDA. They can be used to calculate all kinds of chemical bonds like covalent, ionic, metallic, and hydrogen bridges.

However DFT using GGA fails to describe van der Waals interactions properly.

246

Improving the van der Waals interactions

Grimme (2006) introduced an additional term:

In this empirical correction are parameters and is the distance between two ions.

247

16

6 61 1

/ 1

1

1 ij r

ijN N

disp dmp iji j i ij

dmp ij d R R

CE s f R

R

f Re

,,6 6 rij Rs C

ijR

Implementation of wave functions

For the implementation of the wave functions, the single particle wave functions are often expanded in an orthonormal basis system with some cutoff.

In solid state physics a plane wave basis is often used (plane wave functions are not localized, as the electrons in crystals).

j jk kk

c

248

ikxk e

Number of wave functions

For 16 water molecules a cutoff of about 70 Ry is needed. This corresponds in this case to about 15000 plain waves per electron.

249

Localized basis sets

For localized basis sets so called „Pulay forces“ appear. These are numerical forces coming from the finite basis. These Pulay forces can be up to an order of magnitude bigger than the physical forces and have to be corrected.

250

2

( ) l rl lr c r e Example: Gaussian type orbitals

Using localized basis sets can help to decrease the number of basis functions for chemical problems.

Wave functions are then dependent on the ion positions.

Paul Pulay1969

Forces on external potential

The Hellmann-Feynman theorem tells how to calculate the forces of the external potential (moving the ions in the ion-electron potential).

2

2| |

d Rm

dt R

H

Hellmann-Feynman theorem

251

Adding the time dependence

Introducing the time dependence is now easy:

Born-Oppenheimer:

- Solve the electronic problem.

- Move the ions due to Hellmann-Feynman using classical MD methods.

Problem: solving the electronic problem every time step is very expensive.

252

iterate

253

Born Oppenheim Method

Simulation of two guanidium molecules using Born-Oppenheimer technique in the CP2K code.

Institute for Computational Molecular Science, Temple University

254

Car – Parrinello method

Roberto Car Michele Parrinello

(1984)

Ab initio method to calculate molecular structures once then use Hellmann-Feynman for ion forces and an artificial force on the electron single particle wave functions.

Car-Parrinello method

R.Car and M. Parrinello came up with the following Hamiltonian:

21, , ,

2 2

,

n n ICP i i i i

I iI

KS n Ion ni

PH R R

M

E R E R

ion-kinetic energy

Kohn-Sham Energy

artificial mass

plus the constraint that the single particle wave functions must be orthonormal.

255

ion-ion energyartificial kinetic energy of the wave functions.

NOT QM-KINETIC ENERGY

Car-Parrinello method

Using Euler-Lagrange we get the equation of motion:

21 1[{ }, ]

2 2

KS

I I ij i jijI I

KS Ion

i ij jji i

KS N Ionsconst I I I i i i

I i

EM R

R R

E E

E M R E R E

forcing orthonormality of the wave functions

1) Look for wave functions minimizing Kohn-Sham energy.

2) Proceed with Car-Parrinello equation of motion.256

257


The fictious mass μ of the electrons is a tunable

parameter of the method describing how stiff

the electron motion is coupled to the nuclei.

If μ→ 0 the electronic response is very rapid

and the electrons remain to a sufficiently high

degree in the ground state. But numerically one

wants to avoid too large accelerations. So electron

dynamics is made artificially slower than reality.


Since the electronic configuration is now following an equation of motion, the integration step has to be small enough to resolve the electronic motion. Since μ is usually smaller than the mass of the ions, the CP time steps are usually smaller than the time steps of the direct Born-Oppenheimer approach. Often time steps of the order of a tenth of a femtosecond are used.

258

( )i r IR The electrons have to

adiabatically follow the nuclei

If there is an energy transfer from the nuclei to the electrons

the electronic temperature increases

and the system leaves the BO surface

( ) ( )CP BOI IR t R t

E

Necessary condition for CP-MD

(Pastore, Smargiassi and Buda, Phys. Rev. A 44, 6334, 1991)

CP: continous line

BO: points CP BOF F

The difference is small and

oscillatory

Comparison of CP and BOMD forces (Si crystal )

Advantages of Car-Parrinello

Comparing Car-Parrinello with „classical“ Born-Oppenheimer:

+ Car-Parrinello equations need much less computer time per time step. But they need smaller time steps. There is still speed gain in real time.

+ Smaller energy-fluctuations since always the same bindings are considered.

- Problems with light ions for big μ

262

263

Simplicity of plane-waves (PW) basis, requires pseudopotentials

The Fourier transform implies :

- Details in real space are described if their characteristic length is larger than the inverse of the largest wavevector.

- Quality of a plane wave basis set can be systematically increased by increasing the cut-off energy.

Problem : huge number of PWs is required to describe localizedfeatures (core orbitals, oscillations of other orbitals close to the nucleus)

Pseudopotentials (or, in general, « pseudization »)to eliminate the undesirable small wavelength features.

264

Pseudopotentials

Electrons in the inner shellsdo not contribute to valencebonding – so they are frozenin the state they have in anisolated atom.

Neglecting the frozen coredoes not add anycontribution to the energy(von Barth and Gelatt,1980).

The choice of pseudopotentials

265

CPMD works with several types of pseudopotentials:

The most commonly used for non-metallic atoms are Martin-Troullier(MT) [Phys. Rev. B, 43, 1993 (1991)]:- Work well for light main group elements: C, N, O, S, P etc.- However, if one changes the functional one must also change the pseudopotentials.- They are not appropriate for transition metals.- Typical cut-off: 70 RyWith transition metals you need to use Goedecker-Hutter or Vanderbilt ultra soft:- GH require large cutoff, cover most elements.- Vanderbilt: low cut-off ~ 40Ry, more calculations required, in CPMD.

266


This method is applied in solid state physics,

atomic and molecular physics and can study

crystalline structure, liquids and amorphous

materials. The method can be substantially

improved by using tight-binding potentials

and by representing part of the electronic

wave functions in Fourier space using FFT.

Too big artificial mass

Larger μ implies larger time steps. How large might μ be? What effect does a too large μ have?

267

For water it seems that μ should be at most 1/5 of the lightest ion (H, D or T).

Grossman et al. J. Chem. Phys. 120, 300 (2004)

Effect on different GGA‘s

268

One of the approximations was the GGA.

What is its effect?

Changing the GGA has a big effect on g(r) of water.

VandeVondele et al.

J. Chem. Phys. 122, 014515 (2005)

2H O

Further possibilities

Krack et al. J. Chem. Phys. 9404 117, (2002)

Since the electronic structure is known, new properties like the X-ray scattering intensity can be measured.

269

270


Simulation of 32

water molecules

and one hydronium

ion (H3O+) at 400K

and density 1 g/cm3

during 500 ps.

by Axel Kohlmeyer

CP2K Molecular Dynamics

271

- Linear combination of Gaussian-type orbitals (GTO) for Kohn-Sham orbitals.

- Auxiliary plane wave (PW) basis set for electronic charge density, and long term Coulomb interaction.

Implementation of the Gaussian and plane waves (GPW) basis:

( )PW GTO GTO PW

Removing the PW contribution of the GTO

Comparing PW with Gaussians in Car-Parrinello method

272

The Advantages of PWHas a better scaling than Gaussian based methods-without loss of accuracy.Plane-waves are easy to program-fast math libraries. No Pulay Forces.Can do solids, polymers and molecules all in same framework.Many properties are easy to calculate with PW.

The Disadvantages of PWCare must be taken with real space grid and energy cut-off.Ripple noise makes it hard to converge structures.To do isolated molecules requires a lot of work: screening of images.Need lots of plane-waves.Chemists have to learn Solid state physics lingo-culture gap.

The Problems of PWExchange is non-trivial and very expensive to include in this formalism.Pseudo-potentials!

CP2K: Advantages and Disadvantages

273

Advantage:- Benefiting combination of the best of the two

basis set.

Disadvantages:- Computational overhead due to

transformation steps.- More complicated code.- Combination of advantages implies possibly

also the combination of disadvantages.

CP2K: Larger simulations possible

274

Left: Spin density illustrates the extent of the hole in liquid water just after

ionization. Right: Three views of the DNA crystal.- A hybrid Gaussian and plane wave density functional scheme. Lippert G, Hutter J, Parrinello M: mol. phys. 92 (3): 477-487 1997.- An efficient orbital transformation method for electronic structure calculations. VandeVondele J, Hutter J: j.

chem. phys. 118 (10) : 4365-4369 2003.

BigDFT: Wavelets

275

Wavelets as a basis set for Kohn-Sham system.

1. Localized both in real and in Fourier space.

2. Allow for adaptivity (for internal electrons).

3. Systematic basis set.

J. of Chem. Phys., 129 014109 (2008), L. Genovese, A. Neelov, S. Goedecker, T. Deutsch, S. A. Ghasemi, A. Willand, D. Caliste, O. Zilberberg, M. Rayson, A. Bergman and R. Schneider, "Daubechies wavelets as a basis set for density functional pseudopotential calculations".

276

Wavelets and Scaling function

, ,

( , , ) ( ) ( ) ( )i lmn l m nl m n

x y z c x y z wavelets

Kohn-Sham orbital expansion coefficients

Scaling function: low resolution levels = linear combination of high resolution levels:

( ) (2 )m

l j lj m

x h x j

277

Wavelets and Scaling function

It is possible to reconstruct the missing information for the coarseness of the resolution.

(2 ) ( ) ( )m m

l j l j lj m j m

x h x j g x j

Another wavelet

Increase the resolution without modifying grid space!

BigDFT: Resolution

Fine grid: High resolution Coarse grid: Low resolution

Problem: Cubic scaling

For systems bigger than 500 atoms:orthonormalization operation is predominant (N3).

Path Integral CPMD

Simulation of proton transfer in FI-FA dimer using PIMD to improve the behaviour of the H-atoms.

by Anatole von Lilienfeld

279

280

Questions of last semester

• Critical behaviour of the Ising model

• M(RT)2 canonical Monte Carlo

• Detailed balance

• Finite size scaling

281

15 relevant questions

• Fluctuation-dissipation theorem for M

• Dynamic correlations and dynamic scaling

• Glauber and Kawasaki dynamics

• Creutz demons

• Binder cumulants

• First order transitions (Potts model)

• Swendsen-Wang cluster algorithm

282

15 relevant questions

• Verlet and leap frog schemes• Verlet tables and linked cells• Particle – mesh method• Constraint method with Lagrange

multipliers• Rigid bodies, quaternions• Nosé – Hoover thermostat• Event driven simulations• Inelasticity and finite time singularity

event driven programming - eth z...event driven programming 1 as opposed to „batch“ programming...

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