computer modeling and simulation of natural phenomena · 2019-01-08 · phenomenon ! pde!...
TRANSCRIPT
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Scientific
Computer modeling and simulation of naturalphenomena
Bastien Chopard
Computer Science Department,
University of Geneva, Switzerland
Lugano, Oct 18, 2016
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Examples of models and methods
I N-body systems, molecular dynamics
I Mathematical equations : ODE, PDE
I Monte-Carlo methods (equilibrium, dynamic, kinetic)
I Cellular Automata and Lattice Boltzmann method
I multi-agent systems
I Discrete event simulations
I Complex network
I L-systems.......
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What is a model ?
Many definitions :
I Simplifying abstraction of reality
I containing only the essential elements with respect to theproblem
I A mathematical or rule-based representation of a phenomenaI But a model may also be :
I A fit of dataI An animal (medical study)I ...
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What is a good model ?
A Einstein :Everything should be made as simple as possible, but not
simnpler
I In silico simulations : understand, predict and control a process
I Allows scientists to formulate new questions that can beaddressed experimentally or theoretically
I Adapt the model to the problem
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Discrete Event SimulationsDo ants find the shortest path between nest and food ?
Nest Food
0 1
0 1fraction of ant in the shortest trail
0
0.5
freq
uen
cy o
ver
10
0 s
imu
lati
on
s
histogram: distribution of ants
longest path shortest path
0 300time
0
50
nu
mb
er o
f an
ts
number of ants in a short and long trail
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Magritte’s apple
I A model is only a model, not reality
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Magritte’s apple
I A model is only a model, not reality
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Same reality, di↵erent models, di↵erent languages
Hydrodynamics
@t
u+ (u ·r)u = �1
⇢rp + ⌫r2u
phenomenon ! PDE! discretization ! numerical solution
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From PDEs to virual universe
One defines a discret universe as an abstraction of the real world
phenomenon ! computer model
Collision Propagation
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Multi-Agent Model
I Set of bacteria moving in space with concetration ⇢(x , y) ofnutrient
I Let ⇢i
(t) be the concentration seen by bacteria b
i
at time t
I if ⇢i
(t) � ⇢i
(t � �t), the bacteria move straight withprobability 0.9
I if ⇢i
(t) < ⇢i
(t � �t), the bacteria move straight withprobability 0.5
I Otherwise it makes a random turn
I Movie
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Beyond the physical space : complex network
A model of opinion propagation in a social network
(Lino Velasquez, UNIGE)
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Voter model : time evolution
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L-systems F ! F [+F ]F [�F ]F , � = 25o.F [ + F ] F [ - F ] F [ + F [ + F ] F [ -
F ] F ] F [ + F ] F [ - F ] F [ - F [ + F ]
F [ - F ] F ] F [ + F ] F [ - F ] F [ + F [
+ F ] F [ - F ] F [ + F [ + F ] F [ - F ]
F ] F [ + F ] F [ - F ] F [ - F [ + F ] F [
- F ] F ] F [ + F ] F [ - F ] F ] F [ + F ]
F [ - F ] F [ + F [ + F ] F [ - F ] F ] F [
+ F ] F [ - F ] F [ - F [ + F ] F [ - F ]
F ] F [ + F ] F [ - F ] F [ - F [ + F ] F [
- F ] F [ + F [ + F ] F [ - F ] F ] F [ +
F ] F [ - F ] F [ - F [ + F ] F [ - F ] F ]
F [ + F ] F [ - F ] F ] F [ + F ] F [ - F ]
F [ + F [ + F ] F [ - F ] F ] F [ + F ] F
[ - F ] F [ - F [ + F ] F [ - F ] F ] F [ +
F ] F [ - F ] FIterations 3,4 and 5
Code for 3 iterations
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Cellular Automata
B. Chopard et M. Droz : Cellular Automata Modeling of PhysicalSystems, Cambridge University Press, 1998.
B. Chopard, Cellular Automata and lattice Boltzmann modeling of
physical systems, Handbook of Natural Computing, Rozenberg,Grzegorz ; Back, Thomas ; Kok, Joost N. (Eds.) Springer, pp.
287–331, 2013
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Definition
What is a Cellular Automata ?
I Mathematical abstraction of the real world, modelingframework
I Fictitious Universe in which everything is discrete
I But, it is also a mathematical object, new paradigm forcomputation
I Elucidate some links between complex systems, universalcomputations, algorithmic complexity, intractability.
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Example : the Parity Rule
I Square lattice (chessboard)
I Possible states sij
= 0, 1
I Rule : each cell sums up the states of its 4 neighbors (north,east, south and west).
I If the sum is even, the new state is sij
= 0 ; otherwise s
ij
= 1
Generate “complex” patterns out of a simple initial condition.
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Pattern generated by the Parity Rule
t=0 t=31 t=43
t=75 t=248 t=292
t=357 t=358 t=359
t=360 t=511 t=571
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CA Definition
I Discrete space A : regular lattice of cells/sites in d dimensions.
I Discrete time
I Possible states for the cells : discrete set S
I Local, homogeneous evolution rule � (defined for aneighborhood N ).
I Synchronous (parallel) updating of the cells
I Tuple : < A, S ,N ,� >
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Neighborhood
I von Newmann
I Moore
I Margolus
I ...
ll lr
ul ur
(b)(a)
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Boundary conditions
I periodic
I fixed
I reflexive
I ....
b a b
periodic
1 a
fixed
a a
adiabatic
b a b
reflection
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Generalization
I Stochastic CA
I Asynchronous update : loss of parallelism, but avoidoscillations
I Non-uniform CA
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Implementation of the evolution rule
m states per cells, k neighbors.
I On-the-fly calculation
I Lookup table
I Finite number of possible universes : mm
k
possible rules wherem is the number of states per cell and k the number ofneighbors.
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Historical notes
I Origin of the CA’s (1940s) : John von Neumann and S. Ulam
I Design a better computer with self-repair and self-correctionmechanisms
I Simpler problem : find the logical mechanisms forself-reproduction :
I Before the discovery of DNA : find an algorithmic way(transcription and translation)
I Formalization in a fully discrete world
I Automaton with 29 states, arrangement of thousands of cellswhich can self-reproduce
I Universal computer
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Langton’s CA
I Simplified version (8 states).
I Not a universal computer
I Structures with their own fabrication recipe
I Using a reading and transformation mechanism
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Langton’s CA : basic cell replication
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Langton’s Automaton : spatial and temporal evolution
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Langton’s CA : some conclusions
I Not a biological model, but an algorithmic abstraction
I Reproduction can be seen from a mechanistic point of view(Energy and matter are needed)
I No need of a hierarchical structure in which the morecompicated builds the less complicated
I Evolving Hardware.
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CA as a mathematical abstraction of reality
I Several levels of reality : macroscopic, mesoscopic andmicroscopic.
I The macroscopic behavior depends very little on the details ofthe microscopic interactions.
I Only “symmetries” or conservation laws survive. Thechallenge is to find them.
I Consider a fictitious world, particularly easy to simulate on a(parallel) computer with the desired macroscopic behavior.
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A Caricature of reality
t=4 t=10 t=54
What is this ?
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The real thing
Wilson Bentley, 1902
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Snowflakes model
I Very rich reality, many di↵erent shapes
I Complicated true microscopic description
I Yet a simple growth mechanism can capture some essentialfeatures
• A vapor molecule solidifies (!ice) if one and only onealready solidified molecule is in its vicinity• Growth is constrained by 60o angles
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Examples of CA rulesCooperation models : annealing rule
I Growth model in physics : droplet, interface, etcI Biased majority rule : (almost copy what the neighbors do)
Rule :
sumij
(t) 0 1 2 3 4 5 6 7 8 9
s
ij
(t + 1) 0 0 0 0 1 0 1 1 1 1
The rule sees the curvature radius of domains
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Cells di↵erentiation in drosophila
In the embryo all the cells are identical. Then during evolution theydi↵erentiate
I slightly less than 25% become neural cells (neuroblasts)
I the rest becomes body cells (epidermioblasts).
Biological mechanisms :
I Cells produce a substance S (protein) which leads todi↵erentiation when a threshold S0 is reached.
I Neighboring cells inhibit the local S production.
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CA model for a competition/inhibition process
I Hexagonal lattice
I The values of S can be 0 (inhibited) or 1 (active) in eachlattice cell.
I A S = 0 cell will grow (i.e. turn to S = 1) with probabilityp
grow
provided that all its neighbors are 0. Otherwise, it staysinhibited.
I A cell in state S = 1 will decay (i.e. turn to S = 0) withprobability p
decay
if it is surrounded by at least one active cell.If the active cell is isolated (all the neighbors are in state 0) itremains in state 1.
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Di↵erentiation : results
(b)(a)
The two limit solutions with density 1/3 and 1/7, respectively.
I CA produces situations with about 23% of active cells, foralmost any value of p
anihil
and p
growth
.
I Model robust to the lack of details, but need for hexagonalcells
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Excitable Media, contagion models
I 3 states : (1) normal (resting), (2) excited (contagious), (3)refractory (immuned)1. excited ! refractory2. refractory! normal3. normal ! excited, if there exists excited neighbors (otherwise,
normal ! normal).
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Greenberg-Hastings Model
Is 2 {0, 1, 2, ..., n � 1}
I normal : s = 0 ; excited s = 1, 2, ..., n/2 ; the remaining statesare refractory
I contamination if at least k contaminated neighbors.
t=5 t=110
t=115 t=120
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Belousov-Zhabotinski (tube worm)
The state of each site is either 0 or 1 ; a local timer with values 0,1, 2 or 3 controls the 0 period.
(i) where the timer is zero, thestate is excited ;
(ii) the timer is reset to 3 for theexcited sites which have two, ormore than four, excited sites intheir Moore neighborhood.
(iii) the timer is decreased by 1unless it is 0 ;
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Forest fire
(1) a burning tree becomes anempty site ;
(2) a green tree becomes a burningtree if at least one of its nearestneighbors is burning ;
(3) at an empty site, a tree growswith probability p ;
(4) A tree without a burningnearest neighbor becomes aburning tree during one timestep with probability f
(lightning).
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Complex systems
Rule of the Game of Life :
I Square lattice, 8neighbors
I Cells are dead or alive(0/1)
I Birth if exactly 3living neighbors
I Death if less than 2or more than 3neighbors
t t+10 t+20
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Complex Behavior in the game of life
Collective behaviors develop (beyond the local rule)“Gliders” (organized structures of cell) can emerge and can movecollectively.
t=0 t=1 t=2 t=3 t=4
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Complex Behavior in the game of life
A glider gun (image : Internet)
I There are more complex structures with more complexbehavior : a zoology of organisms.
I The game of life is a Universal computer
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Langton’s ant
This is an hypothetical animal moving on a 2D lattice, acoring tosimple rules, which depend on the color of the cell on which theant sits.
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The rules
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Some evolution steps
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Some evolution steps
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Some evolution steps
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Some evolution steps
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Some evolution steps
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Some evolution steps
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Some evolution steps
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Some evolution steps
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Some evolution steps
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Where does the ant go in the long run
I Animation...
I
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Where does the ant go in the long run
I Animation...
I t=6900 t=10431 t=12000
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The ants always escape to infinity
for any initial coloration of the cells
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What about many ants ?
I Adapt the “change ofcolor” rule
I Cooperative anddestructive e↵ects
I The trajectory can bebounded or not
I Past/futur symmetryexplains periodic motion
t=2600 t=4900 t=8564
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Impact on the scientific methodolgy
I The laws are perfectly known
I But we cannot predict the details of the movements (whendoes a highway appears)
I Microscopic knolwdge is not enough to predict themacroscopic behavior
I Then, the only solution is the observe the behavior
I The only information we have on the trajectory are the reflectof the symmetries of the rule
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Prediction means to compute faster than reality
t=0
t=31
t=43
t=75
t=248
t=292
t=357
t=358
t=359
t=360
t=511
t=571
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Prediction means to compute faster than reality
(a) (b) (c)
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Wolfram’s rules256 one-dimensional, 3 neighbors Cellular Automata :
(a) (b) (c) (d)
Coombs, Stephen 2009, The Geometry and Pigmentation of Seashells
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Wolfram’s rules : complexity classes
(a) (b) (c) (d)
I Class I Reaches a fixed point
I Class II Reaches a limit cycle
I Class III self-similar, chaotic attractor
I Class IV unpredicable persistent structures, irreducible,universal computer
Note : it is undecidable whether a rule belongs or not to a givenclass.
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Wolfram’s rules : 1D, 5 neighbors
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Other simple rules
I time-tunnel
Sum(t) = C (t) + N(t) + S(t) + E (t) +W (t)
C (t + 1) =
⇢C (t � 1) if Sum(t) 2 {0, 5}1� C (t � 1) if Sum(t) 2 {1, 2, 3, 4}
I random
C (t + 1) = (S(t).and .E (t)).xor .W (t).xor .N(t).xor .C (t)
I string : a one-dimensional spring-bead system
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Tra�c Models
A vehicle can move only when the downstream cell is free.
time t
time t+1
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Flow diagram
The car density at time t on a road segment of length L is definedas
⇢(t) =N(t)
L
where N is the no of cars along L
The average velocity < v > at time t on this segment is defined as
< v >=M(t)
N(t)
where M(t) is the number of car moving at time t
The tra�c flow j is defined as
j = ⇢ < v >
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Flow diagram of rule 184
0 1 car density
0
1
Tra
ffic
flo
w
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Tra�c in a Manhattan-like city
a b
cd
e
f g
h
(a) (b)
(a)
0 1car density
0
1
<v
>
free rotary
road spacing=4road spacing=32road spacing=256
(b)
0 1car density
0
0.35
Tra
ffic
flo
w
traffic-lightfree rotaryflip-flop
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Case of the city of Geneva
I 1066 junctions
I 3145 roadsegments
I 560886 road cells
I 85055 cars
Origin
Destination
1
3
2
43
14
2
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Travel time during the rush hour
I3
I3
2
time
0 I
insertion probability
p2
p1
Average travel time
0 5 10 15 20 25 30 35 40 45
10
15
20
25
30
35
Departure time [minutes]
Tra
vel t
ime
[m
inu
tes]
Trip 2
Average travel time
0 5 10 15 20 25 30 35 40 45
10
15
20
25
30
35
Departure time [minutes]
Tra
vel t
ime
[m
inu
tes]
Trip 3
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Lattice gases
Fully discrete molecular dynamics
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Example : HPP model collision rules
I HPP : Hardy, Pomeau, dePazzis, 1971 : kinetic theory ofpoint particles on the D2Q4lattice
I FHP : Frisch, Hasslacher andPomeau, 1986 : first LGAreproducing a (almost) correcthydrodynamic behavior(Navier-Stokes eq.)
(a)
(b)
(c)
time t time t+1
Exact mass and momentum conservation : that is what reallymatters for a fluid ! ! !
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FHP model
p=1/2
p=1/2
Stochastic rule with Conservation of mass and momentum.
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Flow past an obstacle (FHP)
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Why can such a simple model work ?
I At a macroscopic scale, the detail of the interaction does notmatter so much
I Only conservation laws and symmetries are important
I We can invent our own fluid, especially one adapted tocomputer simulation
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Demos
I Pressure/density wave : aniotropy
I Reversibility
I Spurious invariants : momentum along each line and column,checkerboard invariant
I Di↵usion, DLA, reaction-di↵usion models
I Snow transport by wind
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Lattice Boltzmann (LB) models
I Lattice Gases implement an exact dynamics
I But they require large simulations, statistical averages andhave little freedom to adjust problem parameters
I In the early 1990s, the discrete Boltzmann equation describingthe average dynamics of a lattice Gas was re-interpreted (withimprovements) as a flow solver
I ! Lattice Boltzmann models
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The lattice Boltzmann (LB) method : the historical way
I Historically, LB was born from Lattice Gases, discrete kineticmodels of colliding particles
I Now the LB method is often derived by a discretizationprocedure (in velocity, space and time variables) of thestandard Boltzmann equation
@t
f (v , r , t) + v · @r
f (v , r , t) = ⌦(f )
I where f (v , r , t) is the density distribution of particles atlocation r , time t, with velocity v .
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The Lattice Boltzmann scheme : definitions
v1
v2
v3
v4
v5
v6 v7 v8
I Possible particle velocities : vi
, i = 0, 1, ..., q � 1
I Lattice spacing : �x , time step : �t, |vi
| = �x/�t.
If
in
i
(r, t) is the density of particle entering site r with velocityvi
, at time t.
I Density : ⇢(r, t) =P
i
f
in
i
;
I Velocity : ⇢u =P
i
f
in
i
vi
I Momentum tensor ⇧↵� =P
i
f
in
i
(r, t)vi↵vi�
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The Lattice Boltzmann scheme : definitions
Collision Propagation
I Possible particle velocities : vi
, i = 0, 1, ..., q � 1
I Lattice spacing : �x , time step : �t, |vi
| = �x/�t.
If
in
i
(r, t) is the density of particle entering site r with velocityvi
, at time t.
I Density : ⇢(r, t) =P
i
f
in
i
;
I Velocity : ⇢u =P
i
f
in
i
vi
I Momentum tensor ⇧↵� =P
i
f
in
i
(r, t)vi↵vi�
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The Lattice Boltzmann scheme : definitions
Collision Propagation
I Possible particle velocities : vi
, i = 0, 1, ..., q � 1
I Lattice spacing : �x , time step : �t, |vi
| = �x/�t.
If
in
i
(r, t) is the density of particle entering site r with velocityvi
, at time t.
I Density : ⇢(r, t) =P
i
f
in
i
;
I Velocity : ⇢u =P
i
f
in
i
vi
I Momentum tensor ⇧↵� =P
i
f
in
i
(r, t)vi↵vi�
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The Lattice Boltzmann scheme : definitions
Collision Propagation
I Possible particle velocities : vi
, i = 0, 1, ..., q � 1
I Lattice spacing : �x , time step : �t, |vi
| = �x/�t.
If
in
i
(r, t) is the density of particle entering site r with velocityvi
, at time t.
I Density : ⇢(r, t) =P
i
f
in
i
;
I Velocity : ⇢u =P
i
f
in
i
vi
I Momentum tensor ⇧↵� =P
i
f
in
i
(r, t)vi↵vi�
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The Lattice Boltzmann scheme : definitions
Collision Propagation
I Possible particle velocities : vi
, i = 0, 1, ..., q � 1
I Lattice spacing : �x , time step : �t, |vi
| = �x/�t.
If
in
i
(r, t) is the density of particle entering site r with velocityvi
, at time t.
I Density : ⇢(r, t) =P
i
f
in
i
;
I Velocity : ⇢u =P
i
f
in
i
vi
I Momentum tensor ⇧↵� =P
i
f
in
i
(r, t)vi↵vi�
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The Lattice Boltzmann scheme : dynamics
Collision Propagation
I Collision : f outi
= f
in
i
+ ⌦i
(f )
I Propagation : f ini
(r +�tvi
, t +�t) = f
out
i
(r, t)
Collision and Propagation :
f
i
(r +�tvi
, t + ⌧) = f
i
(r, t) + ⌦i
(f ) (1)
where f = f
in
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The single relaxation time LB scheme (BGK)
The collision term ⌦i
is a relaxation towards a prescribed localequilibrium distribution
⌦i
(f ) =1
⌧(f eqi
(⇢,u)� f
i
) (2)
where
f
eq
i
= ⇢wi
(1 +vi
· uc
2s
+1
c
4s
Q
i↵�u↵u�) (3)
contains the desired physics (here hydrodynamics) and Q
i↵� is
Q
i↵� = v
i↵vi� � c
2s
�↵�
⌧ is a constant called the relaxation time
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Choice of the vi
and lattice weight wi
The “microscopic” velocities vi
must be such that there existsconstants w
i
and c
2s
so that :
X
i
w
i
= 1
X
i
w
i
vi
= 0
X
i
w
i
v
i↵vi� = c
2s
�↵�
X
i
w
i
v
i↵vi�vi� = 0
X
i
w
i
v
i↵vi�vi�vi� = c
4s
(�↵���� + �↵���� + �↵����)
X
i
w
i
v
i↵vi�vi�vi�vi✏ = 0 (4)
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Lattice Geometries DdQq
d is the space dimension and q the number of microscopicvelocities
I D2Q9 : 2D, square lattice with diagonals and rest particles.
I D3Q19 : 3D, with rest particles
have enough symmetries.
v1
v2
v3
v4
v5
v6 v7 v8
w0 = 4/9 w1 = w3 = w5 = w7 = 1/9
w2 = w4 = w6 = w8 = 1/36
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Continuous limit
Up to order O(�x
2) and O(�t
2), and provied that Ma << 1, theLB eq.
f
i
(r +�tvi
, t + ⌧) = f
i
(r, t) +1
⌧(f eqi
� f
i
) (5)
is equivalent to Navier-Stokes equations
⇢@t
⇢+ @↵⇢u↵ = 0@t
u+ (u ·r)u = �1⇢rp + ⌫r2u
(6)
for ⇢ =P
i
f
i
and ⇢u =P
i
f
i
u.
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Properties :
Viscosity :⌫ = c
2s
�t(⌧ � 1/2)
Pressure :p = ⇢c2
s
Thus, LB-fluids are compressible
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Relations between the fi
’s and the hydrodynamic quantites
Hydrodynamic quantitiesfrom the f
i
f
i
from the hydrodynamic quantities
I ⇢ =P
i
f
i
I ⇢u =P
i
f
i
vi
I ⇧↵� =P
i
v
i↵vi�fi
If = f
eq + f
neq
If
eq
i
= ⇢wi
(1 + vi
·uc
2s
+ 12c4
s
Q
i↵�u↵u�)
If
neq
i
= ��t⌧ w
i
c
2s
Q
i↵�⇢S↵�
whereS↵� = (1/2)(@↵u� + @�u↵)
andQ
i↵� = v
i↵vi� � c
2s
�↵�
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Boundary conditions
(a) (b) (c)
(a) Specular reflection, (b) bounce back condition and (c) trappingwall conditionThe Bounce Back rule implements a no-slip condition. It is themost common choice :
f
out
i
= f
in
�i
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Boundary conditions : beyond bounce-back
f1
f2f3
f4
f5
f6f7f8
Compute the missing population so as to have the desired physicalproperties
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Pros and cons on the LB method
+ Closer to physics than tomathematics
+ Quite flexible to newdevelopments, intuitive,multiphysics
+ Complicated geometries,cartesian grids
+ no need to solve a Poissonequation
+ Parallelization
- Recent methods
- No e�cient unstructuredgrids
- Intrinsically a timedependent solver
- Not always so easy
- Still some work to have afully consistentthermo-hydrodynamicalmodel.
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More advantages...
I Streaming is exact
I Non-linearity is local
I Numerical viscosity is negative
I Extended range of validity for larger Knudsen numbers
I Palabos open source LB software (http ://www.palabos.org)
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Wave equation
v1
v2
v3
v4
f1
f0
f1
f2
f3
f4
(a) (b)
f
i
(r + ⌧vi
, t + ⌧) = f
i
(r, t) + 2(f eqi
� f
i
) (7)
f
eq
i
= a⇢+ bu · vi
Conservation of ⇢, its current u and time reversibility. Note thatPf
2i
is also conserved.This is equivalent to
@2t
⇢+ c
2r2⇢ = 0
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CA for Reaction-Di↵usion processes
p0
p p2
p
A B C A B
C
ν=1 ν=0
Di↵usion Reaction
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LB Reaction-Di↵usion
f
i
(r +�tvi
, t + ⌧) = f
i
(r, t) + !(f eqi
� f
i
) +�t
2dR (8)
with R the reaction term (for instance R = �k⇢2). and
f
eq =1
2d⇢
This is equivalent to
@t
⇢ = Dr2⇢+ R
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Demos
http ://cui.unige.ch/⇠chopard/CA/Animation/root.html
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Palabos : an Open-Source solver (UNIGE)Multiphysics, same code from laptop to massively parallelcomputer : (www.palabos.org)Droplet Pumps Washing machines
Energy converter Air conditioning sedimentation
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Simulation of river Rhone in Geneva
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How to treat cerebral aneuryms : flow diverters
I The stentreducesbloodflow inthe aneurysm
I Clotting isinduced in theaneurysm
Our goal is to elucidate the mechanisms leading to thrombusformation from biological knowledge and numerical modeling
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Fully resolved simulation with a flow diverter
Pipeline flow diverter from EV3-COVIDIEN�x �t diameter # fluid nodes Re
25 µm 1 µs 3.7 mm 40 millions ⇡ 300
CPU time : 10 days (on 120 WestmereIntel cores)
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Spatio-temporal Thrombosis Model
I Low shear : creation of TF, thenthrombin from endothelial cells
I Fibrinogen and anti-thrombin are insuspension, brought by fresh blood
I thrombin+fibrinogen ! fibrin(=clot)
I thrombin+anti-thrombin ! 0
I Platelets attach to the fibrin,compact the clot and allowre-endothelialization
I Clot stops to grow when allthrombin molecules have beenconsumed
Need clever multiscale solutions for the numerical implementation
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Thrombosis Model
Pulsatile versus steady flow
0 0.2 0.4 0.6 0.8 1 1.2 1.40.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
s
m/s
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Simulation of the thrombus in giant aneurysm
0 0.5 1 1.5 2 2.50.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
t [s]
u [m
/s]
Mean velocity (two periods)
I movie
I accelerated for2200 heart cycles
⌫ ⇢ inlet diam. aneurysm size inlet flow3.7e-6 m
2/s 1080 kg/m3 0.8 mm 8 cm 4⇥ 10�6m
3/s
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Validation with a patient
Blue : patient Red : simulation
Another case
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Vertebroplasty
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Palabos Simulation
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Experiment versus simulation
After 6 ml After 7 ml
Good agreement within experimental errors
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Dynamical load balancing on PalabosDomains reallocation at regular time intervals
Performance with andwithout data migra-tion
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Exercices
I Play with a python code producing a 2D flow around a sphere(d2q9.py). For instance, change the Reynolds number RE
I Play with a python code modeling the movement of bacteriain a field of nutrients (bacteria.py). Try to add a source anddi↵usion of nutrients, and the change in concentration wheneaten by the bacteria
http://cui.unige.ch/
~
chopard/FTP/USI/
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Acknowledgments
I Jonas Latt
I Yann Thorimbert
I Orestis Malaspinas