Introduction to Lattice-Boltzmann Method

Introduction to

Introduction to

the Lattice Boltzmann Methods:

the Lattice Boltzmann Methods:

From Theory to Practice

From Theory to Practice

Wanho Choi

Graphics & Media Lab. at Seoul National Univ.

A Recent Result using LBM

History of LBM

History of LBM

Cellular Automata

Lattice Gas Automata

Lattice Boltzmann Methods

Graphics & Media Lab. at Seoul National Univ.

Weak Definitions for the Cellular Automata

Weak Definitions for the Cellular Automata

 _{Regular arrangements of single cells of the same kind}  _{Finite number of discrete states per each cell.}

 _{Simultaneous (synchronous) update for cell states}  Deterministic and uniform update rules

What is the Lattice Boltzmann Methods?

What is the Lattice Boltzmann Methods?

 _{Relatively new simulation technique}

 _{Particle based formulation to simulate fluid flows}  _However, mesoscopic _{model rather than microscopic}

– Microscopic model

– Mesoscopic kinetic equations

※ Microscopic model: molecular dynamics (MD), etc.

※ Macroscopic model: finite [difference|element|volume] method (FDM, FEM, FVM), etc.

 It considers particle distributions rather than individual particles.

Graphics & Media Lab. at Seoul National Univ.

Comparisons

Comparisons

 Pros.

– Straightforward implementation

– Complex boundary treatment

– Easy to parallelize (fully explicit)

 Cons.

– Weakly compressible

– Diffusive motions

Parallelism

Parallelism

 Nine PS3 connected via a Gigabit Ethernet switch

 Fedora Core 6

Graphics & Media Lab. at Seoul National Univ.

Particle Distribution Function

Particle Distribution Function

)

,

,

(

t

Particle Distribution Function

Particle Distribution Function

 f: Particle distribution function in phase space (x,

ξ

)

)

,

,

(

t

Graphics & Media Lab. at Seoul National Univ.

Particle Distribution Function

Particle Distribution Function

 f: Particle distribution function in phase space (x,

ξ

)

 The probability of finding a particle(=molecule) traveling with velocity

ξ

at position x & time t.

)

,

,

(

t

Particle Distribution Function

Particle Distribution Function

 f: Particle distribution function in phase space (x,

ξ

)

 The probability of finding a particle(=molecule) traveling with velocity

ξ

at position x & time t.

 The amount of particles traveling with a velocity

ξ

at position x & time t.

)

,

,

(

t

Graphics & Media Lab. at Seoul National Univ.

Particle Distribution Function

Particle Distribution Function

 f: Particle distribution function in phase space (x,

ξ

)

 The probability of finding a particle(=molecule) traveling with velocity

ξ

at position x & time t.

 The amount of particles traveling with a velocity

ξ

at position x & time t.

)

,

,

(

t

f

x

ξ

1

e

5

e

1

e

2

e

6

e

3

e

8

e

7

e

e₁ e₅ e₈ e₃ e₂ e₆ e₇ e₄ f₁ f₃ f₂ f₄ f₅ f₆ f_{7 f} 8

Particle Distribution Function

Particle Distribution Function

 f: Particle distribution function in phase space (x,

ξ

)

 The probability of finding a particle(=molecule) traveling with velocity

ξ

at position x & time t.

 The amount of particles traveling with a velocity

ξ

at position x & time t.

 Macroscopic properties:

)

,

,

(

t

f

x

ξ

1

e

5

e

1

e

2

e

6

e

3

e

8

e

7

e

e₁ e₅ e₈ e₃ e₂ e₆ e₇ e₄ f₁ f₃ f₂ f₄ f₅ f₆ f_{7 f} 8

Graphics & Media Lab. at Seoul National Univ.

Particle Distribution Function

Particle Distribution Function

 f: Particle distribution function in phase space (x,

ξ

)

 The probability of finding a particle(=molecule) traveling with velocity

ξ

at position x & time t.

 The amount of particles traveling with a velocity

ξ

at position x & time t.

 Macroscopic properties: – Density:

)

,

,

(

t

f

x

ξ



 x ξ ξ x,t) f ( , ,t)d (  1

e

5

e

1

e

2

e

6

e

3

e

8

e

7

e

e₁ e₅ e₈ e₃ e₂ e₆ e₇ e₄ f₁ f₃ f₂ f₄ f₅ f₆ f_{7 f} 8

Particle Distribution Function

Particle Distribution Function

 f: Particle distribution function in phase space (x,

ξ

)

 The probability of finding a particle(=molecule) traveling with velocity

ξ

at position x & time t.

 The amount of particles traveling with a velocity

ξ

at position x & time t.

 Macroscopic properties: – Density: – Velocity:

)

,

,

(

t

f

x

ξ



 x ξ ξ x,t) f ( , ,t)d ( 



  ξ x ξ ξ x x u f t d t t ( , , ) ) , ( 1 ) , ( 

e

1 5

e

1

e

2

e

6

e

3

e

8

e

7

e

e₁ e₅ e₈ e₃ e₂ e₆ e₇ e₄ f₁ f₃ f₂ f₄ f₅ f₆ f_{7 f} 8

Graphics & Media Lab. at Seoul National Univ.

LBM Models

LBM Models

 _{Lattice is composed of many cells on Cartesian grid.}

 _DdQq

– d: the number of dimension

Boltzmann Equation

Boltzmann Equation

 _{It approximates the fluid by a dilute gas of particles.}

 _{In a dilute gas, molecules move freely as particles most of time except for}

two-body collisions.

 _{Such a dilute gas can be described by the} Boltzmann Equation_.

Graphics & Media Lab. at Seoul National Univ.

Boltzmann Equation

Boltzmann Equation

 _{It approximates the fluid by a dilute gas of particles.}

 _{In a dilute gas, molecules move freely as particles most of time except for}

two-body collisions.

 _{Such a dilute gas can be described by the} Boltzmann Equation_.

(Ludwig Boltzmann, 1872)

)

( f

Ω

f

f

t

f

_





















ξ

F

x

ξ

Boltzmann Equation

Boltzmann Equation

 _{It approximates the fluid by a dilute gas of particles.}

 _{In a dilute gas, molecules move freely as particles most of time except for}

two-body collisions.

 _{Such a dilute gas can be described by the} Boltzmann Equation_.

(Ludwig Boltzmann, 1872)

– f: particle distribution function

– ξ: microscopic velocity (=particle velocity)

– F: external force field

– Ω: collision operator (due to the interaction of molecules)

)

( f

Ω

f

f

t

f

_





















ξ

F

x

ξ

Graphics & Media Lab. at Seoul National Univ.

BGK Collision Model

BGK Collision Model

 _{Recall the Boltzmann equation.}

) ( f f f t f _{ }           ξ F x ξ

BGK Collision Model

BGK Collision Model

 _{Recall the Boltzmann equation.}

 The collision operator(Ω) is fully non-linear, and hard to be described.

– BGK(Bhatnagar-Gross-Krook) approximation ) ( f f f t f _{ }           ξ F x ξ

Graphics & Media Lab. at Seoul National Univ.

BGK Collision Model

BGK Collision Model

 _{Recall the Boltzmann equation.}

 The collision operator(Ω) is fully non-linear, and hard to be described.

– BGK(Bhatnagar-Gross-Krook) approximation



f

f

f

eq BGK







(

)

) ( f f f t f _{ }           ξ F x ξ

BGK Collision Model

BGK Collision Model

 _{Recall the Boltzmann equation.}

 The collision operator(Ω) is fully non-linear, and hard to be described.

– BGK(Bhatnagar-Gross-Krook) approximation

– feq: equilibrium distribution (Maxwell-Boltzmann distribution)

– It means the difference from the equilibrium during the relaxation time λ.

– Standard model for practical use

– P. L. Bhatnagar, E. P. Gross, and M. Krook. A model for collision processes in gases. Phys. Rev., 94. 511-525, 1954



f

f

f

eq BGK







(

)

) ( f f f t f _{ }           ξ F x ξ

Graphics & Media Lab. at Seoul National Univ. RT m D eq

_e

RT

m

f

2 | | 2 2 2 2

2

u ξ 



















Maxwell

 D: dimension

 m: mass of a particle (We will use the normalized mass, m=1.)  _{R: Boltzmann constant}

 _{T: temperature}

 ξ: microscopic velocity (particle velocity)

 u: macroscopic velocity (mean particle velocity, fluid velocity)  ρ: fluid density RT m D eq

_e

RT

m

f

2 | | 2 2 2 2

2

u ξ 



















Maxwell

Graphics & Media Lab. at Seoul National Univ.

 D: dimension

 m: mass of a particle (We will use the normalized mass, m=1.)  _{R: Boltzmann constant}

 _{T: temperature}

 ξ: microscopic velocity (particle velocity)

 u: macroscopic velocity (mean particle velocity, fluid velocity)  ρ: fluid density

 Approximation (Taylor expansion in u up to the 2nd _order)

– Assumption: small velocities, and low Mach number

RT m D eq

_e

RT

m

f

2 | | 2 2 2 2

2

u ξ 



















Maxwell

 D: dimension

 m: mass of a particle (We will use the normalized mass, m=1.)  _{R: Boltzmann constant}

 _{T: temperature}

 ξ: microscopic velocity (particle velocity)

 u: macroscopic velocity (mean particle velocity, fluid velocity)  ρ: fluid density

 Approximation (Taylor expansion in u up to the 2nd _order)

– Assumption: small velocities, and low Mach number

RT m D eq

_e

RT

m

f

2 | | 2 2 2 2

2

u ξ 



















Maxwell

Maxwell

-

-

Boltzmann Distribution

Boltzmann Distribution





 

 

























 

RT

RT

RT

e

RT

f

RT D eq

2

2

1

2

2 2 2 2 2 /

u

u

ξ

u

ξ

ξ ξ





Graphics & Media Lab. at Seoul National Univ.

Discretization

Discretization

of the Equilibrium Distribution Function

of the Equilibrium Distribution Function













_

_

_

_

_

_



(

)

2

3

)

(

2

9

)

(

3

1

)

,

(

u

₂

u

₂

u

2 ₂

u

u

c

e

c

e

c

w

f

eq _{i i i} i





Discretized

Discretized

Boltzmann Equation

Boltzmann Equation

) ( f f f t f _{ }           ξ F x ξ : Boltzmann eqn.

Graphics & Media Lab. at Seoul National Univ.

Discretized

Discretized

Boltzmann Equation

Boltzmann Equation

 f f f f t f eq _            ξ F x ξ ) ( f f f t f _{ }           ξ F x ξ : Boltzmann eqn.

Discretized

Discretized

Boltzmann Equation

Boltzmann Equation

 f f f t f _ eq        x ξ  f f f f t f eq _            ξ F x ξ ) ( f f f t f _{ }           ξ F x ξ : Boltzmann eqn.

: linearized Boltzmann eqn.

Graphics & Media Lab. at Seoul National Univ.

Discretized

Discretized

Boltzmann Equation

Boltzmann Equation

 f f f t f _ eq        x ξ              x ξ f t f dt df  f f f f t f eq _            ξ F x ξ ) ( f f f t f _{ }           ξ F x ξ : Boltzmann eqn.

: linearized Boltzmann eqn.

Let’s ignore the external force term is ignored for the derivation.



f f

dt

df eq _

Discretized

Discretized

Boltzmann Equation

Boltzmann Equation

 f f f t f _ eq        x ξ              x ξ f t f dt df  f f f f t f eq _            ξ F x ξ ) ( f f f t f _{ }           ξ F x ξ : Boltzmann eqn.

: linearized Boltzmann eqn.

Let’s ignore the external force term is ignored for the derivation.



f f

dt

df eq _

 : ordinary differential equation (ODE) form Assume that δ_t is small enough and feqis smooth enough.



( ) ( )

 

1 ( ) ( )



) ( ) (t f t f t f t f t f t f t eq eq t          

Graphics & Media Lab. at Seoul National Univ.

Discretized

Discretized

Boltzmann Equation

Boltzmann Equation

 f f f t f _ eq        x ξ              x ξ f t f dt df  f f f f t f eq _            ξ F x ξ ) ( f f f t f _{ }           ξ F x ξ : Boltzmann eqn.

: linearized Boltzmann eqn.

Let’s ignore the external force term is ignored for the derivation.



f f

dt

df eq _

 : ordinary differential equation (ODE) form Assume that δ_t is small enough and feqis smooth enough.



( ) ( )

 

1 ( ) ( )



) ( ) (t f t f t f t f t f t f t eq eq t          



(

,

,

)

(

,

,

)



1

)

,

,

(

)

,

,

(

t

f

t

f

t

f

t

f

x



_t

ξ

ξ



_t



x

ξ



eq

x

ξ



x

ξ







LBM Procedure

LBM Procedure



( , , ) ( , , )



1 ) , , ( ) , , ( t f t f t f t f eq t tξ ξ x ξ x ξ x ξ x       

Graphics & Media Lab. at Seoul National Univ.

LBM Procedure

LBM Procedure

 Per each discrete direction

– Streaming – Collision



( , , ) ( , , )



1 ) , , ( ) , , ( t f t f t f t f eq t tξ ξ x ξ x ξ x ξ x       

LBM Procedure

LBM Procedure

 Per each discrete direction

– Streaming – Collision



( , , ) ( , , )



1 ) , , ( ) , , ( t f t f t f t f eq t tξ ξ x ξ x ξ x ξ x       

)

,

(

)

,

(

t

f

t

t

f

_iin

x



_iout

x



e

_i









i eq i in i in i out i

t

f

t

f

t

f

t

F

f

(

x

,

)



(

x

,

)



1

(

x

,

)



(

x

,

)





Graphics & Media Lab. at Seoul National Univ.

LBM Procedure

LBM Procedure

Graphics & Media Lab. at Seoul National Univ.

Pseudo Code for D3Q19

Boundary Condition

Graphics & Media Lab. at Seoul National Univ.

Derivation of Transport Equations

Derivation of Transport Equations

   , ) ( , ) ( , ) ( , ) ( t t t f t f t f t f i i i eq i i i i i i x x x e x     

Derivation of Transport Equations

Derivation of Transport Equations

   , ) ( , ) ( , ) ( , ) ( t t t f t f t f t f i i i eq i i i i i i x x x e x     



3 3



2 ) ( , ) ( . . ) ( ) ( 2 1 ) ( ) ( ) , ( t H OT t t t f t f t t f t f f t t t f i _i i i i i i i i i i     e  e  x e x e x                              3 3 2 , . . ! 2 1 ) , ( ) , ( expansion by y x T O H y y f x x f y y f x x f y x f y y x x f Taylor                                    

Graphics & Media Lab. at Seoul National Univ.

Derivation of Transport Equations

Derivation of Transport Equations

   , ) ( , ) ( , ) ( , ) ( t t t f t f t f t f i i i eq i i i i i i x x x e x     



3 3



2 ) ( , ) ( . . ) ( ) ( 2 1 ) ( ) ( ) , ( t H OT t t t f t f t t f t f f t t t f i _i i i i i i i i i i     e  e  x e x e x                              3 3 2 , . . ! 2 1 ) , ( ) , ( expansion by y x T O H y y f x x f y y f x x f y x f y y x x f Taylor                                     t f t t f t t t f t f here i i i i i i i    _       _{ }                     e e x e x( ) ( ) ,

Derivation of Transport Equations

Derivation of Transport Equations

   , ) ( , ) ( , ) ( , ) ( t t t f t f t f t f i i i eq i i i i i i x x x e x     



3 3



2 ) ( , ) ( . . ) ( ) ( 2 1 ) ( ) ( ) , ( t H OT t t t f t f t t f t f f t t t f i _i i i i i i i i i i     e  e  x e x e x                              3 3 2 , . . ! 2 1 ) , ( ) , ( expansion by y x T O H y y f x x f y y f x x f y x f y y x x f Taylor                                     t f t t f t t t f t f here i i i i i i i    _       _{ }                     e e x e x( ) ( ) ,





     i eq i i i i i i f f t t T O H t f t t f t          _{ }          _{ }    2 3 3 2 ) ( , ) ( . . ) ( 2 1 e e e

Graphics & Media Lab. at Seoul National Univ.

Derivation of Transport Equations

Derivation of Transport Equations

   i eq i i i i i f f t f t t f t         _{ }          _{ }   2 2 ) ( 2 1 e e                                ) , ( ) , ( ) , ( ) , ( ) , ( ; ; ; Thus, . ; expansion Enskog -Chapman by ) 2 ( 2 ) 1 ( ) 0 ( 0 ) ( 0 0 0 1 0 0 1 0 t f t f t f t f t f t where t t t t t t i i i n n i n i x x x x x x x x x       

Derivation of Transport Equations

Derivation of Transport Equations

                               ) , ( ) , ( ) , ( ) , ( ) , ( ; ; ; Thus, . ; expansion Enskog -Chapman by ) 2 ( 2 ) 1 ( ) 0 ( 0 ) ( 0 0 0 1 0 0 1 0 t f t f t f t f t f t where t t t t t t i i i n n i n i x x x x x x x x x       

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         ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( 2 ) 2 ( 2 ) 1 ( ) 0 ( ) 2 ( 2 ) 1 ( ) 0 ( 2 0 1 0 2 0 1 0 t f t f t f t f t f t f t f t t t t i i i eq i i i i i i x x x x x x x e e                                               i eq i i i i i f f t f t t f t         _{ }          _{ }   2 2 ) ( 2 1 e e

Graphics & Media Lab. at Seoul National Univ.

Derivation of Transport Equations

Derivation of Transport Equations

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         ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( 2 ) 2 ( 2 ) 1 ( ) 0 ( ) 2 ( 2 ) 1 ( ) 0 ( 2 0 1 0 2 0 1 0 t f t f t f t f t f t f t f t t t t i i i eq i i i i i i x x x x x x x e e                                           

Derivation of Transport Equations

Derivation of Transport Equations

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         ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( 2 ) 2 ( 2 ) 1 ( ) 0 ( ) 2 ( 2 ) 1 ( ) 0 ( 2 0 1 0 2 0 1 0 t f t f t f t f t f t f t f t t t t i i i eq i i i i i i x x x x x x x e e                                            , of powers different of ts coefficien Collecting       ) 2 ( ) 0 ( 2 0 0 ) 1 ( 0 0 1 ) 0 ( 2 ) 1 ( ) 0 ( 0 0 ) 0 ( 1 ) 0 ( 0 2 1 : : : i i i i i i i i i i eq i i f f t f t t f f f t f f f                                     e e e · · · ② · · · ③ · · · ①

Graphics & Media Lab. at Seoul National Univ.

Derivation of Transport Equations

Derivation of Transport Equations

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         ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( 2 ) 2 ( 2 ) 1 ( ) 0 ( ) 2 ( 2 ) 1 ( ) 0 ( 2 0 1 0 2 0 1 0 t f t f t f t f t f t f t f t t t t i i i eq i i i i i i x x x x x x x e e                                            , of powers different of ts coefficien Collecting       ) 2 ( ) 0 ( 2 0 0 ) 1 ( 0 0 1 ) 0 ( 2 ) 1 ( ) 0 ( 0 0 ) 0 ( 1 ) 0 ( 0 2 1 : : : i i i i i i i i i i eq i i f f t f t t f f f t f f f                                     e e e · · · ① · · · ② · · · ③ 

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              i i i i i i i i i i i i f w f w f w t w ) 1 ( ) 0 ( 0 ) 0 ( 0 directions all over summing and weights g Multiplyin : e ②

Derivation of Transport Equations

Derivation of Transport Equations

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         ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( 2 ) 2 ( 2 ) 1 ( ) 0 ( ) 2 ( 2 ) 1 ( ) 0 ( 2 0 1 0 2 0 1 0 t f t f t f t f t f t f t f t t t t i i i eq i i i i i i x x x x x x x e e                                            , of powers different of ts coefficien Collecting       ) 2 ( ) 0 ( 2 0 0 ) 1 ( 0 0 1 ) 0 ( 2 ) 1 ( ) 0 ( 0 0 ) 0 ( 1 ) 0 ( 0 2 1 : : : i i i i i i i i i i eq i i f f t f t t f f f t f f f                                     e e e 

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              i i i i i i i i i i i i f w f w f w t w ) 1 ( ) 0 ( 0 ) 0 ( 0 directions all over summing and weights g Multiplyin : e ② · · · ① · · · ② · · · ③ 1 , 0 0 1 1 s constraint ) ( ) 0 (                              n f w f w i i n i i i i i i e u e   eqn. Continuity : 0 ) ( 0         u t

Graphics & Media Lab. at Seoul National Univ.

Laplace & Poisson Equation Solver

Laplace & Poisson Equation Solver

Laplace & Poisson Equation Solver

Graphics & Media Lab. at Seoul National Univ.

Laplace & Poisson Equation Solver

Laplace & Poisson Equation Solver

 _{Laplace equation:}

– The solution of a time-dependent diffusion process:

0  X X t X _{ }   _{ }      _{  }   0 then , 0 if X t X 

Laplace & Poisson Equation Solver

Laplace & Poisson Equation Solver

 _{Laplace equation:}

– The solution of a time-dependent diffusion process:

 _{Poisson equation:} 0  X X t X _{ }   _{ }      _{  }   0 then , 0 if X t X  h  X

Graphics & Media Lab. at Seoul National Univ.

Laplace & Poisson Equation Solver

Laplace & Poisson Equation Solver

 _{Laplace equation:}

– The solution of a time-dependent diffusion process:

 _{Poisson equation:}

– The solution of a time-dependent diffusion process:

0  X X t X _{ }   _{ }      _{  }   0 then , 0 if X t X  h  X h      X t X _{ }      _{  }   h X t X then , 0 if 

Laplace & Poisson Equation Solver

Laplace & Poisson Equation Solver

 _{Laplace equation:}

– The solution of a time-dependent diffusion process:

 _{Poisson equation:}

– The solution of a time-dependent diffusion process:

0  X X t X _{ }   _{ }      _{  }   0 then , 0 if X t X  h  X h      X t X _{ }      _{  }   h X t X then , 0 if 

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) , ( u  _i  _i e_i u  _i e_i u 2  _i uu eq i A B C D f     _i eq i A f ( ,u) 

Graphics & Media Lab. at Seoul National Univ.

Results

Results

Graphics & Media Lab. at Seoul National Univ.

Results