lattice-boltzmann methods: from basics to fluid...

Folie 1ECCOMAS School Adv. Comp. Methods for FSI – Lattice Boltzmann Methods4.05.2006

LatticeLattice--BoltzmannBoltzmann MethodsMethods::fromfrom Basics to Basics to FluidFluid--StructureStructure--InteractionInteraction

Manfred KrafczykManfred KrafczykInstitute for Computer Applications in Civil Engineering

[email protected]://www.cab.bau.tu-bs.de

with contributions fromSebastian Geller & Jonas Tölke

http://www.cab.bau.tu-bs.de/

05.2006 ECCOMAS School Adv. Comp. Methods for FSI – Lattice Boltzmann Methods

overview

a simple kinetic approach„The“ Lattice-Boltzmann method ?efficiency considerations

MRTgrid refinement

fluid-structure-interactionconclusion


continuous fluid particle ensemble(real gas)

ρν t),,r(v ,p, rrt),r(v ,,m ii

rrω

t),r(eirr


( ) ( ) i,, Ω=−++ txntttexn iiirrr δδ

Evolution equation for ‚digital‘ particles:

Coupling to ‚real world properties‘ via moments:

∑−

=

=1

0

m

iinρ i

m

iienu rr ∑

−

=

=1

1

1ρ

)( 02 ρρ −=∞ scp

Chapman-Enskog expansion shows:

Dynamics of velocity and pressure is (for Ma 0) equivalent to that given by the Navier-Stokesequations:


„The“ Lattice-Boltzmann method ?!Boltzmann equation

Taylor- + Chapman-Enskog-Expansion

Chapman-Enskog-Expansion

Navier Stokes equations

continuity equation

BGK-Approximation (Bhatnagar, Gross, Krook)

discretization in space and time

Lattice Boltzmann equation (LBGK)


( ) ( ) ( ) ( ) ( ) ( )( )txftxfx

ttxftttxft

txfttxf eqii

iiiii ,,1,,,, rrrrrrr

−−=Δ

Δ+−Δ+Δ++

Δ−Δ+

τξξ

xt x

Discretization: Finite Differences

Δ=Δ 1ξ

( ) ( ) ( ) ( )( )txftxftttxftttxf eqiiiii ,,,, rrrrr

−Δ

−=Δ+−Δ+Δ+τ

ξTaylor expansion + Chapman-Enskog Analysis:

)2

(2 tsc Δ

−= τρμ ρξρ3

22

1xscp == ∑⎟

⎠⎞

⎜⎝⎛ −Δ

∝i

neqiii ftS )(1

2 βααβ ξξτ


LBGK

Typical claims (prejudices ??) found in the literature:

suited for flows in complex geometries

efficient for complex fluids (especially multiphase problems)

easy to vectorize

very good scaling on parallel machines

no numerical scheme („cellular automata“)

easy to program

The LB-equation


Does LBGK do a better job than ‚conventional‘ approaches ?or

is it a computationally efficient method ?

structural advantages:linear and exact advection operatorconservative scheme for mass and momentumno numerical viscosity

structural disadvantages:conditionally stablecartesian isotropic gridsinherently transient scheme


computational aspects• basic cells: squares and cubes• coupled space and time resolution• convergence properties:

LBE second-order accurate with respect to the corresponding solution of incompressible Navier-Stokes flow

• because of their explicit nature and local stencil LBE models are very well suited for vectorization and parallelization

• stress tensor locally available

• weakly compressible scheme (no Poisson equation is solved for the pressure)• no numerical viscosity• very efficient explicit time stepping scheme (high cell Reynolds-number)• hydrodynamic boundary conditions are introduced for distributions• conservative scheme for mass and momentum


„Classical“ approach: direct discretization of the Navier-Stokes equations

Second-order symmetric Finite Difference in space, explicit first order in timeunconditionally unstable for the advection equationmax. cell Reynolds number <2 (Navier-Stokes)

Finite Volume second order in space, explicit in timeFinite Differences implicit in space and timeSpectral methodsFinite Elements (h-, p-, hp-version, discontinuous Galerkin)meshless methods (SPH, X-FEM, ....)…

A general statement concerning the efficiency of LBGK in comparison to allof the above methods in general is by no means justifiable !


summary standard LBGK

decent method for CFD

easy to understand, simple programming

no numerical viscosity

substantial improvements possible:improved data structures for non-matrix codes

grid refinement

second order boundary conditions

non-equilibrium initial conditions

multi relaxation time models (MRT)

adaptivity


LBGK for grid level l:

continuity of density/pressure and momentum requires for two grids of level l={c, f }:

Additionally stresses have to be continuous:

( )),(),(

213

),(),( )0(,,

2

,, txftxft

ttxftttxf lili

l

llillili

rrrrr−

Δ+

Δ−=−Δ+Δ⋅+

ξνξ

),(),(),( )0()0()0(

,, txftxftxf ificirrr

==

∑∝∏⎟⎟⎠

⎞⎜⎜⎝

⎛−

Δ=

i

neqiii

neq

l

l ftS )()(12 βααβαβ ξξτ

grid refinement (Filippova 1998, Krafczyk 1998, Yu, 2002, Crouse, 2002)


Linear approximation:

( )( )txftDt

DftOtxft

txfi

l

eqieq

ii

eqi ,

213

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

)()( rrrr

δνξ

+−==Δ+∇+

∂∂

Identity of total derivativesimplies for non-equilibrium parts:

f

c

fi

ci

t

t

ff

δν

δν

213213

)1(,

)1(,

+

+−=

Taylor-expansion delivers equivalent PDE:

( ) ( )( )txftxftDt

DftOtxft

txf eqii

l

ii

i ,,

213

1)(),(),( )( rrrrr−

+−==Δ+∇+

∂∂

δνξ


nested timestepping on non uniform grids

collision

collision

collision

collision collision

collision collision collision collision collision

t=+0 t=+4t=+2t

1

3

2

t=+3t=+1

level

adaptive grid refinement after coarsest time step

propagation


boundary conditions – no slip

),(~),( txfttxf DirectionctioninversDire =Δ+

• simple bounce back

simple bounce back implies second order spatial accuracy only for q=0.5

• second order bounce back (Bouzidi, Firdaouss, Lallemand 2001)

wall

wall

ff~

q

ff~


other boundary conditions

01 uff invers

t ∗+=+ φ

• inflow (velocity):

• outflow (reference pressure):

eqeqinversinvers

t ffff ++−=+1

• multi-reflection BCs [2.4]

2)(2 swii cuewρ−

• moving walls [2.6]:

q < 0,5

2)( swii cquewρ−q >= 0,5

additional terms in case of

•Activation of „new fluid“ nodes / Deactivation of „old fluid“ nodes

(Eulerian grid): interpolation + local Poisson-type iteration

•automatic update of q-values after mesh adaptation


computation of subgrid distances q

NNW

W

SW SE

E

NE

S

qEqSE

qS

qSW

P

SR

Q

T

A

B

NNW

W

SW SE

E

NE

S

qEqSE

qS

qSW

P

SR

Q

T

A

B

D2Q9-Model D3Q19-Model


non-equilibrium initial conditions

Initialization based on1. equilibrium distributions based on constant velocity or2. inconsistent pressure fields may result in undesired accoustic modes and a

degradation of stability / accuracy (e.g. Taylor vortex)

solution for 2):In case of given initial velocity field a consistent pressure field can beobtained by applying successive collision/propagation cycles based on fixed nodalvelocities while the pressure field is allowed to evolve until convergence.

)0,( =txu r

solution for 1):Given a consistent initial velocity and pressure field, compute the correspondingequilibrium distributions and corrections

),(),( )0()0(

txft

txff iiineq

irrr

∇+∂

∂−≅ ξτ


( ) ( ) ( ) ( )( )txmtxmSMtxftttxf eq ,,ˆ,, 1 rrrrr−−=−++ −δδξα

and

( ) ( )txmMtxf ,, 1 rr −=

0≠is for non-conserved moments

optimize to minimize hyperviscosity and other model artefactsis

MRT models [4.x]

),,,,,,0,0,0(ˆ 1epsheatheatvisvise ssssssdiagMSMS == −


Re=2000(52^3 nodes)

GLBE

LBGK

efficiency gainup to ~ 1:100

[4.4]


putting it all together:

MRTsecond order BClocal grid refinementefficient data structures


Benchmark stationary (Ergun Re=1) and transient (Ergun Re=200)

Re 1: pressure plots for y=0.5/0.625H

drag & lift coefficients for obstacle „A“

Re 200: drag & lift for obstacle „D“

FE-solver: „Featflow“, Prof. Turek, TU Dortmundwww.featflow.de

A B C D


LBE grid: level 6 to 9

FE-mesh: level 5


results for the transient case (Ergun Re = 200, cylinder D):

scheme #dof cd[%] cl[%] Tref [%] CPU-time[s] / Tref

LB (6-8)LB (6-9)

199.656243.774

2.40.3

0.41.6

0.50.5

3046

FEM(4)FEM(5)

113.264450.528

1.30.1

0.31.5

9.20.1

22265

CFXCFXCFX

385.485917.616

1.807.428

1.60.5

0

2.51.3

0

0.20.2

0

2856659413440

Ma = 0.02reference: cd = 2.0548

cl = 0.9150Tref = 4.2327

red: FEM, blue: LBE

Geller et al.,Computers & Fluids, in print


summary MRT

excellent method for CFD (demonstrated for transient laminar flows)

complex programming (conservation law of programming complexity ?)

no numerical viscosity

improved boundary conditions via optimization of specific moments(no viscosity dependent slip layer for Stokes flow)

computational effort comparable to LBGK (+~20 %)


alternative discretizations of the discrete Boltzmann equation

Keeping in mind that the Lattice-Boltzmann method can be interpreted as afirst order explict FD discretization, alternative approaches can be considered:

higher order FD methodsFinite Volume methodsFinite Element methodsspectral methods…

utilization of multigrid solvers may provide optimum algorithmic complexity(J. Tölke, M. Krafczyk, E. Rank, J. Stat. Phys, Vol. 107, Nos.1/2, pp. 573-591, 2002, D. Mavriplis, accepted for publication in Comp. & Fluids)

unstructured grids with anisotropic elements


reference results:

Mittal & Kumar, Int. J. Numer. Meth. Fluids 31: 1087-1120 (1999)

rigid body example::Oscillator Re = 325

dimensionless mass = 4,7273,

damping coefficient = 0,00033structural frequency Fs = 0,53

fluid-structure-interaction


LB-solver “VirtualFluids”

(TU Braunschweig)p-FEM solver “AdhoC”

(TU Munich)

fluid-structure-interaction: elastic structure

structure solver Fluid solver

Moving Surface mesh:• flat triangles• loads on nodes• displacements on nodes• linear interpolation


txtxfttxfvF DirinversDir

iii Δ

Δ⋅+Δ+⋅=∑

2

)],(~),([rr

)),(~),((0 txfttxfcdI DirinversDir +Δ+=

momentum exchange (Ladd, 1992, 2002) :

Mapping of loads on surface mesh / method1 (conservative)

forces:


Mapping of loads on surface mesh / method 2 (profile preserving)

stress tensor (local):βααβαβ δρ ,,

2 )2

1( kkk

neqk

viss vvfscP ∑−+=

BCBCCBABABBAB lnPPlnPPF21)

41

43(

21)

43

41( rrr

+++=forces:

linear extrapolation of stress tensor depending on configuration of active nodes:


LB-Fluid-Solver Fe-Structure-Solver

step=0 step=0

step=1

Δts

step=n

nΔtFstep=1

loads

displacements

new surface geometry

internstep 1

internstep 2level

Interpolated structureSurface geometry

internstep 1Interpolated structuresurface geometry

internstep 2level

coupling algorithm


self-induced oscillating beam in a fluid


LB-Fluid-Solver Fe-Structure-Solver

Coupling algorithm (implicit)

Iterate until convergence

U displacements

F loads


Conclusions

LB models are either based on the „evolution“ equation or other numerical discretizations of the discrete Boltzmann equations

both approaches can be utilized to develop efficient tools for complex flows and complexfluids (for Navier-Stokes in the preasymptotic range)

general statements / comparisons of the computational efficiency of LB vs. „classical“ CFD methods are usually meaningless.

suitable boundary conditions (conservative, no Knudsen-layers for arbitrary orientation, strong locality) are available, yet non-trivial

LB methods offer a suitable framework for modeling and simulating multi-scale, multi-physics problems

work in progress: FSI in 3D including turbulence and free surface flow (wave impact)


other LBGK benchmarks:

[1.1] W.F.C. van Wageningen, D. Kandhai , R. F. Mudde, and H.E.A. van den AkkerDynamic Flow in a Kenics Static Mixer: a Comparison between the Lattice Boltzmann methodFLUENT and LDA ExperimentsAIChE J., InPress

[1.2] L.-S. Luo, D. Qi, L.-P. WangApplications of the Lattice Boltzmann Method to Complex and Turbulent FlowsLecture Notes in Computational Science and Engineering 21, pp. 123-130, 2002

[1.3] M. Artoli, D. Kandhai, H.C.J. Hoefsloot, A.G. Hoekstra, P.M.A. SlootLattice BGK simulations of flow in a symmetric bifurcationFGCS, 2004

[1.4] A. Alajbegovic, C. Teixeira, D. Hill, A. AnagnostThe Study of Benchmark Laminar Flows Using DIGITAL PHYSICS

[1.5] M. Rottensteiner, B. Hupertz, H. Fogt, L. LührmannComparison of different CFD tools based on a standardized post-processingSAITS 01164, 2001

[1.6] R. Shock, S. Mallick, H. Chen, V. Yakhot, R. ZhangRecent simulation results on 2D NACA airfoils using a lattice Boltzmann based algorithm


second order boundary condition publications:

[2.1] P. Lallemand, L.-S. Luo Lattice Boltzmann method for moving boundariesJournal of Computational Physics 184, 406–421, 2003

[2.2] I. Ginzburg, D. d’HumiéresMulti-reflection boundary conditions for lattice Boltzmann modelsPhysical Review E Stat Nonlin Soft Matter Physics, 68(6 Pt 2):066614, 2003

[2.3] D. Yu, R. Mei, L.-S. Luo, and W. ShyyViscous flow computations with the method of lattice Boltzmann equationProgress in Aerospace Sciences 39(5):329-367, 2003

[2.4] M. Bouzidi, M. Firdaouss, P. LallemandMomentum transfer of a Boltzmann-lattice fluid with boundariesPhysics of Fluids, Volume 13, Issue 11, pp. 3452-3459, 2001

[2.5] I. Ginzburg, D. d'HumièresLocal second-order boundary methods for lattice Boltzmann modelsJ. Stat. Physics 84, 927, 1996

[2.6] R. Verberg, A.J.C. LaddA lattice-Boltzmann model with sub-grid scale boundary conditionsPhysical Review Letters, 84:2148-2151, 2000


multi relaxation time model publications:

[4.1] D. d’HumieresGeneralized lattice Boltzmann equations. Rarefied gas dynamics: theory and simulationsProg. Aeronaut. Astronaut. 159, 450-458, 1992

[4.2] P. Lallemand and L.-S. LuoTheory of the lattice Boltzmann method: Dispersion,dissipation, isotropy, Galilean invariance, and stabilityPhysical Review E 61, No. 6 (2000) 6546–6562, 2000

[4.3] M. Bouzidi, D. d’Humieres, P. Lallemand, L.-S. LuoLattice Boltzmann Equation on a two-dimensional Rectangular GridJournal of Computational Physics 172:704-717, 2001

[4.4] D. d’Humieres, I. Ginzburg, M. Krafczyk, P. Lallemand, L.-S. LuoMultiple-relaxation-time Lattice Boltzmann Models in 3DPhilosophical Transactions of Royal Society of London A 360(1792):437-451, 2002

lattice-boltzmann methods: from basics to fluid...

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