Introduction:Introduction:Lattice Boltzmann Method for Lattice Boltzmann Method for Non-fluid ApplicationsNon-fluid Applications

Ye ZhaoYe Zhao

Lattice Boltzmann MethodLattice Boltzmann Method Microscopic numerical solver initially Microscopic numerical solver initially

desinged for fluid dynamicsdesinged for fluid dynamics Simple, explicit and parallel schemeSimple, explicit and parallel scheme Parallel scheme for hardware Parallel scheme for hardware

accelerationacceleration Graphics hardware: GPU, GPU cluster Graphics hardware: GPU, GPU cluster

LBM in Computer GraphicsLBM in Computer Graphics Natural phenomena with fluidsNatural phenomena with fluids

Fluid FlowsFluid Flows Smoke and fireSmoke and fire Ink dispersionInk dispersion Snowing Snowing Liquid mixture Liquid mixture (Wang et al. 2006)(Wang et al. 2006)

More …More … Non-fluids Non-fluids

Lighting Lighting (Geist et al. 2004)(Geist et al. 2004)

Image ProcessingImage Processing

Navier-Stokes EquationNavier-Stokes Equation

Model the macroscopic behavior of Model the macroscopic behavior of fluidfluid

Fupuut

u

ut

21

0

Microscopic DynamicsMicroscopic DynamicsMicroscopic particles

inside fluids

Fictitious particles moving along lattice links

Lattice Boltzmann EquationLattice Boltzmann Equation

f : f : probability distribution function of particle populations on each linkprobability distribution function of particle populations on each link

Recover the Navier-Stokes equationRecover the Navier-Stokes equation At the limit of low Mach number fluids At the limit of low Mach number fluids

Discretize Boltzmann equation on Discretize Boltzmann equation on discrete moving directions discrete moving directions ii (1988)(1988)

)(),(),( ftrfttetrf iiii

Lattice StructureLattice Structure

Single-relaxation-time LBMSingle-relaxation-time LBM

)),(),((1

),(),( * uftrftrftrf eqiiii

CollisionCollision

BGK model BGK model (1954) for equilibrium(1954) for equilibrium

))()((),( 22 ueDCuueBAuf iieq

i

Streaming along a link to a neighborStreaming along a link to a neighbor),()1,( *trfterf iii

Macroscopic PropertiesMacroscopic Properties

Density, velocity and viscosityDensity, velocity and viscosity

6

12

ii

i

ii

efuf

Body ForcesBody Forces

ieFB

2

12)),(),((

1),()1,( trftrftrfterf eq

iiiii

Graphics HardwareGraphics Hardware Graphics processing unitsGraphics processing units (GPU)(GPU)

Low priceLow price Inherently parallelInherently parallel ProgrammableProgrammable

Booming growth rate on speedBooming growth rate on speed 3.0 GHz dual-core Pentium4: 24.6 GFLOPs3.0 GHz dual-core Pentium4: 24.6 GFLOPs NVIDIA GeForce FX 7800: 165 GFLOPsNVIDIA GeForce FX 7800: 165 GFLOPs

Ideal general-purpose applicationsIdeal general-purpose applications Large data setsLarge data sets High parallelismHigh parallelism Minimal dependencies between data Minimal dependencies between data

elementselements

Computation on GPUComputation on GPU

Texture

Images Fragment

Processing

Fragments

Fragment w/ Colors

Graphics:

Data

Numerical

Calculation

Fragments

Results

Computation:

LBM Acceleration on GPULBM Acceleration on GPU Simple addition, subtraction Simple addition, subtraction

and multiplicationand multiplication

Computation on cluster Computation on cluster Large-scale simulationLarge-scale simulation Domain decompositionDomain decomposition

LBM for Non-fluids LBM for Non-fluids ApplicationsApplications

Actually LBM is a special numerical Actually LBM is a special numerical solver for partial differential solver for partial differential equations (PDE)equations (PDE)

PDEs are widely used in graphics and PDEs are widely used in graphics and visualization applicationsvisualization applications Image processingImage processing Surface processingSurface processing Volume graphics and visualizationVolume graphics and visualization Computer vision …Computer vision …

Use LBM in these fields?Use LBM in these fields? Pros:Pros:

LBM is simple to implement LBM is simple to implement both CPU and GPUboth CPU and GPU

Flexible and easy to modify its schemeFlexible and easy to modify its scheme Computational speed very fast with Computational speed very fast with

hardware accelerationhardware acceleration Cons:Cons:

Difficult understanding at the beginningDifficult understanding at the beginning Memory usageMemory usage

Worth a try!Worth a try!

Research TopicsResearch Topics

How to modify LBM scheme in order to How to modify LBM scheme in order to theoretically recover the preferred theoretically recover the preferred PDEsPDEs

How to improve hardware acceleration How to improve hardware acceleration performance performance New-generation GPU architectureNew-generation GPU architecture Memory use optimizationMemory use optimization

How to incorporate with particular How to incorporate with particular applicationsapplications

Modify LBM Equilibrium Modify LBM Equilibrium Equation for DiffusionEquation for Diffusion

For fluid dynamics For fluid dynamics BGK model BGK model (1954) for equilibrium(1954) for equilibrium

))()((),( 22 ueDCuueBAuf iieq

i

For diffusion onlyFor diffusion onlyAuf eq

i ),(

Recover Diffusion EquationRecover Diffusion Equation Chapman-Enskog expansion, Chapman-Enskog expansion, εε

represents a small expansion represents a small expansion parameter (Knudsen number)parameter (Knudsen number)

Use this and Taylor expansion on Use this and Taylor expansion on LBM equationLBM equation

Recover Diffusion Equation Recover Diffusion Equation (Continue)(Continue)

We get expanded equation :We get expanded equation :

Constraints Constraints

Summation on expanded equation Summation on expanded equation over iover i

Recover Diffusion Equation Recover Diffusion Equation (Continue)(Continue)

Diffusion tensor defined by lattice Diffusion tensor defined by lattice structurestructure

For a particular lattice structure we For a particular lattice structure we obtainobtain

For D3Q19For D3Q19

ApplicationsApplications

Volume SmoothingVolume Smoothing

Volume SmoothingVolume Smoothing

Volume Volume SmoothingSmoothing

Surface SmoothingSurface Smoothing

Surface SmoothingSurface Smoothing

Image De-noisingImage De-noising

Poisson Image EditingPoisson Image Editing

Poisson Equation with Dirichlet Poisson Equation with Dirichlet boundaryboundary

is a computed guidance field is a computed guidance field computed from origin imagecomputed from origin image

PerformancePerformance

GPU NVidia 8800GTX with 768MB GPU NVidia 8800GTX with 768MB memorymemory

More ApplicationsMore Applications Apply to others beyond diffusionApply to others beyond diffusion Use level set equation as a pivot Use level set equation as a pivot

exampleexample It has many application fieldsIt has many application fields Combine several different types of PDEsCombine several different types of PDEs

Propagation interface as zero level Propagation interface as zero level setset

d is distance to interfaced is distance to interface Surface motionSurface motion

F = Signed speed in direction of normalF = Signed speed in direction of normal

Level SetLevel Set

F Speed FunctionF Speed Function

Curvature flow Curvature flow diffusion on distance diffusion on distance fieldfield

Advected propagation by a external fieldAdvected propagation by a external field Constant speed expansionConstant speed expansion

advcurvprop FFFF

),,(0 tyxUFt

LBM for Level SetLBM for Level Set Use distance field as the Use distance field as the

computational primitive in LBMcomputational primitive in LBM Play the role as density in fluidsPlay the role as density in fluids Design appropriate equlibrium Design appropriate equlibrium

function for PDEsfunction for PDEs

ThanksThanks