the lattice-boltzmann model for the visual...

THE LATTICE-BOLTZMANN MODEL FOR THE

VISUAL SIMULATION OF SMOKE

by

Usman Raza Alim

B.S., University of Rochester, 2001

a thesis submitted in partial fulfillment

of the requirements for the degree of

Master of Science

in the Department

of

Computer Science

c© Usman Raza Alim 2007ROCHESTER INSTITUTE OF TECHNOLOGY

Spring 2007

All rights reserved. This work may not be

reproduced in whole or in part, by photocopy

or other means, without the permission of the author.

APPROVAL

Name: Usman Raza Alim

Degree: Master of Science

Title of thesis: The Lattice-Boltzmann Model for the Visual Simulation of

Smoke

Examining Committee: Dr. Joe Geigel

Chair

Dr. Sidney Marshall, Reader

Prof. Sean Strout, Observer

Dr. Hans-Peter Bischoff, Graduate Coordinator

Date Approved:

ii

Abstract

The modeling and simulation of smoke, like other fluid phenomena, is an important and

challenging problem in Computer Graphics. In order to achieve high levels of visual realism,

physically based methods are usually used. Smoke is treated as a non-reactive substance

that is transported by an incompressible fluid. The dynamics of such a fluid is described by

a set of non-linear partial differential equations known as the Navier-Stokes (NS) equations.

Numerical methods for solving the NS equations can broadly be classified into two

categories, top-down and bottom-up. Top-down methods are well-studied and have been

successfully used in Computer Graphics to produce highly realistic animations of smoke

and other fluid phenomena. On the other hand, bottom-up methods such as the Lattice-

Boltzmann Method (LBM) are still being improved and have only been recently investigated

by the Computer Graphics community.

In this thesis, we adapt the Lattice-Boltzmann Method for the purpose of smoke simu-

lation in both two and three dimensions, and qualitatively compare the simulation results

with those obtained through a top-down method. In order to achieve visual realism, we

use physically based buoyancy and vorticity forces to drive the flow and employ a semi-

Lagrangian method to advect smoke densities along the flow. Our simulation results reveal

that the LBM retains much of the realism characteristic of the top-down methods, and at

the same time, offers considerable advantages in terms of simplicity and efficiency.

iii

To the memory of my grandfather, Muhammad Alimuddin

iv

Acknowledgments

The journey to the culmination of this work has been a long and intersting one. Along the

way, many people extended their support in various ways and my gratitude goes to them.

I would like to thank:

• Dr. Joe Geigel, my supervisor in Rochester, for his continued support and guidanceand for introducing me to the fascinating area of Computer Graphics.

• Dr. Torsten Möller, my supervisor in Vancouver, for his cooperation and understand-ing, for proofreading the manuscripts and for providing an excellent workspace.

• All the students at SFU with whom I had some very fruitful discussions, in particular,Yonas Tesfezghi Weldesselassie and Alireza Entezari.

• Sadia, my wife, for her unending and unflinching support.

• Dr. Nabila Babur, my mother, for teaching me to be patient during stressful times.

• Dr. Babur Zahiruddin, my father, for his financial support. My brothers, Salman andZeeshan, for their camaraderie.

• Dr. Muhammad Alimuddin, my late grandfather whose spirit thrives in me and con-tinues to provide ecouragement and inspiration.

v

Contents

Approval ii

Abstract iii

Dedication iv

Acknowledgments v

Contents vi

List of Tables ix

List of Figures x

1 Introduction 1

1.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Procedural Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Top-down Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Bottom-up Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Evaluation Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Top-down Model 7

2.1 Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

vi

2.1.3 Advection of Smoke . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 External Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Vorticity Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 The Lattice-Boltzmann Model 12

3.1 The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 The BGK Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 The Lattice-Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1 Discrete Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.2 LBGK Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.3 Macroscopic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.4 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 17

3.2.5 Advection of Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.6 External Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Two and Three Dimensional Lattices . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 D2Q7 and D2Q9 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.2 D3Q15, D3Q19 and D3Q27 Lattices . . . . . . . . . . . . . . . . . . . 21

3.3.3 Computing the Equilibrium Distributions . . . . . . . . . . . . . . . . 22

3.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.1 Solution to Navier-Stokes equations . . . . . . . . . . . . . . . . . . . 24

3.4.2 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Implementation 27

4.1 The Method of Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.1 A note on Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 30

4.1.2 Computing the Intermediate Velocity Field . . . . . . . . . . . . . . . 31

4.1.3 Solving the Discrete Poisson Problem . . . . . . . . . . . . . . . . . . 33

4.2 Physically Based Rendering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Scattering Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.2 Phase Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.3 Volumetric Photon Mapping . . . . . . . . . . . . . . . . . . . . . . . 40

vii

5 Results and Discussion 42

5.1 The Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 2D Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2.1 Top-Down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2.2 Bottom-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 3D Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.3.1 Top-Down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.3.2 Bottom-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 Conclusion and Future Work 61

Appendices 63

A Supplemental Material 63

Bibliography 64

viii

List of Tables

3.1 Coefficients for the D2Q9 Lattice . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Coefficients for the D3Q15 Lattice . . . . . . . . . . . . . . . . . . . . . . . . 24



5.1 2D simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2 2D LBM simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 3D simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.4 3D LBM simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 56

ix

List of Figures

3.1 The D2Q7 and D2Q9 lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Three Dimensional Sublattices . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1 The halfway boundary ∂Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 A schematic representation of the method of projection . . . . . . . . . . . . 32

5.1 2D top-down results with no vorticity confinement . . . . . . . . . . . . . . . 46

5.2 2D top-down results with vorticity confinement . . . . . . . . . . . . . . . . . 47

5.3 2D LBM results with no vorticity confinement, RE = 50 . . . . . . . . . . . . 50

5.4 2D LBM results with no vorticity confinement, RE = 100 . . . . . . . . . . . 51

5.5 2D LBM results with vorticity confinement, RE = 100 . . . . . . . . . . . . . 52

5.6 3D top-down results with vorticity confinement . . . . . . . . . . . . . . . . . 55

5.7 D3Q19 results with vorticity confinement, RE = 100 . . . . . . . . . . . . . . 57




x

Chapter 1

Introduction

The level of realism found in computer generated imagery nowadays is stunning. Anima-

tion studios, movie directors, video game developers and artists are increasingly relying on

computers to create breath-taking visual effects. Their goal is to achieve a degree of realism

that makes their imagery virtually indistinguishable from reality. Bringing digital imagery

to life is a challenging task and has been the focus of research in Computer Graphics (CG)

for the past couple of decades or so. In particular, the modeling and simulation of fluid

phenomena such as fire, water and smoke has received a great deal of attention. In this

thesis, we wish to explore one such problem namely, the visual simulation of smoke.

Smoke is an amorphous substance with a highly complex and turbulent fluid motion.

Producing animation sequences of smoke is a two-stage process. The first stage models the

geometry and motion of smoke while the second stage renders the smoke onto the screen. In

order to produce realistic visual simulations, it is important that a physically-based model

be employed for the two stages. This implies that both the dynamics of smoke and its

interaction with light must be accurately modeled.

In this chapter, we shall present an overview of some of the methods that have already

been used to address this problem and discuss how they motivate the goals of this thesis.

The organization of the rest of the thesis is described at the end of this chapter.

1.1 Previous Work

A good model for the simulation of fluid phenomena such as smoke must be efficient, easy

to use and be capable of producing highly realistic results. However, there is a tradeoff

1

1.1. PREVIOUS WORK

between these goals and it is no surprise that most of the existing simulation strategies have

concentrated on one goal while compromising on the others. The different approaches that

have so far been used to animate gaseous phenomena can be broadly classified into three

different categories: procedural, top-down and bottom-up.

1.1.1 Procedural Methods

In the early days when the problem of the simulation of gaseous phenomena started gaining

interest, researchers were concerned more with the appearance of smoke than with accuracy.

Consequently, these models were not physically-based and did not account for the dynamics

of fluid flow. Both the geometry and the motion of smoke were procedurally generated.

Ebert et al. used solid spaces to animate gaseous phenomena [9]. A solid space is a

three dimensional space that associates with each point, one or more attributes. In their

work, the density of the gas is represented as a procedurally generated solid space. In order

to mimic the small scale turbulent detail that exists within the geometry of smoke, they

used a turbulence function [22] that computes a pseudorandom noise value for a point. The

motion of the gas is also procedurally generated by animating the solid space that the gas

resides in. They made use of helical paths to create interesting visual effects. In order to

render the volume densities, they employed a volume rendering procedure that accounted

for the attenuation of light as it traverses a gas volume.

Procedural methods, while being simple in their description, have the obvious drawback

that they fail to capture the physical dynamics of fluid flow. The procedures are dependent

on a set of parameters which can be hard to fine tune as they don’t have any physical

counterparts.

1.1.2 Top-down Methods

The dynamics of smoke and other gaseous phenomena is governed by the Navier-Stokes (NS)

equations of fluid flow. At any instant, the dynamics of an incompressible fluid (constant

mass density) in two or three dimensions can be described by a velocity field and a pressure

field. Smoke is described as a scalar density field that is injected into the fluid and is

advected by it. Advection is the name given to the motion of a non-reactive substance

along the velocity field. Provided that the initial velocity and pressure fields are known, the

dynamics of these fields over time is completely determined by the Navier-Stokes equations.

2

1.1. PREVIOUS WORK

Navier-Stokes equations and other non-linear partial differential equations are usually

solved through a top-down approach where the spatial domain is discretized through finite

difference, finite volume or finite element methods. This is the conventional way fluid

flow problems are solved and there is immense literature on these methods in the field of

Computational Fluid Dynamics (CFD).

The goal of a simulation in Computer Graphics is visual realism rather than physical

accuracy. For this reason, computational grids are coarse as compared to their CFD coun-

terparts and time steps are usually taken to be large in order to reduce the complexity

of the simulation. Foster et al. [14] provided a finite difference solution to the NS equa-

tions. Because of the coarse discretization and an explicit integration scheme, their work

was prone to instabilities at large time steps. Later, Stam proposed a fluid solver based on

a semi-Lagrangian numerical scheme [26]. This method is unconditionally stable and can

incorporate large time steps.

Smoke in air is an example of turbulent flow where small scale vorticity (curl of the

velocity field) features are prominent. A coarse spatial sampling is unable to capture these

small scale details and any turbulence that is initially present quickly decays over time.

In order to remedy this problem, Fedkiw et al. [12] proposed the addition of a physically

consistent vorticity confinement force that injects small scale turbulent details back into the

velocity field. They also used a physically-based photon map renderer to produce highly

detailed and realistic animations of smoke. This model is considered to be the state of the

art top-down model for smoke simulation.

1.1.3 Bottom-up Methods

Recently, bottom-up approaches have been introduced and consist of the Lattice Gas Cel-

lular Automata (LGCA) and the Lattice Boltzmann Method (LBM). The starting point of

these approaches is a discrete microscopic model which by construction, yields the Navier-

Stokes equations of fluid flow. LGCA consist of a discrete lattice in two or three dimensions.

Fluid behavior is a self-organizing process that evolves according to microscopic collisions of

particles. These collisions are defined so that, in the macroscopic limit, the fluid conserves

mass and momentum. In LGCA, the collisions are described as boolean operations. This

all-or-none nature of the collisions can give rise to statistical noise in the computed flows.

The Lattice-Boltzmann Method remedies this problem by replacing the boolean operations

with real-valued particle distributions that move along each link of the lattice. The LBM

3

1.2. OBJECTIVES

updates these ddistributions at each node based on two simple and local rules: collision

and propagation. Collision describes the redistribution of particle packets at each node.

Propagation describes the motion of the particle packets to the nearest neighbors along the

velocity directions. Macroscopic quantities such as density and velocity are calculated as

ensemble averages from the packet distributions. A significant advantage that the LBM has

over the top-down approaches is its simplicity. Computation of the LBM consists of inex-

pensive addition, subtraction and multiplication operations. Furthermore, these operations

are performed locally which can potentially improve calculation speed by parallelization.

Both LGCA and the LBM have recently been introduced to computer graphics as an

alternative to the top-down CFD based techniques. Dobashi et al. [8] have used LGCA

to produce realistic animations of clouds. The LBM has been used successfully to com-

pute velocity fields for the purpose of fire simulation [29], creating wind fields [30] and the

simulation of gaseous phenomena [32].

The LBM based simulation of gaseous phenomena [32] is largely aimed at producing

real-time fluid effects. For this reason, the authors have used relatively low resolution grids

which are unable to simulate small scale turbulent details. Real-time computation speed is

achieved by texture hardware acceleration which is readily available in modern GPUs. Small

scale turbulent details are added by employing high resolution textures which attempt to add

the vortices and swirls that are characteristic of smoke. These textures are then rendered

by texture mapping hardware available in the GPU. Even though this method achieves real-

time computation speeds, the low resolution lattices and texture mapping greatly limit its

visual realism.

1.2 Objectives

The focus of this thesis is the improvement of the LBM used in [32] to simulate smoke. The

goal is to incorporate sufficient detail in the model so that the resulting simulation possesses

a level of visual realism that is qualitatively comparable to the top-down approach of Fedkiw

et al. [12]. In particular, the following three aspects will be considered.

1. Lattice Geometry

Wei et al. [32] use the D3Q19 lattice which is a three dimensional lattice consisting

of 19 discrete velocity vectors. According to the authors, this lattice structure is

4

1.3. EVALUATION CRITERIA

a compromise between computational efficiency and reliability. A denser lattice with

more velocity vectors will be better able to simulate the effects of a continuous velocity

field. We hope to improve their results by incorporating the D3Q27 lattice.

2. Small Scale Detail

Real smoke swirls and curls in air because of the vorticity of the flow field. Coarse CG

grids and the even coarser lattices in the LBM are unable to capture these small scale

details. The LBM can be improved by including a vorticity confinement force similar

to the one proposed by Fedkiw et al. [12].

3. Physically-Based Rendering

Since our goal is visual realism, it is important that the rendering algorithm takes

into account, the physical interaction of light with smoke. We will therefore employ a

photon map based renderer so that the resulting animation sequences are as realistic

as possible.

1.3 Evaluation Criteria

The top-down approach described in [12] is considered to be the best technique for the simu-

lation of smoke. In order to evaluate the effectiveness of our improved LBM, we will compare

and contrast it with this top-down approach. For this purpose, the simulation technique

outlined in [12] will be implemented so that it can be used as a basis for comparison.

Since the top-down and bottom-up approaches differ only in the first stage of the simu-

lation, i.e the stage that solves the NS equations, we shall use the same rendering algorithm

for both methods. Our evaluation consists of a qualitative comparison of the visual realism

of the two different approaches. We hope that our model will be simpler to implement, will

be highly comparable to the top-down approach and will produce results with a similar or

greater level of visual realism.

1.4 Thesis Organization

The remainder of this thesis is organized as follows. In chapter 2, we describe the top-down

model in detail and discuss how the equations of fluid flow can be used to simulate the

5

1.4. THESIS ORGANIZATION

dynamics of smoke. Chapter 3 is devoted to a detailed discussion of the bottom-up Lattice-

Boltzmann method. Chapter 4 describes some implementation specific details pertinent to

our evaluation system. In particular, it describes the photon-map algorithm used. Results

are provided in chapter 5 while the concluding remarks are presented in chapter 6.

6

Chapter 2

Top-down Model

At the microscopic level, smoke is a particulate matter that consists of tiny solid particles

suspended in a gas such as air. The concentration of these particles is described by a density

field and their motion is dependent on the motion of the gas that they are suspended in. As

the gas flows, particles of smoke are hurled along the flow, a process known as advection.

The motion of the smoke particles in turn influences the motion of the gas, giving rise

to a complex smoke-air dynamical system. A visually realistic simulation must take into

consideration, the dynamics of fluid flow as well as the interaction of smoke with air. In this

chapter, we shall take a look at the CFD based top-down model that attempts to capture

this complex dynamics of smoke [12].

2.1 Fluid Flow

Fluid dynamics is a rich and well-established field and numerous journals and books are

dedicated to it (see e.g [5]). The flow of a Newtonian fluid is described by the Navier-Stokes

(NS) equations which characterize the dynamics in terms of macroscopic fluid properties

such as density, viscosity, velocity and pressure. Like other classical dynamical systems,

these equations are based on conservation laws namely, the conservation of mass and the

conservation of momentum. The law of conservation of mass states that mass is neither

created nor destroyed during the process of fluid flow. The law of conservation of momentum

is nothing but Newton’s second law of motion and states that the change in momentum in

any portion of the fluid is proportional to the applied forces.

7

2.1. FLUID FLOW

2.1.1 Assumptions

The NS equations describe the motion of a general class of fluids. For the purpose of smoke

simulation, the fluid we are concerned with is air. There are certain assumptions that we

can make regarding air that will enable us to simplify the equations. In particular, we

assume that the fluid velocity is well below the speed of sound. Under this assumption,

compressibility of the fluid becomes negligible allowing us to treat air as a constant density

fluid. Also, we shall assume that the fluid has negligible viscosity, i.e it is inviscid .

2.1.2 Euler equations

When the fluid is inviscid and incompressible, the NS equations reduce to the incompressible

Euler equations , which are:

∇ · u = 0 (2.1)∂u

∂t= −(u ·∇)u−∇p + G (2.2)

In the above equations, u is the velocity of the fluid in two or three dimensions, p is its

pressure and G accounts for external body forces acting on the fluid. All of these quantities

are in general time-dependent.

Equation (2.1) is a mathematical formulation of the principle of conservation of mass.

It states that the fluid velocity is solenoidal (divergence-free) at all times. The solenoidal

nature of the velocity field ensures that there is no change in the density of the fluid within

any infinitesimal volume. Equation (2.2) is a restatement of the principle of conservation

of momentum. The interested reader is referred to [5] for the derivation of these equations.

Note that the density of the fluid does not explicitly appear in the equations. Since the

fluid is incompressible, the constant mass density is taken to be one.

In order to completely specify the dynamics of the fluid, we also need to supplement

the Euler equations with initial and boundary conditions. Let us suppose that the above

equations are defined on the domain Ω ⊂ Rd, d ∈ {2, 3}. The fluid velocity and pressureas well as the external body forces are given by the time-dependent real valued functions

u : Ω × R+ → Rd, p : Ω × R+ → R and G : Ω × R+ → Rd respectively. These functionssatisfy the Euler equations alongwith the initial condition

u(x, 0) = u0, x ∈ Ω (2.3)

8

2.2. EXTERNAL FORCES

and the boundary condition

u(x, t) · n̂ = 0, x ∈ ∂Ω, t ∈ R+ (2.4)

where u0 is a given functions,∂Ω ⊂ Rd is the boundary of the domain Ω and n̂ is the unitoutward normal to the boundary. Equation (2.4) is a a natural boundary condition for the

Euler equations in the sense that it guarantees that the problem is well-posed. Physically,

it corresponds to the scenario that the fluid is not allowed to leak through the boundary.

However, the fluid is free to move tangential to the boundary. For this reason, this boundary

condition is also sometimes known as the free-slip boundary condition.

2.1.3 Advection of Smoke

We assume that smoke is injected into the fluid as a non-reactive substance that is advected

by the flow. The density and temperature of the smoke are given by the scalar fields

P (x, y, z, t) and T (x, y, z, t) respectively. The motion of these two scalar fields is governed

by the advection-diffusion equation:

∂P

∂t= −u ·∇P (2.5)

∂T

∂t= −u ·∇T (2.6)

Another simplification that has been incorporated into the model is that the diffusion

term is set to zero in the above equations. Since the smoke particles are much bigger as

compared to the molecules of air, their diffusive behavior can be ignored. The model also

assumes that heat flow is described entirely by advection (equation 2.6). Even though

this assumption is not physically sound, it is adequate for our purposes since we are more

interested in a visual simulation rather than a physical one.

2.2 External Forces

As smoke moves through air, it interacts with it causing the flow to change. Hot smoke is

buoyant and tends to rise while heavy smoke tends to fall due to gravity. In reality, this

behavior is quite complex. It is not just the smoke exerting a force on the fluid, the fluid,

by virtue of its pressure is also exerting a force on the smoke. Fedkiw et al. ( [12]) have

9


used a simple linear equation to model this behavior as an external force acting on the fluid.

Their results show that a linear model gives convincing results. They have also proposed

the addition of a vorticity confinement force that attempts to preserve small scale details.

2.2.1 Buoyancy

Simply stated, the force of buoyancy used in this model is proportional to the temperature

and density of smoke, two quantities that are treated as being mutually independent. This

can be translated into the following linear equation:

fbuoy = [−αρ + β (T − Tamb)] k̂ (2.7)

Here, k̂ is the unit vector that points in the upward direction, Tamb is the temperature

of the ambient air and α and β are proportionality constants that control the amount of

upward or downward force to exert. This is a plausible model since when T = Tamb and

ρ = 0, the force exerted is zero, what we would expect when there is no smoke present.

2.2.2 Vorticity Confinement

In fluid mechanics, vorticity is defined as the curl of the velocity field (∇×u) and describesthe rotational behavior of a fluid. A flow with a high degree of vorticity is said to be

turbulent. Anyone who has carefully observed smoke in air knows that the flow is turbulent

and contains fine vortex structures. As the smoke rises, it swirls and curls because of the

vorticity of the flow. A good model for smoke simulation should attempt to preserve the

turbulent structures. Unfortunately, the grids used in Computer Graphics simulations are

usually coarsely sampled. Any vorticity that may be initially present in the flow decays

over time as a result of numerical dissipation. In order to preserve vortex structures, very

high resolution grids must be used so that the discrete solution to the fluid flow equations

is close to the continuous one. This, however, is not a practical solution as we would like to

maintain the efficiency of the simulation.

Fedkiw et al. [12] have remedied this problem by borrowing from CFD, a technique

known as vorticity confinement. It attempts to locate areas in the flow where small scale

vorticity features should be generated. The vorticity of an incompressible flow is given by:

ω = ∇× u (2.8)

10


The magnitude of the vorticity field gives a measure of the amount of turbulence present

in the flow. From this field, normalized vorticity location vectors N , that point from lower

vorticity concentrations to higher vorticity concentrations, are computed.

η = ∇ ||ω|| (2.9)

N =η

||η|| (2.10)

The vorticity confinement force is finally computed as

f conf = ǫh (N × ω) (2.11)

The positive parameter ǫ is used to control the amount of the confinement force to add

and h is the spatial discretization of the grid. As the resolution of the grid is increased, h

becomes less guaranteeing that the confinement force would be zero when the resolution is

infinite, thus giving the correct solution.

So, in summary, the top-down model attempts to simulate the dymanics of smoke by solving

the incompressible Euler equations. These equations assume that air is an inviscid and

incompressible fluid. Smoke is present in air as a particulate substance that is simply

advected by the flow. The model also incorporates a temperature field that is advected in a

similar manner. The interaction of smoke with air is modelled as a simple linear body force

that is added externally to the euler equations. The problem of numerical dissipation due

to coarse discretizations is solved by employing a physically consistent vorticity confinement

force. We have not described how the euler equations are discretized and numerically solved,

nor have we talked about the interaction of light with smoke. These two topics are the

subject of chapter 4.

11

Chapter 3

The Lattice-Boltzmann Model

In contrast to the top-down Computational Fluid Dynamics models that describe macro-

scopic fluid properties, the Lattice-Boltzmann Model (LBM) is a bottom-up approach that

describes the behavior of a fluid at a mesoscopic level. At the microscopic level, fluid flow

is described in terms of molecular dynamics. The particles of a fluid move about freely and

interact with each other giving rise to properties such as velocity, vorticity and pressure

that are observed at macroscopic scales. It is quite a challenging task to model the complex

molecular interactions that take place at microscopic scales. Since most applications of fluid

flow in Computer Graphics are concerned with the average behavior of a fluid, top-down

models are usually preferred over the microscopic ones.

Mesoscopic models such as the LBM lie in between the microscopic and the macroscopic

levels. They attempt to describe the behavior of a fluid by modelling the evolution of

averaged distributions of particles. This has the advantage that the complex motion of

individual fluid particles is replaced with averaged quantities that are much simpler to deal

with and yet accurately predict the macroscopic behavior of the fluid. As compared to

top-down models, the LBM has the advantage that the evolution rules are linear and local

thus making the method simpler and faster. The LBM is therefore a well-suited candidate

for solving fluid flow problems in Computer Graphics.

This chapter is intended to be a self-contained introduction to the Lattice-Boltzmann

Model as it pertains to the problem of visual simulation of smoke. A detailed derivation

of the model and the recovery of the incompressible Navier-Stokes equations is beyond

the scope of this thesis. Where appropriate, references to more thorough treatments shall

be provided for the sake of the interested reader. Since the LBM is a relatively recent

12

3.1. THE BOLTZMANN EQUATION

contribution to CFD, it has has its limitations which we shall point out as necessary.

3.1 The Boltzmann Equation

As the name suggests, the LBM has its roots in the kinetic theory of gases where distribution

functions such as the Maxwell-Boltzmann distribution ( [31]) play a key role. Consider a

domain Ω ⊂ Rd where d ∈ {2, 3}. A single species distribution function f(x,v, t) is afunction of the form

f : Ω× Rd × R+ → R+ (3.1)

and gives the continuous time-dependent probability of finding a particle within an in-

finitesimal volume located at x and having velocity v. It is assumed that all particles are

identical and have the same mass which is chosen to be unity for the sake of simplicity.

The kinetic theory of gases deals with the time evolution of such distribution functions and

relates macroscopic quantities such as fluid density, velocity and pressure to the underlying

microscopic particle distributions.

Under constant temperature conditions and when no external forces are acting on the

fluid, the distribution function evolves according to the Boltzmann Equation

∂f

∂t+ v ·∇f = Q(f, f) (3.2)

Here v is the velocity of a particle at the location x and at time t. The gradient term ∇f

is the spatial gradient of f given by:

∇f =d∑

i=1

∂f

∂xi(3.3)

The Boltzmann Equation (3.2) might seem daunting at first glance but is quite intuitive

when one considers the underlying particle dynamics. Essentially, there are two processes

that govern the evolution of the distribution function namely collision and propagation.

The term Q(f, f) on the right hand side term of (3.2) is a two particle collision term and

models the way in which the distribution changes as a result of two particles colliding at

the location x at time t. The incoming velocities {v,v1} of the two particles are changed tothe outgoing velocities {v′,v′

1} as a result of the collision thereby causing an overall change

in the distribution function.

The term v·∇f on the left side of (3.2) says that the the distribution function at the location

13

3.1. THE BOLTZMANN EQUATION

x changes due to the propagation of particles. Within an infinitessimal time interval dt,

a particle at the location x having velocity v propagates to the new location x + vdt and

hence changes the overall distribution function.

The behavior of the fluid at the macroscopic level can be recovered by ensemble averaging

the underlying particle dynamics. In particular, the fluid density at the location x at time

t is given by averaging the distribution function over the entire microscopic velocity space

ρ(x, t) =

∫f(x,v, t)dv (3.4)

whereas the fluid velocity is recovered through the first velocity moment of the distribution

function.

u(x, t) =1

ρ

∫vf(x,v, t)dv (3.5)

The Boltzmann Equation therefore completey describes the dynamics of a fluid both

at the microscopic as well as the macroscopic scales. In order to numerically solve for

the distribution function, the equation is discretized in the spatial, velocity and temporal

domains and an appropriate choice for the collision term is made.

3.1.1 The BGK Approximation

The collision term in the countinuous Boltzmann equation (3.2) in general has a complex

integral form. In most applications of Kinetic theory to fluid dynamics, the collision integral

is approximated by simpler expressions. One such approximation is the BGK approximation

proposed by Bhatnagar, Gross and Krook and incorporated into the so-called Lattice BGK

models by Qian et al. [25].

The BGK approximation is motivated by the fact that the large amount of detail of the

two-particle collision integral Q(f, f) does not significantly influence the values of quantities

at the macroscopic scale. Consequently, the two-particle term Q(f, f) is replaced by the

simpler term J(f) which models the affect of collisions as a relaxation of the distribution

towards a Maxwellian equilibrium distribution f eq(x,v, t):

J(f) =1

τ[f eq(x,v, t)− f(x,v, t)] (3.6)

The parameter τ has dimensions of time and controls the frequency with which the distri-

bution function relaxes to equilibrium. The BGK approximation is also sometimes referred

to as the single time relaxation approximation and is quite popular in the LBM literature.

14

3.2. THE LATTICE-BOLTZMANN EQUATION

Although there are other forms of the collision integral in use, our present work employs

the BGK approximation and we shall not be making use of any other approximations of the

collision term. Lattice BGK models are suitable for use in Computer Graphics since they

satisfy the incompressible Navier-Stokes equations within appropriate macroscopic limits.

3.2 The Lattice-Boltzmann Equation

In computational applications, a discrete form of the continuous Boltzmann equation (3.2)

is used. Since the contiuous distribution function depends on space, velocity and time, the

velocity space is first truncated to a discrete lattice leading to the discrete Boltzmann equa-

tion which is then spatially and temporally discretized giving rise to the Lattice Boltzmann

Equation.

3.2.1 Discrete Boltzmann Equation

In the continuous version of the Boltzmann equation, the velocity space of a particle (the

space of all allowable velocities that a particle can have) is also continuous. The first

step towards a discrete approximation to the Boltzmann equation is a truncation of the

continuous velocity space to a finite set of velocities {ci} (i ∈ {0, 1, 2, ..., n − 1}). Thismeans that a particle located at x that was previously allowed to move about freely with

any velocity v is now restricted to have one of the n velocities ci. With the truncated

velocity space, the distribution function reduces from a function that describes distributions

over arbitrarily many velocities (f(x,v, t)) to a function that describes distributions over a

finite lattice (fi(x, t)). The Boltzmann equation with the BGK approximation becomes the

discrete Boltzmann equation given by:

∂fi∂t

+ ci ·∇fi =1

τ(f eqi − fi) (3.7)

3.2.2 LBGK Equation

The discrete Boltzmann equation (3.7) is still not suitable for computational use since it is

continuous is space and time. When this equation is discretized spatially and temporally, the

domain spacing ∆x and the time step ∆t are chosen such that the lattice velocity ci =∆x∆t .

This ensures that particles located at the lattice location x move within a time step ∆t to

a neighboring location x + ci∆t. With this discretization scheme, the discrete Boltzmann

15


equation with the BGK approximation (3.7) becomes the lattice BGK equation:

fi(x + ci∆t, t + ∆t)− fi(x, t) =1

τ(f eqi (x, t)− fi(x, t)) (3.8)

A simpler interpretation of this equation over a two or three dimensional discrete lattice

is as follows:

1. Collision

At time t, particles at lattice location x collide with each other and in the process,

change the distribution function to f∗i (x, t) where:

f∗i (x, t) = fi(x, t) +1

τ(f eqi (x, t)− fi(x, t)) (3.9)

The equilibrium particle distribution f eqi (x, t) is actually an implicit function of space

and time. Its value depends on the fluid density and velocity and is of the form

f eqi (ρ(x, t),u(x, t)). It is computed by taking the velocity moments of the Maxwell-

Boltzmann distribution and equating it with the respective moments of the equilib-

rium distribution f eqi (ρ,u) [16]. For certain two and three dimensional lattices, this

procedure results in closed form expressions for the equilibrium particle distribution

in terms of the fluid density and velocity (section 3.3). This particular form of the

collision step also ensures that mass and momentum are locally conserved.

2. Propagation

After the collision step, the new distribution values f∗i (x, t) simply propagate to their

neighboring lattice locations within a time step ∆t. More formally:

fi(x + ci∆t, t + ∆t) = f∗

i (x, t) (3.10)

This step does not alter the distributions as they are propagating. Therefore, by

construction, mass is conserved.

Equations (3.9) and (3.10) are also sometimes written in a combined form eliminating the

intermediate post-collisional distribution function f∗i (x, t).

fi(x + ci∆t, t + ∆t) = (1−1

τ)fi(x, t) +

1

τf eqi (x, t) (3.11)

It is worth pointing out that the collision and propagation rules are simple and linear.

The collision step at a particular lattice site is entirely local and does not need information

16


from any other lattice sites. During the propagation step, distribution values propagate

to the lattice neighbors. This step therefore updates the distribution function at atmost n

lattice sites. Owing to the simplicity and locality of these rules, the LBM is readily amenable

to parallelization.

3.2.3 Macroscopic Quantities

Analogous to the continuous case, the density and velocity of the fluid are given as ensemble

averages of the distribution function. Integrals over the velocity space (equations (3.4) and

(3.5))become summations over the truncated velocities:

ρ(x, t) =

n−1∑

i=0

fi(x, t) (3.12)

u(x, t) =1

ρ

n−1∑

i=0

fi(x, t)ci (3.13)

3.2.4 Initial and Boundary Conditions

Since the variables of interest are the macroscopic quantities, the LBM needs be able to

handle initial and boundary conditions that are prescribed in terms of the fluid density and

velocity. The general approach for handling initial conditions is to set the initial distribution

function f0i (x) equal to the Maxwellian distribution that satisfies the initial fluid density

ρ0(x) and the initial fluid velocity u0(x).

f0i (x) = feqi (ρ0(x),u0(x)), x ∈ Ω (3.14)

Starting from an initial particle distribution over the lattice, the system evolves accord-

ing to the collision and propagation rules defined by equations (3.9) and (3.10). While the

collision rule is well-defined for all lattice sites, the propagation rule is not well-defined for

lattice sites that have one or more neighbors outside the domain Ω. In order to satisfy

Dirichlet boundary conditions (fluid velocity ub specified on the boundary ∂Ω), the prop-

agation rule has to be modified at lattice sites close to the boundary so that the velocity

obtained as a result of averaging the propagated distributions equals the Dirichlet boundary

value ub. This can be a hard problem for boundaries with complex geometry but for some

simple boundaries, a straightforward bounce back scheme has been proposed that is capable

of satisfying a Dirichlet boundary condition up to a second order accuracy [2, 18]. Since

17


our domains are simple, we have made use of the bounce-back scheme to satisfy a no-slip

boundary condition.

3.2.5 Advection of Scalar Fields

Our discussion so far has dealt with fluids in general. In order to make use of the LBM for

the purpose of simulating smoke, we also need a way to transport scalar quantities that are

injected into the fluid. This can be achieved in one of two different ways.

1. Bottom-Up

For a scalar field that is advected by the fluid, we can define a distribution function

similar to the fluid distribution function. This distribution function is then propa-

gated between different lattice sites in a manner akin to the propagation step of the

LBM. The scalar value at each node is then recovered by an ensemble average of the

distribution function corresponding to the scalar field. This procedure is known as the

Moment-Propagation Method and has been used in the LBM community for solving

the scalar transport problem [19,21].

As shown in [21], the moment propagation method approximates to second order, the

advection-diffusion equation

∂S

∂t= −u ·∇S + µ∇2S (3.15)

where S(x, t) is the scalar field being transported and µ is the coefficient of diffusion.

Compare these equation with the temperature and density advection equations (2.5

and 2.6). The equations are the same in the limit µ→ 0.

2. Top-Down

The top-down approach solves the advection equation directly using a semi-Lagrangian

integration scheme. Once the velocity field u has been computed by the LBM, it can

be used to advect scalar quantities such as smoke density and temperature. This

method is direct and does not make use of the underlying distribution functions.

We prefer to employ the more direct top-down approach for the following reasons. There

is extra computation overhead involved in keeping track of additional distribution functions

in the moment propagation method. It is also not possible to completely eliminate the

diffusion of scalar quantities in the moment propagation method. As the diffusion coefficient

18


is lowered, the method becomes more and more unstable giving rise to negative values of

the scalar field P . On the other hand, a direct semi-Lagrangian solution of the advection

equation does not suffer from these problems and is unconditionally stable. A direct solution

will also ensure that any differences in the visual simulation of smoke between the top down

model (chapter 2) and the LBM are entirely due to the stages that solve for the unknown

fluid velocity. This will further ensure a more faithful comparison between the two models.

3.2.6 External Forces

The scalar fields injected into the fluid exert forces on it thus causing the flow to change.

The buyonacy of hot air pushes air upwards while heavy smoke tends to fall due to gravity.

Small scale details are added into the fluid in the form of a vorticity confinement force that

cause the flow to swirl. Wei et al. [32] have used a simple and linear temperature field

to model the effect of a buoyancy force acting on the display primitives. Their force term

only changes the motion of the display primitives, it does not affect the underlying fluid

particle distributions or the flow field. This approach might be sufficient for an interactive

simulation on graphics hardware. However, it is inadequate for our purposes since our aim

is to produce visually realistic simulations of smoke. For this reason, we need to incorporate

the effect of external forces into the LBM.

When there are external forces acting on a fluid, a new force term appears in the Boltz-

mann equation (3.2). The external forces do not alter the collision operator Q(f, f) but

they do change the distribution function such that particles tend to get more concentrated

in the direction of the force. This redistribution must also satisfy the laws of conservation

of mass and momentum so that the appropriate macroscopic equations of fluid flow can be

receovered.

Various methods have been proposed for incorporating external forces into the LBM

(see e.g [3, 33]). Researchers in the LBM community seem to favor those methods that are

second order accurate, satisy the Navier-Stokes equations and don’t require a modification

of the ensemble averages for fluid density and velocity (equations (3.12) and (3.13)). Since

these factors are also in line with our goals, we shall employ the force model used by Junk

et al. [18]. According to this model, an external force term G(x, t) influences the collision

step as follows:

f∗i (x, t) = fi(x, t) +1

τ(f eqi (x, t)− fi(x, t)) + 3h3Wici ·G(x, t) (3.16)

19

3.3. TWO AND THREE DIMENSIONAL LATTICES

The factor h is related to the lattice spacing and the time step (∆t = h2 = ∆x2). This

particular relation between the space and time is necessary to obtain convergence towards

the incompressible Navier-Stokes equations. The factor Wi is a lattice dependent quantity

that is also used in the computation of the equilibrium distribution function. It will be

introduced in section 3.3.3.

The propagation step remains unchanged, i.e after collision and forcing, the new distribution

values f∗i propagate along the lattice velocities ci to their respective neighboring lattice sites.

The fluid density and velocity are still given by the ensemble equations (3.12) and (3.13).

3.3 Two and Three Dimensional Lattices

The notation DdQn is usually used to refer to LBM lattices where d is the spatial dimension

and n is the number of velocities in the truncated velocity space. The arrangement of the

lattice velocities must respect certain isotropy and symmetry requirements [31], so that the

lattice can yield the Navier-Stokes equations in the macroscopic limit. In practice, the D2Q7

and the D2Q9 lattices are commonly used in two dimensions and the D3Q15, D3Q19 and

D3Q27 lattices are used in three dimensions.

In the remainder of this thesis, the base unit for lattice spacing and the time step are

taken to be unity (∆x = 1,∆t = 1). This is merely a notational convenience, the lattice

spacing and the time step can be scaled to any appropriate units.

3.3.1 D2Q7 and D2Q9 Lattices

The lattice sites in the D2Q7 lattice are arranged in a triangular fashion giving the lattice

hexagonal symmetry. Each lattice site (or node) is connected to six other nodes that are

arranged in a regular hexagon around the node (Fig. 3.1). The lattice velocities are:

c0 = (0, 0)

ci =

(cos

2πi

6, sin

2πi

6

)i ∈ {1, ..., 6}

c0 corresponds to the distribution of particles that are at rest at the node while the other

velocities correspond to particles moving with unit speed in six different directions around

the node. The D2Q7 lattice is included in our discussion for the sake of completeness. Our

two dimensional simulations are performed on the D2Q9 lattice which has the geometry of

20


a regular cartesian grid (Fig. 3.1). Each node has a rest velocity c0 and is connected to eight

nearest neighbors along the remaining eight lattice velocities. The nine lattice velocities are

given by:

c0 = (0, 0) (3.17)

ci =

(cos

2π(i − 1)8

, sin2π(i − 1)

8

)i ∈ {1, 3, 5, 7} (3.18)

ci =√

2

(cos

2π(i− 1)8

, sin2π(i− 1)

8

)i ∈ {2, 4, 6, 8} (3.19)

In comparison to the D2Q7 lattice, D2Q9 is a multispeed lattice. Particles moving horizon-

tally or vertically have unit speed whereas particles moving diagonally travel with a speed

of√

2.

b

c0

bb

b

b b

b

c1c2

c3

c4 c5

c6

(a) The D2Q7 Lattice

b

c0b

bbb

b

b b b

c1

c2c3c4

c5

c6 c7 c8

(b) The D2Q9 Lattice

Figure 3.1: Two dimensional lattices. The arrows indicate the different lattice velocities.Both lattices have a rest velocity c0. (a) The D2Q7 lattice has hexagonal symmetry. Eachnode has six neighbors. (b) The D2Q9 lattice has eight neighbors and possesses rectangularsymmetry. It is geometrically similar to a regular cartesian grid.

3.3.2 D3Q15, D3Q19 and D3Q27 Lattices

The three dimensional lattices D3Q15, D3Q19 and D3Q27 have cubic symmetry and there-

fore have the same geometry as a regular three dimensional cartesian grid. On a 3D grid,

the velocity space can be truncated to four different speeds, each speed defining its own

sublattice (Fig. 3.2).

21


• Sublattice 0This sublattice consists of the zero velocity particles (0, 0, 0) that stay at the lattice

site during the propagation step.

• Sublattice 1This consists of particles moving with unit speed given by the 6 nearest neighbors

(±1, 0, 0), (0,±1, 0) and (0, 0,±1).

• Sublattice 2This is formed by the 12 second-nearest neighbors given by (±1,±1, 0), (±1, 0,±1)and (0,±1,±1). Particles in this sublattice move with a speed of

√2.

• Sublattice 3Finally sublattice 3 is composed of the 8 third-nearest neighbors given by (±1,±1,±1).This corresponds to particles moving with speed

√3.

The D3Q15 lattice is made up of sublattices 0, 1 and 3 having a total of (1 + 6 + 8)

fifteen velocity directions. The D3Q19 lattice consists of sublattices 0, 1 and 2 having a

total of (1 + 6 + 12) nineteen velocities. The D3Q27 lattice consists of all four sublattices

and propagates distributions to all 26 neighbors of a node. As pointed out by Mei et al. [20],

D3Q15 is the least isotropic and more vulnerable to numerical instabilities. D3Q19 is more

efficient as compared to D3Q27 and has better stability than D3Q15. Although D3Q27

is the most isotropic, it does not always give the best results. We shall not be making

use of the D3Q15 lattice. Wei et al. [32] have favored the D3Q19 lattice for all their tests

because of its computational efficiency. Since our aim is visual realism, we shall use both

the D3Q19 and D3Q27 lattices in our simulations and compare the results in terms of their

visual quality.

3.3.3 Computing the Equilibrium Distributions

In order to compute the equilibrium distribution of particles, it is assumed that particles

relax towards a Maxwellian equilibrium which is a function of fluid density and velocity

alone. It is derived by applying the maximum entropy principle under the constraints of

mass and momentum conservation [16, 31]. For a multi-speed LBM lattice, this procedure

22


z

y

x

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

Figure 3.2: The four cubic sublattices. Sublattice 0 consists of the rest velocity and is shownas a black node in the center. Sublattice 1 consists of the 6 unit speed velocity vectors formedby the blue nodes. The 12 green nodes form the velocity vectors of sublattice 2 (speed

√2)

while the 8 red nodes form the velocity vectors of sublattice 3 (speed√

3).

yields the following expression for the equilibrium distribution function:

f eqi (ρ,u) = ρWi

[1 + 3ci · u +

9

2(ci · u)2 −

3

2u · u

](3.20)

The coefficients Wi are obtained by taking the velocity moments of the continuous Maxwell-

Boltzmann distribution and setting them equal to the corresponding discrete velocity mo-

ments. For a multi-speed LBM lattice, particle distributions having the same speed have

the same value of Wi. Equation (3.20) can be used to compute the equilibrium distributions

for the lattices D2Q9, D3Q15, D3Q19 and D3Q27. The different values of the coefficients

Wi are tabulated below along with the ordering of the velocity vectors ci.

23

3.4. LIMITATIONS

Speed Velocity Vectors ci Wi0 0 4/91 1,3,5,7 1/9√2 2,4,6,8 1/36

Table 3.1: Coefficients for the D2Q9 Lattice

Speed Velocity Vectors ci Sublattice Wi0 0 0 2/91 1,. . . ,6 1 1/9√3 7,. . . ,14 3 1/72


3.4 Limitations

The Lattice Boltzmann Model with the BGK approximation imposes some limitations on

the kinds of flows that can be accurately simulated. Even though the LBM is a second-order

method and recovers the Navier-Stokes equations, it does so under some constraints.

3.4.1 Solution to Navier-Stokes equations

Through an analytical technique known as the Chapman-Enskog expansion [25, 31], it can

be shown that the fluid density (3.12) and fluid velocity (3.13) obtained through the LBM

solve the incompressible Navier-Stokes equations

∇ · u = 0 (3.21)∂u

∂t= −u ·∇u + ν∇2u−∇p + G (3.22)

where ν is the viscosity of the fluid, p is the scalar pressure and G is an external body

force. Note that the constant mass density of the fluid is taken to be unity and does

not appear in the Navier-Stokes equations. The LBM on the other hand explicitly keeps

track of the evolution of the mass density ρ as other quantities such as the equilibrium

distribution function depend on it. The BGK approximation to the collision operator and

the propagation rules of the LBM are designed to ensure that the density of the fluid does not

deviate much from its initial value. The LBM therefore solves the Navier-Stokes equations

24

3.4. LIMITATIONS

Speed Velocity Vectors ci Sublattice Wi0 0 0 1/31 1,. . . ,6 1 1/18√2 7,. . . ,18 2 1/36


Speed Velocity Vectors ci Sublattice Wi0 0 0 8/271 1,. . . ,6 1 2/27√2 7,. . . ,18 2 1/54√3 19,. . . ,26 3 1/216


in the incompressible limit. More formally, when the fluid speed |u| is sufficiently small ascompared to the speed of sound, the velocity field computed through the LBM converges to

the velocity field that solves the incompressible Navier-Stokes equations. The incompressible

limit is also sometimes known as the small Mach number limit and is forumalted as

Ma :=|u|cs≪ 1 (3.23)

where cs is the speed of sound of the lattice. For the D2Q9 lattice, cs = 1/√

2 whereas for

the 3D D3Q15, D3Q19 and D3Q27 lattices, cs = 1/√

3 [20].

3.4.2 Viscosity

Since the LBM solves the Navier-Stokes equations in the incompressible limit, diffusion

phenomena that give rise to viscosity cannot be ignored. The viscosity parameter ν appear-

ing in equation (3.22) is related to the relaxation time scale τ that appears in the BGK

approximation (3.6) as follows:

ν =1

3(τ − 1

2) (3.24)

Since the viscosity cannot be negative, τ must always be greater than 12 . This relationship

between the relaxation time and the fluid viscosity gives the LBM the flexibility to simulate

flows with varying degrees of turbulence. Highly viscous flows are in general less turbulent

25

3.4. LIMITATIONS

than less viscous ones. The degree of turbulence in a flow is characterized by a dimensionless

constant known as the Reynolds Number (Re := vsLν

where vs is the average fluid speed, L

is the characteristic length and ν is the viscosity). The Reynolds number of the flow can

therefore be controlled by adjusting the relaxation time τ . Lattice Boltzmann models tend

to be more stable at low Reynolds numbers thus limiting the degree of turbulence that can

be achieved.

3.4.3 Stability

As a final remark, we would like to mention that the stability of Lattice Boltzmann methods

is very much an active area of research (see e.g [1, 6]). Like any other numerical scheme,

there is a certain region in parameter space where the LBM is stable. Stability decreases

as we go closer to the boundary of the parameter space. As a rule of thumb, the LBM

is stable as long as we are close to the incompressible limit (3.23). Increasing the fluid

velocity or the magnitude of the external force decreases stability. Stability also decreses

with increasing Reynolds numbers. The structure of the lattice also affects stability in the

sense that more isotropic lattices tend to be more stable than less isotropic ones for the

same set of parameter values.

26

Chapter 4

Implementation

The Euler equations are usually solved using finite difference, finite element or finite volume

methods. In most applications of fluid dynamics in Computer Graphics, finite difference

methods are largely used [12, 14, 26]. This particular discretization scheme is inline with

our goals since the spatial discretization of the discrete Boltzmann equation (3.7) is also

based on finite differences. We shall therefore conduct our simulations on regular two and

three dimensional grids where the grid spacing h is the same in all directions. This also

ensures that, within the domain of interest Ω, there is a one-to-one correspondence between

a Cartesian grid point and an LBM lattice node.

Our domain Ω is taken to be a rectangle in two dimensions and a hexhadron in three

dimensions. The following description applies to a square in 2D and a cube in 3D but can

easily be generalized to axis aligned domains.

The location of a grid point x is given by x := (xi, yj) = (ih, jh) in two dimensions and

x := (xi, yj, zk) = (ih, jh, kh) in three dimensions where h is the spacing in each direction

and i, j, k ∈ {1, 2, . . . ,M − 1,M}. The boundary ∂Ω is located halfway between the interiorgrid points defined by i, j, k ∈ {1, 2, . . . ,M − 1,M} and the exterior points defined byi, j, k = 0 and i, j, k = M +1. It corresponds to the four sides of a square in two dimensions

and the six faces of a cube in three dimensions (Figs. 4.2(a) and 4.2(b)). As we shall see

later, a halfway boundary is a useful construct and simplifies the handling of boundary

conditions in numerical methods. The exterior points are actually outside the domain and

are not included in the simulations. They are also sometimes referred to as dry points. The

interior therefore has a resolution of M2 in 2D and M3 in 3D. We denote the total number

of interior grid points as N irrespective of the dimensionality. Quantities such as velocity

27

4.1. THE METHOD OF PROJECTION

and smoke density are defined at the grid points and interpolation is used to approximate

their values at non-grid locations.

b b b b

b b b b

b b b b

b b b b

bc

bc

bc

bc

bc

bc

bc bc bc bc bc bcbc

bc

bc

bc

bc

bcbc bc bc bc bc bc

(a) 2D domain (b) 3D domain

Figure 4.1: A schematic representation of the simulation domains and their boundaries.The interior points are indicated in blue, the dry points are indicated by dashed lines. Theboundary resides halfway between the interior points and the dry points.

In this chapter, we describe the top-down method of projection that is commonly used

in Computer Graphics to solve fluid flow problems. An important step in this method is

the solution of a Poisson problem. Computer Graphics literature often tends to gloss over

this step and we have tried to provide a detailed analysis of the problem. The outputs of

both the top-down and bottom-up procedures consist of two and three dimensional smoke

density grids. While displaying 2D grids is straightforward, it is not a trivial task to obtain

realistic 2D projections from 3D smoke density grids. In the latter part of this chapter, we

describe how photon-mapping can be used to render 3D smoke datasets.

4.1 The Method of Projection

One of the popular numerical methods for solving the Euler equations as well as the incom-

pressible Navier-Stokes equations is based on the method of projection that was proposed

by Chorin in 1967 [4]. At the core of this method is the Helmholtz-Hodge Decomposition

Theorem which states that:

Helmholtz-Hodge Decomposition Theorem. Let Ω be a region in space with a smooth

28


boundary ∂Ω. A vector field w on Ω can be uniquely decomposed in the form

w = u + ∇p (4.1)

where u has zero divergence and is parallel to ∂Ω; i.e u · n̂ = 0 on ∂Ω.

We shall not prove this theorem here and refer the reader to [5] for an elegant proof. It

is a very powerful theorem in the area of vector calculus and is also sometimes known as

the fundamental theorem of vector calculus. It allows any smooth vector field to be broken

up into its constituents, a solenoidal vector field that is tangential to the boundary and a

scalar function p.

Let us denote the operator that orthogonally projects a vector field w onto its divergence-free

part as P. If w is as given in (4.1), then the operator P acts on w as follows:

Pw = u, (4.2)

where u is divergence-free and parallel to ∂Ω. Also, given a vector field u that is divergence-

free and parallel to ∂Ω, it projects onto itself, i.e.

Pu = u. (4.3)

Furthermore, if p is a scalar function that satisfies (4.1), then

P(∇p) = 0. (4.4)

The projection operator allows one to eliminate the pressure term from the momentum

equation (2.2) yielding a partial differential equation that is easier to solve. Applying P to

both sides of (2.2) gives

P

(∂u

∂t

)= P [−(u ·∇)u−∇p + G] . (4.5)

Since we are dealing with the incompressible Euler equations, the fluid velocity u is always

divergence-free. If we further impose the boundary condition that u always remains parallel

to the boundary ∂Ω, then (4.5) reduces to

∂u

∂t= P [−(u ·∇)u + G] , (4.6)

owing to the facts that P(∂u∂t

) = ∂u∂t

and P(∇p) = 0.

29


Equation (4.6) expresses the momentum equation in terms of the fluid velocity u alone

and allows one to devise a method to solve for the unknown fluid velocity without having to

factor the fluid pressure into the partial differential equation. Suppose that the fluid velocity

is known at a time t and one is interested in finding the velocity at a later time t+∆t. One

first solves for an intermediate velocity field w according to (4.6) and then projects it onto

its divergence free part. The intermediate velocity field w satisfies the differential equation

∂w

∂t= −(w ·∇)w + G, (4.7)

with the initial condition that w = u at time t. It is not necessarily divergence free and has

to be projected according to (4.1) to obtain the divergence free velocity that is a solution

to the Euler equations at time (t + ∆t).

Let w(x, t + ∆t) be the velocity field that has been computed to satisfy (4.7). A

divergence-free velocity field u(x, t + ∆t) is obtained from w by taking the divergence

of both sides of (4.1),

∇ ·w = ∇ · u + ∇ ·∇p.

Since u is divergence-free, this reduces to a Poisson equation for the pressure p.

∇2p = ∇ ·w. (4.8)

Thus, applying the projection operator P to the intermediate velocity field w is tantamount

to solving a Poisson equation for pressure. Once the pressure p has been computed, the

divergence-free velocity field u(x, t + ∆t) is given by

u = w −∇p. (4.9)

4.1.1 A note on Boundary Conditions

The Poisson equation for pressure (4.8) is an elliptic partial differential equation whose so-

lution depends on the nature of boundary conditions. In most fluid dynamics problems,

boundary conditions for the velocity field u are known and an appropriate boundary condi-

tion for the pressure p is derived from them so that the velocity field computed through (4.9)

satisfies the given boundary conditions. It is important to handle boundary conditions cor-

rectly since solid boundaries often give rise to rotational and turbulent behavior.

In our simulations, we have imposed the free-slip boundary condition (2.4) on u. As

mentioned earlier, this is a natural boundary condition for Euler equations and ensures that

30


the fluid is not allowed to move orthogonal to the walls. The portion of the fluid in contact

with the walls is free to move in a tangential direction. When the walls of the container are

stationary (which is the case we are considering), the free-slip boundary condition can be

formulated as

u(x, t) · n̂ = 0, x ∈ ∂Ω, (4.10)

where n̂ is the unit outward normal to the wall. The free-slip boundary condition is also

needed to satisfy the requirements of the Helmholtz-Hodge decomposition theorem (4.1).

A boundary condition for the Poisson equation (4.8) in terms of the intermediate velocity

field w can be obtained by applying the free-slip condition to (4.9). Thus, on the boundary

∂Ω,

∇p · n̂ = w · n̂, (4.11)

which means that the boundary condition for w governs the boundary condition for p. Since

w(x, t) is initially equal to u(x, t), it is natural to impose the condition that w remains

parallel to the boundary, i.e. w ·n̂ = 0 on ∂Ω. Under this condition, the boundary conditionfor p becomes a Neumann boundary condition and the complete Poisson problem (4.8) is

given by {∇2p = ∇ ·w in Ω

∇p · n̂ = 0 on ∂Ω.(4.12)

The solution 1 to the Poisson problem (4.12) ensures that the resultant velocity u(x, t+

∆t) is divergence-free and satisfies the free-slip boundary condition. A numerical method

for solving the discretized Poisson problem is presented in Section 4.1.3.

4.1.2 Computing the Intermediate Velocity Field

The intermediate velocity field w(x, t + ∆t) is a solution to the partial differential equa-

tion (4.7) over a time step ∆t with the initial condition that w0(x) = u(x, t) and the

boundary condition that w · n̂ = 0. It is computed using a two step procedure (Fig. 4.2) asexplained by Stam [26].

1. Adding Forces

In the first step, the forces acting on each grid point (buoyancy, vorticity etc.) at

1 In general, the Neumann problem for Poisson’s equation (∇2p = f in Ω, ∇p · n̂ = g on ∂Ω) has asolution unique up to a constant if and only if the compatibility condition

R

Ωf dx =

R

∂Ωg dS is satisfied. In

our case, f = ∇ ·w and g = w · n̂ = 0; and the divergence theorem ensures thatR

Ω∇ ·w dx =

R

∂Ωw · n̂ dS.

31


u(x, t) = w0(x)add force︷︸︸︷

=⇒ w̃(x)advect︷︸︸︷=⇒ w(x)

project︷︸︸︷=⇒ u(x, t + ∆t)

Figure 4.2: The Method of Projection for solving the incompressible Euler equations. Theknown velocity at time t sequentially undergoes three steps, namely, the addition of forces,advection and projection to yield the velocity field at a later time (t + ∆t).

the current time t are computed through the corresponding force equations (2.7 and

2.8). Assuming that the time step ∆t is sufficiently small so that the force G(x, t) is

effectively constant over the time interval, it can be added to the velocity field through

a simple first-order Euler integration scheme, giving the resulting velocity field

w̃(x) := w0(x) + G(x, t)∆t. (4.13)

2. Advection

The second step accounts for the advection term −(w ·∇)w. This is done through aLagrangian approach based on the method of characteristics. The effect of advection

is to transport a quantity along the velocity field of the fluid. In this case, the quantity

being transported is the fluid velocity itself; i.e. each component of the fluid velocity

is transported along the vector field. Assuming that the time step is sufficiently small

so that the fluid velocity does not change much, the method of characteristics reveals

that information is propagated along the streamlines of the vector field.

A streamline of the velocity field w̃ is given by the parametric curve that is a solution

to the system of ordinary differential equations given by

d

ds(x(s)) = w̃(x(s)). (4.14)

Let r(x0, s) denote the parameterized streamline that starts at location x0 when

s = 0 and solves the above system of ordinary differential equations. The method

of characteristics tells us that the velocity at x0 at time (t + ∆t) is the same as the

velocity at the location given by r(x0,−∆t). The advected velocity field w that solvesequation (4.7) is therefore given by:

w(x) = w̃ (r(x,−∆t)) . (4.15)

32


In order to compute the new velocity at a grid location x, we perform a second-order

Runge-Kutta particle trace to obtain the location r(x,−∆t) of the particle a time ∆t ago.We do this at each of the N interior grid points. The velocity is taken to be zero at the

boundary ∂Ω so that it satisfies the free-slip condition. As a result of this integration, the

particle might end up either in a non-grid location or completely outside the domain Ω.

In the former case, we use interpolation to determine the velocity value at the non-grid

location. Linear interpolation is a good choice since it is fast and gives reasonable results.

For better results, cubic interpolation can be used but it is much more expensive. In the

latter case, we clip the trace against the boundary and set the velocity equal to zero to

ensure that the free-slip boundary condition is satisfied. Both Runge-Kutta integration and

interpolation are standard numerical schemes and we refer the reader to [24] for details.

4.1.3 Solving the Discrete Poisson Problem

The Poisson problem (4.12) when spatially discretized over the domain Ω becomes a sparse

linear system. There are many methods for solving such systems of linear equations (see

e.g. [7,24]), each with its pros and cons depending on the nature of the sparse matrix. These

methods can broadly be classified into two categories, direct and iterative. Direct methods

get an exact solution to the linear system whereas iterative methods refine a solution at

each step of the iteration until some convergence criterion is met. For arbitrary domains,

these methods are often quite expensive but when the domains are simple, fast and direct

solvers can be used. In fact, using rectangular domains in our simulations allows us to use a

rapid fast Fourier transform (FFT) based solver that has a time complexity of O(N log N).

We shall first describe how the FFT based method solves a one-dimensional Poisson

problem and then generalize the recipe to two and three dimensions. Let us consider the

following 1D Poisson problem

− d2p(x)

dx2= f(x),

1

2≤ x ≤M + 1

2, (4.16)

with the Neumann boundary conditions that px(12 ) = px(M +

12 ) = 0. As mentioned before,

we discretize the interior of the interval into M equally spaced points xi,

(xi = ih, i ∈ {1, 2, . . . ,M−1,M}) with the grid spacing h being unity. With this discretiza-tion, a second-order central difference approximation to the Poisson equation (4.16) is given

bypi+1 − 2pi + pi−1

h2= fi, (4.17)

33


where pi = p(xi) and fi = f(xi). When we evaluate the above equation (4.17) at all M grid

points, we get the matrix equation

−1 11 −2 1

. . .. . .

. . .

1 −2 1. . .

. . .. . .

1 −2 11 −1

p1...

pi−1

pi

pi+1...

pM

= h2

f1...

fi−1

fi

fi+1...

fM

, (4.18)

which can also be written as

TM · p = h2f . (4.19)

The M ×M sparse matrix TM is tridiagonal (consists of zeros everywhere except for thediagonal and off-diagonal entries), the M × 1 column vector p is made up of the unknownvalues of the function p(x) at the discretized locations; and the M × 1 column vector fconsists of the known values of the function f(x).

The Neumann boundary conditions cause the first and last entries of the main diagonal to

be -1. From (4.17), the equation for p1 is

p2 − 2p1 + p0h2

= f1.

Since the Neumann boundary condition is specified at the halfway grid point, a central

difference approximation to the first derivative gives the relation

px(1

2) =

p1 − p0h

= 0,

from which we can conclude that p0 = p1. Substituting this back into the equation for p1

above, we getp2 − p1

h2= f1.

A similar reasoning applies to the other boundary location and explains why the first and

last entry on the main diagonal of TM is -1 and not -2. This also explains our motivation

behind using a halfway boundary at the beginning of this chapter.

The sparse matrix TM has a special form that allows us to use the FFT to solve the linear

system (4.19). In the continuous realm, the eigenfunctions of the second derivative operator

34


( d2

dx2) are sines are cosines. As one would expect, the matrix TM also has eigenvalues and

eigenvectors that are trigonometric functions. In particular, the eigenvalues of TM are given

by

λj := 2− 2 cos[π(j − 1)

M

], j ∈ {1, . . . ,M}, (4.20)

and the corresponding eigenvectors are

zj(i) := aj cos

[(2i− 1)(j − 1)π

2M

], (4.21)

aj =

√1M

if j = 1√2M

if j 6= 1.

Note that when j = 1, λ1 = 0. This makes the matrix singular with a null-space of dimension

one spanned by the eigenvector z1 = (1, . . . , 1)T . Consequently, the solution p is unique

upto a constant, i.e. if p is a solution to (4.19), then so is p + c(1, . . . , 1)T . This comes as

no surprise since the solution to the continuous Poisson problem with Neumann boundary

conditions is also unique upto a constant. In practice, the zero eigenvalue λ1 is usually

replaced by a small positive number ǫ to avoid division by zero.

Let Z = {z1, . . . , zM} be the orthogonal matrix formed by the eigenvectors and Λ bethe diagonal matrix formed by the corresponding eigenvalues. An eigen-decomposition of

TM is given by

TM = ZΛZT , (4.22)

which when applied to (4.19) gives the solution to the discrete Poisson problem as

p = (ZΛ−1ZT ) · (h2f). (4.23)

There is a very close connection between the eigenvector matrix Z and the fast Fourier

transform [7]. In fact, multiplication by ZT is the same as computing the discrete cosine

transform (DCT). Similarly, multiplication by Z computes the inverse discrete cosine trans-

form (IDCT).

Let us denote the discrete cosine transform of a column vector f as f̂ and its individual

components as f̂i. The right hand side of (4.23) can be computed using the following simple

algorithm.

35


Algorithm 1: 1D discrete Poisson problem with Neumann boundary conditions.Input: The column vector f .

Output: The solution p to the 1D Poisson problem.

1DPoisson(f)

(1) f̂ ← DCT(h2f).(2) for i = 1 to M

(3) f̂i ← (f̂i)/(λi).(4) p← IDCT(̂f ).(5) return p

The discrete cosine transform can be computed from the fast Fourier transform in con-

stant time. Since the above algorithm makes use of the transform twice (once forward

and once backward), the overall time complexity of this algorithm is the same as the time

complexity of FFT, i.e. O(M log M).

Extension to Higher Dimensions

The above recipe can easily be extended to two and three (and possibly higher) dimensions.

In two or three dimesions, a central difference approximation to the Laplacian operator (∇2)yields a linear system similar to (4.18). The analog of the matrix TM can be obtained by

using matrix Kronecker products as explained in [7]. We follow a slightly different approach

that is a generalization of the 1D DCT based algorithm presented above.

Let us remind the reader that the Poisson problem we are interested in solving is given

by: {−∇2p = f in Ω

∇p · n̂ = 0 on ∂Ω,(4.24)

which is the same as the original problem (4.12) when f = −∇ · w. Suppose that thediscretized function is given by f which has the components fi,j in 2D and fi,j,k in 3D

(i, j, k ∈ {1, . . . ,M}). Again, let us denote the discrete cosine transform (which is now twoor three dimensional) of f as f̂ and its individual components as f̂i,j in 2D (or f̂i,j,k in 3D).

The eigenvalues of the 2D linear system that arises as a result of the discretization

of (4.24) are

λi,j := λi + λj = 4− 2 cos[π(i− 1)

M

]− 2 cos

[π(j − 1)

M

]. (4.25)

36


In 3D, the eigenvalues are

λi,j,k := λi + λj + λk = 6− 2 cos[π(i− 1)

M

]− 2 cos

[π(j − 1)

M

]− 2 cos

[π(k − 1)

M

]. (4.26)

Denoting a d-dimensional discrete cosine transform as DCTd (d ∈ {2, 3}), and the corre-sponding inverse cosine transform as IDCTd, Algorithms 2 and 3 listed below are the two

and three dimensional analogues of Algorithm 1.

Algorithm 2: 2D discrete Poisson problem with Neumann boundary conditions.Input: A 2D grid f .

Output: The 2D grid p that is a solution to the 2D Poisson problem (4.24).

2DPoisson(f)

(1) f̂ ← DCT2(h2f).(2) for i = 1 to M

(3) for j = 1 to M

(4) f̂i,j ← (f̂i,j)/(λi + λj).(5) p← IDCT2(̂f ).(6) return p

Algorithm 3: 3D discrete Poisson problem with Neumann boundary conditions.Input: A 3D grid f .

Output: The 3D grid p that is a solution to the 3D Poisson problem (4.24).

3DPoisson(f)

(1) f̂ ← DCT3(h2f).(2) for i = 1 to M

(3) for j = 1 to M

(4) for k = 1 to M

(5) f̂i,j,k ← (f̂i,j,k)/(λi + λj + λk).(6) p← IDCT3(̂f ).(7) return p

Algorithm 2 has a time complexity of O(M2 log M2) while Algorithm 3 has a time

complexity of O(M3 log M3). In general, if the total number of grid points is N , a solution

to a d-dimensional Poisson problem with Neumann boundary conditions can be computed

in O(N log N) time. With minor modifications, this recipe can also be used for grids with

37

4.2. PHYSICALLY BASED RENDERING

unequal number of grid points in each dimension as well as for the Poisson problem with

Dirichlet 2 boundary conditions.

4.2 Physically Based Rendering

The top-down Euler equations as well as the bottom-up LBM simulate the complex motion

of smoke in two and three dimensions. In order to achieve visual realism, the density grids

must be rendered using an algorithm that takes into account, the physical interaction of light

with smoke. Gaseous phenomena such as smoke do not interact with light the same way

opaque surfaces do. They are transparent substances that allow light to pass through and at

the same time, cause it to attenuate and scatter. The name, participating media, is usually

used in Computer Graphics to refer to such substances. Photon mapping, an extension

of ray tracing, is a photorealistic rendering technique that makes it possible to efficiently

simulate global illumination in scenes with participating media. We have therefore made

use of photon mapping to render the three dimensional smoke datasets generated through

the top-down and bottom-up approaches. In this section, we present the relevant equations

that model the interaction of light with smoke. A detailed account of how photon mapping

is used to solve the rendering problem can be found in [17,23].

4.2.1 Scattering Media

Smoke, like other participating media, scatters and attenuates an incident beam of light.

This means that a ray of light that has travelled a certain distance through smoke in a

straight line, will have its intensity reduced because of the interaction of

the lattice-boltzmann model for the visual...

Documents