Fluid Simulation for Computer Graphics.pdf

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Contents

Preface

I: The Basics

1. The Equations of Fluids

1.1 Symbols

1.2 The Momentum Equation

1.3 Lagrangian and Eulerian Viewpoints

1.4 Incompressibility

1.5 Dropping Viscosity

1.6 Boundary Conditions

2. Overview of Numerical Simulation

2.1 Splitting

2.2 Splitting the Fluid Equations

2.3 Time Steps

2.4 Grids

3. Advection Algorithms

3.1 Semi-LagrangianAdvection

3.2 Boundary Conditions

3.3 Time Step Size

3.4 Dissipation

3.5 Reducing Numerical Dissipation

4. Making Fluids Incompressible

4.1 The Discrete PressureGradient

4.2 The Discrete Divergence

4.3 The Pressure Equations

4.4 Projection

4.5 More Accurate Curved Boundaries

4.6 The Compatibility Condition

II: Different Types of Fluids

5. Smoke

5.1 Temperature and Smoke Concentration

5.2 Buoyancy

5.3 Variable Density Solves

5.4 Divergence Control

6. Water

6.1 Marker Particles and Voxels

6.2 Level SetMethods

6.3 Extrapolation

6.4 More Accurate Pressure Solves

7. Fire

7.1 Thin Flames

7.2 Volumetric Combustion

8. Viscous Fluids

8.1 Stress

8.2 Applying Stress

8.3 Strain Rate and Newtonian Fluids

8.4 Boundary Conditions

8.5 Implementation

III: More Algorithms

9. Turbulence

9.1 Vorticity

9.2 Vorticity Confinement

9.3 Procedural Turbulence

10. Hybrid Particle Methods

10.1 Particle Advection

10.2 Secondary Particles

10.3 Vortex Particles

10.4 Particle-in-CellMethods

10.5 The Particle Level Set Method

11. Coupling Fluids and Solids

11.1 One-Way Coupling

11.2 Weak Coupling

11.3 The Immersed Boundary Method

11.4 General Sparse Matrices

11.5 Strong Coupling

12. Shallow Water

12.1 Deriving the Shallow Water Equations

12.2 The Wave Equation

12.3 Discretization

13. Ocean Modeling

13.1 Potential Flow

13.2 Simplifying Potential Flow for the Ocean

13.3 Evaluating the Height Field Solution

13.4 Unsimplifying the Model

13.5 Wave Parameters

13.6 Eliminating Periodicity

A. Background

A.1 Vector Calculus

A.2 Numerical Methods

B. Derivations

B.1 The Incompressible Euler Equations

B.2 The Pressure Problem as a Minimization

Bibliography

i i i i Fluid Simulation for Computer Graphics i i i i Download at www.Pin5i.Com

i i i i Fluid Simulation for Computer Graphics Robert Bridson A K Peters, Ltd. Wellesley, Massachusetts i i i i Download at www.Pin5i.Com

i i i i Editorial, Sales, and Customer Service Oﬃce A K Peters, Ltd. 888 Worcester Street, Suite 230 Wellesley, MA 02482 www.akpeters.com Copyright c 2008 by A K Peters, Ltd. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechani- cal, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner. Library of Congress Cataloging-in-Publication Data Bridson, Robert, 1976-- Fluid simulation for computer graphics / Robert Bridson. p. cm. Includes bibliographical references and index. ISBN 978-1-56881-326-4 (alk. paper) 1. Computer graphics--Mathematics. 2. Computer animation. 3. Three-dimensional display systems. I. Title. T385.B744 2008 006.6 96--dc22 2008021768 Front cover: “The Foamy Brine” (ink on paper), Robert Bridson, 2006. Printed in India 12 11 10 09 08 10 9 8 7 6 5 4 3 2 1 i i i i Download at www.Pin5i.Com

i i i i For my wife, Rowena, and children Jonathan, Elliot, Arthur and Eleanor i i i i Download at www.Pin5i.Com

i i i i Contents Preface I The Basics 1 The Equations of Fluids 1.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Momentum Equation . . . . . . . . . . . . . . . . . . 1.3 Lagrangian and Eulerian Viewpoints . . . . . . . . . . . . 1.4 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . 1.5 Dropping Viscosity . . . . . . . . . . . . . . . . . . . . . . 1.6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 2 Overview of Numerical Simulation 2.1 Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Splitting the Fluid Equations . . . . . . . . . . . . . . . . 2.3 Time Steps . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Advection Algorithms 3.1 Semi-Lagrangian Advection . . . . . . . . . . . . . . . . . 3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 3.3 Time Step Size . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Reducing Numerical Dissipation . . . . . . . . . . . . . . 4 Making Fluids Incompressible 4.1 The Discrete Pressure Gradient . . . . . . . . . . . . . . . 4.2 The Discrete Divergence . . . . . . . . . . . . . . . . . . . 4.3 The Pressure Equations . . . . . . . . . . . . . . . . . . . 4.4 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . vii xi 1 3 3 4 6 10 13 13 17 17 19 21 21 27 27 31 32 35 37 41 42 44 47 61 i i i i Download at www.Pin5i.Com

i i i i viii Contents 4.5 More Accurate Curved Boundaries . . . . . . . . . . . . . 4.6 The Compatibility Condition . . . . . . . . . . . . . . . . II Different Types of Fluids 5 Smoke 5.1 Temperature and Smoke Concentration . . . . . . . . . . 5.2 Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Variable Density Solves . . . . . . . . . . . . . . . . . . . 5.4 Divergence Control . . . . . . . . . . . . . . . . . . . . . . 6 Water 6.1 Marker Particles and Voxels . . . . . . . . . . . . . . . . . 6.2 Level Set Methods . . . . . . . . . . . . . . . . . . . . . . 6.3 Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 More Accurate Pressure Solves . . . . . . . . . . . . . . . 7 Fire 7.1 Thin Flames . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Volumetric Combustion . . . . . . . . . . . . . . . . . . . 8 Viscous Fluids 8.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Applying Stress . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Strain Rate and Newtonian Fluids . . . . . . . . . . . . . 8.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . Implementation . . . . . . . . . . . . . . . . . . . . . . . . 8.5 III More Algorithms 9 Turbulence 9.1 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Vorticity Conﬁnement . . . . . . . . . . . . . . . . . . . . 9.3 Procedural Turbulence . . . . . . . . . . . . . . . . . . . . 10 Hybrid Particle Methods 10.1 Particle Advection . . . . . . . . . . . . . . . . . . . . . . 10.2 Secondary Particles . . . . . . . . . . . . . . . . . . . . . . 10.3 Vortex Particles . . . . . . . . . . . . . . . . . . . . . . . . 62 71 73 75 75 78 78 80 83 83 87 94 95 101 102 105 107 107 109 111 115 116 125 127 127 131 133 137 139 141 144 i i i i Download at www.Pin5i.Com

i i i i Contents 10.4 Particle-in-Cell Methods . . . . . . . . . . . . . . . . . . . 10.5 The Particle Level Set Method . . . . . . . . . . . . . . . 11 Coupling Fluids and Solids 11.1 One-Way Coupling . . . . . . . . . . . . . . . . . . . . . . 11.2 Weak Coupling . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The Immersed Boundary Method . . . . . . . . . . . . . . 11.4 General Sparse Matrices . . . . . . . . . . . . . . . . . . . 11.5 Strong Coupling . . . . . . . . . . . . . . . . . . . . . . . 12 Shallow Water 12.1 Deriving the Shallow Water Equations . . . . . . . . . . . 12.2 The Wave Equation . . . . . . . . . . . . . . . . . . . . . 12.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 13 Ocean Modeling 13.1 Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Simplifying Potential Flow for the Ocean . . . . . . . . . 13.3 Evaluating the Height Field Solution . . . . . . . . . . . . 13.4 Unsimplifying the Model . . . . . . . . . . . . . . . . . . . 13.5 Wave Parameters . . . . . . . . . . . . . . . . . . . . . . . 13.6 Eliminating Periodicity . . . . . . . . . . . . . . . . . . . . A Background A.1 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . A.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . B Derivations B.1 The Incompressible Euler Equations . . . . . . . . . . . . B.2 The Pressure Problem as a Minimization . . . . . . . . . . Bibliography Index ix 147 150 155 155 158 159 160 163 169 170 174 176 179 179 182 187 190 193 193 195 195 203 207 207 209 213 221 i i i i Download at www.Pin5i.Com

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