animating Fluids Like Water, Smoke, And Fire Using Physics-based Simulation Is Increasingly Important In Visual Effects, In Particular In Movies, Like the Day After Tomorrow , And In Computer Games. This Book Provides A Practical Introduction To Fluid Simulation For Graphics. The Focus Is On Animating Fully Three-dimensional Incompressible Flow, From Understanding The Math And The Algorithms To The Actual Implementation. Contents......Page 6 Preface......Page 9 I: The Basics......Page 10 1.1 Symbols......Page 11 1.2 The Momentum Equation......Page 12 1.3 Lagrangian and Eulerian Viewpoints......Page 14 1.4 Incompressibility......Page 18 1.6 Boundary Conditions......Page 21 2.1 Splitting......Page 25 2.2 Splitting the Fluid Equations......Page 27 2.4 Grids......Page 29 3.1 Semi-LagrangianAdvection......Page 34 3.2 Boundary Conditions......Page 38 3.3 Time Step Size......Page 39 3.4 Dissipation......Page 42 3.5 Reducing Numerical Dissipation......Page 44 4. Making Fluids Incompressible......Page 47 4.1 The Discrete PressureGradient......Page 48 4.2 The Discrete Divergence......Page 50 4.3 The Pressure Equations......Page 53 4.4 Projection......Page 67 4.5 More Accurate Curved Boundaries......Page 68 4.6 The Compatibility Condition......Page 77 II: Different Types of Fluids......Page 79 5.1 Temperature and Smoke Concentration......Page 80 5.3 Variable Density Solves......Page 83 5.4 Divergence Control......Page 85 6.1 Marker Particles and Voxels......Page 88 6.2 Level SetMethods......Page 92 6.3 Extrapolation......Page 99 6.4 More Accurate Pressure Solves......Page 100 7. Fire......Page 105 7.1 Thin Flames......Page 106 7.2 Volumetric Combustion......Page 109 8.1 Stress......Page 111 8.2 Applying Stress......Page 113 8.3 Strain Rate and Newtonian Fluids......Page 115 8.4 Boundary Conditions......Page 119 8.5 Implementation......Page 120 III: More Algorithms......Page 129 9.1 Vorticity......Page 130 9.2 Vorticity Confinement......Page 134 9.3 Procedural Turbulence......Page 136 10. Hybrid Particle Methods......Page 140 10.1 Particle Advection......Page 142 10.2 Secondary Particles......Page 144 10.3 Vortex Particles......Page 147 10.4 Particle-in-CellMethods......Page 150 10.5 The Particle Level Set Method......Page 153 11.1 One-Way Coupling......Page 157 11.2 Weak Coupling......Page 160 11.3 The Immersed Boundary Method......Page 161 11.4 General Sparse Matrices......Page 162 11.5 Strong Coupling......Page 165 12. Shallow Water......Page 171 12.1 Deriving the Shallow Water Equations......Page 172 12.2 The Wave Equation......Page 176 12.3 Discretization......Page 178 13.1 Potential Flow......Page 180 13.2 Simplifying Potential Flow for the Ocean......Page 183 13.3 Evaluating the Height Field Solution......Page 188 13.4 Unsimplifying the Model......Page 191 13.6 Eliminating Periodicity......Page 194 A.1 Vector Calculus......Page 196 A.2 Numerical Methods......Page 204 B.1 The Incompressible Euler Equations......Page 208 B.2 The Pressure Problem as a Minimization......Page 210 Bibliography......Page 214
