1. Efficient Simulation of Large Bodies of Water by Coupling Two and Three Dimensional Techniques SIGGRAPH 2006 Geoffrey Irving Eran Guendelman Frank Losasso Ronald Fedkiw

2. Abstract ◇ effective simulation of a boat in a large box ◇ Similar to traditional 2D height field, the water is represented by tall cells ◇ Accurate representation of bottom topography

3. Introduction ◇ Motivation : Stormy seas, Sudden floods, cascading rapids ◇ Very expensive effects ◇ Appearance is governed by a surface layer ◇ Interior flow is not visible

4. Octree and Height field ◇ Octree’s drawback - Too much node, Increased numerical dissipation - Large octree cells can’t represent bottom topography ◇ Height field’s merit - Detail refinement in the horizontal direction - Capturing the effects of complex bottom topography well - Not support overturning or other interesting 3D behavior

5. Our method requirement (1) ◇ Detailed 3D behavior near the interface ◇ “Optical depth” is greater than the visible optical depth  uniform MAC grid Navier-Stokes solver near the interface

6. Our method requirement (2) ◇ Represent without losing plausible bulk motion ◇ Represent important details such as bottom topography  Coarsen in the vertical direction obtaining tall thin cells  reach down to the bottom of the domain

7. Advantage ◇ The Octree method suffers from increased dissipation ◇ RLE (Run-Length-Encoded) [Houston et al. 2004; Breen et al. 2004; Wiebe and Houston 2004; Nielsen and Museth 2005; Houston et al. 2005; Houston et al. 2006]  only in the air ◇ Our method greatly reduces the computational cost

8. Previous Work ◇ 3D Navier-Stokes equations - Foster and Metaxas 1997b - Stam 1999 - Fedkiw et al. 2001 ◇ Method for simulating a water surface - Foster and Metaxas 1996 - Foster and Metaxas 1997a - Foster and Fedkiw 2001 ◇ Particle level set - Enright et al. 2002

9. Previous Work ◇ Conservative height field fluid model - Baxter et al. 2004b ◇ Viscoelastic fluid - Goktekin et al. 2004 ◇ Solid fluid coupling - Calson et al. 2004 - Guendelman et al. 2005 ◇ Control - McNamara et al. 2004 - Shi an Yu 2005 ◇ Two phase flow -Hong and Kim 2005

10. Previous Work ◇ Large and deep water simulation - Fournier and Reeves 1986 - Peachey 1986 - Mastin et al. 1987 - Ts’o and Barsky 1987 - Thone et al. 2000 - Hinsinger et al. 2002 ◇ Navier-Stokes Equation update for large bodies of water - Chen and Lobo 1994 - Thon and Ghazanfarpour 2001 - Neyret and Praizelin 2001

11. Grid Structure ◇ Uniform cells and Tall cells ◇ Scalars are stored at cell centers ◇ Velocity components are stored on their respective faces ◇ Tall cells contain uppermost and bottommost pressure values ◇ Pressure values of tall cells can be interpolated

12. Grid Structure ◇ Horizontal pressure derivative ◇ Co-locate horizontal velocities ◇ Vertical pressure derivative ◇ Co-locate vertical velocities

13. Refinement and Coarsening ◇ When change the structure, Interpolate velocities in new grid ◇ The vertical velocities at new tall cells are set to the average value ◇ Those for uniform cells are defined as their interpolated value ◇ This conserves vertical momentum

14. Refinement and Coarsening ◇ For horizontal velocities ◇ Let u(j a ), u(j a +1), …, u(j b )  We desire bottom and top values u a and u b

15. Refinement Analysis and Comparison ◇ With a sufficient optical depth satisfactory results are obtained (middle : ¼ refined case) ◇ 1/16 refined case obtain nonphysical results (right) ◇ Since the computational cost of a tall cell is independent of its height, the tall cells are equivalent in cost regardless of the depth of the water

16. Refinement Analysis and Comparison ◇ Right shows a octree simulation of the splash ◇ Our method (left) is faster and larger simulation

17. Uniform Three-Dimensional Method ◇ Navier-Stokes Equations ◇ Mass conservation ◇ Linear Momentum

18. Mass Conservation

19. Linear Momentum

20. Update u using N-S Eqns

21. Advection ◇ We chose conservative method ◇ The conservation does not cure large truncation errors ◇ it only makes the results more physically plausible

22. Advection ◇ Ignoring the pressure and force terms, we focus on the u velocity ◇ Velocity is updated based on the fluxes across faces ◇ The flux are computed by averaging the velocities to the faces

23. Advection ◇ CFL condition on the time step for stability ◇ We could refine the tall cells into a uniform grid, and re-coarsen ◇ Instabilities appear sharp changes ◇ compute the total weight

24. Laplace Equation ◇ After advection, we solve for the pressure and make the divergence free ◇ We assume that the density is spatially constant

25. Laplace Equation ◇ Consider two adjacent tall cells extending from j 1 to j 2 and j 3 to j 4 ◇ Intersecting in a minimal face from j a = max(j 1, j 3 ) to j b = min(j 2, j 4 ) ◇ Component of discrete gradient is linear with value (P x ) a and (P x ) b ◇ given by interpolating pressure to j a and j b

26. Laplace Equation ◇ Derivative Py = (p 2 -p 1 )/((j 2 -j 1 )dy) vertical pressure p 1 at j 1 and p 2 at j 2 ◇ The discrete volume weighted divergence of a uniform cell ◇ +- sign is chosen the face, A f is the area of the face

27. Laplace Equation ◇ Total flux contribution ◇ We substitute (P x ) a and (P x ) b for u a to u b

28. Examples

29. Example

30. Conclusion ◇ Novel method for the simulation of large bodies of water ◇ The bulk of the water volume is represented with tall cell ◇ Capturing detailed surface motion ◇ Representing detailed bottom topography