1. 1 Modeling Highly- Deformable Liquid Chih-Wei Chiu Computer Graphics and Geometry Modeling Laboratory National Chiao Tung University June 25, 2002 Advisors: Cheng-Chung Lin and Jung-Hong Chuang

2. 2 Agenda Introduction Introduction Previous Work Previous Work Method Method Result Result Conclusion Conclusion

3. 3 Modeling Fluids Particles Particles A particle represent a fixed- mass fluid element A particle represent a fixed- mass fluid element There may be interaction forces between particles There may be interaction forces between particles Volume Volume Partition the scene into many cells Partition the scene into many cells Every cell carries a density function representing the proportion of air and liquid Every cell carries a density function representing the proportion of air and liquid Surface can be inferred from density functions Surface can be inferred from density functions

4. 4 Computational Aspect Particles Particles Number grows with cube of resolution Number grows with cube of resolution Computationally expensive Computationally expensive Volume Volume Only one equation for each cell to evolve the density function (more on this later) Only one equation for each cell to evolve the density function (more on this later) Computationally economical Computationally economical

5. 5 Geometry Aspect Particles Particles No straightforward way to extract a smooth surface No straightforward way to extract a smooth surface Resolve details independent of grid resolution Resolve details independent of grid resolution Volume Volume Iso-surface can be extracted from the volume data Iso-surface can be extracted from the volume data Potentially under-resolve details if the grid is too coarse Potentially under-resolve details if the grid is too coarse smooth not smooth

6. 6 Proposed Method Since volume method and particles have nearly “complemented” strength and weakness… Since volume method and particles have nearly “complemented” strength and weakness… Combination of particles and volume method Combination of particles and volume method Track initial fluid by volume Track initial fluid by volume Resolve highly-deformed regions by particles Resolve highly-deformed regions by particles

7. 7 Previous Work Wave Simulation Wave Simulation Surface is a parametric function can be animation over time Surface is a parametric function can be animation over time Fourier synthesis [Mastin ’87] Fourier synthesis [Mastin ’87] Wave-tracing [Ts’o ‘87] Wave-tracing [Ts’o ‘87] Particle Systems Particle Systems Very heuristic, Cheap computation Very heuristic, Cheap computation Ship wakes [Goss ‘90] Ship wakes [Goss ‘90] Thin-film water splash [Ashraf ’99] Thin-film water splash [Ashraf ’99] Splash “subsystem” – particles are generated when the water surface is under impact [O’Brien ’95] [Mould ’97] Splash “subsystem” – particles are generated when the water surface is under impact [O’Brien ’95] [Mould ’97]

8. 8 Molecular Dynamics Molecular Dynamics Simulate elastic and inelastic objects and viscous fluids Simulate elastic and inelastic objects and viscous fluids Computational expensive for large volume of fluid Computational expensive for large volume of fluid Lack of coupling between velocity and pressure Lack of coupling between velocity and pressure SPH (Smoothed Particle Hydrodynamics) SPH (Smoothed Particle Hydrodynamics) Deformable object [Desbrun’96] Deformable object [Desbrun’96] Lava [Stora ‘99] Lava [Stora ‘99] Advantages Advantages More economical than molecular dynamics More economical than molecular dynamics Couple pressure and velocity Couple pressure and velocity Previous Work That’s why we choose it!

9. 9 3-D Navier-Stokes Equations 3-D Navier-Stokes Equations Produce the most realistic motion but are very computationally expensive Produce the most realistic motion but are very computationally expensive Initiated by Foster ’96 Initiated by Foster ’96 Used MAC formulation dated by 1965 Used MAC formulation dated by 1965 Semi-Lagrangian integration allows large time step without harming the accuracy [Stan’99] Semi-Lagrangian integration allows large time step without harming the accuracy [Stan’99] Implicit Surface Implicit Surface Represent fluid volume by a density function [Kunimatsu ’01] Represent fluid volume by a density function [Kunimatsu ’01] Hybrid model [Foster ’01] Hybrid model [Foster ’01] Dynamic implicit surface Dynamic implicit surface Allow particles to locally correct surface value Allow particles to locally correct surface value Previous Work

10. 10 Outline Fluid dynamics solver p n & v n Partition the scene by a grid (only in the first time-step) grid Evolve density functions v n+1 Generate particles in the drastic motion region F n+1 Move particles by SPH existing particles new particles Performed per time-step Subdivision Performed during rendering Interpolation high resolution grid high resolution F n+1 Convert particle volumes to F particles positions high resolution F n+1 with particle volumes Extract iso-surface Rendered using a ray- tracer polygon mesh Our Methods !!! FnFn

11. 11 Modeling Splash Problem Problem Grid may be too coarse to resolve the detail features in a surface cell Grid may be too coarse to resolve the detail features in a surface cell Put particles in all surface cell is too computational expensive Put particles in all surface cell is too computational expensive Solution Solution Introduce particles in the cell with drastic motion Introduce particles in the cell with drastic motion

12. 12 Drastic Motion Criterion Seed particles in the surface cell with drastic motion Seed particles in the surface cell with drastic motion Limit the maximum number of particles in a cell Limit the maximum number of particles in a cell Existing particles provides more accurate approximation Existing particles provides more accurate approximation Fluid particle should not travel more than one cell size in a time step; 0<γ<1 Fluid traveling distance in a time step Lower tends to produce more particles Lower γ tends to produce more particles

13. 13 Volume Fractions

14. 14 Particle Seeding Particles are randomly positioned inside the interface (fluid side) Particles are randomly positioned inside the interface (fluid side) Particles velocities are linearly interpolated in the cell Particles velocities are linearly interpolated in the cell u v Reconstructed interface

15. 15 Interface Reconstruction Approximate the interface by a plane ax+by+cz+d=0 Approximate the interface by a plane ax+by+cz+d=0 (a,b,c) VfVf Normal is easy to compute, e.g. by spatial gradients Normal is easy to compute, e.g. by spatial gradients Find d so that the “cut volume” given by the interface is V f [Scardovelii ‘00] Find d so that the “cut volume” given by the interface is V f [Scardovelii ‘00] Cut volume

16. 16 Partition the scene by a grid (only in the first time-step) grid Evolve density functions F n+1 Move particles by SPH Subdivision Performed during rendering Interpolation high resolution grid high resolution F n+1 Convert particle volumes to F particles positions high resolution F n+1 with particle volumes Extract iso-surface Rendered using a ray- tracer polygon mesh Next Slide… My own method !!!

17. 17 Subdivision Volume Fractions Coarse grid Fine grid Smooth surface Subdivision & interpolation iso-surface extraction Convert particles volumes to F Non-smooth surface Without subdivision & interpolation Without converting particles to F iso-surface extraction Small particles are directly rendered as blobs

18. 18 Why Blobs Doesn’t Work? Non-smooth surface

19. 19 3x3x3 subdivision Convert particles to volume fractions

20. 20 Result Computational grid - 28x24x28 Computational grid - 28x24x28 Simulation time per frame – 8 ~ 60 sec Simulation time per frame – 8 ~ 60 sec Cycles per frame - 5 Cycles per frame - 5 Average number of particles - 500 Average number of particles - 500 Rendering time per frame – 1.5 ~ 4 min Rendering time per frame – 1.5 ~ 4 min Subdivision – 3x3x3 Subdivision – 3x3x3 AMD Athlon 1000 MHZ AMD Athlon 1000 MHZ 512 MB DRAM 512 MB DRAM

21. 21 Control the Number of Particles γ=0.2γ=0.1

22. 22 Comparison 1 O'Brien,J.F.O'Brien,J.F.; Hodgins,J.K., "Dynamic Simulation of Splashing Fluids", in Proceedings of Computer Animation '95, pp.198-205, 1995.Hodgins,J.K. Mould,D.Mould,D.; Yang,Y.H., "Modeling Water for Computer Graphics", Computers & Graphics, vol.21, no.6, pp.801-814, 1997.Yang,Y.H. Generate particles when the vertical velocities of a column exceed a predefined threshold Extend O’Brien et al. Allow droplets to split.

23. 23 Comparison 2 Enright,D.Enright,D.; Marschner,S.; Fedkiw,R., "Animation and Rendering of Complex Water Surfaces", in Procceeding os SIGGRAPH 2002, 2002 (to appear).Marschner,S.Fedkiw,R. Foster,N.Foster,N.; Fedkiw,R., "Practical Animation of Liquids", Computer Graphics, pp.23-30, 2001.Fedkiw,R. Hybrid model: Dynamic level set implicit surface. Marker particles locally correct surface. Extend Foster ’01: Put particles on both sides of the surface.

24. 24 Comparison 3 Heuristic Approach Marker Particles Identify drastic motion region and generate particles. Place particles near the surface to capture drastic motion. Autonomous particles moves independently of the fluid. Passive particles velocities are interpolated from the grid node velocities.

25. 25 Conclusion A hybrid approach integrating particles (Lagrangian method) and volume fractions (Eulerian method) A hybrid approach integrating particles (Lagrangian method) and volume fractions (Eulerian method) A scheme to identify drastic motion area and seed particles A scheme to identify drastic motion area and seed particles Smooth surface by converting particles to volume fractions Smooth surface by converting particles to volume fractions