1. More Accurate Pressure Solves

2. Solid Boundaries  Voxelized version works great if solids aligned with grid  If not: though the error in geometry is O(∆x), translates into O(1) error in velocities!  The simulation physics sees stair-steps, and gives you motion for the stair-step case

3. Quick Fix  Measure accurate normals  Before pressure solve, on boundary, fix  Then do voxelized pressure, set up to not modify boundary

4. Quick Fix Fails  This is equivalent for grid-aligned solids  Works great for highly dynamic splashing etc.  Fails miserably in steadier situations

5. Quick Fix Fails  This is equivalent for grid-aligned solids  Works great for highly dynamic splashing etc.  Fails miserably in steadier situations GRAVITY

6. Quick Fix Fails  This is equivalent for grid-aligned solids  Works great for highly dynamic splashing etc.  Fails miserably in steadier situations BOUNDARY FIX

7. Quick Fix Fails  This is equivalent for grid-aligned solids  Works great for highly dynamic splashing etc.  Fails miserably in steadier situations PRESSURE SOLVE

8. Quick Fix Fails  This is equivalent for grid-aligned solids  Works great for highly dynamic splashing etc.  Fails miserably in steadier situations  Fictitious currents emerge and unstably grow

9. Rethinking the problem  See Batty et al. (tomorrow)  If we keep our fluid blobs constant volume, incompressibility constraint means: blobs stay in contact with each other (no interpenetration, no gaps)  Staying in contact == inelastic, sticky collision

10. Inelastic, sticky collisions  Take two particles. Interaction force update:  (F is like pressure gradient)  Contact constraint is: (like divergence-free condition)

11. Inelastic, sticky collisions (2)  Can solve for F to satisfy kinematic constraint  Equivalently, find F that minimizes kinetic energy of system:  Kinematics comes for free…

12. Variational Pressure Solve  Pressure update is fluid particle interaction  Incompressible means no stored energy: fully inelastic  Thus it must minimize kinetic energy + work with constraint p=0 on free surface  Can prove it’s equivalent to regular PDE form!

13. Variational goodness  The solid wall boundary condition vanishes! (automatically enforced at minimum)  Discretizing kinetic energy integral much simpler:  Just need average fluid density in each cell, and volume fraction of fluid inside cell

14. Linear System  Plug in discrete pressure update in discrete KE  Quadratic in pressures  Find discrete minimum == solve linear system  Linear system guaranteed to be symmetric, positive definite  In fact, it’s exactly the same as voxelized – except each term is weighted by volume fractions

15. Benefits  Actually converges! (error is O(∆x) or better)  Handles resting case perfectly: KE is minimized by zeroing out velocity, which we get from hydrostatic pressure field  Can handle sub-grid-scale geometry  E.g. particles immersed in flow, narrow channels, hair…  Just need to know volume displaced!

16. Extra goodness  Can couple solids in flow easily:  Figure out formula for discrete pressure update to solid velocity  Add solid’s kinetic energy to minimization  Automatically gives two-way “strong” coupling between rigid bodies and flow, perfectly compatible velocities at boundary, no tangential coupling…

17. Free Surfaces  The other problem we see with voxelization is free surface treatment  Physics only sees voxels: waves less than O(∆x) high are ignored  At least position errors will converge to zero… but errors in normal are O(1)! (rendering will always look awful)

18. Ghost Fluid Method  Due to Fedkiw and coauthors pipi p i+1 p=0

19. Ghost Fluid Method  Voxelized version: pipi p i+1 p=0

20. Ghost Fluid Method  GFM version: pipi p i+1 p=0

21. GFM with solids  Complementary to variational solve: GFM just changes the pressure update  However, for triple junctions (solid+liquid+air) it gets difficult to make this just right

22. Reinterpret GFM  The multiple fluid (“two-phase”) jump conditions:  So take finite difference for  Use average fluid density for pressure update (average between liquid and air)

23. Variational free surface  Simply need volume fractions per cell (how much of cell is liquid+air), and average densities per cell (mix between liquid and air)  Use average density for pressure update, volume fraction for KE estime  Speed things up: all-air cells eliminated (set p=0 there)  This is the discrete free surface approximation!