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More Accurate Pressure Solves

More Accurate Pressure Solves. 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. Quick Fix.

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More Accurate Pressure Solves

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  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 + workwith 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 pi+1 pi p=0

  19. Ghost Fluid Method • Voxelized version: pi+1 pi p=0

  20. Ghost Fluid Method • GFM version: pi+1 pi 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!

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