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

Measure accurate normals

Before pressure solve, on boundary, fix

Then do voxelized pressure, set up to not modify boundary

vu ← vu+ n ng(vuSOLID −vu)( )

Quick Fix Fails

This is equivalent for grid-aligned solids

Works great for highly dynamic splashing etc.

Fails miserably in steadier situations

Quick Fix Fails

This is equivalent for grid-aligned solids

Works great for highly dynamic splashing etc.

Fails miserably in steadier situationsGRAVITY

Quick Fix Fails

This is equivalent for grid-aligned solids

Works great for highly dynamic splashing etc.

Fails miserably in steadier situationsBOUNDARY FIX

Quick Fix Fails

This is equivalent for grid-aligned solids

Works great for highly dynamic splashing etc.

Fails miserably in steadier situationsPRESSURE SOLVE

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

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

Inelastic, sticky collisions

Take two particles. Interaction force update:

(F is like pressure gradient) Contact constraint is:

(like divergence-free condition)

vu1 ←

vu1 +1m1

vF

vu2 ← vu2 −1m2

vF

vu1 −

vu2 =0

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…

F =argmin

12

m1vu1

2 +12

m2vu2

2⎛⎝⎜

⎞⎠⎟

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!

p =argmin 1

2 ρ unew2

∫∫∫ − p(−n)gΔtusolidsolid∫∫

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

KE ≈

12ρi+ 1

2 jkVoli+1/2 jk ui+ 12 jk( )

2

ijk∑ +K

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

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!

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…

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)

Ghost Fluid Method

Due to Fedkiw and coauthors

pi pi+1

p=0

Ghost Fluid Method

Voxelized version:

pi pi+1

p=0

∂p∂x≈pi+1 − piΔx

Ghost Fluid Method

GFM version:

pi pi+1

p=0

∂p∂x≈pi+1 − piθΔx

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

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)

[p]=0

vugn[ ] =0 ⇒1ρ

∂p∂n

⎡

⎣⎢

⎤

⎦⎥=0

1

ρ∂p∂n

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!