pressure solve

7/29/2019 Pressure Solve

1/23

More Accurate Pressure Solves


2/23

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


3/23


4/23

Quick Fix Fails

This is equivalent for grid-aligned solids

Works great for highly dynamic splashing etc.

Fails miserably in steadier situations


5/23

Quick Fix Fails



Fails miserably in steadier situationsGRAVITY


6/23

Quick Fix Fails



Fails miserably in steadier situationsBOUNDARY FIX


7/23

Quick Fix Fails



Fails miserably in steadier situationsPRESSURE SOLVE


8/23

Quick Fix Fails



Fails miserably in steadier situations Fictitious currents emerge and unstably grow


9/23

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


10/23

Inelastic, sticky collisions

Take two particles. Interaction force update:

(F is like pressure gradient) Contact constraint is:

(like divergence-free condition)

u1 vu1

1

m1

vF

vu2

vu2

1

m2

vF

u1 u2 0


11/23

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 argmin1

2m1

vu1

21

2m2

vu2

2


12/23

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 its equivalent to regular PDE form!

p argmin 12unew

2

p(n)gtusolidsolid


13/23

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

KE1

2

i 12jkVoli1/2 jk ui 1

2jk

2

ijk

K


14/23

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, its exactly the same as voxelized

except each term is weighted by volume fractions


15/23

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!


16/23

Extra goodness

Can couple solids in flow easily:

Figure out formula for discrete pressure update to

solid velocity

Add solids kinetic energy to minimization

Automatically gives two-way strong couplingbetween rigid bodies and flow, perfectly

compatible velocities at boundary, no tangential

coupling


17/23

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)


18/23

Ghost Fluid Method

Due to Fedkiw and coauthors

pi p

i+1

p=0


19/23


20/23

Ghost Fluid Method

GFM version:

pi p

i+1

p=0

px

pi1 pix


21/23

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


22/23

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 1

p

n

0

1

pn


23/23

pressure solve

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