A Fast Variational Framework for Accurate Solid-Fluid Coupling Christopher Batty Florence Bertails Robert Bridson University of British Columbia.

A Fast Variational Framework for Accurate Solid-Fluid Coupling

A Fast Variational Framework for Accurate Solid-Fluid Coupling

Christopher Batty

Florence Bertails

Robert Bridson

University of British Columbia

MotivationMotivation

• Goal: Simulate fluids coupled to objects.

• Extend the basic Eulerian approach:

– Advect fluid velocities

– Add forces (eg. gravity)

– Enforce incompressibility via pressure projection

• See eg. [Stam ‘99, Fedkiw et al. ‘01, Foster & Fedkiw ‘01, Enright et al. ‘02, etc.]

MotivationMotivation

• Cartesian grid fluid simulation is great!

– Simple

– Effective

– Fast data access

– No remeshing needed

• But…

MotivationMotivation

• Achilles’ heel: Real objects rarely align with grids.

OverviewOverview

Three parts to our work:

1) Irregular static objects on grids

2) Dynamic & kinematic objects on grids

3) Improved liquid-solid boundary conditions

Previous WorkPrevious Work

• First solution Voxelize

– [Foster & Metaxas ’96]

• Easy!

• “Stairstep” artifacts

• Artificial viscosity

• Doesn’t converge under refinement!

Previous WorkPrevious Work

• Better solution Subdivide nearby

– [Losasso et al. ‘04]

• Stairs are smaller

• But problem remains

Previous WorkPrevious Work

• Better yet Mesh to match objects

– [Feldman et al. ‘05]

• Accurate!

• Needs remeshing

• Slower data access

• Trickier interpolation

• Sub-grid objects?

And now… back to the future?And now… back to the future?

• We’ll return to regular grids

– But achieve results like tet meshes!

Pressure ProjectionPressure Projection

• Converts a velocity field to be incompressible (or divergence-free)

• No expansion or compression

• No flow into objects

Images courtesy of [Tong et al. ‘03]

Pressure ProjectionPressure Projection

• We want the “closest” incompressible velocity field to the input.

• It’s a minimization problem!

Key Idea!Key Idea!

• Distance metric in the space of fluid velocity fields is kinetic energy.

• Minimizing KE wrt. pressure is equivalent to the classic Poisson problem!

fluid

nn 211

2

1KE u

Minimization InterpretationMinimization Interpretation

• Fluid velocity update is:

• Resulting minimization problem is:

ptn

uu ~1

2

~2

1minarg

fluidpp

t

u

What changes?What changes?

• Variational principle automatically enforces boundary conditions! No explicit manipulation needed.

• Volume/mass terms in KE account for partial fluid cells.

– Eg.

• Result: Easy, accurate fluid velocities near irregular objects.

22/12/12/12

1KE iii uvol

Measuring Kinetic EnergyMeasuring Kinetic Energy

Discretization DetailsDiscretization Details

• Normal equations always give an SPD linear system.

– Solve with preconditioned CG, etc.

• Same Laplacian stencil, but with new volume terms.

Classic:

Variational:

x

uu

x

ppp iiiii

2/12/12

11 )1()1()1()2()1(

x

uVuV

x

pVpVVpV iiiiiiiiiii

2/12/12/12/12

12/12/12/112/1 )()()()()(

Object CouplingObject Coupling

• This works for static boundaries

• How to extend to…

– Two-way coupling?

• Dynamic objects fully interacting with fluid

– One-way coupling?

• Scripted/kinematic objects pushing the fluid

Object Coupling – Previous WorkObject Coupling – Previous Work

• “Rigid Fluid” [Carlson et al ’04]

– Fast, simple, effective

– Potentially incompatible boundary velocities, leakage

• Explicit Coupling [Guendelman

et al. ’05]

– Handles thin shells, loose coupling approach

– Multiple pressure solves per step, uses voxelized solve

Object Coupling – Previous WorkObject Coupling – Previous Work

• Implicit Coupling [Klingner et al ‘06, Chentanez et al.’06]

– solves object + fluid motion simultaneously

– handles tight coupling (eg. water balloons)

– requires conforming (tet) mesh to avoid artifacts

A Variational Coupling FrameworkA Variational Coupling Framework

Just add the object’s kinetic energy to the system.

Automatically gives:

– incompressible fluid velocities

– compatible velocities at object surface

fluid

solidVMVu *

2

1

2

1KE

2

A Coupling FrameworkA Coupling Framework

Two components:

1) Velocity update:

How does the pressure force update the object’s velocity?

2) Kinetic energy:

How do we compute the object’s KE?

Example: Rigid BodiesExample: Rigid Bodies

1) Velocity update:

2) Kinetic Energy:

Discretize consistently with fluid, add to minimization, and solve.

solid

CM

solid

p

p

nXx

n

ˆ)(Torque

ˆForce

22

2

1

2

1KE Iωv m

Sub-Grid Rigid BodiesSub-Grid Rigid Bodies

Interactive Rigid BodiesInteractive Rigid Bodies

One Way CouplingOne Way Coupling

• Conceptually, object mass infinity

• In practice: drop coupling terms from matrix

Paddle VideoPaddle Video

Wall SeparationWall Separation

• Standard wall boundary condition is u·n = 0.

– Liquid adheres to walls and ceilings!

• Ideally, prefer u·n ≥ 0, so liquid can separate

– Analogous to rigid body contact.

Liquid Sticking VideoLiquid Sticking Video

Wall Separation - Previous WorkWall Separation - Previous Work

• If ũ·n ≥ 0 before projection, hold u fixed.– [Foster & Fedkiw ’01, Houston et al ’03, Rasmussen et al ‘04]

• Inaccurate or incorrect in certain cases:

Wall SeparationWall Separation

• Two cases at walls:

– If p > 0, pressure prevents penetration (“push”)

– If p < 0, pressure prevents separation (“pull”)

• Disallow “pull” force:

– Add p ≥ 0 constraint to minimization

– Gives an inequality-constrained QP

– u·n ≥ 0 enforced implicitly via KKT conditions

Liquid Peeling VideoLiquid Peeling Video

Future DirectionsFuture Directions

• Robust air-water-solid interfaces.

• Add overlapping ghost pressures to handle thin objects, à la [Tam et al ’05]

Future DirectionsFuture Directions

• Explore scalable QP solvers for 3D wall-separation.

• Extend coupling to deformables and other object models.

• Employ linear algebra techniques to accelerate rigid body coupling.

SummarySummary

• Easy method for accurate sub-grid fluid velocities near objects, on regular grids.

• Unified variational framework for coupling fluids and arbitrary dynamic solids.

• New boundary condition for liquid allows robust separation from walls.

Thanks!Thanks!