A Fast Variational Framework for Accurate Solid-Fluid Coupling Christopher Batty Florence Bertails...
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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!