A Fast Variational Framework for Accurate Solid-Fluid Coupling

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A Fast Variational Framework for Accurate
Solid-Fluid Coupling

• Christopher Batty
• Florence Bertails
• Robert Bridson
• University of British Columbia

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Motivation

• 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.

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Motivation

• Cartesian grid fluid simulation is great!
• Simple
• Effective
• Fast data access
• No remeshing needed
• But

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Motivation

• Achilles heel Real objects rarely align with
  grids.

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Overview

• Three parts to our work
• Irregular static objects on grids
• Dynamic kinematic objects on grids
• Improved liquid-solid boundary conditions

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Previous Work

• First solution ? Voxelize
• Foster Metaxas 96
• Easy!
• Stairstep artifacts
• Artificial viscosity
• Doesnt converge under refinement!

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Previous Work

• Better solution ? Subdivide nearby
• Losasso et al. 04
• Stairs are smaller
• But problem remains

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Previous Work

• Better yet ? Mesh to match objects
• Feldman et al. 05
• Accurate!
• Needs remeshing
• Slower data access
• Trickier interpolation
• Sub-grid objects?

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And now back to the future?

• Well return to regular grids
• But achieve results like tet meshes!

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Pressure 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

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Pressure Projection

• We want the closest incompressible velocity
  field to the input.
• Its a minimization problem!

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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!

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Minimization Interpretation

• Fluid velocity update is
• Resulting minimization problem is

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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.

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Measuring Kinetic Energy
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Discretization Details

• Normal equations always give an SPD linear
  system.
• Solve with preconditioned CG, etc.
• Same Laplacian stencil, but with new volume
  terms.
• Classic
• Variational

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Object 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

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Object 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

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Object 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

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A Variational Coupling Framework

• Just add the objects kinetic energy to the
  system.
• Automatically gives
• incompressible fluid velocities
• compatible velocities at object surface

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A Coupling Framework

• Two components
• Velocity update
• How does the pressure force update the objects
  velocity?
• Kinetic energy
• How do we compute the objects KE?

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Example Rigid Bodies

• Velocity update
• Kinetic Energy
• Discretize consistently with fluid, add to
  minimization, and solve.

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Sub-Grid Rigid Bodies
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Interactive Rigid Bodies
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One Way Coupling

• Conceptually, object mass ? infinity
• In practice drop coupling terms from matrix

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Paddle Video
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Wall Separation

• Standard wall boundary condition is un 0.
• Liquid adheres to walls and ceilings!
• Ideally, prefer un 0, so liquid can separate
• Analogous to rigid body contact.

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Liquid Sticking Video
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Wall Separation - Previous Work

• If un 0 before projection, hold u fixed.
• Foster Fedkiw 01, Houston et al 03,
  Rasmussen et al 04
• Inaccurate or incorrect in certain cases

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Wall Separation

• Two cases at walls
• If p gt 0, pressure prevents penetration (push)
• If p lt 0, pressure prevents separation (pull)
• Disallow pull force
• Add p 0 constraint to minimization
• Gives an inequality-constrained QP
• un 0 enforced implicitly via KKT conditions

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Liquid Peeling Video
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Future Directions

• Robust air-water-solid interfaces.
• Add overlapping ghost pressures to handle thin
  objects, à la Tam et al 05

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Future 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.

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Summary

• 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.

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Thanks!