Practical Animation of Liquids

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Practical Animation of Liquids
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What it does

• Modelling and animating liquids
• Mainly designed for computer animation
• 3D environment
• Handle water and viscous liquids
• Interaction with graphics primitives
• Parametric curves
• Stationary or moving polygons

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What it does

• Allows control over the motion
• Conserves realistic behaviour
• Stays visually coherent
• Looks realistic
• ? Achieved by trading off engineering
  correctness for computational efficiency

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Example
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How does it work?

• The algorithm is divided into 6 steps
• Model the static environment as a voxel grid
• Model the liquid volume using a combination of
  particles and an implicit surface
• Then, for each simulation time step
• Update the velocity field by solving
  Navier-Stokes equations (pressure and velocity)
  using finite differences combined with a
  semi-Lagrangian method
• Apply velocity constraints due to moving objects
• Enforce incompressibility by solving a linear
  system built from the Navier-Stokes equation
  (pressure)
• Update the position of the liquid volume
  (particles and implicit surface) using the new
  velocity field

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Static environment

• Grid of voxels which usually has a rectangular
  shape
• Each cell has a pressure variable at its centre
  and shares a velocity variable with each of its
  adjacent neighbours
• Each cell is either tagged as empty or filled

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Static environment Example
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Liquid representation

• Two visuals are used
• Particles
• Provide details where the surface starts to
  splash
• Their position evolve smoothly over time by
  simple convection (cf. cells)
• Their velocity is computed directly from the
  velocity grid using tri-linear interpolation and
  each particle is moved according to an
  inertialess equation
• Particles have a low computational overhead
• But no straightforward way to extract a smooth
  polygonal (or parametric) description of the
  actual liquid surface

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Liquid representation

• Isocontour
• Provides a smooth surface to regions of liquid
  that are well resolved compared to the grid
• The surface is represented using an implicit
  function
• To look good, a high density of particles is
  required
• Uses Dynamic Level Set to conform (visually) to
  the physics of liquids
• However, suffer from severe volume loss
  especially on the relatively coarse grids
  commonly used in computer graphics
• An hybrid surface model is used to solve this
  problem (particles)

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Liquid representation Example
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Velocity field

• The process occurs in 3 stages
• Compute the Courant-Friedrichs-Levy (CFL)
  Condition
• First order Semi-Lagrangian method is used to
  update the convective component
• Central differencing is then used on the viscous
  terms of Navier-Stokes equations (pressure and
  velocity)
• ? The result of the previous calculations are
  added together to update the velocity filed
• The Semi-Lagrangian method allows to use large
  time steps
• However, large time steps increase the
  dissipation and thus, ruin the visual effect in
  liquid animation

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Velocity field

• To avoid the dissipation and save computational
  power
• Alternate between a large time step for updating
  the Navier-Stokes equations and a series of small
  time steps (that add up to the large time step)
  for the particles and the level set

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Moving objects

• Previous methods have difficulties to deal with
  moving objects
• Require highly resolved computational grids
• Or a grid mechanism that can adapt itself to the
  moving object surface
• ? Not well suited to animation because of the
  additional time complexity involved

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Moving objects

• Method used
• Based on three basic rules
• Liquid should not flow into the object. At any
  point of contact, the relative velocity between
  the liquid and object along the objects surface
  normal should be greater than or equal to zero
• We directly set the component of liquid velocity
  normal to the object
• ? We know the object surface normal and can
  calculate the liquid velocity relative to that
  surface in a given cell
• Tangential to the surface, the liquid should flow
  freely without interference.
• Neither the particles nor the level set surface
  should pass through any part of the surface of
  the object
• We know where the polygons that comprise the
  object surface are and in what direction they are
  moving
• ? We simply move the particles so that they are
  always outside the surface of the object

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Moving objects

• Solution
• As a boundary condition, any cell within a solid
  object has its velocities set to that of the
  moving object
• The velocity field is updated using Navier-Stokes
  equations (pressure and velocity). No special
  consideration is given to cells containing an
  object (they are all allowed to change freely as
  if they contain liquid)
• Each cell that intersects an object surface gets
  the component of the object velocity along its
  normal set explicitly as outlined above
• Cells internal to the object have their
  velocities set back to the object velocity
• During the mass conservation step, the velocity
  for any grid cell that intersects the object is
  held fixed
• However, the grid resolution is the limiting
  factor

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Mass conservation

• The calculations used in the steps III and IV
  lead to mass dissipation
• Ensure that the incompressibility condition is
  satisfied for every grid cell
• Use the Laplacian operator to couple local
  pressure changes to the divergence in each cell
• Static object and empty cells dont disrupt this
  structure
• It can be solved quickly and efficiently using a
  Preconditioned Conjugate Gradient (PCG) method
• Further efficiency gains can be made by using an
  Incomplete Choleski preconditioner to accelerate
  convergence
• The resulting velocity field conserves mass (is
  divergence free) and satisfies the Navier-Stokes
  equations

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Conclusion

• Method
• Good looking liquid
• Correct physics of liquids
• Interaction with polygonal objects within the
  liquid
• Control of the animation

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Examples