3D Wave Motion and Fluids

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2D Wave Motion and Fluids

• Raylene Deck, Jamie Drummond, Jessica Ho Tyler
  Johnston

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Overview

• Basic wave formation
• Fun stuff with fluids
• Implementation
• Demo
• Questions

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Motivation

• Why do you want to create realistic waves?
• Accurate wave simulation is vital in many
  applications
• Tsunami impacts
• Oil rig platforms
• Engineering of ships
• Nuclear coolant

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Basic Wave Formation

• Basic Concepts
• Goals of Liquid Motion Animation
• Navier-Stokes Equations
• CFL Stability Condition
• 3D Realistic Animation of Liquids
• Mesh (Grid) Layout

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Basic Concepts

• When simulating waves there is a general tradeoff
  between being visually realistic versus
  physically accurate.
• Provide enough physical realism to obtain goals,
  but in reasonable computational time
• 2D implementation most common with Navier-Stokes
  implementation.

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Goals of Liquid Motion Animation

• Model and animate quickly on computer
• Model how it interacts with traditional computer
  graphics environment
• Allow model manipulation to break boundaries of
  real world physics
• Create great looking pictures

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Navier-Stokes Equations

• Describes the motion of fluids
• Take into account
• Viscosity
• Convection
• Gravity
• Pressure

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Navier-Stokes Equations (contd)

• Assumes that the fluid is continuous
• Solution solves for the velocity of the fluid at
  a given point in space and time
• Direct implementation of Navier-Stokes is not
  possible since Navier-Stokes is computationally
  intensive.
• Discretize implementation of Navier-Stokes models
  are frequently used

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Courant-Friedrichs-Levy (CFL) Stability Condition

• Constraint of implementation where the time step
  must be within a certain factor in order for the
  algorithm to be stable
• Allows the algorithm to converge to a solution
• Large time steps will lead to more erroneous
  results.

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3D Realistic Animation of Liquids

• Goal Realistically animate liquid phenomena
• Foundation Navier-Stokes with momentum and mass
  conservation to describe fluid motion
• 3D modeling also takes into account
• Buoyancy, convection, mass transport, submerged
  obstacles

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3D Realistic Animation of Liquids (contd)

• u, v, w are velocities
• p is local pressure
• g is gravity
• v is viscosity of the fluid

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3D Realistic Animation of Liquids (contd)

• Discretizing Navier-Stokes in 3D
• Solve for NS across the entire environment
• Cells can either be
• solid obstacle
• full of fluid
• surface cell
• empty
• Computationally expensive!!!

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Grids and Meshes

• Fluid surfaces and areas can be modeled on
  several bases
• Quadtrees
• Octrees
• Polygon meshes

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Quadtree

• A point is directly effected by the four points
  around it, one in each direction
• Primarily used for 2D animation
• http//www.cs.wustl.edu/suri/cs506/projects/quad.
  html

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Octree

• A point is directly effected by the 8 points
  around it, one in each direction
• Can be 2D or 3D based

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Comparison

• Quadtree is requires less computability and
  storage space, but is less visually accurate than
  octree which requires more computation and
  storage
• In 2D, both grids only allow the points to move
  up and down, not side-to-side
• 4 vs. 8 grid demo

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Polygon Meshes

• Useful for simulating changed in the density of
  gases, more dense, more polygons
• The polygons change shape, location and size with
  each time step
• Generalized Semi-Lagrangian step
• Trace back from the position where velocity is
  stored in new mesh, xi, interpolate velocity
  using old mesh and velocity field, and update
  velocity in new mesh

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Paddle Through Smoke Demo
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Rendering Natural Waters

• Simon Premoze, Michael Ashikhmin
• Department of Computer Science
• University of Utah

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What to Consider?

• One of the most daunting tasks in graphics
• 3 Components to create realistic water
• Atmospheric conditions
• Wave generation
• Light transport
• We will only address the last 2 points

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More To Consider

• Reflectivity!
• Water's reflectivity will vary between 5 and
  100
• Reflected light from the bottom of the water
• Scattered light from water volume itself
• Scattering must be approximated well enough to
  recreate colours

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Wave Generation

• Assume that the surface waves are assembled from
  many linear waves generated by wind
• The height of the water surface at the location x
  on the grid and time t is

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Whitecaps and Foam

• Whitecaps the foamy part of actively breaking
  waves
• Total foam area depends on the temperature
  difference between the air and the water and on
  water chemistry

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Whitecaps and Foam

• Proper treatment of whitecaps and foam is really
  difficult! But we can approximate...
• Let f be the fractional area of the wind-blown
  water surface that is covered by foam. Then..

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Light Transport

• Must consider the interaction of light with the
  water body
• Split this into 2 major parts
• Events on the surface
• Light transport inside the water volume

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Light Transport Across the Surface

• If a ray hits a surface, it is split into
  reflected and refracted rays
• Snell's Law
• Set n 1 for air and n 4/3 for water

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Light Transport Within the Water

• Once photons from the sky and sun pass through
  the air-water surface,they initiate a complex
  chain of scattering and absorption events within
  the water body
• The behavior of radiance within natural water
  bodies is governed by the radiance transfer
  equation
• Finding radiance in water body extremely
  difficult. Even brute force requires enormous
  amounts of computation.
• Need a different approach!

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Light Field

• Light field function that describes the amount
  of light traveling in every direction through
  every point in space
• Assume existence of 2 light fields
• Diffuse Field Radiance due to combined effect
  of light scattering through the media
• Directional Radiance what we are interested in
  rendering

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Radiance without so much Math

• Equation gives apparent radiance just below the
  air/water surface L(0) of the target at depth z
  having its own radiance L(z)
• Is the total path
  length
• Is the diffuse radiance just below
  the sea surface

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Turbidity

• Turbidity cloudy or muddy appearance due to
  particles of matter suspended in water
• Affects penetration of sunlight into a body of
  water
• Ranges from 0 to 1
• Can obtain chromaticity values for the
  transmittance per meter of water

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Practical Animation of Liquids

• Nick Foster
• PDI/DreamWorks
• Ronald Fedkiw
• Stanford University

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Goal

• Present a general method of animating liquids as
  they move through a 3D environment and interact
  with graphics such as moving polygons and
  parametric curves

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Algorithm

• 1) Model the static environment as a voxel grid
• 2) Model the liquid volume using a combination of
  particles and implicit surface
• For each simulation time step
• 3) Update the velocity field
• 4) Apply velocity constraints due to moving
  objects
• 5) Enforce incompressibility by solving a linear
  system
• 6) Update the position of the liquid volume

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Static Environment

• Represent the environment that we want the liquid
  to move in as a rectangular grid of voxels with
  side length
• Each cell has a pressure variable at its center
  and shares a velocity variable with each of its
  adjacent neighbours

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Liquid Representation

• Derived from a combination of inertialess
  particles (provides detail where the surface
  starts to splash) and dynamic isocontour
  (provides a smooth surface to regions of liquid)
• Particles placed into the grid according to some
  liquid distribution
• Their positions then evolve over time by a simple
  convection

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Hybrid Surface Model

• If a particle is more than a few grid cells away
  from, and inside the surface, then the particle
  is deleted
• Increased efficiency since particles are only
  needed locally near the surface of a liquid as
  opposed to the whole volume
• If cells are sparsely populated, extra particles
  are introduced
• Improves surface resolution

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Hybrid Surface Model

• Next, determine if the surface is smooth
• Smooth regions have low curvature and the
  particles are ignored
• High curvature regions indicate splashing so we
  use particles

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Hybrid Surface Model

• Some particles will escape the inside of the
  liquid layer since the grid is too coarse to
  represent them
• These particles can be rendered as small liquid
  drops
• Some stray particles can be used as fine spray or
  mist

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Non-Liquid Cells

• Cells that do not contain particles are either
  empty (open air) or part of an object
• If it is empty, its pressure is set to
  atmospheric pressure and velocity on each of its
  faces shared with another empty cell is set to 0
• An object cell can have velocities and pressures
  set to many different combinations to approximate
  liquid flowing into an out of an environment or
  to approximate different object material
  properties

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Liquid Surface

• We need to make sure that the air does not mix or
  inhibit the motion of a liquid, while allowing it
  to flow freely into empty cells
• Done by explicitly enforcing incompressibility
  within each cell that contains part of the liquid
  surface
• Velocities adjacent to a liquid filled cell are
  left alone, whereas the others are set directly

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Conservation of Mass

• Conservation of mass is only satisfied in
  surface cells where we have explicitly enforced
  it
• To do this, we solve the linear system of
  equations given by using the Laplacian operator
  to couple local pressure changes to the
  divergence of each cell

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Moving Objects

• Conditions that must be respected
• 1)Liquid should not flow into the object
• 2)Liquid should flow freely without interference
• 3) The particles should never pass through any
  part of the surface of the object

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Moving Objects

• 3) is easily solved since we know where the
  polygons are that make the object surface and
  what direction they are moving
• So we simply move the particles so they are
  always outside the surface of the object
• To take care of 1), we directly set the component
  of liquid velocity normal to the object
• 2) is taken care of by implementing an algorithm
  which the liquid is both pushed along by an
  object while being allowed to flow freely around
  it

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BUBBLES
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Simulation

• Simulation is based off of Navier-Stokes
  equations for incompressible viscous fluids
• u is velocity, p is pressure, v is viscosity, ?
  is density
• f is an external force such as gravity, buoyancy,
  or surface tension

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Traditional Level Set Method

• Typical fluid simulators use a signed distance
  function ø to represent the interface
• Magnitude of ø equals the distance from given
  point to an interface
• ø gt 0 if in the gas region
• ø lt 0 if in the liquid region
• ø 0 if on the interface
• Used in formulas for v and ?

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Problem

• The coloured portion is the
  actual film region, and the
  dashed line is the level
  set
  representation
• The thinnest film is lost
• Can only represent features
  whose smallest length
  scale is larger than size
  of grid cell

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Regional Level Set Method

• Region 1,2,... instead of liquid and gas
• Regional scalar function
• øR(X) ltR(X), S(X)gt
• R(X) returns region code
• S(X) returns distance to interface
• Replace all ø(X) with øR(X)

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Bubble Life Cycles

• For bubbles transited from other materials, such
  as beer bubbles or boiling water, directly insert
  the bubble with a new region code

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Bubble Life Cycles

• Bubbles will persist until the liquid film
  ruptures
• Change the region code of points contained in the
  two regions separated by the rupturing liquid
  film into the same region

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Bubble Life Cycles

• Liquid film ruptures when its thickness is too
  thin to resist the surface tension force
• Drainage of liquid can occur down the film due to
  gravity
• Stretching and shrinking of the film occur over
  time
• Film thickness is based on region thickness of
  each bubble

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ImplementationAlgorithm

• 1. Define a grid for water height computation
• 2. Apply initial deflections to some grid points
• 3. Compute new water height values using wave
  profiles at two previous time moments
• 4. Render the water surface
• 5. If necessary apply height excitations at grid
  points (rain drops...)
• 6. Go to step 3.

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Implementation Details Wave Model

• 128x128 Grid for Water Surface calculations -
  (from fast water animation corrected formula
  errors)
• Physically accurate model with damping (waves
  decrease over time similar to concept of spring
  constant which resists the motion in the spring)
• Lowering the constant increases the waters
  ability to move freely

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Implementation Details Wave Model (contd)

• For instance by lowering the damping constant by
  25 you could achieve the effect of going from
  water to a molasses type material in terms of
  viscosity and density (resists motion more,
  appears thicker)
• Each point on the grid is calculated by taking
  its 8 neighboring points into account (including
  corner points) and computing a pseudo average of
  the heights at each neighbor on the grid over the
  previous two time steps.

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Implementation Details Wave Model (contd)

• After each time step the newly computed height
  value is drawn.
• However, only 0.5 Corner point value is
  considered (circle of influence)
• Discrete version of the Navier Stokes equation,
  which also takes into the CFL condition that a
  wave cannot move more than 1 unit per time
  iteration (1 grid point per time step).

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ImplementationRain Model

• Rain Grid of size (Water Grid / 2) x (Water Grid
  / 2) for ease of calculation
• Rain grid is shifted to even and odd Grid Points
  at each time step to reduce the number of
  calculations required for rain grid computation
  (could probably reduce the grid by Water Grid / 4
  and still produce realistic simulation results).
• Each rain drop is affected by gravity using the
  2D gravity model for distance calculations
  (calculated at each time step dt)

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ImplementationRain Model (contd)

• 2D Velocity and distance Vt (dTG V0), Dt
  (dtG V0)dt
• If wanted/needed a wind vector could be used in
  the 3rd dimension and the resulting formulas
  would just result in a 3d vector computation.
• Initial velocities V0 of the Rain model are
  uniformly distributed from 0 - 1 and then scaled
  for the appropriate model (Wave currently) being
  implemented

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ImplementationRain Model (contd)

• This initial velocity probability allows for
  "normal" rain motion, otherwise a normalized
  (probability) Cartesian plane of water would fall
  and not be distributed properly over time (looks
  cool but not very natural)
• Probabilities of a rain "falling" from the cloud
  is uniformly calculated, pair this with
  randomized initial velocities and you can
  simulate realistic cloud shapes by altering the
  probabilities.

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ImplementationRain Model (contd)

• Higher velocities come from cloud points higher
  from the Wave model when a certain reference
  point is used
• Not a realistic model for physical accuracy
  (except under heavy rain conditions with no
  wind), but since it is currently being used to
  just test the wave model (reactions of) this is
  not really a problem.

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Demo Time!!!

• (The Original Standford Bunny)

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References

• Discretizing the Wave Equation.
  lthttp//www.mtnmath.com/whatrh/node66.htmlgt
• Efficient Simulation of Large Bodies of Water by
  Coupling Two and Three Dimensional Techniques.
  Irving, Guendelman, et al. ACM Transactions on
  Graphics (TOG) , ACM SIGGRAPH 2006 Papers
  SIGGRAPH '06. Vol. 25, Issue 3, July 2006.
• Fast Water Animation Using the Wave Equation
  with Damping. Y. Nishidate and G. P. Nikishkov.
  lthttp//www.u-aizu.ac.jp/niki/papers/200520Fast
  20water20animation.pdf. gt
• Fluid Animation with Dynamic Meshes. Klingner,
  Bryan. 2006.
• Fluids in Deforming Meshes. Feldman, OBrien,
  et al. Proceedings of the 2005 ACM SIGGRAPH. 2005
  pp. 255-259.
• Modeling Water for Computer Animation. Foster,
  Nick and Metaxas, Dimitri. Communications of the
  ACM. Vol. 43, Issue 7, July 2000.
• Navier-Stokes Equations. lthttp//en.wikipedia.org/
  wiki/Navier-Stokes_equationsgt
• Practical Animation of Liquids. N. Foster, R.
  Fedkiw, et al. ACM Siggraph 2001.
• Realistic Animation of Liquids. Foster, Nick
  and Metaxas, Dimitri. GRAPHICAL MODELS AND IMAGE
  PROCESSING Vol. 58, No. 5, September, pp.
  471483, 1996 ARTICLE NO. 0039
• Rendering Natural Waters. S. Premoze and M.
  Ashikhmin. Eighth Pacific Conference on Computer
  Graphics and Applications. 2000 p. 23
• Simulation and Rendering for Bubbles in
  Liquid.Matsumura, Tomoya.
• Simulation of Bubbles. Zheng, Wen. 2006.
• Water Drops on Surfaces. Wang, Huamin.