Animating smoke with dynamic balance

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Animating smoke with dynamic balance

• Jin-Kyung Hong
• Chang-Hun Kim
• ?? ???

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Contents

• Abstract
• Introduction
• contribution
• Related Work
• Computing Errors in the Advection Term
• The Equations of Flow
• Errors Compensation Scheme
• Vortex Advection Based on Vorticity Confinement
• Vorticity Confinement
• Vortex Advection Scheme
• Implementation
• Results and Discussion
• Conclusion

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Abstract

• Numerical method for avoiding dissipation
• Compensation for the errors induced by
  semi-Largangian scheme
• New advection term
• Vortex advection based on a vorticity confinement
  force

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Introduction (1/3)

• Numerical error caused by discretization is as
  small as possible
• fundamentally improve accuracy of the simulation,
  without additional computation
• dynamic balance to maintain the coherence of the
  field

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Introduction (2/3)

• Compensation for losses in the energy of the
  velocity field by an advection step
• Focus on maintaining the vorticity
• Separate the vorticity field from main velocity
  field
• In new advection step, estimate the error during
  each time interval and compensate

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Introduction (3/3)

• Reduces the numerical dissipation which
  necessarily results from the linear interpolation
  of a semi-Lagrangian scheme
• Unique type of vortex
• fully Eulerian

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contribution

• 1. Improving the method for solving the
  differential equation for the advection step by
  using error compensation
• 2. Allowing the smoke model to remain dynamic
  near the center of a vortex by the use of vortex
  advection

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Related work (1/2)

• Realistic animation of liquids Foster Metaxas
  1996
• Stable Fluids Stam 1999
• Visual simulation of smoke Fedkiw 2001

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Related work (2/2)

• Back and forth error compensation and correction
  methods for removing errors induced by uneven
  gradients of the level set function Dupont 2003
• A vortex particle method for smoke, water and
  explosions Selle 2005

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Computing Errors in the Advection Term

• The Equations of Flow
• Error compensation Scheme

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The Equations of Flow

• Navier-Stokes equation
• Velocity vector field u (u, v, w)
• 
  (1)
• (2)

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The Equations of Flow
Standard fluid simulation process
Mass Conservation to counteract dissipation
Force
diffusion
Self-advection of the velocity vector field
projection
Density advection along the velocity vector field
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The Equations of Flow

• Advection step
• (3)

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Error Compensation Scheme

• Semi-Lagrangian scheme with error compensation
  that considers the time intervals before and
  after the advection step

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Semi-Lagrangian advection
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Semi-Lagrangian advection
Let d be any of the components of the fluid
velocity.
A first order accurate backwards Euler time-step
To find the velocity of a given voxel at
time t?t, we trace the velocity field backwards
in time to time t, and take the velocity from
there.
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Numerical Dissipation

• Semi-Lagrangian advection (or in fact, just about
  any usable Eulerian method) has a
  flawnumerical dissipation
• When we advect a field, the new values are
  smoothly interpolated at various points from the
  old values
• That interpolation smoothes the field

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Dissipation Example (1/3)

• Start with a function nicely sampled on a grid

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Dissipation Example (2/3)

• The function moves to the left(perfect
  advection) and is resampled

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Dissipation Example (3/3)

• And now we interpolate new sample values, and
  ruin it all!

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The problem

• In the limit ?x-gt0, this error goes to zero
• Problem we cant or wont take the limit
• Ideally we want a grid with only just enough
  resolution to represent details we care about
• We may be forced to use something even coarser if
  computer resources too limited
• Numerical dissipation very quickly smoothes them
  away

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The Symptoms

• For velocity fields
• It looks like fluids are too sticky (molasses) or
  implausible length scale (scale model)
• Swirly turbulent detail quickly decays, left with
  just boring bulk motion
• For smoke concentration
• Smoke diffuses into thin air too fast,nice thin
  features vanish

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Error Compensation Scheme

• (4)
• (5)
• (6)
• (7)

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Error Compensation
t
t
t 1
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Backward error compensation algorithm
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Forward error correction method
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Forward error correction method
t
t
t 1
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Vortex Advection Based on Vorticity Confinement

• Using vortex advection based on vorticity
  confinement, we propose modified equations for
  developing a fluid simulation with a continuous
  vortex.
• Vorticity Confinement
• Vortex Advection Scheme

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Vorticity Confinement (1/4)

• (8)
• (9)
• (10)

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Vorticity Confinement (2/4)
h height of grid E (constant) control the
amount of small scale detail
(Slide by Jos Stam, SIGGRAPH 2003)
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Vorticity Confinement (3/4)
h height of grid E (constant) control the
amount of small scale detail
(Slide by Jos Stam, SIGGRAPH 2003)
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Vorticity Confinement (4/4)
h height of grid E (constant) control the
amount of small scale detail
(Slide by Jos Stam, SIGGRAPH 2003)
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Vortex Advection Scheme (1/4)

• Two important properties of dynamic balance are
• It has an advection step with error compensation
• It uses vortex advection
• Vortex advects along a velocity vector field

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Vortex Advection Scheme (2/4)

• We separate the vorticity field from the main
  velocity field
• New advection term
• (11)
• (12)

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Vortex Advection Scheme (3/4)
Velocity field
Vorticity
Vorticity confinement field
Keeping vorticity
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Vortex Advection Scheme (4/4)
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Implementation -Simulation Steps-
Determine the vorticity field
Equation (7)
Equation (12)
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Results and Discussion
No swirling motion
Vortex is lost quickly
Smoke keeps spinning
w/o VC w/ VC w/ VA
lt Stam 99 gt
lt Fedkiw et al., 01 gt
lt Our method gt
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Conclusion

• We have proposed New method for persistent
  modeling for the unique features of smoke such as
  vortices