A Semi-Lagrangian CIP Fluid Solver Without Dimensional Splitting

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A Semi-Lagrangian CIP Fluid Solver Without
Dimensional Splitting
EUROGRAPHICS 2008

• 2008.09.12
• Doyub Kim
• Oh-Young Song
• Hyeong-Seok Ko
• presented by ho-young Lee

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Abstract

• USCIP a new CIP method
• More stable, more accurate, less amount of
  computation compared to existing CIP solver
• Rich details of fluids
• CIP is a high-order fluid advection

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Abstract

• Two shortcomings of CIP
• Makes the method suitable only for simulations
  with a tight CFL restriction
• CIP does not guarantee unconditional stability
• ? introducing other undesirable feature
• This proposed method (USCIP) brings significant
  improvements in both accuracy and speed

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Introduction

• Attempts for the accuracy of the advection
• Eulerian framework
• Monotonic cubic spline method
• CIP method (CIP, RCIP, MCIP)
• Back and force error compensation and
  correction(BFECC)
• Hybrid method (Eulerian and Largrangian
  framework)
• Particle levelset method
• Vortex particle
• Derivative particles

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Introduction

• This paper develops a stable CIP method that does
  not employ dimensional splitting

Original CIP Rational CIP MCIP
Stability Unstable More stable than Origin CIP More stable than Rational CIP
Computation time lower than MCIP lower than MCIP high
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Related Work

• Visual simulation of smoke, Fedkiw R., Stam J.,
  Jensen H. W. Computer Graphics. 2001
• Monotonic cubic interpolation

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Related Work

• CIP Methods
• A universal solver for hyperbolic equations by
  cubic-polynomial interpolation, Yabe T., Aoki T.
  Computer Physics. 1991.
• Original CIP
• Stable but non-dissipative water, Song O.-Y.,
  Shin H., Ko H.-S. ACM Trans Graph. 2005.
• Monotonic CIP
• Derivative particles for simulating detailed
  movements of fluids, Song O.-Y., Kim D., Ko
  H.-S. IEEE Transactions on Visualization and
  Computer Graphics. 2007.
• Octree data structure with CIP

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Related Work

• Etc..
• Animation and rendering of complex water
  surfaces, Enright D., Lossaso F., Fedkiw R. ACM
  Trans. Graph. 2002.
• To achieve accurate surface tracking in liquid
  animation
• Texure liquids based on the marker level set,
  Mihalef V., Metaxas D., Sussman M. In
  Eurographics. 2007.
• The marker level set method
• Vortex particle method for smoke, water and
  explosions, Selle A., Rasmussen N., Fedkiw R.
  ACM Trans. Graph. 2005.
• Simulating fluids with swirls

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Original CIP Method

• Key Idea
• Advects not only the physical quantities but also
  their derivatives
• The advection equation can be written as
• Differentiating equation (1) with respect to the
  spatial variable x gives

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Original CIP Method

• The value is approximated with the cubic-spline
  interpolation

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Original CIP Method

• 2D and 3D polynomials
• In 2D case

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Original CIP Method

• 2D Coefficients

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Original CIP Method

• Takes x and y directional derivatives
• Two upwind directions
• One starting point
• Not use the derivative information at farthest
  cell corner
• The method is accurate only when
• The back-tracked point falls near the starting
  point of the semi-Lagrangian advection

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Original CIP Method

• Problem for simulations with large CFL numbers
• Stability is not guaranteed

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Monotonic CIP Method

• To ensure stability
• Uses a modified version of the grid point
  derivatives
• Dimensional splitting

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Monotonic CIP Method

• A single semi-Lagrangian access in 2D
• 6 cubic-spline interpolations
• Two along the x-axis for and
• Two along the x-axis for and
• One along the y-axis for and
• One along the y-axis for and
• In 3D, 27 cubic-spline interpolations

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Monotonic CIP Method

• Two drawback of MCIP method
• First, High computation time
• The computation time for MCIP is 60 higher than
  that of linear advection
• Second, Numerical error
• The split-CIP-interpolation requires second and
  third derivatives
• Must be calculated by central differencing
• This represents another source of numerical
  diffusion

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Unsplit Semi-Lagrangian CIP Method

• To develop USCIP
• Go back to original 2D and 3D CIP polynomials
• Make necessary modifications
• Utilize all the derivative information for each
  cell
• 12 known values in a cell
• at the four corners
• 2 additional terms

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Unsplit Semi-Lagrangian CIP Method

• 2 extra terms
• The mismatch between
• The number of known values (12)
• and the number of terms (10)
• To overcome this mismatch
• Leat-squares solution
• Over-constrained problem
• Insert extra terms

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Unsplit Semi-Lagrangian CIP Method

• Three principles for the two added terms
• Not create any asymmetry
• If is added, then must be
  added
• Contain both x and y
• Rotation and shearing
• The lowest order terms should be chosen
• To prevent any unnecessary wiggles
• The terms that pass all three criteria are
  and

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Unsplit Semi-Lagrangian CIP Method

• To guarantee that the interpolated value will
  always be bounded by the grid point values
• A provision to keep the USCIP stable
• When the interpolated result is larger/smaller
  than the maximum/minimum of the cell node values,
• Replace the result with the maximum/minimum value
• Guarantees unconditional stability without
  over-stabilizing
• USCIP works on compact stencils
• No need to calculate high-order derivatives
• Reduce the computation time

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Unsplit Semi-Lagrangian CIP Method

• USCIP requires fewer operations than MCIP
• Unsplit polynomial is more complicated
• But split-CIP involves multiple interpolations
• MCIP 693 operations for a 3D interpolation
• USCIP 296 operations for a 3D interpolation
• Only 43 of the total operation count needed for
  MCIP

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Experimental Results

• Rigid Body Rotation of Zalesaks Disk

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Experimental Results

• Rising Smoke Passing Through Obstacles
• Generate realistic swirling of smoke
• Under complicated internal boundary conditions
• Without the assistance of vortex reinforcement
  mothods

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Experimental Results

• Dropping a Bunny-shaped Water onto Still Water
• Generated complicated small-scale features
• Droplets
• Thin water sheets
• Small waves

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Experimental Results

• Vorticity Preservation Test
• FLIP vs USCIP
• Noisy curl field

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Conclusion

• Presented a new semi-Lagrangian CIP method
• Stable, fast, accurate result
• Two additional fourth-order terms
• Reflect all the derivative information
• Stored at the grid points
• The proposed technique ran more than
• Twice as fast as BFECC or MCIP
• Clearly less diffusive