Geometrical Reconstruction of Material Interfaces with Arbitrary Mesh

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Geometrical Reconstruction of Material Interfaces
with Arbitrary Mesh
Jin Yao
Lawrence Livermore National Laboratory
This work was performed under the auspices of
the U.S. Department of Energy by the University
of California, Lawrence Livermore National
Laboratory under Contract No. W-7405-Eng-48. 
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A high order volume of fluid method is desired
for treating

• T-intersections (corners).
• Very thin regions.
• Curvature of the interface.
• Continuity of reconstructed interface across cell
  walls.
• Arbitrary meshes.

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The Standard Youngs Method

• Second order accurate.
• Locally iterative (bi-section).
• Discontinuities on cell walls.
• Difficulty with T-intersections.
• Extremely complex for arbitrary hexahedral
  meshes.

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The Proposed Method

• Third Order Accurate.
• Local Newton-Iterations.
• Continuous on Cell Walls.
• Purely Geometrical.
• Capable of Sharp Corners.
• Independent on Mesh Regularity.

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Interface Reconstruction The Basic Steps

• Find slope of interface in cells.
• Match partial volume fractions.
• Determine shape of interface using local Taylor
  expansion of the interface.
• Create Continuous Interface (if Needed).

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The Four basic geometrical operations

• . Determining the slope of the interface
  contained in a mixed cell.
• . Finding the intersection of the interface and a
  cell (to construct a planar interface facet).
• . Deriving the shape of the facet (planar,
  curved, or corner).
• . Calculate the cell volume bounded by the
• interface and match the giving volume fraction.

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1. How To Determine Slope

• Least-Squared
• Area Fitting

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Intrinsic Conservations by Least-Squared Area
Fitting
gt Volume Conservation
gt Conservation of Linearly-Distributed
Quantities (can be Mass, Energy, Momentum)
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Determine the Shape of an Interface Facet

• Taylor's expansion of surface in the
  surface-normal intrinsic-coordinate.
• Solution of the least squared problem for
  interface shape.
• Construct corners using nearby facets.

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Normal Intrinsic Coordinate

• Taylor expansion of the interface to 3rd order

Taylor expansion for planar facet i
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The Quadratic Shape Fitting
Again, the solution of the Least-squared problem
conserves the volume under these facets.
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Accuracy of Curvature Fit

• Geometry A cylinder of radius (7.0, 10.25)
  with height 1.
• Mesh unit cubic cells.
• Maximum error in curvature 4.21 .
• L2 error in space 0.001003.
• Inexact input volume by the shape-generator
  contributes to the error.

The gaps between neighbor facets on cell walls
may be invisible (imagine the facets as tiles on
a curved floor. Youngs planar tiles vs. our
curved tiles).
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How to Calculate Polyhedron-Volume Bounded by a
Planar Facet/Corner

• Find the part of a face cut by the given plane
  (the facial-cut).
• Calculate the volume contribution of a facial cut
  (with a point on the plane or the corner tip).
• Sum the contributions from all the facial cuts.

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Facial Cut By a Corner

• Find the intersection of the corner and the plane
  that holds the face.
• Find the intersection of the two polygons (the
  face and the corner of the plane).
• Link the Intersections Orderly.

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How to Calculate Cell Volume Bounded by a Curved
Facet

• Calculate the partial cell-volume cut by the
  planar facet.
• Add the volume between the curved facet and the
  planar facet.
• Integration of a quadratic polynomial of two
  variables on an arbitrary polygon is required.

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Match Volume Fraction

• Newton-Raphson Iteration
• nk1 nk (v(nk) v)/s(nk),
• n is the normal shift s(n) is cross-area v(n)
  is volume below facet v the volume to match.
  (fast, good 1st guess, small tolerance OK).
• ( Youngs used bi-section with a tolerance of
  2)

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Detect Sharp Corners

• Look for mixed cells which contain uncolored
  nodes.
• Looking for mixed cells with its nodes marked all
  by a single color.
• Look for facets with big curvature after curve
  fitting.

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Advection with Interface Reconstruction

• Initial slopes are provided with the
  post-Lagrange facets.
• With facet slopes given, the volume of fluids
  interface-reconstruction can be performed as
  described before.
• No extra degree of difficulty for multiple
  materials because the facet configuration is
  known.
• T-intersections are carried over.

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Interface Remapping

• Collect neighbor facets.
• Define local interface geometry.
• Find Intersection of the interface and the
  relaxed cell.

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Example Diagonal Translation of Polyhedron with
Planar Geometry
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With Curved Geometry (N-Material)
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Reduced Numerical Surface Tension
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Numerical Surface Tension with CALE
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Benefits with the New Method

• Mesh regularity independent.
• Planar interface stays planar.
• Curvature is easily obtained.
• No gaps among neighbor facets.
• Intersections can be carried over.
• Multiple materials can be handled.
• Allow treatment of thin regions/cracks.

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References

• Efficient Volume Calculation, John K. Dukowicz,
  LANL.
• A Geometrically Derived Priority System for
  Youngs Interface Reconstruction, Stewart Messo
  and Sean Clancy, LANL.
• Time-Dependent Multi-Material Flow with Large
  Fluid Distortion, D. L. Youngs, AWE, 1982.
• The Eulerian Interface Advection Scheme in CALE,
  Robert Tipton, LLNL, 1994.
• New VOF Interface Capturing and Reconstruction
  Algorithms, Peter Anninos, LLNL, 1999.
• Split and Un-split Volume of Fluid Methods for
  Interface Advection, Peter Anninos, LLNL, 2000.
• HELMIT A New Interface Reconstruction Algorithm
  and a Brief Survey of Existing Algorithms, R. D.
  Giddings, AWE, 1999.
• Recent Developments in HELMIT, R. D. Giddings,
  AWE, 2004.
• A Simple Advection Scheme for Material Interface,
  Byung-II Jun, LLNL, 2000,
• Improved Mix Interface Reconstruction with Small
  Stencils, Jeff Grandy, LLNL.