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. 

Geometrical Reconstruction of Material Interfaces with Arbitrary Mesh

Geometrical Reconstruction of Material Interfaces with Arbitrary Mesh

A high order volume of fluid method is desired for treating:

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.

The Standard Youngs MethodThe Standard Youngs Method

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

meshes.

The Proposed MethodThe Proposed Method

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

Interface Reconstruction: The Basic StepsInterface 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).

The Four basic geometrical operations:The Four basic geometrical operations:

1) . Determining the slope of the interface contained in a mixed cell.

2) . Finding the intersection of the interface and a cell (to construct a planar interface facet).

3) . Deriving the shape of the facet (planar, curved, or corner).

4) . Calculate the cell volume bounded by the

interface and match the giving volume fraction.

1. How To Determine Slope1. How To Determine Slope

Least-Squared Area Fitting:

.)( ,)(

)(2

1

2

dxybaxbJdxybaxxa

J

dxybaxJ

Intrinsic Conservations by Least-Squared Area FittingIntrinsic Conservations by Least-Squared Area Fitting

.)( 0 dxydxbaxbJ

=> Volume Conservation

dx.yxb)dxx(ax 0aJ

=> Conservation of Linearly-Distributed Quantities (can be Mass, Energy, Momentum…)

Determine the Shape of an Interface FacetDetermine 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.

Normal Intrinsic CoordinateNormal Intrinsic Coordinate

Taylor expansion of the interface to 3rd order: .)(

2

1 22 dcban

Taylor expansion for planar facet i:

iii wvun

The Quadratic Shape FittingThe Quadratic Shape Fitting

ddwvudcbaJ 222 )]()(2

1[

Again, the solution of the Least-squared problem conserves the volume under these facets.

Accuracy of Curvature FitAccuracy 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).

How to Calculate Polyhedron-Volume Bounded by a Planar Facet/CornerHow 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.

Facial Cut By a CornerFacial 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.

How to Calculate Cell Volume Bounded by a Curved FacetHow 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.

Match Volume FractionMatch Volume Fraction

Newton-Raphson Iteration*:

nk+1 = 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%)

Detect Sharp CornersDetect 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.

Advection with Interface Reconstruction 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.

Interface RemappingInterface Remapping

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

Example: Diagonal Translation of Polyhedron with Planar GeometryExample: Diagonal Translation of Polyhedron with Planar Geometry

With Curved Geometry (N-Material)With Curved Geometry (N-Material)

Reduced Numerical Surface TensionReduced Numerical Surface Tension

Numerical Surface Tension with CALENumerical Surface Tension with CALE

Benefits with the New MethodBenefits with the New Method

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

ReferencesReferences

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.