Interface reconstruction with VOF (volume of fluids) is an essential phase in an ALE (Arbitrary Lagrange & Eulerian) simulation. Some historical issues associated with the Youngs method have not been well addressed. One is that the slope estimate is not always accurate. A more serious issue is that the interface is discontinuous on cell faces with a curved interface geometry. Plus, with an existing VOF method a corner cannot be reconstructed in general. We propose a new VOF approach in order to address these issues. We treat the case of a single material and void, and the partial volumes in mixed cells exactly given. We assume the nodes not owned by any pure elements are sparsely distributed in a mesh. This is generally true for a moderate curvature. Then, most mixed cell nodes can be coloured with pure elements, and can provide orientation of interface facets. For a given interior mixed zone in two-dimensions, we find its mixed neighbours and three partial volumes are given. A linear facet has two degrees of freedom therefore can be reconstructed with a local optimization with volume matching. A quadratic facet (with three degrees of freedom) can be computed exactly that matches the known partial volumes. This is to say a planar interface geometry can be locally exactly reconstructed away from a corner, and interface curvature can be calculated with neighbour volume fractions. The case of a corner can be identified with a sudden slope change and high curvature with noticeable gaps between neighbour facets. Then, a local optimization for matching volumes can again be performed and the corner is reconstructed. We have implemented this algorithm in 2D at the Lawrence Livermore National Laboratory. Preliminary tests show that exact planar geometries can be reconstructed, and corners can be accurately defined. In the case of a curved geometry, the gaps between neighbour facets are of high order. When necessary, the gaps between facets can be eliminated by a local modification of a facet without affecting the order of accuracy. The proposed algorithm is by nature applicable to an arbitrarily given mesh because the only necessary requirement is a function to accurately compute the partial volume bounded by a facet (linear, corner, circular, or conic). A similar approach is expected to work in 3D provided an accurate method to compute volumes.
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