Multi-material interface reconstruction on generalized polyhedral meshes

We describe multi-material (more than two materials) interface reconstruction methods for 3D mesh of generalized polyhedrons. Basic information used in interface reconstruction is volume fractions of each material in mixed cell containing several materials. All methods subdivide mixed cell into set of pure non-overlapping sub-cells each containing just one material that have reference volume fraction. We describe three methods. First two methods represent extension of standard piece-wise linear interface construction (PLIC) methods into 3D and use information only about volume fractions. First method is first-order accurate and based on discrete gradient of volume fraction as estimate to normal to interface. Second method is planarity preserving (second-order accurate) and is extension to 3D of least squares volume of fluid interface reconstruction algorithm (LVIRA, see [46, 45] for 2D case). The third method is extension to 3D of so-called moment of fluid (MoF) method, [14, 15]. MoF method is second-order accurate. This method uses information not only about volume fraction but also about position of the centroids for each material. In contrast to standard PLIC methods, MoF method uses only information from the cell where reconstruction is performed, no information from neighboring cells is needed. Also MoF method provides automatic ordering of the materials in the process of interface reconstruction. Optimal ordering is based on comparing positions of the reference and actual (centroids of reconstructed pure sub-cells) positions of the centroids. Performance of the methods is demonstrated on numerical examples. In appendices we describe in detail all geometrical and optimization algorithms needed for implementation of interface reconstruction methods.