Parallel Mesh Refinement Without Communication

This paper presents a technique for the adaptive refinement of tetrahedral meshes. What makes it unique is that no neighbor information is required for the refined mesh to be compatible everywhere. Refinement consists of inserting new vertices at edge midpoints until some tolerance (geometric or otherwise) is met. For a tetrahedron, the six edges present 2 = 64 possible subdivision combinations. The challenge is to triangulate the new vertices (i.e., the original vertices plus some subset of the edge midpoints) in a way that neighboring tetrahedra always generate the same triangles on their shared boundary. A geometric solution based on geometric properties (edge lengths) was developed previously, but did not account for geometric degeneracies (edges of equal length). This paper provides a solution that works in all cases.