Study on Delaunay tessellations of 1-irregular cuboids for 3D mixed element meshes

Mixed elements meshes based on the modified octree approach contain several co-spherical point configurations. While generating Delaunay tessellations to be used together with the finite volume method, it is not necessary to partition them into tetrahedra; co-spherical elements can be used as final elements. This paper presents a study of all co-spherical elements that appear while tessellating a 1-irregular cuboid (cuboid with at most one Steiner point on its edges) with different aspect ratio. Steiner points can be located at any position between the edge endpoints. When Steiner points are located at edge midpoints, 24 co-spherical elements appear while tessellating 1-irregular cubes. By inserting internal faces and edges to these new elements, this number is reduced to 13. When 1-irregular cuboids with aspect ratio equal to $\sqrt{2}$ are tessellated, 10 co-spherical elements are required. If 1-irregular cuboids have aspect ratio between 1 and $\sqrt{2}$, all the tessellations are adequate for the finite volume method. When Steiner points are located at any position, the study was done for a specific Steiner point distribution on a cube. 38 co-spherical elements were required to tessellate all the generated 1-irregular cubes. Statistics about the impact of each new element in the tessellations of 1-irregular cuboids are also included. This study was done by developing an algorithm that construct Delaunay tessellations by starting from a Delaunay tetrahedral mesh built by Qhull

Tessellations Of Cuboids With Steiner Points

This paper presents a study of dierent 1-irregular cuboids (cuboids with at most one Steiner point on each edge) that can appear when meshes are generated using extensions of the modied octree approach [5], and then gives a recommendation how to handle them. The study is divided into two parts depending on the type of renement used: First, for the bisection based approach (Steiner points are midpoints of the cuboid edges), the 1-irregular cuboids are classied into equivalence classes (each element of the class is partitioned in the same way) and the exact value of the number of equivalence classes is computed. As this value is not too big, all 1-irregular cuboids can be handled using a hash table, and then a tessellation can be always found in constant time. Second, for the intersection based approach (Steiner points can be located at any position along a cuboid edge), the total number of 1-irregular cuboids, and upper and lower bounds for the number of equivalence classes are comp..