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

##### David Contreras, Nancy Hitschfeld-Kahler

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.

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