Multithread Lepp-Bisection Algorithm for Tetrahedral Meshes

Longest edge refinement algorithms were designed to deal with the iterative and local refinement of triangulations for finite element applications. In 3-dimensions the algorithm locally refines a tetredra set Sref and some neighboring tetraedra in each iteration. The new points introduced in the mesh are midpoints of the longest edge of some tetrahedra of either of the input mesh or of some refined nested meshes. All the tetrahedra are refined by its longest-edge (bisection by the plane defined by the midpoint of the longest edge and the two opposite vertices). In 2-dimensions the longest edge bisection guarantees the construction of refined triangulations that maintain the quality of the input mesh [18, 21, 3]. Even when the extension of this property to 3-dimensions is an open problem yet, empirical evidence shows that the 3D algorithm behaves analogously to the 2-dimensional algorithm in practice. Lepp-bisection algorithm is an efficient reformulation of the longest edge algorithm with the following advantages: (a) only local refinement operations are performed which always maintain a conforming mesh (the intersection of pairs of adjacent tetrahedra is either a common vertex, or a common edge or a common face); (b) the use of the Lepp concept allows to easily design parallel algorithms.