MESH REFINEMENT BASED ON THE 8-TETRAHEDRA LONGEST-EDGE PARTITION ´ Angel Plaza 1 Mar´ ıa-Cecilia Rivara 2 1 University of Las Palmas de Gran Canaria, Spain, aplaza@dmat.ulpgc.es 2 DCC, University of Chile, Santiago de Chile, Chile, mcrivara@dcc.uchile.cl ABSTRACT The 8-tetrahedra longest-edge (8T-LE) partition of any tetrahedron is defined in terms of three consecutive edge bisections, the first one performed by the longest-edge. The associated local refinement algorithm can be described in terms of the polyhedron skeleton concept using either a set of precomputed partition patterns or by a simple edge- midpoint tetrahedron bisection procedure. An effective 3D derefinement algorithm can be also simply stated. In this paper we discuss the 8-tetrahedra partition, the refinement algorithm and its properties, including a non-degeneracy fractal property. Empirical experiments show that the 3D partition has analogous behavior to the 2D case in the sense that after the first refinement level, a clear monotonic improvement behavior holds. For some tetrahedra a limited decreasing of the tetrahedron quality can be observed in the first partition due to the introduction of a new face which reflects a local feature size related with the tetrahedron thickness. Keywords: mesh refinement, longest-edge bisection, longest-edge algorithms, tetrahedral meshes 1. INTRODUCTION Skeleton algorithms for local mesh refinement /dere- finement of triangular and tetrahedral meshes have been proposed by Plaza and Carey [10, 11, 12]. In two dimensions, the algorithm is an alternative formula- tion of the 4-triangles longest-edge algorithm [14, 15]. The 2-dimensional skeleton algorithm [10, 11] works over the edges wireframe mesh affected by the refine- ment (target triangles and some neighbors to assure the construction of a conforming mesh) by perform- ing midpoint bisection of the involved edges. Then this information is used to select the appropriate tri- angle partition pattern (between a set of three pat- terns) to refine each individual triangle. This idea was then generalized to 3-dimensions [11, 12] by in- troducing an 8-tetrahedra partition which induces the 4-triangles partition of its faces. The 3-dimensional skeleton algorithm performs: (1) the refinement of the 3-dimensional edges wireframe mesh affected, (2) the refinement of the faces surface mesh (by using the 4-triangles partition and associated partial partitions), and (3) the volume refinement of each tetrahedron either by using a simple edge bisec- tion procedure or according to an appropriate pattern, selected between a set of precomputed partition pat- terns. In this paper we study the properties of the 8- tetrahedra partition showing that each full partition pattern is equivalent to a sequence of seven tetrahe- dron edge bisections by the midpoint of the tetrahe- dron edges, the first one being performed by the tetra- hedron longest-edge. Then we take advantage from the improvement and fractal properties of the 4-triangles longest-edge partition to show some non-degeneracy properties in 3-dimensions. We also show that for the meshes globally refined by using the 8-tetrahedra par- tition, the asymptotic average number of tetrahedra sharing a fixed vertex is equal to 24. An empirical study about the behavior of the 8- tetrahedra partition is also included. This shows that consistently, from the second refinement level, both