Finite Elements in Analysis and Design 44 (2008) 748--758 Contents lists available at ScienceDirect Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel The seven-triangle longest-side partition of triangles and mesh quality improvement Alberto M ´ arquez a , Auxiliadora Moreno-Gonz ´ alez a, ∗ , ´ Angel Plaza b , Jos ´ e P. Su ´ arez c a Department of Applied Mathematics I, University of Seville, Spain b Department of Mathematics, University of Las Palmas de Gran Canaria, Spain c Department of Cartography and Graphic Engineering, University of Las Palmas de Gran Canaria, Spain ARTICLE INFO ABSTRACT Article history: Received 27 October 2006 Received in revised form 9 April 2008 Accepted 26 April 2008 Available online 2 July 2008 Keywords: Refinement Longest-edge based algorithms Improvement of mesh quality A new triangle partition, the seven-triangle longest-edge partition, based on the trisection of the edges is presented and the associated mesh quality improvement property, discussed. The seven-triangle longest-edge (7T-LE) partition of a triangle t is obtained by putting two equally spaced points per edge. After cutting off three triangles at the corners, the remaining hexagon is subdivided further by joining each point of the longest-edge of t to the base points of the opposite sub-triangle. Finally, the interior quadrangle is subdivided into two sub-triangles by the shortest diagonal. The self-improvement property of the 7T-LE partition is discussed, delimited and compared to the parallel property of the four-triangle longest-edge (4T-LE) partition. Global refinement strategies, combining longest-edge with self-similar partitions, are proposed, based on the theoretical geometrical properties. © 2008 Elsevier B.V. All rights reserved. 1. Introduction In the context of finite element methodology, the adaptability of the mesh and the analysis of the approximation error are important issues to be addressed [1]. In recent years, many partitions and as- sociated refinement and coarsening algorithms have been proposed and studied [2--7]. In the area of adaptive finite element methods, mesh refinement algorithms that maintain the non-degeneracy of the elements and the conformity and smoothness of the grid are cer- tainly desirable. Non-degeneracy means that the minimum angle of the triangles is bounded away from zero when the partition or re- finement is applied. Conformity refers to the requirement that the intersection of non-disjoint triangles is either a common vertex or a common edge. The smoothness condition states that the transition between small and large elements should be gradual. Non-degeneracy, conformity and smoothness are also desirable properties in adaptive tessellation of NURBS surfaces [8]. In this sense, Delaunay meshes have been widely used, since they avoid long, `skinny' triangles and produce the maximum possible smallest- internal angle of any triangle [9]. Refinement techniques are also used for enhancement of mesh obtained from trimmed NURBS sur- faces. See an application of this in [10]. The number of triangles can be further increased/decreased depending on the application re- quirements. ∗ Corresponding author. Tel.: +34 954486482; fax: +34 954488160. E-mail addresses: almar@us.es (A. Márquez), auxiliadora@us.es (A. Moreno-González), aplaza@dmat.ulpgc.es (Á. Plaza), jsuarez@dcegi.ulpgc.es (J.P. Suárez). 0168-874X/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2008.04.007 Longest-edge based algorithms have been used with Delaunay triangulation for the quality triangulation problem [11,12], with de- tails of the fractal properties of the meshes obtained by these algo- rithms also given [5,13,14]. Some refinement methods have had exact angle counts since they first existed [2,3,6,15] and, consequently, the non-degeneracy of the triangulation is proved. The four-triangle longest-edge (4T-LE) refinement algorithm proposed by Rivara [16], given that it is based on the longest-edge bisection, never produces an angle smaller than half the minimum original angle [16,17], whilst, moreover, revealing remarkable mesh quality improvement between certain limits, as recently studied in [18]. However, this mesh quality improvement depends on the geometry of the initial triangle as will be underlined herein. In search of a better mesh quality improvement by iterative par- tition of the mesh, in this paper, we have introduced a new triangle partition, the seven-triangle longest-edge (7T-LE) partition. This par- tition, first, positions two equally spaced points per edge and, then, the interior of the triangle is divided into seven sub-triangles in a manner compatible with the subdivision of the edges. Three of the new sub-triangles are similar to the original, two are similar to the new triangle also generated by the 4T-LE, and the other two triangles are, in general, better shaped. We compare the evolution of a stan- dard quality measurement for the iterative application of the 7T-LE partition to an initial triangle, first with a Delaunay-type partition, and then using the 4T-LE partition. The paper is organized as follows: the 4T-LE partition and the self- improvement property achieved via its application is summarized in Section 2. We go on to present the 7T-LE partition in Section 3,