The 4-triangles longest-side partition of triangles and linear refinement algorithms

Abstract

Abstract. In this paper we study geometrical properties of the iterative 4-triangles longest-side partition of triangles (and of a 3-triangles partition), as well as practical algorithms based on these partitions, used both directly for the triangulation refinement problem, and as a basis for point insertion strategies in Delaunay refinement algorithms. The 4-triangles partition is obtained by joining the midpoint of the longest side with the opposite vertex and the midpoints of the two remaining sides. By means of simple geometrical arguments we show that the iterative partition of obtuse triangles systematically improves the triangles (while they remain obtuse) in the following sense: the sequence of smallest angles monotonically increases while the sequence of largest angles monotonically decreases in an amount (at least) equal to the smallest angle of each iteration. This allows us to improve the known bound on the smallest angle (without making use of previous results), and to obtain a better a priori bound on the number of similarly distinct triangles, as a function of the geometry of the initial triangle. Numerical evidence, showing that the practical behavior of the 4-triangles partition is in complete agreement with this theory, is included. A 4-triangles refinement algorithm is also discussed and illustrated. Furthermore, we show that the time cost of the algorithm is linear independently of the size of the triangulation.

There are simple and robust refinements (almost) as good as Delaunay

“…A well accepted acronym used to name this class of mesh subdivision is nT-LE, where n is the number of new triangles (T) produced after a single subdivision and LE stands for longest edge. So, we found in the literature well studied longest-edge partition as 2T-LE, 3T-LE, 4T-LE and 7T-LE, see [14,15,9] and the references therein. It should be noted that the iterative application of these partitions yields good-quality meshes, in the sense that they do not degenerate.…”

“…Longest edge-based refinement has become popular in last decade in the context of mesh refinement [9,11,14]. A well accepted acronym used to name this class of mesh subdivision is nT-LE, where n is the number of new triangles (T) produced after a single subdivision and LE stands for longest edge.…”

On a Bisection Algorithm That Produces Conforming Locally Refined Simplicial Meshes

“…Various properties of partitions generated by such algorithms were proved in a number of works in the 70-th [7,16,18,19]. Later, in the 80-th, mainly due to efforts of M. C. Rivara, bisection-type algorithms became popular also in the FEM community for mesh refinement/adaptation purposes [12,13,14,15]. Several variants of the algorithm suitable for standard FEMs were also proposed, analysed and numerically tested in [1,2,3,8,10,11] (see also references therein).…”

A local adaptive refinement method with multigrid preconditionning illustrated by multiphase flows simulations.

“…Bey [4] and Zhang [24] proposed a generalisation of this method to three dimensions. Other methods based only on bisection have been introduced by Rivara [21,22] or Mitchell [18] in two dimension and Bänsch [3] or Maubach [17] in three dimensions. All these methods depend on the choice of the element and on the dimension.…”