Simple algorithm for adaptive refinement of three-dimensional finite element tetrahedral meshes

Iribarren

1996

Math. Comp.

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.

Section: Figurementioning

confidence: 90%

Section: Introduction: the Triangulation Refinement Problemmentioning

confidence: 99%

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

Iribarren

1996

Math. Comp.

“…See e.g. the applications of Nambiar et al [31], Muthukrishnan et al [30]. Based on the longest edge idea over Delaunay meshes, Lepp-Delaunay algorithms for triangulation improvement have been also developed [39,41,44] 2.…”

Section: Introductionmentioning

confidence: 99%

“…The longest edge algorithms were generalized for 3-dimensional mesh refinement [38,30], as well as for the derefinement or coarsening of the mesh [37]. Improved longest edge algorithms based on using the concepts of terminal edges and longest edge propagating paths were also developed [40,39,43].…”

Section: Introductionmentioning

confidence: 99%

Multithread parallelization of Lepp-bisection algorithms

Applied Numerical Mathematics

Rodríguez

Montenegro

et al. 2012

Longest edge (nested) algorithms for triangulation refinement in two dimensions are able to produce hierarchies of quality and nested irregular triangulations as needed both for adaptive finite element methods and for multigrid methods. They can be formulated in terms of the longest edge propagation path (Lepp) and terminal edge concepts, to refine the target triangles and some related neighbors. We discuss a parallel multithread algorithm, where every thread is in charge of refining a triangle t and its associated Lepp neighbors. The thread manages a changing Lepp(t) (ordered set of increasing triangles) both to find a last longest (terminal) edge and to refine the pair of triangles sharing this edge. The process is repeated until triangle t is destroyed. We discuss the algorithm, related synchronization issues, and the properties inherited from the serial algorithm. We present an empirical study that shows that a reasonably efficient parallel method with good scalability was obtained. Multithread parallelization of lepp-bisection algorithms AbstractLongest edge (nested) algorithms for triangulation refinement in two dimensions are able to produce hierarchies of quality and nested irregular triangulations as needed both for adaptive finite element methods and for multigrid methods. They can be formulated in terms of the longest edge propagation path (Lepp) and terminal edge concepts, to refine the target triangles and some related neighbors. We discuss a parallel multithread algorithm, where every thread is in charge of refining a triangle t and its associated Lepp neighbors. The thread manages a changing Lepp(t) (ordered set of increasing triangles) both to find a last longest (terminal) edge and to refine the pair of triangles sharing this edge. The process is repeated until triangle t is destroyed. We discuss the algorithm, related synchronization issues, and the properties inherited from the serial algorithm. We present an empirical study that shows that a reasonably efficient parallel method with good scalability was obtained.

“…Bank [5] , Rivara et al [6,7,18,8,13], Nambiar et al, [11], Muthukrishnan et al [10], Morin et al, [9]. In particular, the methods based on the longest edge bisection of triangles guarantee that refined meshes of geometrical quality analogous to the input mesh are produced in 2-dimensions [6,7].…”

Section: Introductionmentioning

confidence: 99%

A Study on Delaunay Terminal Edge Method

Proceedings of the 15th International Meshing Roundtable

Summary. The Delaunay terminal edge algorithm for triangulation improvement proceeds by iterative Lepp selection of a point M which is midpoint of a Delaunay terminal edge in the mesh. The longest edge bisection of the associated terminal triangles (sharing the terminal edge) can be seen as a first step in the Delaunay insertion of M . The method was introduced as a generalization of non-Delaunay longest edge algorithms but formal termination proof had not been stated until now. In this paper termination is proved and several geometric aspects of the algorithm behavior are studied.