Miguel A. Padrón

A 3D refinement/derefinement algorithm for solving evolution problems

Abstract In the present study, a novel three-dimensional refinement/derefinement algorithm for nested tetrahedral grids based on bisection is presented. The algorithm is based on an adaptive refinement scheme and on an inverse algorithm introduced by the authors. These algorithms work first on the skeleton of the 3D triangulation, the set of the triangular faces. Both schemes are fully automatic. The refinement algorithm can be applied to any initial tetrahedral mesh without any preprocessing. The non-degeneracy of the meshes obtained by this algorithm has been experimentally shown. Similarly, the derefinement scheme can be used to get a coarser mesh from a sequence of nested tetrahedral meshes obtained by successive application of the refinement algorithm. In this case, the algorithm presents a self-improvement quality property : the minimum solid angle after derefining is not less than the minimum solid angle of the refined input mesh. The refinement and derefinement schemes can be easily combined to deal with time dependent problems. These combinations depend only on a few parameters that are fixed into the input data by the user. Here we present a simulation test case for these kind of problems. The main features of these algorithms are summarized at the end.

Mesh quality improvement and other properties in the four-triangles longest-edge partition

The four-triangles longest-edge (4T-LE) partition of a triangle t is obtained by joining the midpoint of the longest edge of t to the opposite vertex and to the midpoints of the two remaining edges. The so-called self-improvement property of the refinement algorithm based on the 4-triangles longest-edge partition is discussed and delimited by studying the number of dissimilar triangles arising from the 4T-LE partition of an initial triangle and its successors. In addition, some geometrical properties such as the number of triangles in each similarity class per mesh level and new bounds on the maximum of the smallest angles and on the second largest angles are deduced.

Refinement based on longest-edge and self-similar four-triangle partitions

The triangle longest-edge bisection constitutes an efficient scheme for refining a mesh by reducing the obtuse triangles, since the largest interior angles are subdivided. One of these schemes is the four-triangle longest-edge (4T-LE) partition. Moreover, the four triangle self-similar (4T-SS) partition of an acute triangle yields four sub-triangles similar to the original one. In this paper we present a hybrid scheme combining the 4T-LE and the 4T-SS partitions which use the longest-edge based refinement. Numerical experiments illustrate improvement in angles and quality. The benefits of the algorithm suggest its use as an efficient tool for mesh refinement in the context of Finite Element computations.