Ángel Plaza

Local refinement of simplicial grids based on the skeleton

In this paper we present a novel approach to the development of a class of local simplicial refinement strategies. The algorithm in two dimensions first subdivides certain edges. Then each triangle, if refined, is subdivided in two, three or four subelements depending on the previous division of its edges. Similarly, in three dimensions the algorithm begins by subdividing the two-dimensional triangulation composed by the faces of the tetrahedra (the skeleton) and then subdividing each tetrahedron in a compatible manner with the division of the faces. The complexity of the algorithm is linear in the number of added nodes. The algorithm is fully automatic and has been implemented to achieve global as well as local refinements. The numerical results obtained appear to confirm that the measure of degeneracy of subtetrahedra is bounded, and converges asymptotically to a fixed value when the refinement proceeds.

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.

On k-Fibonacci numbers of arithmetic indexes

In this paper, we study the sums of k-Fibonacci numbers with indexes in an arithmetic sequence, say an þ r for fixed integers a and r. This enables us to give in a straightforward way several formulas for the sums of such numbers.

An improved derefinement algorithm of nested meshes

Abstract In this paper we present a new version of the derefinement algorithm developed by Plaza et al . (in Numerical Methods in Engineering , Elsevier Science, Amsterdam, 1992, pp. 225–232; in Algorithms, Software, Architecture , Elsevier Science, Amsterdam, 1992, pp. 409–415; A. Plaza, PhD thesis, University of Las Palmas de Gran Canaria, 1993; Commun. Numer. Meth. Engng. , 1994, 10 , 403–412). 1–4 The purpose is to achieve a better derefinement algorithm with a lesser degree of complexity. We present the theoretical study of this improved derefinement algorithm and of the inverse one for refinement. Firstly, our initial version of the derefinement algorithm is summarized. Then we present the refinement algorithm associated with the improved derefinement one. Finally, automatic control of the sequences of irregular nested triangulations is shown by means of the resolution of an unsteady problem. In this problem the initial mesh has only nine nodes and a combination of refinements and derefinements have been applied to approach both the circular domain and the initial solution.

APPLICATION OF A NONLINEAR EVOLUTION MODEL TO FIRE PROPAGATION

The numerical simulation of fire in forest has been an important objective in recent researches, The rate of spread and shape of a forest fire front is af8ecte.d by many factors. The most important of these are as follows: fuel type and moisture content, wind velocity and variability, forest topography, fire spread mechanism, fuel continuity and the amount of spotting (cf.[ l-21). The development of Geographic Information Systems allows the incorporation of these data to the developed models, The first models took into account constant factors, continuous uniform fuel type, constant wind velocity, moisture and slope. Under these conditions, a fire ignited at a single point reaches a quasi-steady state and progresses toward the down wind direction and expands at a constant rate. These data cannot give precise predictions under variable conditions but are very useful in order to the intuition of the fire controller. Models capable of being incorporated into the computer simulations of fires under variable conditions have been developed, based on cellular automata (cf. [3-7]), and stochastic process [8]. These models can give useful indicators as to fire behavior under such conditions. Combustion phenomena has been extensively studied [9], unsteady flame propagation has been analyzed [lo]. Models based on combustion theories are very difficult to develop because of the diversity of the fuel type and varied chemical composition within a given fuel type. Because of the complexity of the problem, models based rigorously on combustion theory have not been completely developed. In this preliminary work, a first attempt is done to design a computer code for numerical simulation of forest fire spread in landscapes. Basically a convection-diffusion model for temperature and a mass-consistent model for wind field simulation will be assumed. A two-steps chemical mechanism is simplified in order to obtain the heat source. This proposed 2-D model take into account the convection phenomena due to temperature gradients in vertical direction. A numerical solution of the former model is presented using a finite difference method together with the study of stability. This numerical method is contrasted with an adaptive finite element method using reflnementiderefinement techniques (cf. [ 1 1 - 143).

A geometric diagram and hybrid scheme for triangle subdivision

We introduce a geometrical diagram to study the improvement in shape of triangles generated by iterative application of triangle subdivision. The four Triangles Longest Edge (4TLE) subdivision pattern and a new hybrid 4T Longest-Edge/Self-Similar (hybrid 4TLE-SS) scheme are investigated in this way. The map diagram provides a convenient way to visualize the evolution and migration of element shapes leading to a better understanding of the improvement process and the effect of recursive subdivision schemes. A complex variable mapping analysis supports the diagram and similarity class specifications. Numerical comparisons confirm the superiority of the new hybrid scheme.

Binomial Transforms of the k-Fibonacci Sequence

In this paper, we apply the binomial, k-binomial, rising, and falling transforms to the k-Fibonacci sequence. Many formulas relating the so obtained new sequences are presented and proved. Finally, we define and find the inverse transforms of the sequences previously obtained.

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.

Four-triangles adaptive algorithms for RTIN terrain meshes

We present a refinement and coarsening algorithm for the adaptive representation of Right-Triangulated Irregular Network (RTIN) meshes. The refinement algorithm is very simple and proceeds uniformly or locally in triangle meshes. The coarsening algorithm decreases mesh complexity by reducing unnecessary data points in the mesh after a given error criterion is applied. We describe the most important features of the algorithms and give a brief numerical study on the propagation associated with the adaptive scheme used for the refinement algorithm. We also present a comparison with a commercial tool for mesh simplification, Rational Reducer, showing that our coarsening algorithm offers better results in terms of accuracy of the generated meshes.