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Papers by Angel Plaza
Nonlinear Analysis-theory Methods & Applications, 1997
Derefinement Algorithms of Nested Meshes
Google, Inc. (search), Subscribe (Full Service), Register (Limited Service, Free), Login. Search:... more Google, Inc. (search), Subscribe (Full Service), Register (Limited Service, Free), Login. Search: The ACM Digital Library The Guide. ...
Advances in Engineering Software, 1996
In this paper we present a new version of the derefinement algorithm developed by The purpose is ... more In this paper we present a new version of the derefinement algorithm developed by The purpose is to achieve a better derefkment algorithm with a lesser degree of complexity. We present the theoretical study of this improved deretkment 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 derefkrement 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 derefhrements have been applied to approach both the circular domain and the initial solution.
Applied Numerical Mathematics, 2000
In this paper we present a novel approach to the development of a class of local simplicial refin... more 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.
Chaos Solitons & Fractals, 2009
The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of thei... more The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of their properties admit a straightforward proof. Here in particular, we present the derivatives of these polynomials in the form of convolution of k-Fibonacci polynomials. This fact allows us to present in an easy form a family of integer sequences in a new and direct way. Many relations for the derivatives of Fibonacci polynomials are proven.
Chaos Solitons & Fractals, 2008
An extension of the classical hyperbolic functions is introduced and studied. These new k-Fibonac... more An extension of the classical hyperbolic functions is introduced and studied. These new k-Fibonacci hyperbolic functions generalize also the k-Fibonacci sequences, say fF k;n g 1 n¼0 , recently found by studying the recursive application of two geometrical transformations onto C ¼ C [ fþ1g used in the well-known four-triangle longest-edge (4TLE) partition. In this paper, several properties of these k-Fibonacci hyperbolic functions are studied in an easy way. We finalize with the introduction of some curves and surfaces naturally related with the k-Fibonacci hyperbolic functions.
Communications in Numerical Methods in Engineering, 2001
In this paper we discuss a class of adaptive refinement algorithms for generating unstructured me... more In this paper we discuss a class of adaptive refinement algorithms for generating unstructured meshes in two and three dimensions. We focus on Skeleton Based Refinement (SBR) algorithms as proposed by Plaza and Carey and provide an extension that involves the introduction of the graph of the skeleton for meshes consisting of simplex cells. By the use of data structures derived from the graph of the skeleton, we reformulate the Skeleton Based Refinement scheme and devise a more natural and consistent approach for this class of adaptive refinement algorithms. As an illustrative case, we discuss in detail the graphs for 2D refinement of triangulations and for 3D we propose a corresponding new face-based data structure for tetrahedra. Experiments using the two dimensional algorithm and exploring the properties of the associated graph are provided.
Chaos Solitons & Fractals, 2007
The general k-Fibonacci sequence fF k;n g 1 n¼0 were found by studying the recursive application ... more The general k-Fibonacci sequence fF k;n g 1 n¼0 were found by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge (4TLE) partition. This sequence generalizes, between others, both the classical Fibonacci sequence and the Pell sequence. In this paper many properties of these numbers are deduced and related with the so-called Pascal 2-triangle.
Communications in Numerical Methods in Engineering, 1994
An adaptive refinement/derefinement algorithm of nested meshes is presented. Some definitions are... more An adaptive refinement/derefinement algorithm of nested meshes is presented. Some definitions are introduced. The main properties of the derefinement algorithm are remarked upon and its efficiency is shown through two numerical examples: a time-dependent convection-diffusion problems with dominant convection and a quasistationary problem.
Chaos Solitons & Fractals, 2007
We introduce a general Fibonacci sequence that generalizes, between others, both the classic Fibo... more We introduce a general Fibonacci sequence that generalizes, between others, both the classic Fibonacci sequence and the Pell sequence. These general kth Fibonacci numbers fF k;n g 1 n¼0 were found by studying the recursive application of two geometrical transformations used in the well-known four-triangle longest-edge (4TLE) partition. Many properties of these numbers are deduce directly from elementary matrix algebra.
Finite Elements in Analysis and Design, 2005
Two asymptotic properties that arise in iterative mesh refinement of triangles are introduced and... more Two asymptotic properties that arise in iterative mesh refinement of triangles are introduced and investigated. First, we provide theoretical results showing that recursive application of uniform four triangles longest-edge (4T-LE) partition to an arbitrary unstructured triangular mesh produces meshes in which the triangle pairings sharing a common longest edge asymptotically tend to cover the area of the whole mesh. As a consequence, we prove that for a triangle, the induced exterior conforming refinement zone extends on average to a few neighbor adjacent triangles. We determine the asymptotic extent of this propagating path and include results of supporting numerical experiments with uniform and adaptive mesh refinement. Similar behavior and LE propagation from a four triangle self similar (4T-SS) local subdivision alternative is analyzed and compared numerically. Hybrid 4T-LE and 4T-SS LE schemes are also considered. The results are relevant to mesh refinement in finite element and finite volume calculations as well as mesh enhancement in Computer Graphics and CAGD.
Applied Numerical Mathematics, 2000
In the present study, a novel three-dimensional refinement/derefinement algorithm for nested tetr... more 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.
Computer Aided Geometric Design, 2004
The four-triangles longest-edge (4T-LE) partition of a triangle t is obtained by joining the midp... more 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.
Chaos Solitons & Fractals, 2008
The 3-dimensional k-Fibonacci spirals are studied from a geometric point of view. These curves ap... more The 3-dimensional k-Fibonacci spirals are studied from a geometric point of view. These curves appear naturally from studying the k-Fibonacci numbers fF k;n g 1 n¼0 and the related hyperbolic k-Fibonacci functions. In this paper, after a summary of the main properties for the k-Fibonacci numbers, we focus on the geometry features (curvature and torsion) of the 3-dimensional k-Fibonacci spirals. Finally, the Metallic Shofars and their projections on the coordinate planes are also given.
Nonlinear Analysis-theory Methods & Applications, 1997
Derefinement Algorithms of Nested Meshes
Google, Inc. (search), Subscribe (Full Service), Register (Limited Service, Free), Login. Search:... more Google, Inc. (search), Subscribe (Full Service), Register (Limited Service, Free), Login. Search: The ACM Digital Library The Guide. ...
Advances in Engineering Software, 1996
In this paper we present a new version of the derefinement algorithm developed by The purpose is ... more In this paper we present a new version of the derefinement algorithm developed by The purpose is to achieve a better derefkment algorithm with a lesser degree of complexity. We present the theoretical study of this improved deretkment 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 derefkrement 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 derefhrements have been applied to approach both the circular domain and the initial solution.
Applied Numerical Mathematics, 2000
In this paper we present a novel approach to the development of a class of local simplicial refin... more 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.
Chaos Solitons & Fractals, 2009
The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of thei... more The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of their properties admit a straightforward proof. Here in particular, we present the derivatives of these polynomials in the form of convolution of k-Fibonacci polynomials. This fact allows us to present in an easy form a family of integer sequences in a new and direct way. Many relations for the derivatives of Fibonacci polynomials are proven.
Chaos Solitons & Fractals, 2008
An extension of the classical hyperbolic functions is introduced and studied. These new k-Fibonac... more An extension of the classical hyperbolic functions is introduced and studied. These new k-Fibonacci hyperbolic functions generalize also the k-Fibonacci sequences, say fF k;n g 1 n¼0 , recently found by studying the recursive application of two geometrical transformations onto C ¼ C [ fþ1g used in the well-known four-triangle longest-edge (4TLE) partition. In this paper, several properties of these k-Fibonacci hyperbolic functions are studied in an easy way. We finalize with the introduction of some curves and surfaces naturally related with the k-Fibonacci hyperbolic functions.
Communications in Numerical Methods in Engineering, 2001
In this paper we discuss a class of adaptive refinement algorithms for generating unstructured me... more In this paper we discuss a class of adaptive refinement algorithms for generating unstructured meshes in two and three dimensions. We focus on Skeleton Based Refinement (SBR) algorithms as proposed by Plaza and Carey and provide an extension that involves the introduction of the graph of the skeleton for meshes consisting of simplex cells. By the use of data structures derived from the graph of the skeleton, we reformulate the Skeleton Based Refinement scheme and devise a more natural and consistent approach for this class of adaptive refinement algorithms. As an illustrative case, we discuss in detail the graphs for 2D refinement of triangulations and for 3D we propose a corresponding new face-based data structure for tetrahedra. Experiments using the two dimensional algorithm and exploring the properties of the associated graph are provided.
Chaos Solitons & Fractals, 2007
The general k-Fibonacci sequence fF k;n g 1 n¼0 were found by studying the recursive application ... more The general k-Fibonacci sequence fF k;n g 1 n¼0 were found by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge (4TLE) partition. This sequence generalizes, between others, both the classical Fibonacci sequence and the Pell sequence. In this paper many properties of these numbers are deduced and related with the so-called Pascal 2-triangle.
Communications in Numerical Methods in Engineering, 1994
An adaptive refinement/derefinement algorithm of nested meshes is presented. Some definitions are... more An adaptive refinement/derefinement algorithm of nested meshes is presented. Some definitions are introduced. The main properties of the derefinement algorithm are remarked upon and its efficiency is shown through two numerical examples: a time-dependent convection-diffusion problems with dominant convection and a quasistationary problem.
Chaos Solitons & Fractals, 2007
We introduce a general Fibonacci sequence that generalizes, between others, both the classic Fibo... more We introduce a general Fibonacci sequence that generalizes, between others, both the classic Fibonacci sequence and the Pell sequence. These general kth Fibonacci numbers fF k;n g 1 n¼0 were found by studying the recursive application of two geometrical transformations used in the well-known four-triangle longest-edge (4TLE) partition. Many properties of these numbers are deduce directly from elementary matrix algebra.
Finite Elements in Analysis and Design, 2005
Two asymptotic properties that arise in iterative mesh refinement of triangles are introduced and... more Two asymptotic properties that arise in iterative mesh refinement of triangles are introduced and investigated. First, we provide theoretical results showing that recursive application of uniform four triangles longest-edge (4T-LE) partition to an arbitrary unstructured triangular mesh produces meshes in which the triangle pairings sharing a common longest edge asymptotically tend to cover the area of the whole mesh. As a consequence, we prove that for a triangle, the induced exterior conforming refinement zone extends on average to a few neighbor adjacent triangles. We determine the asymptotic extent of this propagating path and include results of supporting numerical experiments with uniform and adaptive mesh refinement. Similar behavior and LE propagation from a four triangle self similar (4T-SS) local subdivision alternative is analyzed and compared numerically. Hybrid 4T-LE and 4T-SS LE schemes are also considered. The results are relevant to mesh refinement in finite element and finite volume calculations as well as mesh enhancement in Computer Graphics and CAGD.
Applied Numerical Mathematics, 2000
In the present study, a novel three-dimensional refinement/derefinement algorithm for nested tetr... more 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.
Computer Aided Geometric Design, 2004
The four-triangles longest-edge (4T-LE) partition of a triangle t is obtained by joining the midp... more 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.
Chaos Solitons & Fractals, 2008
The 3-dimensional k-Fibonacci spirals are studied from a geometric point of view. These curves ap... more The 3-dimensional k-Fibonacci spirals are studied from a geometric point of view. These curves appear naturally from studying the k-Fibonacci numbers fF k;n g 1 n¼0 and the related hyperbolic k-Fibonacci functions. In this paper, after a summary of the main properties for the k-Fibonacci numbers, we focus on the geometry features (curvature and torsion) of the 3-dimensional k-Fibonacci spirals. Finally, the Metallic Shofars and their projections on the coordinate planes are also given.