Original articlesA mathematical proof of how fast the diameters of a triangle mesh tend to zero after repeated trisection
Introduction
Mesh refinement has been a constant area of research in applied mathematics and engineering applications [5], [13], [19], [28]. For example, the longest-edge bisection guarantees the construction of high-quality triangulations [1], [8], [20], [24], [25], [26]. The finite element method requires good-quality meshes (triangulations of surfaces) for the numerical algorithms to run. Although the requirements for meshes largely depend on the algorithm, the sharp angle conditions seem to be a common feature of particular importance in this context. See for example [23] for the introduction to the subject and [21] for the state of art.
More specifically, the objective is to construct a sequence of nested conforming meshes that are adapted to a given criterion. Nested sequences of triangles where each element in the sequence is a child of the parent triangle of the same sequence are of substantial interest in many areas, such as Finite Element Multigrid Methods, Image Multiresolutions, and others. In this sense, the generation of robust, reliable local mesh refinements for the production of meshes for finite element or finite difference methods is a significant area of study, together with the geometric and topological properties of the triangle or tetrahedral partitions [3], [5], [15], [19], [28].
Some of the properties of these longest-edge bisection based partitions and algorithms have already been indicated in the literature [14], [18]. Two critical numerical indicators to guarantee quality meshes in FEM are minimum angle and the longest edge of triangles. For example, Rosenberg and Stenger [20] showed the non-degeneracy property for LE-bisection: if α0 is the minimum angle of an initial given triangle, and αn is the minimum interior angle in new triangles considered at iteration n, then αn ≥ α0/2.
It is already well established that the assumption of regularity over the meshes [3], i.e., the bounded ratio between the outer and inner diameters, leads to the convergence of standard finite element methods. As a consequence of the convergence of the diameters to zero, the bisection method has been proven to be useful in FEM for approximating solutions of differential equations [20], [24], [25].
Therefore, the problem of convergence of the triangulations generated by these methods is of interest and significance. The problem of convergence is one of how fast the diameters of the resulting triangles tend to zero as a repeated partition is performed. Kearfott in [6] proved the convergence of the diameters for the longest-edge bisection, showing that every two iterations of the longest-edge bisection method, the maximum lengths are reduced at least by a factor of . This yields to the bound where δn is the diameter of the mesh after n iterations of the longest-edge bisection. Afterwards, Stynes [24], [25] and Adler [1] independently improved these bounds. They showed that and . It is interesting to note that these bounds cannot be improved as they hold for the case of the equilateral triangle.
Meanwhile, as all this previous background knowledge is well established in the case of the LE-bisection, proving it thus to be a highly robust and reliable method in practice, little evidence has been given with respect to the variant method of LE-trisection. An example of LE-trisection is shown in Fig. 1.
Partitions and local refinement algorithms are related [19]. For example, a local refinement has recently been proposed, based on LE-trisection [15]. Fig. 2 shows a typical refined mesh as obtained by this algorithm. Also the seven-triangle longest edge partition is related to the LE-trisection [9], [16].
It has also been proved [11], [14] that αn ≥ α0/c, with , where, as was previously detailed, αn is the minimum interior angle in the new triangles considered at iteration n, and α0 the minimum angle of the initial given triangle.
Take a triangle with a longest edge length δ0. The subdivision based on the trisection of the longest edge can be applied again to the newly generated triangles. Let δ1 be the maximum length of the newly produced triangles. The repeated application of the partition generates an unstructured mesh of triangles. It can easily be seen that δn, when n extends to a level of refinement n, constitutes a decreasing sequence.
We have recently given numerical studies in [12] which corroborate the convergence of the longest-edge trisection. We studied 10 trisection iterations for 40 triangles with the aforementioned characteristics. The empirical results show the fidelity of the upper and lower bounds presented above, Fig. 3. In the figure, Kearfott-like bound is graphed. This bound may be obtained following a similar reasoning to [6] applied to the LE-trisection. The details for this bound are not given here. Notice that, Kearfott-like bound is not very accurate.
In this paper, we give accurate upper and lower bounds for the convergence speed in terms of diameter reduction. We then establish the rate of convergence for LE-trisection thereby filling the gap in the analysis of the diameter convergence for the LE-trisection of triangles. We answer the question: how fast do the diameters of a triangle mesh tend to zero after repeated trisection? We prove that the longest edge in the given meshes is bounded sharply on the upper limit. It is mathematically demonstrated that the upper bound is attained when an equilateral triangle is trisected. We also prove that for a general initial triangle with longest edge δ0, if δn is the maximum diameter in nth iteration, then and for n ≥ 0.
The structure of the paper is as follows. In Section 2, we introduce a scheme for normalizing triangles. This normalized region has also been used in the literature [2], [17]. In Section 3, we present a hyperbolic geometry that will be used later. The main result of the paper is proved in Section 4. Section 5 is devoted to the proof of a lower bound of the diameters. Finally, a summarized version of some of the conclusions is given.
Section snippets
Introduction to triangle normalization
A method used in the literature of triangle mesh refinement is to normalize triangles [2], [17]. The normalization process consists in applying several possible isometries and dilations to a triangle, matching its longest side with the segment whose endpoints are (0, 0) and (1, 0), and leaving its lower side to the left. Similar triangles are characterized by a unique complex number z in the normalized region, Σ = {0 < x ≤ 1/2, y > 0, (x − 1) 2 + y2 ≤ 1}.
Remember that the LE-trisection of a triangle is
Introduction to hyperbolic distance
We use the results of hyperbolic geometry and particularly the Poincare half-plane model [4], [22], [27] in this paper. The circumferences and straight lines appearing in , and definitions are orthogonal to y = 0, and are geodesics in the Poincare half-plane. The expressions that appear in , and are isometries in the half-plane hyperbolic model since they have the form or with real coefficients verifying a d − b c > 0. As it is known functions
Proof of the upper bound
First, we study the case of the equilateral triangle. The theoretical interest of this case lies in giving the upper bound of the convergence rate for any other triangle. Let δn be the diameter of the triangle mesh after n iterated applications of the LE-trisection of an initial equilateral triangle with longest edge δ0. We seek to obtain the values of δn.
To set up our result, we use an invariant defined for the classes of similar triangles, introduced by Stynes [25] and Adler [1]. For an
Proof of the lower bound
Proposition 6 Let Δ0 be a triangle with longest edge δ0 . Let Δn be any triangle generated in the iteration n with longest edge δn . Then and for n ≥ 0.
Proof Let z0 be the associated complex number to triangle Δ0, and let zn be the associated complex number to Δn. Remember that . To prove that and is equivalent to proving that there are z2n and z2n+1 with for any n ≥ 0. However, it is true that for
Conclusions
In this paper, we have used hyperbolic geometry to prove sharpened bounds for the diameters of the triangles generated by the longest-edge trisection. We then delimit the rate of convergence for this refinement method. This paper is a detailed and rigorous response to the question of how fast the diameters of a triangle mesh tend to zero after repeated trisection is performed, as previously carried out for the longest-edge bisection by Stynes, Adler and Kearfott. The mathematical proofs given
Acknowledgement
This work has been supported in part by the CICYT Project number MTM2008-05866-C03-02/MTM from the Spanish Ministerio de Educación y Ciencia.
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2016, Journal of Computational and Applied Mathematics