1,640 research outputs found

    Local refinement based on the 7-triangle longest-edge partition

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    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. In this paper we specifically introduce a new local refinement for triangulations based on the longest-edge trisection, the 7-triangle longest-edge (7T-LE) local refinement algorithm. Each triangle to be refined is subdivided in seven sub-triangles by determining its longest edge. The conformity of the new mesh is assured by an automatic point insertion criterion using the oriented 1-skeleton graph of the triangulation and three partial division patterns

    Geometric diagram for representing shape quality in mesh refinement

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    summary:We review and discuss a method to normalize triangles by the longest-edge. A geometric diagram is described as a helpful tool for studying and interpreting the quality of triangle shapes during iterative mesh refinements. Modern CAE systems as those implementing the finite element method (FEM) require such tools for guiding the user about the quality of generated triangulations. In this paper we show that a similar method and corresponding geometric diagram in the three-dimensional case do not exist

    There are simple and robust refinements (almost) as good as Delaunay

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    A new edge-based partition for triangle meshes is presented, the Seven Triangle Quasi-Delaunay partition (7T-QD). The proposed partition joins together ideas of the Seven Triangle Longest-Edge partition (7T-LE), and the classical criteria for constructing Delaunay meshes. The new partition performs similarly compared to the Delaunay triangulation (7T-D) with the benefit of being more robust and with a cheaper cost in computation. It will be proved that in most of the cases the 7T-QD is equal to the 7T-D. In addition, numerical tests will show that the difference on the minimum angle obtained by the 7T-QD and by the 7T-D is negligible

    On the adjacencies triangular meshes based on skeleton-regular partitions

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    Abstract For any 2D triangulation , the 1-skeleton mesh of is the wireframe mesh deÿned by the edges of , while that for any 3D triangulation , the 1-skeleton and the 2-skeleton meshes, respectively, correspond to the wireframe mesh formed by the edges of and the "surface" mesh deÿned by the triangular faces of . A skeleton-regular partition of a triangle or a tetrahedra, is a partition that globally applied over each element of a conforming mesh (where the intersection of adjacent elements is a vertex or a common face, or a common edge) produce both a reÿned conforming mesh and reÿned and conforming skeleton meshes. Such a partition divides all the edges (and all the faces) of an individual element in the same number of edges (faces). We prove that sequences of meshes constructed by applying a skeleton-regular partition over each element of the preceding mesh have an associated set of di erence equations which relate the number of elements, faces, edges and vertices of the nth and (n − 1)th meshes. By using these constitutive di erence equations we prove that asymptotically the average number of adjacencies over these meshes (number of triangles by node and number of tetrahedra by vertex) is constant when n goes to inÿnity. We relate these results with the non-degeneracy properties of longest-edge based partitions in 2D and include empirical results which support the conjecture that analogous results hold in 3D

    Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

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    One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

    TetGen, towards a quality tetrahedral mesh generator

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    TetGen is a C++ program for generating quality tetrahedral meshes aimed to support numerical methods and scientific computing. It is also a research project for studying the underlying mathematical problems and evaluating algorithms. This paper presents the essential meshing components developed in TetGen for robust and efficient software implementation. And it highlights the state-of-the-art algorithms and technologies currently implemented and developed in TetGen for automatic quality tetrahedral mesh generation

    eXtended Variational Quasicontinuum Methodology for Lattice Networks with Damage and Crack Propagation

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    Lattice networks with dissipative interactions are often employed to analyze materials with discrete micro- or meso-structures, or for a description of heterogeneous materials which can be modelled discretely. They are, however, computationally prohibitive for engineering-scale applications. The (variational) QuasiContinuum (QC) method is a concurrent multiscale approach that reduces their computational cost by fully resolving the (dissipative) lattice network in small regions of interest while coarsening elsewhere. When applied to damageable lattices, moving crack tips can be captured by adaptive mesh refinement schemes, whereas fully-resolved trails in crack wakes can be removed by mesh coarsening. In order to address crack propagation efficiently and accurately, we develop in this contribution the necessary generalizations of the variational QC methodology. First, a suitable definition of crack paths in discrete systems is introduced, which allows for their geometrical representation in terms of the signed distance function. Second, special function enrichments based on the partition of unity concept are adopted, in order to capture kinematics in the wakes of crack tips. Third, a summation rule that reflects the adopted enrichment functions with sufficient degree of accuracy is developed. Finally, as our standpoint is variational, we discuss implications of the mesh refinement and coarsening from an energy-consistency point of view. All theoretical considerations are demonstrated using two numerical examples for which the resulting reaction forces, energy evolutions, and crack paths are compared to those of the direct numerical simulations.Comment: 36 pages, 23 figures, 1 table, 2 algorithms; small changes after review, paper title change

    Quasiconvex Programming

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    We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
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