USING LONGEST-SIDE BISECTION TECHNIQUES FOR THE AUTOMATIC REFINEMENT OF DELAUNAY TRIANGULATIONS

In this paper we discuss, study and compare two linear algorithms for the triangulation refinement problem: the known longest-side (triangle bisection) refinement algorithm, as well as a new algorithm that uses longest side bisection techniques for refining Delaunay triangulations. We show that the automatic point insertion criterion, taken from the fractal property of optimal (linear) longest-side bisection algorithms, assures the construction of good quality Delaunay triangulations in linear time. Numerical evidence, showing that the practical behaviour of the new algorithm is in complete agreement with the theory, is included. © 1997 by John Wiley & Sons, Ltd.     

Delaunay triangulation of non-convex planar domains

This paper investigates the possibility of integrating the two currently most popular mesh generation techniques, namely the method of advancing front and the Delaunay triangulation algorithm. The merits of the resulting scheme are its simplicity, efficiency and versatility. With the introduction of ‘non-Delaunay’ line segments, the concept of using Delaunay triangulation as a means of mesh generation is clarified. An efficient algorithm is proposed for the construction of Delaunay triangulations over non-convex planar domains. Interior nodes are first generated within the planar domain. These interior nodes and the boundary nodes are then linked up together to produce a valid triangulation. In the mesh generation process, the Delaunay property of each triangle is ensured by selecting a node having the smallest associated circumcircle. In contrast to convex domains, intersection between the proposed triangle and the domain boundary has to be checked; this can be simply done by considering only the ‘non-Delaunay’ segments on the generation front. Through the study of numerous examples of various characteristics, it is found that high-quality triangular element meshes are obtained by the proposed algorithm, and the mesh generation time bears a linear relationship with the number of elements/nodes of the triangulation.     

Delaunay versus max-min solid angle triangulations for three-dimensional mesh generation

The Delaunay triangulation has been used in several methods for generating finite element tetrahedral meshes in three-dimensional polyhedral regions. Other types of three-dimensional triangulations are possible, such as a triangulation satisfying a local max-min solid angle criterion. In this paper, we present experimental results to show that max-min solid angle triangulations are better than Delaunay triangulations for finite element tetrahedral meshes, since the former type of triangulations contains tetrahedra of better shape than the latter type. We also describe how mesh points are generated and triangulated in our tetrahedral mesh generation method.     

Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations

The centroidal Voronoi tessellation based Delaunay triangulation (CVDT) provides an optimal distribution of generating points with respect to a given density function and accordingly generates a high‐quality mesh. In this paper, we discuss algorithms for the construction of the constrained CVDT from an initial Delaunay tetrahedral mesh of a three‐dimensional domain. By establishing an appropriate relationship between the density function and the specified sizing field and applying the Lloyd's iteration, the constrained CVDT mesh is obtained as a natural global optimization of the initial mesh. Simple local operations such as edges/faces flippings are also used to further improve the CVDT mesh. Several complex meshing examples and their element quality statistics are presented to demonstrate the effectiveness and efficiency of the proposed mesh generation and optimization method. Copyright © 2003 John Wiley & Sons, Ltd.     

One machine,one minute,three billion tetrahedra

This paper presents a new scalable parallelization scheme to generate the 3D Delaunay triangulation of a given set of points. Our first contribution is an efficient serial implementation of the incremental Delaunay insertion algorithm. A simple dedicated data structure, an efficient sorting of the points, and the optimization of the insertion algorithm have permitted to accelerate reference implementations by a factor three. Our second contribution is a multithreaded version of the Delaunay kernel that is able to concurrently insert vertices. Moore curve coordinates are used to partition the point set, avoiding heavy synchronization overheads. Conflicts are managed by modifying the partitions with a simple rescaling of the space-filling curve. The performances of our implementation have been measured on three different processors: an Intel core-i7, an Intel Xeon Phi, and an AMD EPYC, on which we have been able to compute three billion tetrahedra in 53 seconds. This corresponds to a generation rate of over 55 million tetrahedra per second. We finally show how this very efficient parallel Delaunay triangulation can be integrated in a Delaunay refinement mesh generator, which takes as input the triangulated surface boundary of the volume to mesh.     

二维任意域约束Delaunay三角化的实现

本文设计了一种逐点加入一局部换边法,提出并证明了二维约束边在约束Ｄｅｌａｕｎａｙ三角化中存在的条件,并据此用中点加点法实现了二维任意域的Ｄｅｌａｕｎａｙ三角剖分,生成的网格均符合Ｄｅｌａｕｎａｙ优化准则,网格的优化在网格生成过程中完成,算法复杂度与点数呈近似线性关系,给出了算法在平面域剖分和包含复杂断层的石油地质勘探散乱数据点集剖分的应用实例。     

Numerical convergence of discrete exterior calculus on arbitrary surface meshes

Discrete exterior calculus (DEC) is a structure-preserving numerical framework for partial differential equations solution, particularly suitable for simplicial meshes. A longstanding and widespread assumption has been that DEC requires special (Delaunay) triangulations, which complicated the mesh generation process especially for curved surfaces. This paper presents numerical evidence demonstrating that this restriction is unnecessary. Convergence experiments are carried out for various physical problems using both Delaunay and non-Delaunay triangulations. Signed diagonal definition for the key DEC operator (Hodge star) is adopted. The errors converge as expected for all considered meshes and experiments. This relieves the DEC paradigm from unnecessary triangulation limitation.     

Selective refinement: A new strategy for automatic node placement in graded triangular meshes

Automating triangular finite element mesh generation involves two interrelated tasks: generatine a distribution of well-placed nodes on the boundary and in the interior of a domain, and constructing a triangulation of these nodes. For a given distribution of nodes, the Delaunay triangulation generally provides a suitable mesh, and Watson's algorithm²⁶ provides a flexible means of constructing it. In this paper, a new method is described for automating node placement in a Delaunay triangulation by seieclive refinement of an initial triangulation. Grading of the mesh is controlled by an explicit or implicit node spacing function. Although this paper describes the technique only in the planar context, the method generalizes to three dimensions as well.     

The construction of improved‐quality mesh on a curved surface using mapping factors

Abstract

The Delaunay triangulation is broadly used on flat surfaces to generate well‐shaped elements. But the properties of Delaunay triangulation do not exist on curved surfaces whose Jacobians are different. In this paper we will present a modified algorithm to improve the shape of triangulation for the curved surface. The experiment results show that making use of “mapping factors” in the Delaunay triangulation and Laplacian method can produce better mesh (most aspect ratios≤3) on a curved surface.     

一种裁剪曲面自适应三角剖分算法

本文介绍了一种裁剪曲面按精度三角剖分算法。三角剖分过程在参数域和曲面空间同时进行，参数域上控制三角片的拓扑关系，曲面空间进行精度检测。算法的核心思想是将裁剪曲面三角剖分视为约束剖分问题，从而使得三角形的细分操作拓展为有效域内插入散乱节点的三角剖分问题。算法简便、实用，三角化结果品质良好，已成功地应用于数控加工刀具轨迹干涉处理等具有精度要求的应用领域。     

‘Ultimate’ robustness in meshing an arbitrary polyhedron

Given a boundary surface mesh (a set of triangular facets) of a polyhedron, the problem of deciding whether or not a triangulation exists is reported to be NP‐hard. In this paper, an algorithm to triangulate a general polyhedron is presented which makes use of a classical Delaunay triangulation algorithm, a phase for recovering the missing boundary facets by means of facet partitioning, and a final phase that makes it possible to remove the additional points defined in the previous step. Following this phase, the resulting mesh conforms to the given boundary surface mesh. The proposed method results in a discussion of theoretical interest about existence and complexity issues. In practice, however, the method should provide what we call ‘ultimate’ robustness in mesh generation methods. Copyright © 2003 John Wiley & Sons, Ltd.     

基于Delaunay三角剖分的层析图像离散数据表面重建算法

提出一种基于Delaunay三角剖分思想的层析图像离散数据的表面重建算法。该算法考虑了组成最优重建表面的三角片的形态特点,以三角片集的内角矢量最大为优化目标,根据Delaunay三角剖分思想,采用局部判定的方法,逐次选取最佳几何形态的三角片,组成最优的重建表面。     

点序对Delaunay三角剖分局部优化的影响

局部变换法和Watson算法是属于逐点添加、局部优化的离散点集Delaunay三角剖分的常用方法,不同的加点次序对这两种算法的局部优化影响较大。研究发现按位置相邻次序加点的方法易产生外接圆较大的扁平三角形,引起较多三角形的局部优化,而按随机次序加点,网格生成过程中网格单元相对匀称,局部优化的三角形较少。以激光点扫描采集的数据为例,统计分析了局部优化三角形的数量及分布特征,点数大于50000时,相邻次序加点方法局部优化三角形的总量是随机次序加点方法的1.6倍以上。建立离散数据的矩形空间索引,按索引轮流加点,点序对局部优化的影响降低,相邻次序加点方法局部优化的三角形总量是随机次序加点方法的1.1~1.3倍,其中随机次序加点与没有空间索引的随机次序相比,局部优化的三角形数量仅增加了约1%。     

UNSTRUCTURED GRID GENERATION AND A SIMPLE TRIANGULATION ALGORITHM FOR ARBITRARY 2-D GEOMETRIES USING OBJECT ORIENTED PROGRAMMING

This paper describes the logic of a dynamic algorithm for a general 2D Delaunay triangulation of arbitrarily prescribed interior and boundary nodes. The complexity of the geometry is completely arbitrary. The scheme is free of specific restrictions on the input of the geometrical data. The scheme generates triangles whose associated circumcircles contain no nodal points except their vertices. There is no predefined limit for the number of points and the boundaries. The direction of generation of the triangles cannot be determined a priori as opposed to the moving front techniques. An automatic node placement scheme reflecting the initial boundary point spacings is used. The successive refinement scheme results in such a point distribution that the triangulation algorithm need not perform any geometric intersection check for overlapped triangles and penetrated boundaries. Further computational saving is provided by using a special binary tree (ADT) in which the points are ordered such that contiguous points in the list are neighbours in physical space. The method consists of a set of simple rules to understand. The dynamic nature of the Object Oriented Programming (OOP) of the algorithms provides efficient memory management on the insertion, deletion and searching processes. The computational effort bears a linear relation-ship between the CPU time and the total number of nodes. Some of the existing methods in the literature regarding triangular mesh generation are discussed in context. © 1997 by John Wiley & Sons, Ltd.     

The boundary recovery and sliver elimination algorithms of three‐dimensional constrained Delaunay triangulation

A boundary recovery and sliver elimination algorithm of the three‐dimensional constrained Delaunay triangulation is proposed for finite element mesh generation. The boundary recovery algorithm includes two main procedures: geometrical recovery procedure and topological recovery procedure. Combining the advantages of the edges/faces swappings algorithm and edges/faces splittings algorithm presented respectively by George and Weatherill, the geometrical recovery procedure can recover the missing boundaries and guarantee the geometry conformity by introducing fewer Steiner points. The topological recovery procedure includes two phases: ‘dressing wound’ and smoothing, which will overcome topology inconsistency between 3D domain boundary triangles and the volume mesh. In order to solve the problem of sliver elements in the three‐dimensional Delaunay triangulation, a method named sliver decomposition is proposed. By extending the algorithm proposed by Canvendish, the presented method deals with sliver elements by using local decomposition or mergence operation. In this way, sliver elements could be eliminated thoroughly and the mesh quality could be improved in great deal. Copyright © 2006 John Wiley & Sons, Ltd.     

An approach to refining three-dimensional tetrahedral meshes based on Delaunay transformations

A technique for refining three-dimensional tetrahedral meshes is proposed in this paper. The proposed technique is capable of treating arbitrary unstructured tetrahedral meshes, convex or non-convex with multiple regions resulting in high quality constrained Delaunay triangulations. The tetrahedra generated are of high quality (nearly equilateral). Sliver tetrahedra, which present a real problem to many algorithms are not produced with the new method. The key to the generation of high quality tetrahedra is the iterative application of a set of topological transformations based on the Voronoi–Delaunay theory and a reposition of nodes technique. The computational requirements of the proposed technique are in linear relationship with the number of nodes and tetrahedra, making it ideal for direct employment in a fully automatic finite element analysis system for 3-D adaptive mesh refinement. Application to some test problems is presented to show the effectiveness and applicability of the new method.     

Constrained boundary recovery for three dimensional Delaunay triangulations

A new constrained boundary recovery method for three dimensional Delaunay triangulations is presented. It successfully resolves the difficulties related to the minimal addition of Steiner points and their good placement. Applications to full mesh generation are discussed and numerical examples are provided to illustrate the effectiveness of guaranteed recovery procedure. Copyright © 2004 John Wiley & Sons, Ltd.     

Tetrahedral mesh generation using Delaunay refinement with non‐standard quality measures

This paper studies the practical performance of Delaunay refinement tetrahedral mesh generation algorithms. By using non‐standard quality measures to drive refinement, we show that sliver tetrahedra can be eliminated from constrained Delaunay tetrahedralizations solely by refinement. Despite the fact that quality guarantees cannot be proven, the algorithm can consistently generate meshes with dihedral angles between 18^circ and 154^°. Using a fairer quality measure targeting every type of bad tetrahedron, dihedral angles between 14^° and 154^° can be obtained. The number of vertices inserted to achieve quality meshes is comparable to that needed when driving refinement with the standard circumradius‐to‐shortest‐edge ratio. We also study the use of mesh improvement techniques on Delaunay refined meshes and observe that the minimum dihedral angle can generally be pushed above 20^°, regardless of the quality measure used to drive refinement. The algorithm presented in this paper can accept geometric domains whose boundaries are piecewise smooth. Copyright © 2011 John Wiley & Sons, Ltd.     

THREE-DIMENSIONAL FINITE ELEMENT MESHING BY INCREMENTAL NODE INSERTION

This paper describes an efficient algorithm for fully automated three-dimensional finite element meshing which is applicable to non-convex geometry and non-manifold topology. This algorithm starts with sparsely placed nodes on the boundaries of a geometric model and a corresponding 3-D Delaunay triangulation. Nodes are then inserted incrementally by checking the tetrahedral mesh geometry and topological compatibility between Delaunay triangulation and the geometric model. Topological compatibility is checked in a robust manner by a method which relies more on a mesh's topology than its geometry. The node placement strategy is tightly coupled to an incremental Delaunay triangulation algorithm, and results in a low growth rate of computational time.     

Efficient surface reconstruction from contours based on two‐dimensional Delaunay triangulation

This paper introduces an efficient method for surface reconstruction from sectional contours. The surface between neighbouring sections is reconstructed based on the consistent utilization of the two‐dimensional constrained Delaunay triangulation. The triangulation is used to extract the parametric domain and to solve the problems associated with correspondence, tiling and branching in a general framework. Natural distance interpolations are performed in order to complete the mapping of the added intermediate points. Surface smoothing and remeshing are conducted to optimize the initial surface triangulations. Several examples are presented to demonstrate the effectiveness and efficiency of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.