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.     

Tetrahedral mesh generation based on space indicator functions

An algorithm for the generation of tetrahedral volume meshes is developed for highly irregular objects specified by volumetric representations such as domain indicator functions and tomography data. It is based on red–green refinement of an initial mesh derived from a body‐centered cubic lattice. A quantitative comparison of alternative types of initial meshes is presented. The minimum set of best‐quality green refinement schemes is identified. Boundary conformity is established by deforming or splitting surface‐crossing elements. Numerical derivatives of input data are strictly avoided. Furthermore, the algorithm features surface‐adaptive mesh density based on local surface roughness, which is an integral property of finite surface portions. Examples of applications are presented for computer tomography of porous media. Copyright © 2012 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.     

Tetrahedral mesh generation based on advancing front technique and optimization scheme

In recent years, demand for three‐dimensional simulations has continued to grow in the field of computer‐aided engineering. Especially, in the analysis of forming processes a fully automatic and robust mesh generator is necessary for handling complex geometries used in industry. For three‐dimensional analyses, tetrahedral elements are commonly used due to the advantage in dealing with such geometries. In this study, the advancing front technique has been implemented and modified using an optimization scheme. In this optimization scheme, the distortion metric determines ‘when and where’ to smooth, and serves as an objective function. As a result, the performance of the advancing front technique is improved in terms of mesh quality generated. Copyright © 2003 John Wiley & Sons, Ltd.     

Model and mesh generation of cracked tubular Y-joints

In this study, the methods for constructing accurate and consistent geometrical and finite element (FE) models for general cracked tubular Y-joints are described. Firstly, geometrical analysis of welded tubular joint is given and it is then extended to the modelling of general cracked Y-joints. The concept of crack surface and a simple mapping approach are suggested to model either through-thickness or surface cracks which can be of any length and located at any position along the brace-chord intersection. Secondly, the geometrical model developed will be used in the generation of consistent FE meshes. The basic concepts used for the design and generation of three-dimensional FE meshes will be described. This will include the meshing procedures for discretization of tubular joints with through-thickness and surface cracks which are frequently regarded as one of the most difficult steps in the construction of tubular joint models. Finally, some mesh generation examples for uncracked and cracked Y-joints will be presented to demonstrate the use of the purposed geometrical model and mesh generation scheme developed.     

Periodic three-dimensional mesh generation for crystalline aggregates based on Voronoi tessellations

In this paper a method for the generation of three-dimensional periodic meshes for the numerical simulation of polycrystalline aggregates is presented. The mesh construction is based on Voronoi and Hardcore Voronoi tessellations of random point seeds. Special emphasis is paid on the periodicity of the mesh topologies which leads to favorable numerical properties for the determination of effective properties using unit cells. The mesh generation algorithm is able to produce high quality meshes at low computational costs. Based on unit cell simulations with different but statistically equivalent microstructures, the effective linear elastic properties of polycrystals consisting of grains with a cubic symmetry are determined. The numerical results are compared with first-, third- and fifth-order bounds and experimental data. Numerical simulations show the efficiency of the proposed homogenization technique.     

Tetrahedral mesh generation in polyhedral regions based on convex polyhedron decompositions

A method using techniques of computational geometry for generating tetrahedral finite element meshes in three-dimensional polyhedral regions is presented. The input to the method consists of the boundary faces of the polyhedral region and possibly internal and hole interfaces, plus the desired number of tetrahedra and other scalar parameters. The region is decomposed into convex polyhedra in two stages so that tetrahedra of one length scale can be generated in each subregion. A mesh distribution function, which is either automatically constructed from the first-stage convex polyhedron decomposition or supplied by the user, is used to determine the tetrahedron sizes in the subregions. Then a boundary-constrained triangulation is constructed in each convex polyhedron, with local transformations being used to improve the quality of the tetrahedra. Experimental results from triangulations of three regions are provided.     

Automatic three-dimensional finite element mesh generation via modified ray casting

Three-dimensional (3-D) finite element mesh generation has been the target of automation due to the complexities associated with generating and visualizing the mesh. A fully automatic 3-D mesh generation method is developed. The method is capable of meshing CSG solid models. It is based on modifying the classical ray-casting technique to meet the requirements of mesh generation. The modifications include the utilization of the element size in the casting process, the utilization of 3-D space box enclosures, and the casting of ray segments (rays with finite length). The method begins by casting ray segments into the solid. Based on the intersections between the segments and the solid boundary, the solid is discretized into cells arranged in a structure. The cell structure stores neighbourhood relations between its cells. Each cell is meshed with valid finite elements. Mesh continuity between cells is achieved via the neighbourhood relations. The last step is to process the boundary elements to represent closely the boundary. The method has been tested and applied to a number of solid models. Sample examples are presented.     

Tetrahedral mesh improvement via optimization of the element condition number

We present a new shape measure for tetrahedral elements that is optimal in that it gives the distance of a tetrahedron from the set of inverted elements. This measure is constructed from the condition number of the linear transformation between a unit equilateral tetrahedron and any tetrahedron with positive volume. Using this shape measure, we formulate two optimization objective functions that are differentiated by their goal: the first seeks to improve the average quality of the tetrahedral mesh; the second aims to improve the worst‐quality element in the mesh. We review the optimization techniques used with each objective function and present experimental results that demonstrate the effectiveness of the mesh improvement methods. We show that a combined optimization approach that uses both objective functions obtains the best‐quality meshes for several complex geometries. Copyright © 2001 John Wiley & Sons, Ltd.     

Tetrahedral mesh improvement using swapping and smoothing

Automatic mesh generation and adaptive refinement methods for complex three-dimensional domains have proven to be very successful tools for the efficient solution of complex applications problems. These methods can, however, produce poorly shaped elements that cause the numerical solution to be less accurate and more difficult to compute. Fortunately, the shape of the elements can be improved through several mechanisms, including face- and edge-swapping techniques, which change local connectivity, and optimization-based mesh smoothing methods, which adjust mesh point location. We consider several criteria for each of these two methods and compare the quality of several meshes obtained by using different combinations of swapping and smoothing. Computational experiments show that swapping is critical to the improvement of general mesh quality and that optimization-based smoothing is highly effective in eliminating very small and very large angles. High-quality meshes are obtained in a computationally efficient manner by using optimization-based smoothing to improve only the worst elements and a smart variant of Laplacian smoothing on the remaining elements. Based on our experiments, we offer several recommendations for the improvement of tetrahedral meshes. © 1997 John Wiley & Sons, Ltd.     

Automatic sizing functions for unstructured surface mesh generation

Accurate sizing functions are crucial for efficient generation of high‐quality meshes, but to define the sizing function is often the bottleneck in complicated mesh generation tasks because of the tedious user interaction involved. We present a novel algorithm to automatically create high‐quality sizing functions for surface mesh generation. First, the tessellation of a Computer Aided Design (CAD) model is taken as the background mesh, in which an initial sizing function is defined by considering geometrical factors and user‐specified parameters. Then, a convex nonlinear programming problem is formulated and solved efficiently to obtain a smoothed sizing function that corresponds to a mesh satisfying necessary gradient constraint conditions and containing a significantly reduced element number. Finally, this sizing function is applied in an advancing front mesher. With the aid of a walk‐through algorithm, an efficient sizing‐value query scheme is developed. Meshing experiments of some very complicated geometry models are presented to demonstrate that the proposed sizing‐function approach enables accurate and fully automatic surface mesh generation. Copyright © 2016 John Wiley & Sons, Ltd.     

Hexahedral mesh generation by medial surface subdivision: Part I. Solids with convex edges

A method is presented for subdividing a large class of solid objects into topologically simple subregions suitable for automatic finite element meshing with hexahedral elements. The technique uses a geometric property of a solid, its medial surface, to define the necessary subregions. The subregions are defined explicitly to be one of only 13 possible types. The subdividing cuts are between parts of the object in geometric proximity and produce good quality meshes of hexahedral elements. The method as introduced here is applicable to solids with convex edges and vertices, but the extension to complete generality is feasible.     

Adaptive tetrahedral mesh generation by constrained Delaunay refinement

This paper presents a tetrahedral mesh generation method for numerically solving partial differential equations using finite element or finite volume methods in three‐dimensional space. The main issues are the mesh quality and mesh size, which directly affect the accuracy of the numerical solution and the computational cost. Two basic problems need to be resolved, namely boundary conformity and field points distribution. The proposed method utilizes a special three‐dimensional triangulation, so‐called constrained Delaunay tetrahedralization to conform the domain boundary and create field points simultaneously. Good quality tetrahedra and graded mesh size can be theoretically guaranteed for a large class of mesh domains. In addition, an isotropic size field associated with the numerical solution can be supplied; the field points will then be distributed according to it. Good mesh size conformity can be achieved for smooth sizing informations. The proposed method has been implemented. Various examples are provided to illustrate its theoretical aspects as well as practical performance. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive triangular–quadrilateral mesh generation

In this paper, we begin by recalling an adaptive mesh generation method governed by isotropic and anisotropic discrete metric maps, by means of the generation of a unit mesh with respect to a Riemannian structure. We propose then an automatic triangular to quadrilateral mesh conversion scheme, which generalizes the standard case to the anisotropic context. In addition, we introduce an optimal vertex smoothing procedure. Application test examples, in particular a CFD test, are given to demonstrate the efficiency of the proposed method. © 1998 John Wiley & Sons, Ltd.     

大规模四面体网格并行生成方法

针对大规模网格串行生成的时间和内存瓶颈问题,阐述了一种非结构化四面体网格并行生成方法,首先对几何体进行初始网格划分,并通过相对体积比及最优分区控制初始网格数量;然后采用图论图划分方法对初始网格进行区域分解;最后采用分裂法进行并行网格生成。提出的基于共享单元的边界判定方法,有效地解决了分区边界节点的匹配问题。通过算例证明了该并行网格生成方法具有良好的并行效率,同时可以产生高质量的四面体网格。     

Hierarchical tree‐based finite element mesh generation

Hierarchical grid generation and its use as a basis for finite element mesh generation are considered in this paper. The hierarchical grids are generated by recursive subdivision using quadtrees in two dimensions and octrees in three dimensions. A numbering system for efficient storage of the quadtree grid information is examined, tree traversal techniques are devised for neighbour finding, and accurate boundary representation is considered. It is found that hierarchical grids are straightforward to generate from sets of seeding points which lie along domain boundaries. Quadtree grids are triangularized to provide finite element meshes in two dimensions. Three‐dimensional tetrahedral meshes are generated from octree grids. The meshes can be generated automatically to model complicated geometries with highly irregular boundaries and can be adapted readily at moving boundaries. Examples are given of two‐ and three‐dimensional hierarchical tree‐based finite element meshes and their application to modelling free surface waves. Copyright © 1999 John Wiley & Sons, Ltd.     

Node placement for triangular mesh generation by Monte Carlo simulation

A new approach of node placement for unstructured mesh generation is proposed. It is based on the Monte Carlo method to position nodes for triangular or tetrahedral meshes. Surface or volume geometries to be meshed are treated as atomic systems, and mesh nodes are considered as interacting particles. By minimizing system potential energy with Monte Carlo simulation, particles are placed into a near‐optimal configuration. Well‐shaped triangles or tetrahedra can then be created after connecting the nodes by constrained Delaunay triangulation or tetrahedrization. The algorithm is simple, easy to implement, and works in an almost identical way for 2D and 3D meshing. Copyright © 2005 John Wiley & Sons, Ltd.     

Automatic mesh generation over intersecting surfaces

A fully automatic general scheme is proposed to determine and analyse the intersections between two groups of surfaces composed of triangular facets. By carefully maintaining the spatial position of the lines of intersection in regenerating mesh locally around surface intersections, pieces of surfaces can be arbitrarily combined and merged. A great variety of objects can be easily created by selectively putting together different surface parts derived from surface intersections. The new algorithm is best applied in conjunction with an existing surface mesh generator to enhance its general capability in dealing with objects built from intersecting surfaces. In fact, it is a powerful surface mesh manipulator, and through the repeated use of the process, complex structures can be rapidly and accurately constructed.