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*We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two- and three-dimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles.*

- Mesh Generation and Optimal Triangulation
- Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator
- Delaunay Mesh Generation
- Delaunay triangulation

In mathematics and computational geometry , a Delaunay triangulation also known as a Delone triangulation for a given set P of discrete points in a plane is a triangulation DT P such that no point in P is inside the circumcircle of any triangle in DT P. Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid sliver triangles. The triangulation is named after Boris Delaunay for his work on this topic from For a set of points on the same line there is no Delaunay triangulation the notion of triangulation is degenerate for this case.

## Mesh Generation and Optimal Triangulation

A computational methodology for automatic two-dimensional anisotropic mesh generation and adaptation. Paulo Roberto M. Lyra I ; Darlan Karlo E. Brazil II dkarlo uol. This paper describes a versatile computational program for automatic two-dimensional mesh generation and remeshing adaptation of triangular, quadrilateral and mixed meshes. The system is flexible to be incorporated into an adaptive global or local remeshing procedure and for generating both, iso and anisotropic meshes.

The main contribution of this work is to extend well established procedures for the generation and adaptation of both, iso and anisotropic triangular meshes, such as local and global remeshing as well as boundary layer mesh generation, to deal with iso and anisotropic quadrilateral and mixed meshes. Several examples are presented to illustrate the quality of the meshes produced, and the flexibilities of the computational system.

Keywords: Iso and anisotropic mesh generation, adaptive remeshing, triangular, quadrilateral and mixed unstructured meshes. The use of an adequate mesh consists in one of the main ingredients for an accurate numerical simulation. In order to obtain such a mesh, a versatile mesh generator and a mesh adaptive procedure must be available. Automatic mesh generation has received much attention from researchers on computational simulation, to minimize manual intervention, to improve mesh quality and to obtain more efficient procedures.

Unstructured methodologies are becoming predominant due to the ability of modeling geometrically complex designs and because they are the natural environment for adaptivity, which may be the only hope for resolving very small scale features e. Most finite element and finite volume codes use unstructured triangulations due to their geometrical flexibility and the low cost of linear triangular elements.

However, certain formulations perform better with quadrilateral elements and it may be necessary to use different types of elements for different physics when solving coupled problems.

In fluid-structure interaction problems, for instance, it is very common to use unstructured triangular meshes for the fluid and quadrilateral meshes for the structural problem. In computational fluid dynamics applications, it may also be interesting to use a mixed mesh, in which quadrilateral elements are used in regions where the flow is essentially one dimensional Hwang and Wu, The utilization of unstructured mesh generation techniques for the simulation of boundary layer problems Hassan, , Marcum, and Thompson et al.

The development of procedures for the generation of anisotropic unstructured meshes of triangles capable of capturing directional features of the physical problem is a topic of intensive research Jansen and Shephard, , but less work has been done regarding anisotropic meshes of quadrilaterals Borouchaki and Frey, Some effort has also been made in order to build unstructured hexahedral meshes.

However, most of the success achieved concentrates on generating quadrilaterals and hexahedra in the vicinity of solids walls, i. It is known that quadrilateral and hexahedra require less cpu time and memory whenever an edge-based data structure is adopted for both finite element FEM and finite volume methods FVM , Borouchaki and Frey, Building good quality quadrilateral and mixed meshes is very challenging because quadrilaterals are very "stiff'' from a geometric point of view, and controlling element distortion in complex domains during analysis that incorporate adaptive procedures is certainly not an easy task.

In this work, an automatic triangular mesh generator based on the advancing front technique Peraire et al. Quadrilateral elements are generated from an original mesh of triangles through a process of merging and splitting.

These approaches allow a good control of the mesh density and gradation, and of the directional stretching and quality of both, triangular and quadrilateral elements. The so called "advancing layers'' technique Hassan et al.

The system also allows for either, a global adaptive remeshing procedure Peraire et al. Here, we extend the local remeshing procedure to deal with anisotropic meshes by computing the mesh parameters, i.

We eliminate the elements within those subregions and re-built the mesh according to the new distribution of mesh parameters. Some relevant issues referring to the auxiliary data structures adopted and other numerical aspects are also discussed.

Unstructured 2-D Mesh Generation. The mesh generation problem consists in subdividing an arbitrary domain into a consistent assembly of elements subregions.

If the generated elements cover the entire domain and the intersection of the elements occurs only on common points or edges, the consistency of the mesh is guaranteed. According to Peraire , the basic requirements for a mesh generator are:. Capacity to handle arbitrary geometries with minimum user intervention;. The necessity of as little input data as possible;. Good control over the spatial variation of size and shape of the elements.

Easy incorporation of adaptive strategies. Triangles are the most flexible elements for automatic mesh generation over complex two dimensional geometries, particularly when mesh grading is required. Several robust and versatile unstructured triangular mesh generators and their extensions for 3-D geometries tetrahedral mesh generators have been developed and are currently used in academic and industrial environments.

There are several well-established triangulation procedures, such as the advancing front Peraire et al. Advancing Front Triangular Mesh Generator. The original advancing front algorithm has been developed over time into a family of programs which are very reliable and flexible for an easy incorporation of mesh adaptation. The advancing front mesh generator can be described as in Algorithm 1.

The computational domain is modeled through the use of cubic splines which are defined by some control points.

Close to singularities extra care must be taken in the definition of these points in order to avoid failure Thompson et al. As a "pre-processing" stage, before the mesh generation begins, we must first build an initial and very coarse triangular background mesh that covers the whole domain. This coarser mesh is used only to provide a piecewise linear spatial distribution of the nodal parameters over the mesh to be constructed. Typically, elements of the generated mesh will have a projected length of d 2 in the direction parallel to a 2 and a projected length of S t d 2 in the direction normal to a 2 see Fig.

During the generation process, the local values of these parameters will be obtained by a linear interpolation over the triangles of the background mesh. The boundary of the domain is represented by the union of boundary segments forming closed loops.

External boundaries are defined in an anti-clockwise fashion while inner boundaries are set in a clockwise manner. As described previously, the generation of a triangular mesh by the advancing front technique begins by the discretization of the boundary of the domain. New points are created according to the mesh parameters which are interpolated from those of the background mesh. At the beginning of the process, the generation front is made by a set of linear segments connecting the boundary nodes.

With the initial front defined, one segment is chosen and, in general, a triangle is created through the insertion of an internal node or by simply connecting existing nodes.

New triangles are built following the same procedure. During the process any segment available to build a new triangle is set as "active" and the others which are set as "non-active" are removed from the generation front. Therefore the boundary segments are not modified during the mesh generation. The procedure continues until the whole domain is discretized. When solving problems which develop some essentially one dimensional features at certain regions e. In these cases, it is important to have the possibility to define a direction and a stretching factor for the elements close to such regions.

At least for linear triangular elements, the use of anisotropic meshes can be extremely important in terms of computational effort and accuracy Rippa, To generate an anisotropic triangulation of the desired domain, it is used a transformation T which is a function of the mesh parameters, i. The effect of this transformation is to map the physical domain into a normalized domain, where a mesh is generated in which the elements are approximately equilateral with unit average size.

This mesh generator provides an accurate geometric modeling and high quality meshes, where the high level of control of the distribution of local mesh parameters eases the incorporation of mesh adaptation strategies. The quality of the meshes is strongly influenced by the mesh optimization stage. A specific mesh improvement strategy for highly anisotropic meshes and the definition of an adequate sequence of mesh enhancement procedures are incorporated into the code.

Several other modifications have been introduced in the original code in order to incorporate the flexibility to deal with predefined multidomains and automatically defined subregions, to build boundary layer meshes, to make possible generating quadrilateral and mixed meshes and the automatic definition of which domains or subregions should be filled up by triangular or by quadrilateral elements.

These features will be fully described in the correspondent sections. Quadrilateral Mesh Generator. Unstructured quadrilateral meshes can be automatically generated in several different ways and do not impose serious topological restrictions on the meshes, being appropriated to deal with complex geometries, naturally allowing local non-uniform mesh refinement. Several different approaches have been proposed to generate unstructured quadrilateral meshes.

These methodologies can be divided into two basic groups: those that try to generate quadrilaterals directly Blacker and Stevenson, , Zhu et al. The conversion of triangular meshes is particularly attractive because these meshes can inherit the properties of the triangular meshes, whose generators are very well developed and once it is always possible to build a triangular mesh over any arbitrary 2-D domain, quadrilateral meshes can be constructed as general as the triangular ones.

It also allows the use of any triangular mesh generator as a "black box", even commercial ones for which source codes are not available, including procedures such as Delaunay methods Weatherill, modified quadtree techniques Schroeder and Shephard, , etc. In the work of Ait-Ali-Yahia et al. In the work of Borouchaki and Frey , anisotropic quadrilateral meshes are generated using a more general approach based on defining an anisotropic discrete metric mapping.

However, apart from the mentioned references, very little have been done with respect to anisotropic fully unstructured quadrilateral meshes. Here, the use of simple strategies during the conversion and mesh quality enhancement steps allows us to get reasonably good anisotropic quadrilateral meshes.

Indirect Approach: Conversion of Triangular Meshes. As we generate a quadrilateral mesh using the conversion strategy, the quadrilateral mesh inherits the characteristics of the initial triangulation.

For both, iso and anisotropic meshes this strategy consists of four main steps, as presented in Algorithm 2. The standard strategy of merging triangles into quadrilaterals consists in eliminating a common edge that belongs to two adjacent triangles. Following the work done by Xie and Ramaekers and Alquati and Groehs , our mesh generator is such that it refrains from merging triangles that would form a non-convex quadrilateral.

Besides, for anisotropic meshes, the merging process will remove a common edge between two adjacent triangles, only if the two quadrilaterals to be created satisfy a quality criteria which is controlled by two geometric parameters, f which is defined in Lee and Lo , and the minimum internal angle q.

We have attempted other strategies, in which the edges are previously grouped in some order Alquati and Groehs, , but our experiments have shown no significant improvement on the final mesh quality. Additionally, when dealing with anisotropic meshes, we redefine the merging procedure step 2 of Algorithm 2 , which is now performed as long as the common edge is one of the two biggest edges of both triangles and we also try to minimize the number of isolated triangles remaining after the merging step, by relaxing the quality criteria, as in general those isolated triangles will lead to bad quality quadrilaterals.

The enhancement of the mesh quality step 4 is slightly different for iso and anisotropic meshes and will be described later on this paper. The adopted procedure generates a quadrilateral mesh with edges that are approximately half of those of the corresponding triangular elements and usually this is not a serious concern, since the user can generate a coarser initial triangulation to obtain the desired mesh density.

The four steps involved in the quadrilateral mesh generation can be seen in Figs. Multi-Domains and Mixed Meshes. By performing the initial triangulation for each domain at a time, keeping, of course, a consistent node numbering through the interfaces of those domains, the mesh generator is capable to build multi-domain meshes.

The multi-domain meshes can either consist of a single element type or different element types in each domain Lyra and Carvalho, Besides, those edges which lie between two domains are identified and none of the mesh enhancement procedures described in the present paper is performed.

The multi-domain flexibility is very important whenever addressing for instance fluid flow problems, such as non-miscible, multi-phase flow, and solid mechanic problems, with different material properties on different portions of the domain. It might also be interesting to use quadrilateral elements in regions where the solution is essentially one dimensional e.

These remarks are valid for both, the finite element and the finite volume method.

## Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator

Meshing quality of the discrete model influences the accuracy, convergence, and efficiency of the solution for fractured network system in geological problem. However, modeling and meshing of such a fractured network system are usually tedious and difficult due to geometric complexity of the computational domain induced by existence and extension of fractures. The traditional meshing method to deal with fractures usually involves boundary recovery operation based on topological transformation, which relies on many complicated techniques and skills. This paper presents an alternative and efficient approach for meshing fractured network system. The method firstly presets points on fractures and then performs Delaunay triangulation to obtain preliminary mesh by point-by-point centroid insertion algorithm. Then the fractures are exactly recovered by local correction with revised dynamic grid deformation approach.

A computational methodology for automatic two-dimensional anisotropic mesh generation and adaptation. Paulo Roberto M. Lyra I ; Darlan Karlo E. Brazil II dkarlo uol. This paper describes a versatile computational program for automatic two-dimensional mesh generation and remeshing adaptation of triangular, quadrilateral and mixed meshes. The system is flexible to be incorporated into an adaptive global or local remeshing procedure and for generating both, iso and anisotropic meshes.

Our book is a thorough guide to Delaunay refinement algorithms that are mathematically guaranteed to generate meshes with high quality, including triangular meshes in the plane, tetrahedral volume meshes, and triangular surface meshes embedded in three dimensions. It is also the most complete guide available to Delaunay triangulations and algorithms for constructing them. We have designed the book for two audiences: researchers, especially graduate students, and engineers who design and program mesh generation software. Exercises are included; so is implementation advice. Delaunay refinement algorithms operate by maintaining a Delaunay or constrained Delaunay triangulation which is refined by inserting additional vertices until the mesh meets constraints on element quality and size.

PDF | We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of.

## Delaunay Mesh Generation

Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of adaptive exact arithmetic to ensure robustness with minimal sacrifice of speed. The problem of triangulating a planar straight line graph PSLG without introducing new small angles is shown to be impossible for some PSLGs, contradicting the claim that a variant of the Delaunay refinement algorithm solves this problem. Unable to display preview.

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Bern and D. Bern , D.

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### Delaunay triangulation

Proceedings of the 21st International Meshing Roundtable pp Cite as. Mesh generation and refinement are widely used in applications that require a decomposition of geometric objects for processing. Longest edge refinement algorithms seek to obtain a better decomposition over selected regions of the mesh by the division of its elements. Until now, these algorithms did not provide theoretical guarantees on the size of the triangulation obtained.

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We survey the computational geometry relevant to nite element mesh generation. We especially focus on optimal triangulations of geometric do- mains in two- and.

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* - Хейл в Третьем узле. Он напал на .*

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Шесть секунд. - Утечка информации. - Никаких изменений.