Paul-Louis George

Automatic mesh generator with specified boundary

Abstract For the purpose of finite element computation for 2D or 3D geometry, one needs an appropriate mesh of the considered domain. A class of full automatic methods, derived from Voronois theory, is suitable to generate the mesh of any shape via a set of points which describes the geometry. Such methods providing triangles in 2D and tetrahedra in 3D can be seen, after an adequate initialization, as the merger of each given point in an existing mesh using an updating process. Unfortunately, this mesh which contains all the given points does not contain, in general, the edges (or the faces) of the boundary which are the natural data to be satisfied. The aim of this paper is, after a brief survey of the different steps of the above method, to point out the problem of the exact fitting of the given boundary and to present a method which guarantees this crucial property.

Delaunay mesh generation governed by metric specifications. Part I algorithms

Abstract This paper proposes a Delaunay-type mesh generation algorithm governed by a metric map. The classical method is briefly established and then the different steps it involves are extended. It will be shown that the proposed method applies in three dimensions. The work is divided in two parts. Part I, i.e. the present paper, is devoted to the algorithmical aspects while Part II will present numerous application examples in the context of finite element computations.

Fully automatic mesh generator for 3D domains of any shape

Abstract Devoted to mesh generation of 3D domains, this paper examines the different approaches actually in progress. A new method is introduced which can be seen as a variant of the Delaunay-Voronoi tesselation coupled with a control of the given boundary used to define the domain under consideration.

3D transient fixed point mesh adaptation for time-dependent problems: Application to CFD simulations

This paper deals with the adaptation of unstructured meshes in three dimensions for transient problems with an emphasis on CFD simulations. The classical mesh adaptation scheme appears inappropriate when dealing with such problems. Hence, another approach based on a new mesh adaptation algorithm and a metric intersection in time procedure, suitable for capturing and track such phenomena, is proposed. More precisely, the classical approach is generalized by inserting a new specific loop in the main adaptation loop in order to solve a transient fixed point problem for the mesh-solution couple. To perform the anisotropic metric intersection operation, we apply the simultaneous reduction of the corresponding quadratic form. Regarding the adaptation scheme, an anisotropic geometric error estimate based on a bound of the interpolation error is proposed. The resulting computational metric is then defined using the Hessian of the solution. The mesh adaptation stage (surface and volume) is based on the generation, by global remeshing, of a unit mesh with respect to the prescribed metric. A 2D model problem is used to illustrate the difficulties encountered. Then, 2D and 3D complexes and representative examples are presented to demonstrate the efficiency of this method.

Parametric surface meshing using a combined advancing-front generalized Delaunay approach

An indirect method for meshing parametric surfaces conforming to a user-specifiable size map is presented. First, from this size specification, a Riemannian metric is defined so that the desired mesh is one with unit length edges with respect to the related Riemannian space (the so-called ‘unit mesh’). Then, based on the intrinsic properties of the surface, the Riemannian structure is induced into the parametric space. Finally, a unit mesh is generated completely inside the parametric space such that it conforms to the metric of the induced Riemannian structure. This mesh is constructed using a combined advancing-front—Delaunay approach applied within a Riemannian context. The proposed method can be applied to mesh composite parametric surfaces. Several examples illustrate the efficiency of our approach. Copyright

Delaunay mesh generation governed by metric specifications. Part II. applications

Abstract This paper gives some application examples resulting from a governed Delaunay type mesh generation method. Isotropic and anisotropic cases are considered, these specifications being given via a metric map. The paper illustrates a study whose algorithmical aspects are described in a report referred to as part 1. Academic examples as well as examples in CFD are discussed.

3D Delaunay mesh generation coupled with an advancing-front approach

This paper describes a fully automatic mesh generation method suitable for domains of any shape in R3. Initially, a so-called boundary mesh is generated wherein internal points are created using advancing-front point placement and inserted using a Delaunay method. The algorithm combines the advantages of efficiency and nice mathematical properties of a Delaunay approach with advancing-front high-quality point-placement strategy. Several three-dimensional examples are presented to demonstrate the overall efficiency and the relevance of the combined procedure. The present method can be extended to isotropic or anisotropic adaptive mesh generation.

ASPECTS OF 2-D DELAUNAY MESH GENERATION

SUMMARY This paper aims to outline the dierent phases necessary to implement a Delaunay-type automatic mesh generator. First, it summarizes this method and then describes a variant which is numerically robust by mentioning at the same time the problems to solve and the dierent solutions possible. The Delaunay insertion process by itself, the boundary integrity problem, the way to create the eld points as well as the optimization procedures are discussed. The two-dimensional situation is described fully and possible extensions to the three-dimensional case are briey indicated. ? 1997 by John Wiley & Sons, Ltd. There exist a large number of papers dealing with 2-D-mesh generation using the Delaunay method (see Reference 1). Nevertheless, we propose a new scheme with ambition to optimize, if possible, the dierent steps used in the practical application of the algorithm. We have tried to develop for each step an optimal solution. The method has been implemented and results show the performance of the algorithm. So we can construct a well-shaped mesh constituted by a million of triangles in a few minutes on a HP 735 workstation. The 2-D case is fully detailed for clarity but most of the results can be extended without diculty in three dimensions. The problem to be solved is to construct a consistent mesh of a domain essentially from its boundary data segments. The resulting mesh will be the support of a nite element computation which can indicate if it is adapted or must be adapted by any method (it is not the goal of this paper). Nevertheless, it is important to begin with a well-shaped mesh knowing that this request can only be based on geometric considerations, since the sole known information is of geometric nature. Thus, the shape and the size of the elements must be consistent with these data.

Optimization of Tetrahedral Meshes

Finite element computations are all the more exact if we start from “good” elements. We are interested in meshes where the elements are tetrahedra and we shall develop utilities allowing us to improve the quality of these meshes.

Improvements on Delaunay-based three-dimensional automatic mesh generator

Abstract This paper describes an automatic mesh generator providing tetrahedral meshes suitable in general for finite element simulations. The mesh generator is of the Delaunay type and the paper focuses on recent improvements relative to this a priori well-known method.