Size gradation control of anisotropic meshes

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Abstract

This paper gives a status on anisotropic mesh gradation. We present two 3D anisotropic formulations of mesh gradation. The metric at each point defines a well-graded smooth continuous metric field over the domain. The mesh gradation then consists in taking into account at each point the strongest size constraint given by all these continuous metric fields. This is achieved by a metric intersection procedure. We apply it to several examples involving highly anisotropic meshes.

Introduction

In many engineering applications, it is desirable to generate anisotropic meshes presenting highly stretched elements in adequate directions. Numerous papers have been published on mesh adaptation for numerical simulations in computational solid or fluid mechanics. Among these papers, some have addressed the problem of creating 3D unstructured anisotropic meshes [5], [6], [10], [12], [15], [17], [20], [21], [23]. In these approaches, different mesh generation methods are considered. Nonetheless all of them are based on the notion of unit mesh in a Riemannian metric space.

In the context of numerical simulations based on finite element or finite volume methods, these works have proved the efficiency of unstructured mesh adaptation to improve the accuracy of the numerical solution as well as for capturing the behavior of physical phenomena. In principle, this technique enables to substantially reduce the number of degrees of freedom, thus impacting favorably (i) the cpu time, (ii) the data storing and (iii) the solution analysis (visualization). Moreover, it has been recently proved in [20] that mesh adaptation impacts positively the order of convergence of numerical scheme by computing the numerical solution with a coherent accuracy in the entire domain.

The size prescription in the generation of anisotropic meshes is achieved thanks to metric fields. However, such metric fields may have huge variations making the generation of a unit mesh difficult or impossible, thus leading to poor quality anisotropic meshes. Generating high-quality anisotropic meshes requires to smooth the metric field by bounding its variations in all directions. To this end, a mesh gradation control procedure was introduced in [4]. It consists in reducing the size prescribed at mesh vertices by checking the metric variations along the mesh edges. The authors first describe an isotropic formulation from which they deduce an extension to anisotropic meshes. They present an homothetic reduction dedicated for surface mesh generation and a non-homothetic reduction. In the context of volume meshing, the homothetic leads to inconsistency in the metric reduction during the size gradation procedure. An anisotropic mesh gradation has also been presented in [16]. This procedure considers spectral decomposition and associate eigenvectors together with ad hoc choices. We prefer a formulation that uses directly well-posed operations on metrics. In [22], an isotropic size gradation control has been applied to the generation of multi-patch parametric surface meshes.

In this paper, we give a status on anisotropic mesh gradation. We present two 3D anisotropic formulations of mesh gradation extending the formulation given in [4] and we apply it to several examples involving highly anisotropic meshes. We formulate the problem mathematically by employing the continuous modeling of a mesh proposed in [19]. The metric at each point defines a well-graded smooth continuous metric field over the domain. The mesh gradation then consists in taking into account at each point the strongest size constraint given by all these continuous metric fields. Numerically, in the context of a mesh with a metric field given at its vertices, the idea consists in imposing at each vertex a size constraint related to all the other vertices of the mesh. To this end, all vertices of the mesh span metric fields in the whole domain by growing their metrics at a rate given by the desired gradation coefficient. Then, the reduced metric at a vertex is the intersection between its metric and all these metrics. Unfortunately, this mesh gradation algorithm is intrinsically of quadratic complexity. We thus establish an approximation to solve it in a linear time.

This paper is outlined as follows. In Section 2 the notion of a metric and the metric-based method to generate anisotropic meshes are described. In Section 3 the isotropic mesh gradation is recalled. Then, two anisotropic formulations of the mesh gradation are presented, Section 4. In the numerical examples, Section 5, we illustrate the efficiency of the presented method for anisotropic mesh adaptation. We also exemplify that anisotropic mesh gradation can be used for other applications such as the generation of well-graded meshes. Finally, the limits of the proposed approaches are exemplified on an analytical example in Section 6. New mesh gradation procedures improving the previous ones are then proposed to remedy these problems.

Section snippets

Metric and mesh generation

In this section, we recall a metric-based method to generate anisotropic meshes. It is based on the notion of Riemannian metric space and on the concept of unit mesh initially introduced in [14]. Well-posed operations on metrics are introduced and we discuss the numerical computation of the length of a given path in a metric space.

Isotropic mesh gradation control

In this section, we recall the main lines of the isotropic mesh gradation introduced in [4].

Let M be an isotropic metric field defined in the entire domain Ω. The metric field can be rewritten M=h-2I with I the identity matrix. Let p and q be two points of the domain with associated sizes hp and hq.

Definition 3.1

The H-shock or size gradation c(pq) related to the segment pq of Ω is the value: c(pq)=maxhphq,hqhp1/M(pq).

The anisotropic H-shock value varies in the interval [1,[. The H-shock in mesh gradation

Anisotropic mesh gradation control

We propose two formulations generalizing in a continuous framework the isotropic mesh gradation to the anisotropic case. In the anisotropic context, the mesh gradation consists in reducing in all directions the size prescribed at any points if the variation of the metric field is larger than a fixed threshold. As we control the variations of the metric in all directions, it is difficult to propose a direct anisotropic extension of the H-shock notion.

The previous analysis leads us to suggest the

Numerical examples

We first discuss the practical implementation of the mesh gradation algorithm. Then, several applications are presented.

Limits of the edge-based mesh gradation algorithm

Although the edge-based approach, presented in Section 5.1, provides good results in most of the practical cases, this algorithm has weaknesses. They are due to the coarse approximation of the quadratic complexity algorithm of the mesh gradation. In the following, these weaknesses are pointed out on a 2D analytical example. We first describe how the mesh adaptation scheme is performed in the context of an analytical case.

We have an initial mesh. The metric is computed at mesh vertices thanks to

Concluding remarks

We have presented and evaluated two anisotropic formulations of the mesh gradation. All approaches generally provide good results for realistic applications. The main problem with these approaches is that we must restrict to linear transitive law for the size variation.

The metric-space-gradation achieve better accuracy and higher anisotropic meshes but it encounters the ray phenomenon in specific cases when anisotropic metrics are prescribed along curves. This is due to the rough approximation

Acknowledgments

I wish to thank A. Loseille and P.L. George for their constructive remarks concerning this work.

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