About Parametric Surface Meshing

2nd Symposium on Trends in Unstructured Mesh Generation, University of Colorado, Boulder, August 1999

MESHING
RESEARCH
CORNER

INRIA, Domaine de Voluceau, Rocquencourt BP 105, 78153 Le Chesnay Cedex, France.
houman.borouchaki@univ-troyes.fr

Abstract
Parametric surface meshing is of utmost importance in many numerical fields including the finite element method. Actually, surface meshing is a necessary step when one wants to construct the mesh of a solid domain in three dimensions.

A wide range of surfaces can be defined by means of composite parametric surfaces. Indeed, most of the surfaces are approximated by polynomial or rational parametric patches as it is in most of the CAD- CAM modelers.

In this paper, we would like to make some remarks about a method suitable to generate a constrained mesh of a parametric patch. The constraint consists of a metric map, which prescribes a size for every direction and also a shape quality about the mesh elements. The aim is then to construct a mesh that conforms to the specifications included in the metric map and such that its elements are as regular as possible.

A mesh of a parametric patch whose element vertices belong to the surface is suitable if the two following properties hold:

• all mesh elements are close to the surface,
• every mesh element is close to the tangent plane related to its vertices.

The first property allows us to bound the gap between the elements and the surface. This gap measures the greatest distance between an element (any point of the latter) and the surface.

The second property means that the surface is locally of order G1, in terms of continuity. To obtain this, the angular gap between the element and the tangent plane at its vertices must be bounded.

In this paper, we show that a mesh satisfying the two above properties conforms to a special metric map, a called geometric map. Then, we introduce a method for surface mesh generation based on a mesh construction in the parametric space in such a way as to follow the above geometric map. The flowchart of the method is as follows:

• A first mesh (in the parametric space) is constructed which is fine enough to capture the local curvature variation. This mesh is then mapped onto the surface.
• The specified metric is defined at all vertices of this surface mesh.
• After a metric transformation step, this metric map is induced in the parametric space. The parametric mesh along with this metric define a Riemaniann structure.
• The parametric space is then meshed with unit length element edges (where the unity is related to the Riemaniann structure). In this way, a so-called unit mesh is created,
• This unit mesh is mapped onto the surface.

Various application examples (including what follows) are provided to illustrate the capabil-ities of the above mentioned method.

(figures omitted)