Keywords: CAD surface, parametric surface meshing, curve discretization, anisotropic meshing, mesh gradation, geometric mesh, conforming mesh, discontinuous metrics.

Scientific computing requires the automatic generation of high quality meshes, in particular isotropic or anisotropic meshes of surfaces defined by a CAD modeller. For this purpose, the two major approaches are called direct and indirect. Direct methods (octree, advancing-front or paving) work directly in the tridimensional space, while indirect methods consist in meshing each parametric domain and mapping the resulting mesh onto the composite surface. Historically, this indirect approach was first used for surface visualization [1] and then for finite element computation [2,3].

In this paper, a general scheme of an indirect approach for generating "geometric" (or geometry-preserving) meshes of a surface constituted by a conformal assembly of parametric patches is proposed. The different steps of the scheme are detailed and, in particular, the definition of the geometric metric at each point of the surface (internal to a patch, belonging to an interface or boundary curve, or extremity of such a curve) as well as its corresponding induced metric in the parametric domains. Isotropic or anisotropic geometric metrics can locally produce significant size variations (internal to a patch or across interface curves) and can even be discontinuous across the interface curves. To control this size variation, various methodologies based on metric reduction have been proposed [4] in the case of a continuous isotropic metric. A novel iterative mesh gradation approach is introduced for discontinuous metrics. The approach uses a particular metric reduction procedure in order to ensure the convergence of the gradation process. In particular, it is shown that in the worst case the anisotropic discontinuous geometric metric map is reduced to an isotropic continuous geometric metric map for which the gradation is controlled. Several application examples are provided to illustrate the capabilities of the proposed method.

1

L.A. Piegl, A.M. Richard, "Tessellating trimmed NURBS surfaces", Computer Aided Design, 27(1), 16-26, 1995. doi:10.1016/0010-4485(95)90749-6

2

J.R. Tristano, S.J. Owen, S.A. Canann, "Advancing front surface mesh generation in parametric space using a Riemannian surface definition", 7th International Meshing Roundtable '98 Proceedings, 429-445, 1998.

3

H. Borouchaki, P. Laug, P.L. George, "Parametric surface meshing using a combined advancing-front - generalized-Delaunay approach", Int J Numer Meth Engng, 49(1-2), 233-259, 2000. doi:10.1002/1097-0207(20000910/20)49:1/2<233::AID-NME931>3.0.CO;2-G

4

H. Borouchaki, F. Hecht, P. Frey, "Mesh gradation control", Int J Numer Meth Engng, 43, 1143-1165, 1998. doi:10.1002/(SICI)1097-0207(19981130)43:6<1143::AID-NME470>3.3.CO;2-9

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