Keywords: surface meshing, adaptive meshing, adaptive refinement, geometrical characteristics, ridge line identification, triangular G1 patch, Bézier curve, Gregory patch, homothetic refinement.

In surface meshing, the accuracy of the geometric approximation with respect to a given tolerance is related to the control of the gap between the mesh and the surface. For each element, this gap value depends on the distance between the element and the surface as well as the deviation of the element from the tangent planes at each corresponding point on the surface. A geometric mesh is a mesh satisfying this geometric requirement. For composite surfaces based on splines or NURBS, Sheng and Hirsh [1], and Piegl and Richard [2], proposed a bound to control this gap during a mesh refinement stage. For other types of surfaces, for instance those involved in biomedical applications, this problem becomes more tedious because the gap value cannot usually be controlled during the mesh generation step.

From a given surface mesh, the geometry can be accurately reconstructed if certain geometrical characteristics are well defined. These characteristics are generally ridges, corners and normals at vertices. The extraction of the geometrical characteristics from a surface mesh is thus a crucial step. In this process, mesh edges and mesh vertices can be classified, depending on the number of smooth surface regions they belong to. If an edge belongs to only one triangle or shares at least three triangles, it is classified as a ridge representing a discontinuity of order 1. Otherwise, a treatment must be established to define smooth regions over mesh elements. A ridge vertex is a corner if it shares at least two different ridge lines. We suppose that all the ridge lines are closed or that their extremities are those of other ridge lines. In other words, all the ridge vertices share at least two ridge edges. This assumption implies that the surface is "piecewise" regular and does not contain any evanescent ridge line.

The identification consists of two steps: initialisation and expansion. It depends on two angular parameters (related to ridges) and (related to corners). The first step, initialisation, consists in identifying potential ridges using a simple sharpness criterion on the edges. Here, we use the second order differences which is the angle between the normals at the two triangles sharing the edge. If this angle is greater than , the corresponding edge is a potential ridge. In the second step, expansion, isolated potential ridges become again regular edges and the other potential ridges are extended, if necessary, to form ridge lines. Isolated potential ridges come from an inaccurate approximation of the surface by the reference mesh. An iterative procedure can be applied to extend an open line composed by potential ridges. At each iteration, a classified regular edge becomes a ridge and is added to one of the extremities of the open line. This procedure is repeated until all extremities of ridge lines share at least two ridge lines.

Then, a smooth geometry interpolating the reference mesh must be defined to "properly" put points on the surface. Several methods exist to define a smooth surface interpolating a triangular surface mesh. The method proposed by Walton and Meek [3] is conceptually simple and more appropriate for remeshing purpose. It consists in constructing at first a network of curve segments interpolating reference mesh vertices and then, another network of triangular patches resting on these segment curves. Originally, this method is suitable for surface mesh without geometrical discontinuities of order 1. We have generalised the method in order to take into account ridges and corners. Besides, a modification is provided which consists in locally unfolding the interpolating surface if it is folded around a mesh vertex.

The purpose of adaptive refinement is to improve the geometrical approximation due to the mesh, using the smooth geometry defined above. Here, we also want to preserve the shape quality of the reference mesh during the refinement. Our procedure consists in iteratively subdividing some mesh elements, precisely those belonging to curved areas. At each iteration of refinement, the "curved" triangles are identified using a geometrical criterion, which for a given triangle represents the maximum angular gap between the normal to the triangle and the normals at its vertices. A triangle is thus considered to be "curved" if the corresponding angular gap is greater than a given threshold. Concerning the refinement, there is only one element subdivision which preserves the element shape quality: the homothetic subdivision into four by adding three new vertices at the middle of the edges. This subdivision applied to a triangle influences the neighbouring triangles and can make the mesh non conforming. Indeed, the new added vertices lie also on the edge of neighbouring triangles. To restore the mesh conformity, the adjacent elements must also be refined.

Some application examples are given to show the efficiency of our approach. The method can also transform a mesh composed of linear elements into a mesh constituted by quadratic elements.

1

X. Sheng and B.E. Hirsch, "Triangulation of trimmed surfaces in parametric space", Computer Aided Design, vol. 24 (8), 437-444, 1992. doi:10.1016/0010-4485(92)90011-X

2

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

3

D.J. Walton and D. S. Meek, "A triangular patch from boundary curves", Computer Aided Design, vol. 28 (2), 113-123, 1996. doi:10.1016/0010-4485(95)00046-1

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