Next: Mesh Optimization Up: Triangulation in the Parametric Space Previous: Advancing Front Technique

## Direct Generation of Anisotropic Meshes

The advancing front algorithm has been in the present work formulated in the stretched parametric space, i.e. the need to transform by the inverse transformation is avoided. This is clearly an advantage, as considerable cost is involved. On the other hand, the technique requires proper attention to be paid to the intricacies of stretched meshes. The crucial issues are:
1. Given an edge with vertices and , compute the location of the third vertex such that the triangle is of optimal'' shape with respect to the prescribed stretch and direction of stretch. This must be done (a) for arbitrarily oriented edge with respect to the prescribed direction of stretch, and (b) efficiently, i.e. with minimum number of computations.
2. Check vertices which are near'' the vertex and use any of them, if the vertex optimizes'' the triangle shape with respect to the shape for all triangles in the neighborhood. This is a very ambitious goal, which can be obviously reached only approximately. The meaning is that the triangle generated in the step should not only be of acceptable shape, but also it should not spoil the shape of the other triangles that will be generated in the following steps.
The first issue is addressed by the following recipe for the computation of the vertex . These parameters are given at the two vertices and : mesh size in the direction of the first principal stretch, the angle between the and axes, and the ratio (). The strategy is based on the simple idea of partitioning the , space into four sectors, . Then the vertex is generated on an ellipse with half-axes corresponding to the mesh sizes . The position on the ellipse is computed from heuristic rules, which were designed to force the triangles to follow the stretch trend of the mesh. The Figure 5 shows on the cases 1-4 the way in which the vertex moves along the ellipse for the edge pointing into the sectors and . To fix ideas consider the Figure 6 which shows an edge with vertices and . The center of the edge is denoted , the point is the vertex of the ellipse centered at , with half-axes being of length and respectively. The ideal location of the vertex should be at for , and close to for very stretched mesh. The heuristic function produces the desired effect (the interpolation is done along the ellipse).

The second issue (the check for extant vertices) uses similar device. The usual strategy for checking existing vertices for acceptability in the generation of the new triangle may pick a vertex, if it is located within a certain circle (compare with Peraire et al. [2]). This idea is modified to consider as acceptable only those vertices which are located inside an ellipse. The axes of this ellipse are aligned with the directions , and the length of the axes is adjusted from so that overly large stretches are avoided.

Next: Mesh Optimization Up: Triangulation in the Parametric Space Previous: Advancing Front Technique

Daniel Rypl
2005-12-03