To demonstrate the performance of the algorithm an academic example is
presented. A set of uniform meshes^{1} over a unit hemisphere
has been generated using 4 different uniform control
grids. The control grids 1, 2, 3, and 4 contain 130, 504, 2094,
and 8222 elements, respectively and have been created for target element size 0.35,
0.175, 0.0875, and 0.04375, respectively. For each control grid, three meshes have been
generated with target element size 0.035, 0.05, and 0.07 using
different selection algorithm (for point to surface projection). The
difference between the ``simple'' and ``enhanced'' algorithm consists
in the fact that ``simple'' algorithm is not using mid-nodes for
sub-triangle selection. The convergence criterion for the projection has
been set to 0.1 % of the target element size.
The achieved results are summarized in
Table 1. For each mesh, the number of generated
elements (Elems) and the total computing time (T-tot) in seconds have
been recorded. Since the algorithm seems to be quite slow (the
same surface mesh generator working with tensor product
polynomial surfaces is about 10 times faster) the total computing time
has been split to a part corresponding strictly to the mesh generation
process (T-mesh) and to a part related to the smoothing phase (T-opt)
(only 2 cycles were considered). It is evident that the computing time is
steadily decreasing with the
decreasing size of the elements in the control grid. Each time the size of
control triangles is halved the total computing time is reduced by a
constant value (specific for a particular mesh size). This can be explained by
reduction of the level of subdivision required by the convergence
criterion by 1 whenever the control mesh size is halved. It can be
also observed that the ``enhanced'' selection algorithm outperforms
(although only slightly) the ``simple''
one on coarser control grids, where a higher probability of
incorrect selection occurs. Note that this performance difference is increasing with the
mesh density.

Since most of the smoothing time (over 85 %) is consumed by the projection algorithm it is be possible to estimate the overhead in the mesh generation phase caused by the projection. The key to an improved performance is to replace the ``accurate'' projection by an ``approximate'' one whenever possible and to reduce the number of ``accurate'' projections to the minimum, ideally to the number of nodes (or surface nodes). The ``approximate'' projection, however, must be accurate enough to guarantee correct behaviour of the mesh generation algorithm. A possible way may be to introduce a (rational) Bezier triangle over each triangle of the original control grid and to adjust its control polygon (and weights) to ``minimize'' its deviation from the limit surface. The parametric space of the (rational) Bezier triangle can be then efficiently used for the implementation of the ``approximate'' projection technique.

- ... meshes
^{1} - All meshes have been generated on a Dell notebook with Intel Pentium II 300 MHz processor and 128 MBytes RAM running under Linux Red Hat 5.2.

*Daniel Rypl
2005-12-03*