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To demonstrate the performance of the algorithm an academic example is
presented. A set of uniform meshes^{2} over a unit hemisphere
has been generated using 3 different uniform control
grids. The control grids 1, 2 and 3 contain 408, 1.680 and 6.744,
elements, respectively. For each control grid, six meshes have been
generated with target element size 0.224, 0.112, 0.056, 0.028, 0.014
and 0.007, respectively. Two sets of results have been produced - the
first one (Fig. 4) with ``exact'' projection (EP) technique to the limit surface and
the second (Fig. 5) with ``approximate'' projection (AP) to Bezier triangular patch
(with ``exact'' projection in last cycle of the mesh smoothing). The
convergence criterion has been set to 0.1 % of the local mesh
size.
Note that the total computational
time in Figs 4 and 5
has been split to a part corresponding strictly to the mesh generation
process and to a part related to 1 cycle of the smoothing phase.

It is quite clear from Fig. 4 that, when using only the
``exact'' projection, 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
the reduction of the level of subdivision required by the convergence
criterion by 1 whenever the control mesh size is halved. It is also
apparent that this results in computational complexity of
the overall algorithm. On the other hand, when using the ``approximate''
projection, the computational complexity of the mesh generation and
smoothing based on the ``approximate'' projection approaches (the
logarithmic contribution of the octree data structure is negligible).

The other example is more realistic. A concrete dam (Fig. 6) has been
discretized using several control grids. Only the enhanced version
with the ``approximate'' projection has been used in this case.
The resulting dependence of
the total computational time on the number of generated elements for
individual control meshes approaches computational complexity and
reveals only negligible sensitivity to the density of the control
grid (clearly caused by the last smoothing cycle with the ``exact'' projection).

#### Footnotes

- ... meshes
^{2}
- 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.

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*Daniel Rypl *

2005-12-03