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To demonstrate the performance of the algorithm an academic example is presented. A set of uniform meshes2 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).


... meshes2
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