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To demonstrate the ability of the developed mesh generation technique
several example meshes are presented in this section. The first example,
the uniform mesh of a hemisphere, has been chosen to compare the overall
quality of the mesh for different decompositions of the hemisphere into
Bezier bicubic patches. Three cases have been investigated. In the first
one, the hemisphere is approximated with a single patch with two strong
singularities as it is displayed in Figure 8. The final mesh is presented
in Figure 9, while the mesh in the parametric space is in Figure 10,
demonstrating the power of modified advancing front technique for
generation of anisotropic meshes with extreme stretches. Shaded
triangles mark the subregions used to eliminate evaluation of infinite
stretches around singular points.
The parametric space with outlined principal directions and associated
stretches is presented in Figure 11. The triangulation
resulting from the approximation by a single patch without any purely
singular point (Fig. 12) is displayed in Figure 13. The associated mesh
in the parametric space is visualized in Figure 14, the principal
stretches in Figure 15. The third case deals with the
hemisphere decomposed into four identical patches cut off from the
hemisphere by four octants (Fig. 16). Each of the patches has a singular
point which corresponds to the top of the hemisphere. The final mesh of the
hemisphere and the intermediate mesh of one of the patches in the parametric
space are depicted in Figures 17 and 18, the latter one demonstrating again
the generation of enormously stretched triangles. The positioning of
mesh control points in the nodes of a quadtree in the parametric space
may me distinct from the figure of principal directions and stretches (Fig. 19).
There are no significant
differences in the mesh quality for the individual decompositions of the
hemisphere into patches. The second example describes a graded mesh on
a hyperbolic paraboloid approximated (only roughly) with a single patch
without any singularity (Fig. 20). Both meshes, the final mesh and the mesh
in the parametric space, are visualized in Figures 21 and 22. In the final
example, a surface close to a twisted rectangle with a hole in the middle
has been triangulated (Fig. 23). The surface has not been decomposed into
more patches, as it could be expected, only a single one has been used. The
opening has been simply specified in the parametric space that has been
subsequently meshed as a domain containing one void subregion, as it is
presented in Figure 25. The resulting mesh is depicted in Figure 24.

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*Daniel Rypl *

2005-12-03