Examples

A set of examples is provided to demonstrate the capability of the presented algorithm for the direct discretization of 3D surfaces and to illustrate its performance from both quantitative and qualitative points of view.

Again, the uniform mesh of a hemisphere is presented as the first
example to compare the quality of the mesh for the three different
decompositions of the hemisphere into rational Bezier
patches. The resulting meshes with the same target element size as
prescribed in Subsection Examples of
Section ** Indirect Triangulation of 3D Surfaces** are depicted in
Figures 34, 35, and 36. The visual inspection
of the results clearly concludes that the meshes are of excellent
quality that is superior to the quality of meshes produced by the
indirect method. This can
be explained by the adopted high quality runtime point placement strategy and
better control over the shape of the isotropic elements in the real
space. Also the employed topological transformation (edge swapping)
influences positively the connectivity and consequently also the
quality of the mesh.

The other example presents a graded mesh of a rotational hyperboloid with prescribed local refinement. Both, the wireframe model and the final mesh, are displayed in Figure 37.

In the next example, a chair (Fig. 38) is subjected to the meshing with
the curvature-based control of the element spacing. A sequence of meshes
with the number of elements varying from 2538 to 373136 has been produced
using the presented algorithm. The dependence of the elapsed mesh
generation time^{4} on the number of generated elements is demonstrated in
Figure 39 and reveals the almost linear computational
complexity of the algorithm. The quality of three selected meshes (the
coarsest with 2538 elements, an intermediate with 120780 elements, and
the finest with 373136 elements) is presented in
Figures 40 and 41 in terms of the
distribution of the element quality and
dihedral angle, respectively. The quality of a triangular element is
assessed as

where is the element area and , , and are the lengths of element sides. The coefficient is used to calibrate the quality to range from 0 to 1 (1 being for the equilateral triangle). Both figures demonstrate the excellent quality of the meshes. In the case of the finest mesh, 90 % of elements have quality rate exceeding 0.95, and 99 % of elements have quality over 0.90. Also 60 % of element dihedral angles do not differ from the optimal angle by more than . These quantities slightly deteriorates with the decreasing number of elements. The coarsest mesh exhibits ``only'' 75 % of elements with quality rate over 0.95 and 99 % of elements are of quality larger than 0.85. The relative number of element dihedral angles ranging from to falls to 46 %. Note however that the dihedral angle distribution reflects the graded nature of the meshes due to the curvature-based mesh density control.

As the last example, the discretization of the Utah teapot is presented. The solid model and the mesh of a quarter of the teapot are depicted together in Figure 42 which demonstrates the capability of the algorithm to handle realistic geometries with the rapidly varying curvature (see the spout of the teapot, for example).

- ... time
^{4} - The timing has been obtained on a Dell notebook with Intel Pentium II 300 MHz processor and 128 MBytes RAM running under Linux Red Hat 5.2.

*Daniel Rypl
2005-12-07*