Next: Triangulation of Discrete 3D Surfaces Up: Direct Triangulation of 3D Surfaces Previous: Computational Complexity


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 3435, 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 time4 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

(87)


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).



Footnotes

... time4
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.


Next: Triangulation of Discrete 3D Surfaces Up: Direct Triangulation of 3D Surfaces Previous: Computational Complexity

Daniel Rypl
2005-12-07