Examples

The actual performance of the proposed algorithm for the generation of quad-dominant and all-quadrilateral meshes is presented on several examples. In the first one, the generation of an initial mixed mesh and the effect of the following mesh optimization and one-level refinement process is demonstrated on a planar domain with square and circular openings (Fig. 66). The top left figure depicts the domain of interest. In the next five figures (top middle, top right, and middle row), the propagation of the front is illustrated. The elements are drawn in grey (quadrilaterals in light grey, triangles in dark grey) to distinguish the front. The completed initial mixed mesh with 22 triangular elements is shown in the bottom left figure. The optimized mixed mesh with only 4 triangular elements is displayed in the bottom middle figure. The final refined all-quadrilateral mesh is shown in the bottom right figure.

In the next example, a locally graded mesh is generated over a complex
planar domain. Two meshes with the same final target element spacing are
presented. The first one, depicted in Figure 67 on the
left, is of a mixed nature and contains 2139 quadrilaterals and 46
triangles. The second one is an all-quadrilateral mesh with 2820
elements (Fig. 67 right). The larger density of the
all-quadrilateral mesh is due to the inability of the coarse mixed
mesh^{12} (later subjected to the
one-level refinement) to properly capture the
prescribed element size gradation, which results in the refinement of
larger areas of the domain and consequently also in the larger number
of elements. This is also apparent from the large number of triangular
elements comprised in the initial mixed mesh (see
Table 4).

The other example presents a uniform mesh of a torus. Again two meshes with the same final target element size have been produced, both displayed in Figure 68. The visual inspection of the meshes reveals no significant differences, which is also evident from almost the same number of generated elements (12 triangles and 1796 quadrilaterals in the mixed mesh and 1744 quadrilaterals in the all-quadrilateral mesh) and the achieved excellent quality (compare Tables 4 and 5). An interesting fact is that, in the case of the all-quadrilateral mesh, all triangular elements have been eliminated during the optimization process (see Table 5). This makes the one-level refinement not necessary if the doubled target element size would be acceptable. In this particular case, however, the torus is modeled by four surfaces which are discretized (including the refinement) independently. Thus the refinement is required to guarantee the conformity of the resulting mesh as a whole.

In the last example, a chair is subjected to the discretization with the combination of the curvature-based mesh density control and prescribed uniform element spacing. Therefore it was difficult to achieve the same final target element size. Nevertheless, similar meshes with almost the same number of elements have been produced. The mixed mesh with 42 triangular and 1669 quadrilateral elements is displayed in Figure 69 on the left. The all-quadrilateral mesh with 1646 quadrilaterals is depicted in the same figure on the right.

The obtained results in terms of the number of generated elements in the initial, optimized, and
all-quadrilateral^{13} mesh, together with the average and worst
element quality and the minimum and maximum dihedral angle in the
final mesh separately for triangular and quadrilateral elements are
summarized in Table 4 for the mixed
meshes and in Table 5 for the all-quadrilateral
meshes. The quality of the triangular elements is assessed
according to Eq. 87. The quality of a convex quadrilateral
element is evaluated as

where , , , and are the qualities of the four different triangles obtained by the two possible diagonals of the quadrilateral. Again, the formula is calibrated to range from 0 to 1 (1 being for the square). Both tables demonstrate a very high quality of the mixed as well as all-quadrilateral meshes.

- ...
mesh
^{12} - Remind that the initial mixed mesh has to be generated with the doubled target element size.
- ...
all-quadrilateral
^{13} - Relevant only for the statistics of the all-quadrilateral meshes.

*Daniel Rypl
2005-12-07*