Civil Engineering Reference
In-Depth Information
20
15
10
5
0
12345678910
Number of ellipses (× 5000)
Figure 4.52
CPU time vs number of ellipses.
CPU time and the number of elements generated can be observed, as shown in Figure 4.52.
The CPU time quoted is just for reference as these examples were done on a relatively slow
machine around 2001 with a speed of 150 MHz and 128-MB RAM.
4.3.3 Examples of surface meshing by ellipse packing
Five examples of anisotropic mesh generation of variable element size over a 2D unbounded
domain by ellipse packing are presented in this section. In two of the examples, the element
size is controlled by a distance function, and in the other three examples, the metric field
for element size control is derived from the surface curvatures, which are mapped to the 3D
space to produce the required surface meshes.
Table 4.2 shows the statistics of the example meshes. M and N are, respectively, the num-
ber of nodes and the number of elements in the mesh.
Max1
and
Avg1
are, respectively, the
maximum and the average of the ratio of the required sizes (hi,
i
, h
j
) at two points Ci
i
and C
j
to
the actual length (d
ij
) of the edge C
i
C
j
.
Max2
and
Avg2
are, respectively, the maximum and
the average of the ratio of the major principal stretch between neighbouring elements.
Max3
and
Avg3
are, respectively, the maximum and the average of the ratio of the minor principal
stretch between neighbouring elements.
Max4
and
Avg4
are, respectively, the maximum
and the average of the ratio of the major principal axis to the minor principal axis of the
packed ellipses. It is noted that the meshes presented in this section are raw meshes without
optimisation by node shifting and diagonal swap.
Example 1 shows an anisotropic mesh whose element size is controlled by the distance to
two crossing lines, and the principal directions of the ellipses are set along the direction of
these two lines. The maximum ratio of the major principal axis to the minor principal axis
is as large as 100 (
Max4
), and the average of this ratio is about 25 (
Avg4
). The difference
in the size between neighbouring elements can be as large as 6.37 (
Max3
), and the average
change between neighbouring elements is 1.61 (
Avg3
). Under the extreme situation, the
error in the required sizes between neighbouring points to the actual length is about 16%
(
Avg1
). The average number of iteration is 9. Figure 4.53a and b shows the ellipse packing
and the associated triangular mesh of the crossing lines. Figure 4.53c and d shows the mag-
nified views at the central part of Figure 4.53a and b.
Figure 4.54a and b shows the ellipse packing and the corresponding mesh of a curve in the
shape of Greek alphabet α, in which the ratio between the major principal axis to the minor
principal axis is about 50 (
Max4
). The average change in size between the neighbouring
ellipse is 1.49 (
Avg3
). The discrepancy in the required sizes between the neighbouring points
to the actual length is about 15% (
Avg1
). One remarkable fact worth mentioning is that by
this frontal mesh generation procedure, the origin or the centre of radiation is not obvious at
all, and indeed this point, in general, cannot be discerned in the mesh by visual inspection.
