Civil Engineering Reference
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average of the ratio of radii between adjacent spheres. Max3 and Avg3 are, respectively, the
maximum and the average gap space between adjacent spheres. With an average gap space of
12%, it can be seen that the spheres are tightly packed together. Max4 and Avg4 are, respec-
tively, the maximum and the average ratio of the actual size of the spheres to the required size.
For the three examples of specified node spacing, the mean deviation is approximately 5%. A
tighter tolerance could have been applied in the sphere packing process to further reduce this
deviation. Max5 and Avg5 are, respectively, the maximum and the average ratio of the edge
length of the tetrahedral elements to the specified value. For the three examples where a node
spacing function is specified, the average deviation is approximately 13%.
Finally, Min6 and Avg6 are, respectively, the minimum and the average γ-quality of the
tetrahedral elements, as defined in Section 5.5.1. The Min6 values are quite low for all the
examples as they are calculated from the raw mesh directly from sphere packing without
mesh quality optimisation to remove sliver elements. In the DT process, the threshold for
accepting a face on the CORE boundary to construct a tetrahedral element was set at γ =
0.001. This could probably be increased slightly to 0.01 to remove potential slivers. The
mesh quality can be further improved by means of any existing optimisation techniques as
described in Chapter 6. Nevertheless, the average γ-quality of the tetrahedral mesh is already
quite high at about 0.78 even without post-generation enhancement. A more detailed break-
down of element γ-quality of the examples is shown in Table 5.7 and Figure 5.90. It can be
seen that most tetrahedral elements are having a γ-value greater than 0.7.
Table 5.7
γ
-quality of tetrahedral meshes of the examples
γ
Random
Planes
Surfaces
Curve 1
Curve 2
0.1
1181
479
348
371
378
0.2
3923
1360
1154
1043
1143
0.3
7774
2467
2019
2004
2100
0.4
12,099
3927
3123
3080
3423
0.5
17,208
6146
5078
5121
5474
0.6
23,717
10,090
8143
8850
9635
0.7
31,538
19,306
15,614
17,158
18,047
0.8
36,610
42,635
35,382
33,741
33,596
0.9
31,849
72,418
61,844
47,256
46,433
1.0
12,552
49,247
42,986
31,422
30,648
(Tetrahedra)
80,000
70,000
60,000
Random
Planes
Surfaces
Curve1
Curve2
50,000
40,000
30,000
20,000
10,000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0 (γ)
Figure 5.90
Histogram of
γ
-quality of the tetrahedral mesh.
