Civil Engineering Reference
In-Depth Information
Example 1 is about a rather difficult case of generating a finite element mesh of random ele-
ment size with a range of 1 to 11, that is, the difference in size between neighbouring spheres
can be as large as 11. A size ratio larger than 11 could have been tested as well. However,
in three dimensions, generation of spheres of random sizes is not a well-defined problem,
because small spheres can penetrate through the cluster of existing spheres between gaps to
any possible locations. For the packing of circles over a 2D unbounded domain, circles of
random size in a range of 1 to 10,000 have been packed (Lo and Wang 2003). The packing
of the first 30,000 spheres of random size is shown in Figure 5.91a, and the corresponding
mesh of 178,451 tetrahedral elements is shown in Figure 5.91b. In this example of packing
spheres of random size, the average size ratio between adjacent spheres is 2.7, and the gap
spacing is approximately 32%. Nevertheless, the average element γ-quality is quite accept-
able at γ = 0.64. Figure 5.91c shows a cross section of the spheres packed, and a layered
structure is revealed when spheres at different distance from the origin are displayed with
different colours. A cut open section of the tetrahedral mesh is shown in Figure 5.91d.
Spheres over three intersecting planes are packed densely in Example 2. As shown in
Figure 5.92a, characteristic lines of small sphere concentration are well recognised on the
surface of the cluster, and the spheres grow into larger size when they are away from the
intersecting planes. A transverse cross section of the corresponding tetrahedral mesh is
depicted in Figure 5.92b. It can be seen that elements of smaller size are generated close to
the intersecting planes. In spite of a strong variation in element size, a high average shape
quality of the mesh with γ = 0.79 was attained without mesh optimisation.
(a)
(b)
(c)
(d)
Figure 5.91
(a) Packing of spheres of random size, (b) tetrahedral elements of random size, (c) a cut section
of the packed spheres and (d) open cut of the tetrahedral mesh.
