Civil Engineering Reference
In-Depth Information
(a)
(b)
Figure 5.92
(a) Packing of spheres on intersecting planes and (b) section of tetrahedral mesh.
(a)
(b)
Figure 5.93
(a) Packing of spheres on intersecting surfaces and (b) section of tetrahedral mesh.
Example 3 is to pack spheres densely over three intersecting curved surfaces in space. This
example is similar to the last example except that the geometry required for dense packing is
different. As shown in Figure 5.93a, characteristic lines of small spheres can be identified on
the surface of the cluster. A transverse cross section of the corresponding tetrahedral mesh
reveals that small elements are generated close to the curved surface, as shown in Figure
5.93b. The average γ-quality of this example is also γ = 0.79.
The last example is to pack spheres along an arbitrary curve in space. As shown in
Figure 5.94a, two lines of spheres of very small size can be distinguished on the surface
of the cluster. The associated tetrahedral mesh is shown in Figure 5.94b. In spite of a large
change in element size moving away from the space curve, a fairly high mean γ-quality of
0.77 can be achieved. A cut section by removing one quarter of the elements to reveal the
internal structure is depicted in Figure 5.94c. It can be seen that there are regions of element
concentration along the path of the spatial curve. If the elements within a short distance
from the curve are retained, elements of consistent small size are generated all along the
curve as shown in Figure 5.94e and f. Similar to the 2D case, it appears that the origin for
sphere packing can be set at any convenient point in space without much impact to the final
results. Figure 5.94c shows the mesh of packing spheres densely over the same curve in space
from an origin placed at the centre of the mesh (curve 1), whereas the mesh shown in Figure
5.94d is generated from an origin outside the mesh (curve 2).
Remarks:
The number of tetrahedral elements or the size of the cluster that could be gen-
erated depends solely on the capacity of the computer and the time allowed for MG. The
