Civil Engineering Reference
In-Depth Information
ADF method operates in order to pack spheres. In fact, what is needed is a point among all
the tetrahedral elements on the surface that is relatively close to the origin. Tight packing of
spheres is achieved by a descending mechanism following the path between existing spheres
by rotation until the lowest point is reached. By its very design, the deepest descending
mechanism by rotation about axes between spheres naturally fulfils the criteria of being the
nearest to the origin, the densest and tangent without overlapping in sphere packing.
5.7.2 Sphere packing and MG algorithm
The idea of MG is to connect centres of tightly packed spheres of variable size by Delaunay
point insertion scheme. Unlike the conventional ADF approach, the procedure does not
start from the object boundary but at a convenient point within the 3D open space. The
initial pack consists of four spheres tangent to each other as shown in Figure 5.83, which
will expand towards the exterior and has to be updated whenever a new sphere is added.
Tetrahedral elements are subsequently generated when the centres of the spheres are con-
nected to form tetrahedra. The data structure and the rules of sphere packing are to be
described in Sections 5.7.2.1 and 5.7.2.2.
5.7.2.1 Data structure
The data structure requirements for MG are very similar to those required by the DT. For a
typical Delaunay point insertion scheme, the four vertices and the four neighbours of each
tetrahedral element have to be stored. In addition, for packing of spheres, the centre, the
radius and the distance from the origin of each inserted sphere have to be stored as well.
As the nodal points of the tetrahedral mesh are the centres of the packed spheres, a neigh-
bouring relationship of the spheres can be defined based on the adjacency of the tetrahedral
elements.
5.7.2.2 Criteria for sphere packing
1.
Nearest.
New spheres have to be generated at locations as close to the origin as possi-
ble. Packing spheres as near to the origin as possible ensures compactness and reduces
the chance of forming holes and voids. By packing spheres close to the origin, the
shape of the cluster of spheres is basically convex and spherical with minor concave
parts.
2.
Densest.
Spheres are to be packed as close to one another as possible. It is best that
spheres are packed tightly together so that the gaps between them are minimised, and
preferably be surrounded by four or five tangential spheres. However, it is sometimes
difficult to pack them densely if we have to fit a sphere of specified size into a given
Origin
S
4
S
3
S
2
S
1
Figure 5.83
Initial pack of four spheres.
