Civil Engineering Reference
In-Depth Information
Augmented zone with
additional layers of cells
marked with red lines
Boundary tetrahedra
Circumspheres
Figure 7.20
Boundary tetrahedra and circumspheres.
Delaunay tetrahedra crossing the zonal boundary are missing, as shown in Figure 7.19.
The next step is to construct all the Delaunay tetrahedra at the boundary between zones.
To do so, boundary cells are added around the zone to all the boundary surfaces of the
zone, as shown in Figure 7.20. For easy visualisation, only those boundary tetrahedra with
the neighbouring zones and their associated circumspheres are shown, whereas tetrahedra
formed with the auxiliary corner points and their associated circumspheres are not shown.
Boundary tetrahedra are defined as those tetrahedra supported on vertex or vertices from
the current zone and vertex or vertices from the neighbouring zone(s), as shown in Figure
7.20. This simple definition is also applicable to Octree and kd-tree spatial partitions.
Layers of cells can be added to the boundary surfaces of the augmented zone until all the
circumspheres of the boundary tetrahedra are bounded to ensure that all boundary tetrahedra
are Delaunay, as shown in Figure 7.20. As each cell contains a roughly equal number of points,
the process converges fairly rapidly and evenly on all the zonal boundary faces in one or two
layers of cells. In this particular example, no additional layer of cells is needed, as all circum-
spheres of the boundary tetrahedra are already bounded by the augmented zone. Similar to
the 2D case, points considered before need not be reconsidered after further point insertions
as the union of circumspheres will always shrink with the introduction of a new point.
7.4.4 Elimination of redundant tetrahedra
Redundant tetrahedra on the zonal boundary can be eliminated concurrently by the generic
minimum vertex allocation scheme introduced in Section 7.3.5. As each node has been given
a zonal label in grouping cells into zones, a tetrahedron with zonal labels z
1
, z
2
, z
3
and z
4
on
its vertices will be assigned to zone z given by
z = min(z
1
, z
2
, z
3
, z
4
)
