Civil Engineering Reference
In-Depth Information
local triangulation of the CORE together with the existing tetrahedra outside the CORE
forms a new triangulation of all the inserted points including the (k + 1)
th
point P, i.e.
TT B
k
+
=−+
1
k
p
p
where
p
= CORE = cavity of non-Delaunay tetrahedra with respect to P
k
2
p
andcircumradiusofT}
p
= ball of tetrahedra = patch of tetrahedra connected to P
p
= set of tetrahedra formed by joining P to each triangle on the CORE boundary
k
=∈ ≤
{
T: OP
r,Oand rare thecircumcentre
2
and
+1
= the DTs of the first k and k + 1 points inserted
k
The enclosing cube of five or six tetrahedra serves as the initial triangulation
0
, and the
DT
n
will be constructed when all the n points are inserted sequentially one after the other
by the point insertion kernel.
5.2.2.5 Adjacency relationship
As the element adjacency is frequently referred to in various steps of the DT, a more delicate
issue is to determine the element adjacency relationship for the new tetrahedra in the CORE
and update those that are attached to the CORE. There is not much a problem in establish-
ing the adjacency relationship between the existing tetrahedra attached to the boundary
facets of the CORE and the new tetrahedra that fill up the interior of the CORE. However,
to determine the adjacency relationship between the new tetrahedra inside the CORE is less
straightforward.
In the triangulation over a 2D domain, the insertion CORE is a polygon, and there is no
difficulty in establishing the adjacency relationship between triangles by creating elements
following the boundary contour of the polygon. In three or higher dimensions, the situation
is quite different. The boundary of the CORE is a surface, and there is no obvious order for
the boundary faces following which the adjacency relationship of the tetrahedra so gener-
ated could be established in a more or less natural manner without much calculation.
From the star-shaped connection of point P with boundary faces of the CORE, each
common face between two tetrahedra within the CORE can be associated with an edge on
the boundary, as shown in Figure 5.4. Tetrahedra ABCP and BADP are neighbours if they
share common edge AB on the boundary of the CORE. Hence, the adjacency relationship
for each tetrahedron can be established by identifying the three edges on the boundary of
the CORE and determining the three neighbours through a matching process of common
Boundary faces
CORE
B
C
D
A
Common
edge
Neighbouring
tetrahedra
P
Figure 5.4
Tetrahedra ABCP and BADP are neighbours.
