Civil Engineering Reference
In-Depth Information
intersected by other edges, even though its three boundary edges have already been restored,
as shown in Figure 5.19. Edges cutting through face ABC can be detected by considering the
three
shells
associated with the edges of the triangle, namely,
shells
generated by rotating
about edges AB, BC and CA, respectively. Penetrating edges can be identified if any edge of
these
shells
cut across the face. These intersecting edges, to a certain extent, can be removed
by proper element swaps. In case there is only one penetrating edge, the
shell
associated with
this edge consists of exactly three elements with a proper connection to the three vertices of
the face, as shown in Figure 5.20. The penetrating edge can be easily removed by a 3-2 ele-
ment swap, which is unconditional and applicable to any
shell
with exactly three tetrahedra.
Suppose that there are more than one penetrating edges, and they have to be considered one
by one. Let PQ be one of the penetrating edges and {R
1
, R
2
,…, R
n
} be the nodes of the asso-
ciated polyhedron opposite to PQ, as shown in Figure 5.21. The ring of n nodes {R
1
, R
2
,…,
R
n
} can be triangulated into n − 2 triangles, and tetrahedra can be formed by joining points
P and Q to the triangles of the triangulated ring. If in any one of the valid triangulations,
there is no internal edge passing through the missing face ABC and each triangle of the ring
is visible to both P and Q, then a swap of a
shell
of n tetrahedra into 2(n − 2) tetrahedra will
remove the penetrating edge PQ. Each penetrating edge and its associated
shell
can be con-
sidered, in turn, for its possible removal by the same procedure. It is noted that George and
C
A
B
Figure 5.19
Face ABC intersected by edges.
P
C
A
B
Q
Figure 5.20
Face ABC intersected by one edge.
P
R
2
C
A
R
1
B
R
n
Q
Figure 5.21
Penetrating edge PQ and its associated shell.
