Civil Engineering Reference
In-Depth Information
Object with
internal partitions
Magnified view of a boundary surface
75
11
2
71
40
1
26
50
67
35
42
22
90
8
5
18
31
46
7
3
87
15
12
77
4
17
51
Figure 8.1
Numbering boundary triangles and non-manifold object.
k
2
?
k
1
Figure 8.2
Check if k
2
is already connected to k
1
.
8.1.3.1 Search for all the edges on boundary surface B
The set of edges is the collection of all the distinct line segments on the boundary surface
B
.
Edges of the triangles are examined in turn. Let k
1
and k
2
be the node numbers of an edge,
with k
1
< k
2
. Before recording k
1
k
2
as a new edge, a check is made on the nodes already con-
nected to k
1
, as shown in Figure 8.2. Let C
k1
be the set of nodes connected to k
1
. Then if k
2
∈ C
k1
, edge k
1
k
2
is ignored; otherwise, edge k
1
k
2
is recorded and C
k1
is updated to C
k1
↦
C
k1
∪ {k
2
}. For a given triangular mesh, the average number of nodes connected to a particular
node is constant and independent of the total number of nodes in the system. Hence, the
efficiency of the scheme for the retrieval of edges is of order N
P
. With little additional care
during the edge-retrieval process, the three edges of each triangle can be identified. This
information is also stored and will be useful later in many topological operations on the ele-
ments in
B
. The algorithm presented in Section 2.5.4 for the retrieval of edges in triangular
meshes on planar domains and over curved surfaces is even simpler; however, the algorithms
just described can work with unoriented triangles, non-manifold objects and non-orientable
surfaces as well, which is an important attribute of the algorithm in the decomposition of
non-manifold objects into closed surfaces for MG.
8.1.3.2 Elements connected to each edge
For each edge L, L = 1,N
L
, where N
L
is the number of edges in
B
, find the list of elements
connected to it {E
1
, E
2
, …, E
n
}. Since the three edges of each triangle are known, the list of
elements connected to each edge can be readily determined by examining each triangular
element in turn. Let L
1
, L
2
and L
3
be the three edges of triangle Δ
if
; then on processing Δ
if
,
L
1
, L
2
and L
3
will include Δ
if
in the list of elements connected to them. By scanning through
all the triangular elements Δ
if
, i = 1,N
B
sequentially, the list of elements connected to an edge
can be established. In fact, the list of elements connected to an edge is the inverse relation-
ship of edges connected to an element, as shown in Figure 8.3.
