Civil Engineering Reference
In-Depth Information
In the sequel, semi-Delaunay triangles will be considered as a special case of non-Delaunay
triangles.
3.7.2.3 Delaunay and non-Delaunay segments
Line segments on the generation front are classified into Delaunay and non-Delaunay segments
according to the Delaunay property of the triangles associated with them. At the beginning of
triangulation, all the line segments on the domain boundary are assumed to be non-Delaunay
since they are not connected to any triangle. Whenever a triangular element is formed, the
Delaunay property of the triangle is examined to see if the circumcircle of this triangle con-
tains any node(s) in its interior or on its circumference. The edges of a Delaunay triangle are
Delaunay segments, and the edges of a non-Delaunay triangle are non-Delaunay segments.
The algorithm presented in the following can generate triangular meshes over general
planar domains that are simply connected or multi-connected. The boundary of the domain
is represented by a disjoint union of simple closed loops of straight line segments. For sim-
ply connected regions, there is only one closed loop, whereas for multi-connected regions,
there can be as many interior loops as the number of openings inside the domain. The nodes
on the exterior boundary are entered in a counter-clockwise order, while the nodes on the
interior boundaries are entered in a clockwise order. The construction and the verification
of the domain boundary are usually done more or less automatically by means of a geomet-
ric modelling interactive module. Nodes on the boundary need not be made following any
particular numerical order; this flexibility allows us later to generate triangular meshes from
one sub-region to another without bothering to identify the common boundaries between
sub-regions. The algorithm first generates additional interior points according to the speci-
fied node spacing requirement. The algorithm then connects the boundary nodes and the
interior nodes in such a way that no elements overlap, and the entire region is covered. The
resulting mesh is a constrained DT, as defined in Section 3.7.2.1.
3.7.2.4 Triangulation process
Suppose that a complete nodal system has been generated according to the specified node
spacing function, the next step is to connect these interior nodes with the boundary node
to form a valid FE mesh. Similar to the classical AFT, at the beginning of triangulation, the
generation front is exactly equal to the domain boundary. While the given domain bound-
ary remains always the same, the generation front evolves continuously throughout the tri-
angulation process and has to be updated whenever a new element is formed.
Let Γ
1
be the set of non-Delaunay line segments and Γ
2
be the set of Delaunay line seg-
ments on the generation front Γ. Since Γ
1
and Γ
2
is a partition of Γ, we always have Γ
1
∪ Γ
2
=
Γ and Γ
1
∩ Γ
2
= Ø. At the beginning of meshing, Γ
1
= Γ = ∂Ω and Γ
2
= Ø. Let Σ be the set
of nodes on the generation front Γ and Λ be the set of interior nodes within the unmeshed
region bounded by Γ. The triangulation is initiated by taking the last segment AB ∈ Γ
1
. A
node C ∈ Σ ∪ Λ has to be found such that triangle ABC lies within the domain Ω and its
associated circumcircle is the smallest, as shown in Figure 3.58.
A node C
ii∈
∈ ∑ ∪ Λ is said to be a candidate node if it satisfies
ii. {Ci,
ii∈
A × C
ii∈
B > 0
ii. {Ci,
ii∈
A ∩ Γ
1
= {Ø, A, {C
ii∈
, A}, C
ii∈
A} and C
ii∈
B ∩ Γ
1
= {Ø, B, {C
ii∈
, B}, C
ii∈
B}
C
ii∈
Σ∪
be the set of candidate nodes. The node to be selected is a node
C
m
∈
such that
CC
Let
= {}
, where
C
ABC
ii∈
is the circumcircle of triangle ABC
ii∈
, which
∈∀∈
,
C
m
BC
ii∈
ii∈
