Civil Engineering Reference
In-Depth Information
Similarly, the intersections between the plane and the other two edges J
2
J
3
and J
3
J
1
are deter-
mined. In case of intersection, the horizontal plane should intersect with exactly two of the three
edges of triangle Δ
i
. Let P and Q be the two points of intersection; then line segment PQ will
be the intersection of the horizontal plane and triangle Δ
i
. Having considered all the triangular
elements in B in turn for intersection with the horizontal plane z = h, the contour line at level h
is given by the set of all these individual line segments, S = {P
j
Q
j
, j = 1, N
S
}.
The order of these line segments of intersection, which appears to be random, in fact related
directly to the numbering of triangular elements Δ
i
in B. However, as the order of the line seg-
ments composing the boundary of the cut section is not important in the generation of interior
nodes, retrieval of structural forms in terms of closed loops for the segments in S by proper
renumbering is not necessary. Points generated on a cut section using the 2D node generation
scheme represent only potential positions at which nodes can be generated. However, whether a
node is actually generated depends also on how close it is to the domain boundary B.
5.5.2.3 Construction of tetrahedral elements
The 3D meshing process is initiated by selecting a triangular facet on the generation front.
The choice is rather arbitrary; however, there are reports suggesting taking the smallest facet
on the front for additional stability (Peraire et al. 1988). Let Σ be the nodal points on the
generation front Γ, Λ be the set of interior nodes remaining inside the generation front and
J
1
J
2
J
3
∈ Γ be the selected base triangular facet on the generation front for the construction
of a new tetrahedral element. To create a tetrahedron at the base triangle, we have to find
a point C from Σ and Λ such that tetrahedron J
1
J
2
J
3
C lies completely within the generation
front, and its γ value is optimised.
1.
Selection of a node on the generation front.
A node C ∈ Σ ∪ Λ is said to be a candidate
node if it satisfies
i. ∈{{Ji,
1
J
2
× J
1
J
2
· J
1
C > 0
and
ii. ∈{{Ji,
i
C ∩ Γ ∈{{J
i
, C}, J
i
C}, i = 1,2,3
Condition (i) ensures that the proposed tetrahedron possesses a positive volume, and
condition (ii) makes sure that the proposed tetrahedron does not cut across the genera-
tion front. Let
=
C
i
Λ
be the set of candidate nodes. The node to be selected
is therefore a node
C
m
∈
such that the γ value of tetrahedron J
1
J
2
J
3
C
m
is maximised,
i.e.
{}
∑∪
γ
(
JJJC
)
≥
γ
(
JJJC
)
∀ ∈
C
123
m
1
23
i
i
2.
Shape optimisation of tetrahedral elements
. The choice of node
C
m
∈
by maximis-
ing the γ value of tetrahedron J
1
J
2
J
3
C
m
is simple enough, but it may not be sufficient
to guarantee the
best
triangulation for domains of very irregular boundaries. Under
such circumstances, a more sophisticated procedure has to be adopted. The idea is that
in the selection of node Ci,
i
, instead of considering merely the γ quality of tetrahedron
J
1
J
2
J
3
C
i
, we ought to consider also the quality of the future tetrahedral elements gener-
ated on the triangular facets J
1
J
2
C
i
, J
2
J
3
C
i
and J
3
J
1
C
i
, as shown in Figure 5.66.
