Civil Engineering Reference
In-Depth Information
and conformity coefficients are given by
PA
ρ
δ
=
min
,
AP
ρ
=
desirablelengthb
etween Aand P
a
AP
ρ
PA
AP
PB
ρ
BP
δ
=
min
,
ρ
=
des
irable length between Band P
b
BP
ρ
PB
BP
PC
ρ
CP
δ
= min
,
ρ
CP
=
desirablelengthbetween Cand P
c
ρ
P
C
CP
and the desirable length between two points P and Q can be estimated simply by
ρ
=
ρ
()()
PQ
ρ
PQ
The conformity of other edges AB, BC and CA is not included as those are existing edges
on the generation front. As all the parameters are between 0 and 1, the overall quality mea-
sure λ will also be within 0 and 1.
5.6.1.7.1 Evaluation of an insertion point
The quality of an insertion point P can be evaluated by considering the tetrahedra deleted
by the point against those created by the point in the Delaunay insertion process. Let {Ti,
i
,
i = 1,NT}
T
} be the non-Delaunay tetrahedra in the CORE and {Ei,
i
, i = 1,NT}
E
} be the new ele-
ments formed with the boundary triangles of the CORE. Then the quality of point P, μ
P
, is
given by
min{ ( }
λ
E
i
i
=
1
,
NE
=
P
min{ ( }
λ
T
i
i
=
1
,
NT
Other convenient norms such as the geometrical mean value can also be used. μ
P
will be greater
than 1 if the minimum quality of the newly created tetrahedra is larger than that of the old tetra-
hedra in the CORE. Greater μ
P
represents better improvement by the introduction of point P, and
hence, we can select an optimal insertion point by comparing the μ-values of the points inserted.
This is a very rigorous scheme for the identification of a strategic insertion point; yet, it is rather
expensive if many points have to be evaluated to obtain high-quality tetrahedral elements.
5.6.1.7.2 Some suggested locations to insert point P
As far as the element shape quality is concerned, the best tetrahedral element that could be
formed with a triangle ABC is to pick a point along the normal passing through the centroid
of the triangle, O, as shown in Figure 5.77. From Section 2.4.10, the optimal point is found
at a height, h, given by
2
9
2
2
2
2
h
=
(
B CCA
+
+
)
Given a node spacing requirement ρ, the overall quality of a tetrahedron depends also
on the conformity coefficients; thus, two more locations at ±20%h relative to the ideal
