Civil Engineering Reference
In-Depth Information
ρ
ρ
2
ρ
1
ρ
2
r
S
S
1
S
m
Figure 2.22
Residue distributed to the segments generated.
Let
m
be the number of line segments generated and
S
i
, i = 1,
m
, be the lengths of the line
segments. Then the residue r can be distributed to the line segments already generated by
m
r
S
∑
*
=
SS
1
+
where
S
=
S
i
i
j
j
=
1
The positions (co-ordinates) of the nodes generated have to be revised using the adjusted
node spacing
S
i
*
.
Remark:
As node spacing can be specified explicitly by means of an analytical function
or implicitly in terms of FE interpolation in adaptive meshing, node spacing is sampled,
for both cases in exactly the same manner, at the nodal points of the boundary for node
regeneration. For the first mesh, when there is no information about the node distribution,
boundary line segments can be divided solely based on the domain geometry by specifying
the appropriate element size at each corner node, and the boundary edges can be subdivided
following a geometric progression, as described in Section 2.4.9.1.
2.4.10
γ
Value of a tetrahedron cannot
exceed the
α
value of its face
In the generation of fully constrained tetrahedral meshes over domains bounded by a given
triangulated surface, one would like to ask, what is the expected quality of the tetrahedral
meshes generated? The following derivation shows that γ
min
is always smaller than or equal
to α
min
, where γ is the γ-quality of a tetrahedron, and α is the α-quality of a triangle.
As shown in Figure 2.23, γ value of tetrahedron ABCD is given by
volume
v
γ=
72 3
=
723
3
2
32
/
s
(
sumofedges squared
)
=
1
3
where v = volume of tetrahedron
ABCD
Ah
.
A = area of triangle ABC and
2
2
2
2
2
2
sAB CCADADBDC
AB
=
+
+
+
+
+
2
2
2
2
2
2
2
=
+
BC
+
CA
+
OA
+
OB
+
O
CCh
+
3
