Civil Engineering Reference
In-Depth Information
0.55
0.5
0.45
Quality Laplace
Local optimisation
GETMe
Combined scheme
0.4
0.35
0.3
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Iterations
Figure 6.34
Convergence characteristics.
Let
a
k
,
b
k
and
c
k
be the vectors emanating from vertex k to the neighbouring vertices
along the edges of the hexahedron, as shown in Figure 6.35. For instance, the transforma-
tion matrix
F
1
of tetrahedron if
1
at vertex
x
1
is given by
F
1
= [
a
1
b
1
c
1
] = [
x
2
−
x
1
x
4
−
x
1
x
5
−
x
1
]
The γ-quality of tetrahedron T
k
is defined by means of
F
k
(k = 1,8) as follows.
33det(F )
γ
k
=
k
⋅ =
Frobenius norm
3
F
k
γ
k
is equal to 1 when vectors
a
k
,
b
k
and
c
k
span the edges of a cube (orthogonal vectors of
equal length), and it will be negative if the volume of tetrahedron if
k
at vertex k, det(
F
k
),
is negative; hence, inverted hexahedral elements can also be detected by checking the
γ-qualities of their vertices. Accordingly, the γ-quality of a hexahedron H can be defined as
18
/
∏
18
γ
()
H
=
γ
and
γ
()
Hif
=
γ
γ
=
min
γ
< 0
k
min
in
k
k
k
=
,
8
7
5
4
c
1
b
1
6
3
1
a
1
2
Figure 6.35
Associated tetrahedron at each vertex of a tetrahedron.
