Civil Engineering Reference
In-Depth Information
Figure 6.5
Dihedral angles of a needle-shaped tetrahedron.
C
h
min
D
A
h
max
B
Figure 6.6
Volume of a tetrahedron ABCD approaches zero when D comes close to triangular face ABC.
6.2.1.5 Minimum dihedral angle is not a valid shape measure
The dihedral angle ϕ at an edge of a tetrahedron between two triangular faces is given by
ϕ = π - acos(
n
1
·
n
2
)
where
n
1
and
n
2
are the unit normals to the faces sharing the same edge under consideration.
The minimum dihedral angle is not a valid shape measure as it cannot detect the degener-
ate case of a needle-shaped tetrahedron, as shown in Figure 6.5. The dihedral angles of the
tetrahedron remain more or less the same as the triangular face opposite to the pointed node
is getting smaller and smaller. Refer to Dompierre et al. (2005) for more degenerate cases,
which cannot be detected by measuring the dihedral angles of a tetrahedron.
6.2.1.6 Edge ratio is not a valid shape measure
The edge ratio of a tetrahedron is defined as the ratio between the shortest edge and the
longest edge of the tetrahedron, i.e.
h
h
max
=
min
edge ratio
The edge ratio is not a valid shape measure as it does not vanish for all the degenerate cases,
as shown in Figure 6.6, for instance, the flattening of a cap element. The edge ratio does not
change much for the flattened tetrahedron ABCD when D reaches face ABC.
Sliver
tetrahedra in
Delaunay triangulation are another example, which cannot be detected by the edge ratio.
6.2.2 Relationship between shape measures
Liu and Joe (1994a) systematically compared shape measures σ, ρ and η and established the
relationship among these shape measures as follows:
η
3
≤ ρ ≤ η
3/4
,
ρ
4/3
≤ η ≤ ρ
1/3
0.2296η
3/2
≤ σ ≤ 1.1398η
3/4
,
0.8399σ
4/3
≤ η ≤ 2.6667σ
2/3
0.2651ρ
2
≤ σ ≤ ρ
1/2
,
σ
2
≤ ρ ≤ 1.942σ
1/2
