Civil Engineering Reference
In-Depth Information
More relationships on shape measures based on the condition number were given by
(Dompierre et al. 2005):
12
/
12
/
13
/
32
/
12
/
−
1
κτ κκηκ ητ ητ
≤≤
3
,
≤≤
3
,
23
/
≤≤
3
,
=
3
/
FF
Liu and Joe (1994a) further proposed an equivalent relationship for shape measures. Let
λ and μ be two different shape measures scaled in the interval [0,1]; λ and μ are equivalent
if there exist positive constants a, b, p and q such that
aμ
p
≤ λ ≤ bμ
q
(6.1)
λ is related to μ(λ ~ μ) if Equation 6.1 holds for λ and μ. This is an equivalence relation since
it is reflexive, symmetric and transitive, i.e. (i) λ ~ λ, (ii) λ ~ μ ⇒ μ ~ λ and (iii) λ ~ μ and μ ~
ν ⇒ λ ~ ν. By this definition of equivalence for shape measures, all valid shape measures are
indeed equivalent in the sense of Equation 6.1. However, equivalence of shape measures
does not result in the same ordering of simplices based on two different shape measures.
Equivalence simply implies that if the shape of a degenerate element approaches zero mea-
sured by one shape measure, so will the others. On the other hand, if one shape measure
gives a value of 1 for a perfect regular element, so do the other measures. As how shape
measures approach these two extremes, the rate and the manner vary from one measure to
another, and not much conclusion can be drawn from meshes evaluated by different shape
measures.
6.2.3 Extension to Riemann space
There were attempts to extend the shape measures defined in the Euclidean space to Riemann
manifold (George and Borouchaki 1998; Formaggia and Perotto 2000; Frey and George
2000; Dompierre et al. 2002, 2005; Sirois et al. 2005). In general, the transformation of
a simplex in the Euclidean space to the Riemann space is non-linear, and the treatment on
shape measures based on the linear algebraic approach cannot be directly applied. Instead,
attentions were paid to the geometrical quantities such as the length, surface area and vol-
ume of the simplex in the Riemann space, which can all be evaluated through the metric ten-
sor. Shape measures can then be defined based on these geometrical quantities evaluated in
the Riemann space. For example, the γ-quality of tetrahedron T measured in the Riemann
space is given by
72 3
um of edgesmeasuredinReimann space
2
)
32
×
signed volume of TinRiemann space
γ
R
()
T
=
(
s
The length of a parametric curve
x
=
x
(t) in the Riemann space is given by
1
1
∫
∫
T
T
T
length of curve
=
(
Fx Fx
⋅
d
) (
⋅ =
d
)
xMx
()
t
() (
t
t
))d
t
where
MFF
=⋅
0
0
∫
∫
−
T
surfacearea
=
(
Fx Fy
⋅ ×⋅
d) (d)
=
det( )
FF
⋅
d
a
with
d
axy
=×
dd
A
A
