Civil Engineering Reference
In-Depth Information
Based on the polar decomposition in the theory of large deformation, deformation tensor
F
can be uniquely decomposed into a rotation part
R
and a pure deformation part
U
such that
F
=
R
·
U
R
T
·
R
=
R
·
R
T
=
I
(
Identity tensor
)
and
U
is symmetric
The deformation
F
between tetrahedron T and the regular element can be measured by
the Green-Cauchy deformation tensor
C
given by
C
=
F
T
·
F
= (
R
·
U
)
T
· (
R
·
U
) =
U
T
·
R
T
·
R
·
U
=
U
2
By this definition,
C
is symmetric and positive definite for non-degenerate tetrahedron T.
Let λ
1
, λ
2
and λ
3
be the eigenvalues of
C
; deformation from the regular element to tetra-
hedron T can be measured by the mean ratio η defined as the ratio between the geometric
mean and the arithmetic mean of the eigenvalues of
C
.
λλλ
3
123
η
=
1
3
(
λλλ
++
)
1
2
3
η attains the minimum value of 0 for degenerated tetrahedra, for which one or more λ values
equal to 0, and it reaches the maximum value of 1 when λ
1
= λ
2
= λ
3
= λ; in such a case, we
have
λ
λ
1
C
=
λ
=
λ
=
λ
I
2
λ
λ
3
For η = 1, tetrahedron T and the regular element are similar and only differ by a scaling fac-
tor λ
1/2
. On the other hand, if not all the λ values are equal, η will be less than 1, i.e. η = 1 only
for regular tetrahedral elements. According to Knupp (2001), among shape measures σ, ρ and
η, only η is an algebraic mesh quality metric. As a result, mean ratio η can also be expressed in
terms of the determinant and the Frobenius norm of transformation matrix
F
as follows:
λλλ
2/
3
2/3
2/3
3
3
3
det( )
()
C
3
(det())
F
3
(det())
:
F
FF
3
(det())
F
123
η
=
=
=
=
=
1
3
T
2
tr
C
tr(
FF
⋅
)
F
(
λλλ
++
)
1
2
3
det(
C
) = det(
F
T
·
F
) = (det(
F
))
2
;
F
:
F
= F
ij
F
ij
(sum over i, j); ‖·‖ = Forbenius norm
In terms of shape measure regularity, η is preferred over σ and ρ in mesh optimisation as
η is more smooth, differentiable everywhere and symmetric about its peak (Dompierre et al.
2005). Alternatively, a more geometrically based quality coefficient γ related to the mean
ratio η was proposed by Lo (1991c),
72 3
signed volume of T
sumofedges
23/2
×
γ()
T
=
(
)
