Civil Engineering Reference
In-Depth Information
where deformation gradient (transformation) tensor
F
maps a vector in the Euclidean space
to the corresponding vector in the Riemann space, vector d
a
is a surface element in the
Euclidean space and A is the area domain of the element in the Euclidean space. The volume
of a simplex K, V, in the Riemann space is given by
∫
∫
∫
V(
=
Fx
⋅ ×⋅
d)(d)( d)
Fx Fx
⋅
⋅
=
det(
F)(x
ddd)
×
xx
⋅
=
det(
F)
v
d
1
2
3
1
2
3
K
K
K
where dv is the volume element spanned by vectors d
x
1
, d
x
2
and d
x
3
. Alternatively, the shape
measure on the Riemann space can be approximately evaluated by taking sample points
over the simplex; in a more rigorous manner, we can take the Gaussian points, and for the
sake of convenience, the vertex points are taken for the evaluation (Alauzet et al. 2003).
A direct extension to Riemann metric may not be consistent with the original idea of
shape measure in the Euclidean space as the element with the highest shape measure may
not be a regular element in the sense that the angles may not be equal and optimal; the faces
may not have the same surface area either, even though all the edges are of the same length.
Regular simplex with equal edges, equal faces and equal angles well defined in the Euclidean
space simply does not exist in the Riemann space, and the shape measure may not attain the
higher value of 1 for simplex with edges of equal lengths; neither can we prove that there is
only a single peak for shape measures evaluated with the Riemann metric. For anisotropic
meshes, the lengths of the edges or distance measure are of primary concern rather than
the shape of the elements as the definition of regular simplex is no longer applicable in the
Riemann space. In other words, anisotropic meshes are generated such that all the edges
ought to be conformable with the specified length requirement measured by the given met-
ric. As the simplex is actually generated in the Euclidean space in which regular element
and shape measure are defined, another approach is to adopt a criterion λ combining shape
measure η in Euclidean metric and an edge conformity coefficient δ to assess the quality of
a tetrahedron in the Riemann space. The conformity coefficient δ
i
of the i
th
edge is given by
r
ρ
i
i
δ
=
min,
rlengthofedgeiin Rie
=
mann metric,
i
ρ=
specified length
i
i
ρ
r
i
i
Combined measure λ for tetrahedral elements can now be defined as
λ = ηδ, where η = mean ratio measured with Eucliden metric, δ = δ
1
δ
2
δ
3
δ
4
δ
5
δ
6
η is included in the combined measure to ensure that the simplex will not be degenerate in
the Riemann space, as a degenerate element in the Euclidean space will map into a degener-
ate element in the Riemann space and vice versa. Shape measure in the Euclidean space is
still meaningful as well-shaped tetrahedra may only be slightly distorted under the trans-
formation of a smooth metric. Combined measure λ is relatively less costly to compute com-
pared to the shape measure evaluated in Riemann metric.
6.2.4 Shape measure for polyhedron
In three dimensions, apart from tetrahedral elements, structured and unstructured solid
FE meshes of hexahedral, pentahedral and pyramid elements are quite common. As these
non-simplicial elements are generated by some automatic processes and are very often mixed
