Civil Engineering Reference
In-Depth Information
needed, whereas in three dimensions, a 3 × 3 matrix is required. In a Riemann space defined
by M on Ω, the length of edge PQ with respect to M in Ω is given by
1
∫
T
PQ
=
PQ
⋅ +⋅
MP
(
tQ
)
PQ dt
(8.5)
M
0
where M(P + tQ) is the metric at point P + tQ, t ∈ [0,1]. ‖PQ‖
M
can be evaluated by numeri-
cal integration using two or more Gaussian points. If r is greater than
2PQ
M
, where r is
the Euclidean distance between P and Q, then edge PQ is bisected. Alternatively, the element
size along a particular direction represented by unit vector
v
can also be given by
h = λ
1
(
v
⋅
u
1
)
2
+ λ
2
(
v
⋅
u
2
)
2
+ λ
3
(
v
⋅
u
3
)
2
(8.6)
where unit vectors
u
1
,
u
2
and
u
3
are the principal stretch directions (eigenvectors) of metric
tensor M, and λ
1
, λ
2
and λ
3
are the required element sizes along the three principal direc-
tions, respectively. In the actual implementation, edge PQ will be divided if r
2
> 2h
1
h
2
, where
h
1
and h
2
are the desirable sizes computed using Equation 8.5 or 8.6 at the two Gaussian
points on edge PQ. The metric M can also be designed to produce the so-called
unit mesh
(Section 4.2.6) such that edges measured by ‖PQ‖
M
should have an ideal lengths of 1 unit,
and edges having length evaluated with respect to the unit metric M greater than
2
ought
to be refined. The anisotropic refinement is tested with the refinement of a standard cube
under different node spacing distributions. A homogeneous metric field is used as the first
test, in which the principal stretch directions are along the principal axis of the cube. Edge
lengths of 3, 1 and 0.2 units are required along the three principal directions. The resulting
mesh depicted in Figure 8.91 consisting of 26,199 node points and 133,766 elements shows
that element sizes are quite different along various directions as specified by the metric
tensor.
The same metric definition is used in the second test. However, in this case, the principal
directions of the metric tensor have been rotated so that one of the principal directions is
now along the diagonal of the cube. The resulting mesh shown in Figure 8.92 consisting
of 42,555 nodes and 218,062 elements shows that the principal directions are along the
diagonal of the faces, which is the result of the projection of the main diagonal onto the
three orthonormal faces. A non-homogeneous size map is employed in the third test. Here,
Figure 8.91
Homogeneous anisotropy.
