Civil Engineering Reference
In-Depth Information
In case of anisotropic element size distribution, the nodal spacing function ρ has to be
extended from a scalar quantity to a metric tensor or matrix such that the element size along
a particular direction specified by unit vector
u
is given by (Section 4.2.11)
ρ =
u
⊤
M
u
= λ
1
(
u
·
u
1
)
2
+ λ
2
(
u
·
u
2
)
2
where unit vectors
u
1
and
u
2
are the principal stretch directions (eigenvectors) of metric
tensor M, and λ
1
and λ
2
are, respectively, the required element size along these directions.
As
u
1
and
u
2
are orthogonal vectors, it is easy to verify that an element size of λ
i
is required
if
u
is in the direction of vector
u
i
. M is a symmetric tensor with positive eigenvalues and
orthogonal eigenvectors. It is, in general, a continuous function over the control space
,
which can be specified analytically, but it is more often that M is defined only at some nodal
points of the control space
for which intermediate values of M are obtained by means of
interpolation.
The nodal spacing function ρ can be specified as a continuous analytical function to
create certain node distribution patterns to test various MG algorithms, for example, line
distribution, cross distribution, spiral distribution and cluster distribution. In practical FE
MG, the nodal spacing function can be specified entirely based on the geometry of the prob-
lem domain. When one does not have much idea about the element size of the first mesh,
the nodal spacing function can be defined based on the length of the boundary edges of the
problem domain, as described in Section 3.5.6.3. The element size can also be estimated
based on previous analyses such that smaller elements will be used in an area of high-
solution gradient so as to capture the characteristics of the solution with a minimum number
of elements, and this procedure is known as FE adaptive refinement analysis presented in
Section 8.9.
3.5.6.3 Element size based on domain boundary
From a geometrical point of view, element size can be related to the line segments of the
boundary of the domain. One has to devise a way to determine the element size at an interior
point of the domain based on the characteristics of the boundary segments. Realising that a
small element size has to be used in areas close to boundary segments of short lengths, the
distance effect from a boundary segment has to be carefully interpreted in order to obtain
a mesh with smooth transition in element size from one end of the domain to the other. For
a planar domain Ω with a boundary composed of N
b
line segments Γ = {A
i
B
i
, i = 1,N
b
}, the
element size or node spacing ρ at any point P(x,y) can be estimated by
ρ
∑
∑
N
b
i
i
r
i
=
1
ρ
(,)
xy
=
(3.1)
1
N
b
r
i
=
1
i
where ρ
i
is the node spacing associated with segment i, which can be computed using
ρ
i
= ||A
i
B
i
||
i = 1,N
b
and r
i
is the distance between point P to the midpoint (centre) Ci
i
of segment A
i
B
i
, i.e.
r
i
= ||PC
i
||
i = 1,N
b
