Civil Engineering Reference
In-Depth Information
Elements around
x
Optimal point
h
u
x
Level surface of η
P
Trial points
Figure 6.25
Shifting
x
along the gradient of
η
P
.
Otherwise, this node is deemed to be optimised since η
P
can hardly be increased any more.
For small Δη
P
< 10
−4
, the shift h
u
can also be reduced from 10% to say 1%. Obviously, the
interval [a, b] and the increment of 0.1λ are rather arbitrary without any theoretical basis,
which can be further refined in the light of more experience. Like the 2D case, each interior
node in the mesh will be processed in turn until all the nodes are processed. A number of
iteration cycles can be performed until η-quality can no longer be improved for mesh
M
.
6.3.3.3 GETMe (3D)
The GETMe is a nodal optimisation method in which nodes are shifted entirely based on
the geometry of the polyhedron in such a way that for each cycle of shifting of nodes, the
polyhedron will become more and more regular. A generic scheme has been proposed for
node shifting of a general polyhedron in which the directions for node shifting are based
on an auxiliary polyhedron formed by connecting the centroids of the faces of the original
polyhedron (Vartziotis and Wipper 2012). However, for a tetrahedron, a simpler procedure
is available in the transformation of a vertex in which the normal of the opposite face repre-
sents the direction of shift (Vartziotis et al. 2009).
6.3.3.3.1 Tetrahedral element
Let {
x
1
,
x
2
,
x
3
,
x
4
} be the vertices of tetrahedron T. The normal vectors to the faces are given by
n
1
=
x
2
x
4
×
x
2
x
3
,
n
2
=
x
3
x
4
×
x
3
x
1
,
n
3
=
x
4
x
2
×
x
4
x
1
,
n
4
=
x
1
x
2
×
x
1
x
3
As shown in Figure 6.26, tetrahedron T is transformed to T′ = {
y
1
,
y
2
,
y
3
,
y
4
}, which is given by
nn
/
1
1
y
y
y
y
x
x
x
x
1
1
nn
/
2
2
2
2
T
==
+λ
nn
/
3
3
3
3
4
4
nn
/
4
4
where parameter λ ≈ 0.8 will only affect the convergence rate of the GETMe transformation.
