Civil Engineering Reference
In-Depth Information
M
3
M
2
M
1
Figure 4.17
Interpolation of metrics over a triangle.
The rotation part for metric M at point P can be defined using area co-ordinates such that
θ = L
1
θ
1
+ L
2
θ
2
+ L
3
θ
3
Again, for the interpolation of the stretch part, we have two choices:
i. Arithmetic progression: λ = L
1
λ
1
+ L
2
λ
2
+ L
3
λ
3
and μ = L
1
μ
1
+ L
2
μ
2
+ L
3
μ
3
L
L
L
L
L
L
ii. Geometric progression:
λλλλ
=
and
=
1
2
3
1
2
3
123
1
23
Knowing θ, λ and μ, the metric M at point P is given by
λ
0
cos( ) in()
sin( )
θ
θ
T
MFFFUR
=
,
=
,
U
=
,
R
=
0
−
θ
cos( θ
The interpolation of metrics M
1
, M
2
and M
3
at the vertices of a triangle by means of area
co-ordinates is shown in Figure 4.17. Since interpolated metrics are computed by the convex
sum formulas, fairly smooth metrics are produced at the interior of the triangle, which are
bounded by the metrics at the vertices.
4.2.9 Lengths controlled by multiple metrics
In adaptive refinement analysis over curved surfaces, the sizes of the finite elements are
controlled not only by the geometry of the curved surfaces but also by the physical require-
ments, e.g. solution errors have to be within a specified amount. Over regions of high stress
or solution gradients, it is necessary to control the size of the finite elements as specified
by the error indicators. Based on the solution errors, a physical metric tensor
M
P
can be
defined to govern the size and shape of the finite elements on Ω.
In case geometrical metric M
Ω
and
M
P
tensors are having the same set of eigenvectors,
or they are defined based on the same set of basis vectors, then the intersection of M
Ω
and
M
P
is simply given by the smaller eigenvalue in each principal direction (eigenvector). In
practice, at the expense of a bit more computation, the length limit, d, of an edge along the
direction of unit vector V governed by a number of metrics M
1
, M
2
, …, M
n
is given by
T
d
=
mi
1
d
whered
=
VMV
n
i
i
i
i
=
If all metrics Mi
i
are defined following the idea of unit length mesh control, then for any
edge of the mesh, we have to ensure that di
i
= V
T
M
i
V ≤ 1. As shown in Figure 4.18, the locus
