Civil Engineering Reference
In-Depth Information
M
1
M
2
Arithmetic progression
M
1
M
2
Geometric progression
Figure 4.15
Intersection of metrics along a straight line.
However, for the interpolation of the stretch part, we have two choices.
i. Arithmetic progression: λ = sλ
1
+ tλ
2
and μ = sμ
1
+ tμ
2
ii. Geometric progression:
λλλ
=
st
and
=
st
12
12
Hence, metric M at point P is given by
0
0
cos( θ
Metric interpolations of M
1
and M
2
along a straight line for arithmetic progression and
geometric progression are plotted in Figure 4.15. It can be seen that the difference between
the two progressions is quite small, and the metrics interpolated by the arithmetic progres-
sion appear to be only slightly larger than those interpolated by the geometric progression.
λ
cos( ) in()
sin( )
θ
θ
T
MFFFUR
=
,
=
,
U
=
,
R
=
−
θ
4.2.8.2 Metric interpolation within a triangular element
By means of polar decomposition of a metric tensor, the metric interpolation within a trian-
gle can be easily defined by means of area co-ordinates of a triangular element. Let M
1
, M
2
and M
3
be, respectively, the metric at the vertices of triangle ABC. By polar decomposition,
metric Mi
i
(i = 1,2,3) at the nodal points of the triangle can be expressed as
λ
0
cos(
θ
) i
n(
θ
)
π
T
i
i
i
∈ 0
2
where λ
i
and μ
i
are the principal stretches, and θ
i
characterises the principal directions of
metric tensor Mi.
i
. Let L
1
, L
2
and L
3
be the area co-ordinates of point P within triangle ABC,
as shown in Figure 4.16. Then the metric at point P is defined as
MFFFUR
=
,
=
,
U
=
,
R
=
θ
,
i
i
i
i
i
i
i
i
i
0
−
sin(
θ
)cos()
θ
i
i
i
Δ
Δ
Δ
Δ
Δ
Δ
1
2
3
L
=
,
L
=
,
L
=
,
Δ
=
areaoftriangleABC
1
2
3
C
∆
1
∆
2
P
A
B
∆
3
Figure 4.16
Interpolation of metric within a triangle.
