Civil Engineering Reference
In-Depth Information
Patch of linear
elements
T3:
P
= [1, x, y]
Q4:
P
= [1, x, y, xy]
Nodal value determined
Sampling point
Patch of quadratic
elements
T6:
P
= [1, x, y, x
2
, xy, y
2
]
Q8:
P
= [1, x, y, x
2
, xy, y
2
]
Figure 8.148
SPR sampling strategy for common 2D elements.
N
N
d
d
∑
∑
2
T
LSF
(
av
+= −+
η
)
σ
s
Pa
(
η
v
)
LSF
(
a
+=
η
v
)
σ
s
−+
Pa
(
η
vPv
)
(
)
α
i
α
α
i
α
α
η
α
=
1
α
=
1
N
N
d
d
∑
∑
T
T
s
s
LSF
(
av
+
η
)
=
σ
−
Pa
(
Pv
)
=
0
σ
−
Pa
P
=
0
α
i
α
α
α
i
α
α
η
η
=
0
α
=
1
α
=
1
N
N
N
N
∑
∑
∑∑
T
s
T
s
T
s
T
σ
−
Pa
P
=
0
P
σ
−
Pa
=
0
P
Pa
=
σ
P
α
i
α
α
ααα
i
ααα
αα
i
α
=
1
α
=
1
α
=
1
α
=
1
N
N
∑
∑
T
−
1
s
T
or
aAb
=
where
A
=
P
P
and
b
=
σ
P
α
α
αα
i
mm
×
m
×
1
m
×××
11
m
m
×
1
α
=
1
α
=
1
Components of smoothed stress
σ
* are computed in turn from the components of stresses
at super-convergent points, and enhanced stresses at nodal points k = 1, n (n = number of
nodes in the element) are evaluated. Thus, a smoothed high-quality stress field can be set up
over the entire mesh by the standard FE interpolation (H
k
) such that
σ
*
*
= H
kk
.
σ
*
σ
Pa
Pa
Pa
Pa
Pa
Pa
1
1
k
1
s
*
*
k
SPR:
σσσ
→→ =
H
σσσσ
*,
*
=
=
or
σ
=
=kn
1,
*
σ
α
α
k
α
2
k
2
2
*
3
k
3
σ
3
