Civil Engineering Reference
In-Depth Information
large discrepancies at nodal points, but somewhere within the element, the results are nearly
exact. Summarising from the one-dimensional example, the strategic points are as follows:
1. Displacements are best sampled at the nodes.
2. Gradients' (or stresses) best accuracy are at Gaussian points corresponding to the poly-
nomial of the field variable used in the approximation.
At such points, the order of convergence of the function or its gradient is one order higher
than that would be anticipated, and thus, these points are referred to as
super-convergent
.
The same concept can be readily extended to elements of higher dimensions. For instance,
the super-convergent points for stress evaluation over a Q8 or L9 element are the 2 × 2
Gaussian points and those for the H20 hexahedral elements are the 2 × 2 × 2 Gaussian
points. However, the super-convergent points for triangular and tetrahedral elements are
less conspicuous and are more difficult to identify.
8.9.3.2 Super-convergent patch recovery
Attempts are generally made to recover the nodal values of stresses from those super-conver-
gent sampling points. Within the element, the recovered (smoothed) stress
σ
* are defined by FE
interpolation from the smoothed stress
()
σ
*
at nodal points, as shown in Figure 8.147. The SPR
procedure for stress recovery will be elucidated by means of 2D examples, as shown in Figure
8.148. To recover stress component by component, we assume for each stress component a
complete second-order polynomial over a patch of Q8 quadratic quadrilateral elements, i.e.
*
σ
i
=
Pa
i
=
1232
,,
for
Delasticity
1
×
m m
×
1
2
2
Polynomial
P
=
1
xyx yy
aaaaaa
T
Coefficients
a
=
1
2
3
4
5
6
By means of the least square fit (LSF) over strategic super-convergent points, we have
N
∑
σ
α
(
)
Pa
2
s
LSF
=
−
where
N
=
numberofsuperconvergent points
i
α
α
=
1
s
PP
α
=
(,),
xy
σ
=
stress at super
-convergent point α
α
α
α
σ
*
σ
s
σ
*
Super-convergent points
Figure 8.147
SPR over a Q8 isoparametric element.
