Civil Engineering Reference
In-Depth Information
p
3
−
α
e
u
≤
N
(8.8)
The main objective of the adaptive analysis is to achieve the above optimal rate of conver-
gence and to reach the desired accuracy with a minimum number of degrees of freedom. It
should be noted that the presence of singularity will significantly affect the size distribution
of an optimal mesh. Small elements have to be placed at these singularities, and the resulting
optimal mesh will be rapidly graded towards these positions (Kelly et al. 1983).
The
a posteriori
error estimate here adopted is based on the Zienkiewicz and Zhu (1987)
(Z
2
) error estimator. The central idea of this error estimate is to obtain a better approxima-
tion of the exact stress from the FE solution following a post-processing procedure. There
are many ways in obtaining an improved stress field from the FE solution (Lee and Lo
1997a,b); however, in terms of simplicity and efficiency, the super-convergent patch recovery
(SPR) technique proposed in Zienkiewicz and Zhu (1992) will be employed for stress recov-
ery, though recovery by equilibrium patches can also be used for triangular and tetrahedral
elements (Boroomand and Zienkiewicz 1997).
8.9.3 Super-convergence and optimal sampling points
It is noted that on many occasions, the displacements or the field variables are more accu-
rately sampled at the nodes defining an element and that the gradients or stresses are best
sampled at some interior points within the element.
8.9.3.1 One-dimensional example
Consider a heat-conduction problem of a second-order equation
d
dx
k
du
dx
++=
β
uQ
0
A typical finite element analysis using linear interpolation is shown in Figure 8.146. The field
variable u is, in general, more accurate at nodal points; however, for the gradient, we observe
Line approximation
to a parabola
u
Analytic solution
Finite element approximation
du
dx
Finite element approximation
of gradient
Gaussian points
x
Figure 8.146
Finite element approximation to the heat conduction problem.
