Civil Engineering Reference
In-Depth Information
P
r
i
A
i
C
i
B
i
Figure 3.34
Determination of element size at point P.
The effect of a short segment is weighted by the distance away from it such that ρ will
be small at the point near the boundary of clustered nodal points, and it will become
larger at the point away from short boundary segments, as shown in Figure 3.34. If a
background grid is used as the control space
, the element size ρ is evaluated by Equation
3.1 at all the grid points of
, and within the cells of
, ρ is computed by interpolation.
On the other hand, if a previous mesh is used as the control space
, element size ρ is
evaluated at all the nodal points of the mesh, and within the elements of
, ρ is computed
by FE interpolation.
3.5.6.4 Element size based on a previous analysis
The adaptive refinement procedure based on an h-version successive mesh refinement is very
popular for its simplicity and efficiency, in which the element size is adjusted for each subse-
quent analysis, while the polynomial of the interpolation function is kept constant (Lo and
Lee 1992). The linear elasticity problem is taken as an example to demonstrate how element
size is to be defined based on a previous analysis.
For a linear elasticity problem, the objective of the adaptive refinement is to achieve a solu-
tion with relative error norm η smaller than some prescribed value η
0
, i.e. η < η
0
. As the exact
energy norm is not available, at any stage of the adaptive refinement process, exact energy
norm ||
u
|| is approximated by
u
such that
≈=+
(
)
2
2
2
2
uuu
ˆ
γ
e
where
u
is the energy norm of the FE solution, and
e
is the estimated error in the energy
norm obtained from the FE stresses and some enhanced more accurate smoothed stresses.
γ is an empirical correction factor suggested by Zienkiewicz and Zhu (1987, 1992), which
is set, respectively, to 1.3 and 1.4 for linear triangular element T
3
and quadratic triangular
element T6. Hence, the relative error norm η is approximated by
η
such that
ηη
γ
≈=
e
u
