Civil Engineering Reference
In-Depth Information
The refined element size
h
i
given by Equation 8.10 is defined only over the i
th
element.
However, the input required by an automatic mesh generator are the element sizes speci-
fied at the nodal points of the mesh. Such a nodal value of the element size can be readily
calculated by averaging the values of all the elements incident to a given node. Usually,
the required element size at the nodal points of the FE mesh are computed and stored for
MG, and within a tetrahedral element (T4, T10 or T20), nodal spacing is evaluated by
FE interpolation. As the problem domain Ω is discretised into FE elements, a piecewise
continuous map of the node spacing function is thus established over the entire problem
domain. A typical module in the determination of the element size in an adaptive refine-
ment analysis will take as input the FE solution of displacements (field variables) at the
nodal points and output the values of the required element size at the corresponding
nodal points. The element size can be smoothed out based on an element area gradient
(Howlett and Zundel 2009).
8.9.5
Examples
Two examples are included in this section to illustrate the adaptive refinement procedure.
The FE sizes of adaptive refinement meshes are computed by the procedure described in
Sections 8.9.1-8.9.4, and the MG is done by the generic refinement algorithm given in
Section 8.4.3. The first example is about an L-shaped object under tension on one of its
faces, and the dimensions, support conditions and loading are shown in Figure 8.149. The
only singularity is along the re-entrant corner line AB, which is of singularity strength of
0.5445. The final adaptively refined FE mesh of 31,607 T10 elements is shown in Figure
8.150. A graphic plot of the adaptive refinement analysis using T4, T10 and T20 elements is
depicted in Figure 8.153.
A notch under uniform tensile force is considered as the second example, as shown in
Figure 8.151a. Owing to symmetry, only one-eighth of the notch is modelled, and the geom-
etry along with the boundary and loading conditions are shown in Figure 8.151b. Again,
only one line (line AB) of singularity is present due to a sudden change in boundary condi-
tion, and it is expected that highly graded elements are needed around this singular line. An
adaptive refinement FE mesh of 30,425 T10 elements for this problem is shown in Figure
8.152. A graphic plot of the adaptive refinement analysis using different orders of tetrahe-
dral elements is given in Figure 8.154.
z
Face z = 1, w = 0
0.5
0.25
Face x = 0, u = 0
B
Face y = 1, v = 0
A
0.5
x
Young's modulus = 1
Poisson's ratio = 0.3
y
Figure 8.149
L-shaped domain subject to a horizontal load.
