Civil Engineering Reference
In-Depth Information
T
1
T
2
C
T
k−1
A
M
T
k
B
Edge L´
D
Figure 8.75
Bisection of the longest edge in a chain of connected triangles.
b. Bisect the LE L′ between triangles T
k
and T
k-1
(L′ ∉
E
as L′ is longer than L).
c. Verify if edges AM, MB, DM and MC should be added to
E
(this step may be
skipped if we do not start with a very coarse mesh to interpolate a rapidly changing
spacing function)
4. Update mesh
S
and its adjacency relationship, and if
E
is not empty, go to step (2)
As we can see by the design of the LEPP algorithm, the mesh will be slightly over-
refined to guarantee the quality of the triangles as some edges not in
E
are also bisected.
If the adjacency relationship is well computed or already available, the time complexity
of the refinement algorithm is virtually linear as in case of a search, T
k
can be located
quite rapidly by adjacency in scanning a few triangles. Take for example the boundary
surface mesh of 450 triangular facets of a solid object, which is to be refined to as small
as 1% in linear scale, as shown in Figure 8.76. The refined mesh of 72,252 triangles
with smaller elements being placed along three orthogonal planes is shown in Figure
8.77. The minimum α-value and the mean α-value of the initial boundary surface are
0.051 and 0.576, respectively. For the refined mesh, upon smoothing by node shifting,
the minimum α-value improved from 0.045 to 0.152, and the mean α-value improved
from 0.825 to 0.840.
Figure 8.76
Initial boundary surface.
