Civil Engineering Reference
In-Depth Information
8.4.2 Refinement of discretised surfaces
Very often, mesh refinements are required for curved surfaces to capture the field variable
in an FE analysis, in an adaptive refinement analysis, and to refine the boundary of a 3D
object for MG. If the curved surface is well defined in analytical form or by means of a
parametric representation, direct MG on the surface, as described in Section 4.4, or MG
on a parametric planar domain, as discussed in Section 4.2, can be applied to generate FE
meshes of the required characteristics. However, in case the curved surface is only available
in a discretised form of a triangular FE mesh, a mesh refinement rather than a complete
mesh regeneration is a very attractive option in producing a refined mesh in compliance with
the specified node spacing function without violating the original geometry represented by
triangular facets.
With a local topology of a planar domain, triangulated curved surfaces can be refined by
the LE bisection and trisection algorithms (Marquez et al. 2008; Plaza et al. 2010, 2012;
Suarez et al. 2012). The LE bisection possesses a number of promising features, and the
two most important characteristics allowing it for efficient mesh refinement are as follows:
(i) the non-degeneracy property, i.e. the minimum angle of the mesh is bounded; and (ii) the
LE bisection always terminates in a finite number of steps. The LE propagation path (LEPP)
algorithm (Suarez et al. 2005, 2008) is one of the most popular techniques to refine a trian-
gular mesh, and the following procedure is an adaptation of the algorithm for the refinement
of a triangulated curved surface with respect to a given node spacing function ρ.
8.4.2.1 Statement of the problem
Given a mesh of triangular elements
S
= {Δ
i
, i = 1, N
s
} and node spacing function ρ(
x
),
x
∈
S
,
refine mesh
S
such that
E
2
≤ 2ρρ
()()
for all edges in
S
joining node points A and B.
A
B
AB
8.4.2.2 Algorithm: Refinement of triangular mesh
{
}
2
1. Build the initial list of edges to be refined,
E
= ∈ ≤
E
S
:
E
2ρρ
(
AB
)()
AB
AB
2. Take the LE L in
E
; identify the two triangles T
1
and T
2
connected to L
3. If L is the LE shared by triangles T
1
and T
2
, then
a. Delete L from the list
E
b. Bisect edge L at the mid-point M as shown in Figure 8.74
c. Verify if edges AM, MB, DM and MC should be added to
E
Otherwise
a. Search for the LE in a chain of connected triangles by the adjacency relationship,
as shown in Figure 8.75, and such a search will terminate, say, in triangle T
k
.
Edge L
T
1
A
C
T
2
M
D
B
Figure 8.74
Bisection of edge L.
