Civil Engineering Reference
In-Depth Information
3.5.6.5.1 Refinement method
By means of the refinement method, additional nodes are created with respect to the existing
triangular elements. Depending on the strategy and the locations for the creation of addi-
tional points, various refinement schemes for 2D and 3D DT have been proposed.
3.5.6.5.1.1 POINT CREATED AT CIRCUMCENTRE
Nodes are created at the circumcentre of those triangles that satisfy certain conditions such
as surface area, in-radius, element aspect ratio, etc., to reduce FE solution error based on
complementary variation principles (Shenton and Cendes 1985; Holmes and Snyder 1988).
3.5.6.5.1.2 POINT CREATED AT CENTROID
Nodes are created at the centroid of those triangles that are considered to be too large;
sometimes, a node can also be positioned at a point that is a weighted sum of the verti-
ces of the triangle (Hermeline 1980) or based on centroidal Voronoi tessellations (Du and
Gunzburger 2002; Secchi and Simoni 2003; Du et al. 2010).
3.5.6.5.1.3 POINTS CREATED ON EDGES
Realising that inserting nodes at the interior of a triangle would very often produce trian-
gular elements of poor quality, Borouchaki and George (1997) proposed the creation of
points on the edges of the triangles. Points are generated along each edge of the triangles
in turn following a geometric progression in compliance with the element size requirement,
and those points that are too close to the existing points will be filtered away. Rivara and
Inostroza (1997) suggested a refinement scheme of DT based on the longest-side bisection.
Remarks:
The refinement methods only suggest where a node can be generated, and a back-
ground grid can be employed to provide information about the desirable element size and
to control whether a proposed point is acceptable or it is rejected as it is too close to some
existing points or simply it is outside the problem domain.
3.5.6.5.2 Use of background grid
The grid points of a background grid are potential positions for point creation in compliance
with the given node spacing requirements. Those points can be rapidly and accurately defined
without ambiguity for MG by means of the Delaunay insertion kernel or alternatively con-
nected up at the interior of the domain by means of standard templates. Uniform rectangular
and triangular grids can be used to generate elements of regular size, and in case of a variable
element size requirement, Quadtree or kd-tree recursive spatial partitions can be applied.
3.5.6.5.2.1 REGULAR GRID
A regular grid of appropriate grid spacing can be superimposed with the given domain, as
shown in Figure 3.35a. Points at grid positions with a sufficient distance from the boundary
edges are inserted one by one to create a mesh of uniform node spacing. The proximity of
a point to an edge can be verified by checking the area of the triangle formed between the
given point and the boundary edge. As grid points of a rectangular grid are at the corner of a
rectangular cell, these cyclic points will cause numerical problems to the Delaunay insertion
