Generation of Computational Mesh for Haemodynamics Analysis - Computational Hemodynamics: Theory, Modelling and Applications - page 161

a node of one cell can be connected to any number of nodes from any number of
other cells. The descriptions of some of these connectivity algorithms (Delauney,
Quadtree/Octree, and Advancing front) are briefly described in this section.
In addition a large number of grid generation software and open source codes
exist and a list is provided in the Appendix. Using a bottom-up approach a meshing
process involves meshing the edges first, followed by filling the enclosed face (Ad-
vancing Front method), and for a 3D geometry filling the interior volume (Delauney
method). Recently polyhedral cells have been used to fill the interior domain as an
alternative to triangular cells. A benefit of applying a polyhedral mesh is that it al-
lows the flexibility of an unstructured mesh to be applied to a complex geometry
without the large computational demands associated with a large tetrahedral mesh.
The application of such cells is relatively new. Nevertheless, polyhedral meshing
has shown thus far to have tremendous advantages over tetrahedral meshing with
regard to the attained accuracy and efficiency of the numerical computations (Kal-
vin and Taylor 1996; Spiegel et al. 2010).
Unstructured meshes can also involve the use of hexahedral, pyramid, and wedge
cells in combination with tetrahedral cells, whereas a structured mesh is reliant on
hexahedral cells or the use of block-structured mesh only. A mesh combining dif-
ferent cell types is a hybrid mesh and typically involves allocating cell elements
that match the boundary surface. For example structured elements are used for wall
boundaries instead of triangular unstructured cells as can resolve the boundary layer
flow field gradients much better. The triangular mesh are difficult to cluster in the
lateral direction due to the underlying triangular structure. In almost all cases, tri-
angular elements are used to fill the surface face (in 2D) or tetrahedral elements are
used to fill the remaining volume (in 3D). Many codes try to automate the genera-
tion of prismatic meshes by allowing the user to define the surface mesh and then
marching off the surface to create the 3D elements. While very useful and effective
for smooth shapes, the extrusion process can break down near regions of high cur-
vature or sharp discontinuities. Hybrid grid methods are designed to take advantage
of the positive aspects of both structured and unstructured grids with the structured
grid in local regions while using unstructured grid in the bulk of the domain.

6.2.5 Delaunay Triangulation

Triangle and tetrahedral meshing are the most common forms of unstructured
mesh generation as they allow maximum flexibility in matching cells with curved
boundaries and increased resolution in flow regions where they matter most. A
common method for triangle and tetrahedral meshing is the Delaunay triangula-
tion criterion which states that no node must be located inside the circumcircle—a
circle circumscribing the mesh element (Shewchuk 2002). In 2D, a circumcircle
of a triangle is the unique circle that passes through all three vertices of its triangle
whereby nodes from other triangles do not exist within the circle. Figure 6.10 shows
that the circumcircle of every Delaunay triangular cell within the mesh generated is
free from other nodes.