Page 216 - Basic Structured Grid Generation

Unstructured grid generation 205

co-ordinate system Oxy. For control purposes, required values of these parameters are
specified at each node of the background grid. The initial background grid is usually
generated manually by the user, and can be quite coarse, even for complex domains. For
example, a background grid consisting of a single element, or two elements (triangles),
can be used to impose the requirement of linear variation of parameters, or of constant
spacing and stretching throughout the domain.
The background grid does not have to align with the boundary of the domain to be
triangulated. In cases where no initial background grid is supplied, a default background
grid is generated, consisting of two triangular elements with a uniform grid-density
requirement, based on empirical rules. The default grid-density value of δ is taken to
be five per cent of the length of the diagonal of the background grid. When adaptive
methods are required, the first grid generated could become the background grid for
the next grid, and then a more detailed specification of the spatial variation of grid
generator parameters can be achieved.
An alternative, or additional, method for grid control, particularly when complex
geometries are involved and there is a need to specify grid parameters in certain
regions, such as the leading and trailing edges of aircraft wings, is provided by a so-
called distribution of sources. In this approach a spatial distribution of grid-cell size is
specified as a function of the distance from a given point to a ‘source’, which could
be a point or a line. The distribution is ‘isotropic’ if it depends only on the distance x
measured in any direction from the source. The isotropic source function for a point
source at S is taken to be

δ 1 if 0 <x <x c

δ(x) = x − x c (8.15)
δ 1 exp ln 2 if x x c ,
D − x c
where δ 1 ,D,and x c are user-specified parameters that can be tuned to control the
variation of triangle size δ about S. A typical graph of δ(x) is shown in Fig. 8.20.

8.3.3 Searching algorithm

To be able to interpolate grid parameters from the background grid (in two dimensions),
it is necessary to locate the triangle of the background grid in which a given point in
the domain lies. This can be achieved by computing the so-called area co-ordinates of
the point. Suppose we have a triangle with vertices labelled 1, 2, 3. The area of this

d(x)

2d 1
d 1
O
x c D x
Fig. 8.20 A possible source function.