Page 227 - Basic Structured Grid Generation

216 Basic Structured Grid Generation

8.3.5 Adaptation and parameter space

If the stretching parameter s at a certain point of the computational domain is equal
to unity, the grid in the neighbourhood of that point should consist of approximately
equilateral triangles. However, as has already been mentioned, grid cells near boundary
segments are often required to be highly stretched (with high aspect ratio). A uniform
distribution of such cells can be created at the boundary, new points (vertices of the
stretched triangles) being situated along perpendicular bisectors of the initial bound-
ary segments in accordance with the required stretching. Stretched cells can then be
extended into the interior of the domain in a marching process, advancing one layer
at a time. When the boundary-layer region has been covered with stretched cells, the
rest of the domain can be covered according to the AFT as described above.
It may be convenient to generate stretched cells by means of a mathematical trans-
formation, such that grid generation takes place in a parameter (or ‘normalized’) space
in which the stretched triangles are transformed to approximately equilateral trian-
gles. The transformation from physical space to parameter space involves rotating and
shrinking an element, in which, referring to Fig. 8.19, a point with position vector x
in physical space is transformed to the point ˜ x,where
1 1
s x s y
s s
˜ x = x, (8.17)
n x n y
the components of s being given by (s x ,s y ) = (cos φ, sin φ) and those of n by
(n x ,n y ) = (− cos φ, sin φ).We alsohave
1
˜ x s · x
˜ x = = s . (8.18)
˜ y n · x
The effect of the transformation on a typical stretched triangle is shown in Fig. 8.40.
It follows that approximately equilateral triangles may be generated in parameter space
and may then be transformed back into stretched triangles in physical space by applying
the inverse transformation to that given in eqn (8.17).

8.3.6 Grid quality improvement

After an unstructured grid has been generated, two procedures may be applied in order
to improve it. These procedures do not change the total number of triangles and nodes
in the grid.

B C
n C
B
D A D
A
s

Fig. 8.40 Transformation from physical space to parameter space.