Page 105 - Basic Structured Grid Generation

94 Basic Structured Grid Generation

Let us now consider the various mappings of the side η = 0 of the unit square. Under
P ξ it is mapped to the straight line AC; under P η it is mapped to the curved boundary
AC; finally under P ξ P η it is mapped to the straight line AC. Similar considerations
applied to each side of the unit square show that the composite map (P ξ + P η − P ξ P η )
is a transformation which maps the entire boundary of the unit square onto the entire
curved boundary ABDC.
This map is called the Boolean sum of the transformations P ξ and P η , and denoted
by P ξ ⊕ P η . Thus
P ξ ⊕ P η = P ξ + P η − P ξ P η . (4.72)
It is clear that P ξ ⊕ P η = P η ⊕ P ξ . The complete formulation is

(P ξ ⊕ P η )(ξ, η) = P ξ (ξ, η) + P η (ξ, η) − P ξ P η (ξ, η)
= (1 − ξ)r(0,η) + ξr(1,η) + (1 − η)r(ξ, 0) + ηr(ξ, 1)
−(1 − ξ)(1 − η)r(0, 0) − (1 − ξ)ηr(0, 1)

−(1 − η)ξr(1, 0) − ξηr(1, 1). (4.73)
This transformation is the basis of transfinite interpolation (TFI) in two dimensions.
A grid will be generated by eqn (4.73) by taking discrete values ξ i , η j of ξ and η with

i − 1 j − 1
0 ξ i = 1and 0 η j = 1, i = 1, 2,..., ˜ı, j = 1, 2,..., ˜,
˜ ı − 1 ˜  − 1
for some choice of ˜ and ˜.

ı
Transfinite interpolation is the most common approach to algebraic grid generation.
It can produce excellent grids quickly in situations where other methods would be
difficult to apply, and it also allows for direct control of the location of grid nodes.
Many two-dimensional regions are easy to grid accurately using TFI. However, there
are some geometries, such as the airfoil, ‘backstep’, and C-grids, where TFI proves to
be unsatisfactory. The main disadvantages are (1) a lack of smoothness in the generated
grids, with any discontinuities in gradient in the boundary curves tending to propagate
into the interior, and (2) a tendency to fold when the geometries are complex.
The method can be extended in many ways. For example, the physical region can
be divided into several parts, with grids being generated in each separate part and
then matched together at the interfaces. This results in discontinuities of slope at the
interfaces, and Hermite polynomial interpolation may be exploited to match slopes
and thus remove the discontinuities. It is also possible to use TFI with higher-order
polynomials as blending functions.

4.3.2 Numerical implementation of TFI

We write eqn (4.73), with reference to Fig. 4.15, as

r(ξ.η) = (1 − ξ)r l (η) + ξr r (η) + (1 − η)r b (ξ) + ηr t (ξ) − (1 − ξ)(1 − η)r b (0)
−(1 − ξ)ηr t (0) − (1 − η)ξr b (1) − ξηr t (1), (4.74)
where the abbreviations l, r, b, t stand for ‘left’, ‘right’, ‘bottom’, ‘top’.