Page 136 - Basic Structured Grid Generation

Differential models for grid generation 125

This result shows that the Jacobian is generally positive as long as α is, and so grids
generated will be non-overlapping. In general the grids will not be orthogonal.

Eliminating η from eqns (5.29), (5.30) leads to the following second-order partial
differential equation for ξ, which is elliptic because of eqn (5.31):
2
2
2
∂ ξ ∂ ξ ∂ ξ ∂α ∂β ∂ξ ∂β ∂γ ∂ξ
α + 2β + γ + + + + = 0. (5.36)
∂x 2 ∂x∂y ∂y 2 ∂x ∂y ∂x ∂x ∂y ∂y
It is straightforward to show that the same equation must be satisfied by η.
By eliminating ∂y/∂η from eqns (5.33) and (5.34) and using eqn (5.31), we obtain
the relation
∂y β ∂x 1 ∂x
= − . (5.37)
∂ξ α ∂ξ α ∂η
2
2
Equating the mixed derivatives ∂ y/∂ξ∂η and ∂ y/∂η∂ξ obtained from eqns (5.33)
and (5.37) now yields the second order p.d.e. for x(ξ, η):

∂ β ∂x 1 ∂x ∂ 1 ∂x β ∂x
− = + .
∂η α ∂ξ α ∂η ∂ξ α ∂ξ α ∂η
This becomes, on performing the differentiations, writing ∂α = ∂α ∂x + ∂α ∂y ,etc.,
∂ξ ∂x ∂ξ ∂y ∂ξ
and simplifying,
2
2

∂ x ∂ x ∂β ∂α √
+ − + g = 0, (5.38)
∂ξ 2 ∂η 2 ∂y ∂x
√
where g is given by eqn (5.35). A similar calculation gives
2
2
∂ y ∂ y ∂β ∂γ √
+ − + g = 0. (5.39)
∂ξ 2 ∂η 2 ∂x ∂y
If we take β = 0 and hence γ = α −1 , eqns (5.33) and (5.34) reduce to the gener-
alized Cauchy-Riemann equations
∂x ∂y

 = α

∂ξ ∂η

(5.40)
∂y 1 ∂x
=− .


∂ξ α ∂η
As in conformal mapping, grid density in the interior of the physical domain is
not controlled by the quasiconformal mapping method. However, stretching func-
tions could be used to overcome this deficiency. More information can be found in
Mastin and Thompson (1984) and in Mastin (1991). Further details on quasiconformal
mapping are contained in Ahlfors (1996).
5.5 Numerical techniques
5.5.1 The Thomas Algorithm

In the numerical solution of the partial differential equations serving as differential
models of grid generation, finite-differencing frequently leads to a set of linear equations