Page 132 - Basic Structured Grid Generation

Differential models for grid generation 121

Adding and using (5.10), we obtain

2 2
2
2
∂ ξ ∂ ξ f 1 ∂ξ ∂ξ f 1 11 f 1 g 22
+ =− + =− g =− . (5.11)
∂x 2 ∂y 2 f ∂x ∂y f f g
1 1 1
Similarly,
2
2
∂ η ∂ η f 2 g 11
+ =− . (5.12)
∂x 2 ∂y 2 f g
2
The effect of introducing the stretching functions is thus equivalent to using the Poisson
system (5.6) with
f 1 g 22 f 2 g 11

P(ξ, η) =− and Q (ξ, η) =− , (5.13)
f g f g
1 2
although of course g 11 ,g 22 ,and g have to be determined as part of the solution.
Using eqn (5.7), the inverse problem is associated with the equations

2
2
2
∂ x ∂ x ∂ x f 1 ∂x f 2 ∂x
g 22 2 − 2g 12 + g 11 2 − g 22 − g 11 = 0,
∂ξ ∂ξ∂η ∂η f ∂ξ f ∂η
1 2
2
2
2
∂ y ∂ y ∂ y f 1 ∂y f 2 ∂y
g 22 2 − 2g 12 + g 11 2 − g 22 − g 11 = 0,
∂ξ ∂ξ∂η ∂η f ∂ξ f ∂η
1 2
(5.14)
where g 11 ,g 12 ,g 22 are given by eqn (1.158).
5.3.1 Orthogonality considerations
Orthogonality is a desirable feature of grids, since the nearer a grid approaches to
orthogonality, the more accurate we generally expect numerical solutions to be. In
three dimensions it is clear from the discussion of surfaces in Chapter 3 that a co-
ordinate line can only be orthogonal at a point to two mutually orthogonal co-ordinate
lines lying in a surface if those lines are in the directions of principal curvature of the
surface at that point. This is a rather restrictive requirement, which makes orthogonality
difficult or impossible to achieve in general. Here we shall confine our discussion to
planar regions.
It is convenient to consider the parameter space above, in which the orthogonality
condition for the χ, σ co-ordinate curves at any point would be given by

∂r ∂r ∂x ∂x ∂y ∂y
· = + = 0. (5.15)
∂χ ∂σ ∂χ ∂σ ∂χ ∂σ
We can now put
∂y ∂x ∂x ∂y
= a r , and =−a r , (5.16)
∂σ ∂χ ∂σ ∂χ