Page 180 - Basic Structured Grid Generation

Variational methods and adaptive grid generation 169

2
Since g = g 11 g 22 − (g 12 ) , this is equal to
1 g 11 g 22
I AO = dξ dη. (6.59)
4 R ϕ
Exercise 5. Show that the Euler-Lagrange equations may be expressed as

g 22 r ξ g 11 r η
+ = 0, (6.60)
ϕ ξ ϕ η
and also in the form (6.48), with

2 2
(x η ) + (y η ) 0
A 11 = ,
2
0 (x ξ ) + (y ξ ) 2

4x ξ x η 2(x ξ y η + x η y ξ )
A 12 =
2(x ξ y η + x η y ξ ) 4x ξ x η
2 2
(x ξ ) + (y ξ ) 0
A 22 = 2 2 ,
0 (x η ) + (y η )
2 2

1 x ξ x η ϕ ξ ((x η ) + (y η ) )
S =− . (6.61)
2
2
ϕ y ξ y η ϕ η ((x ξ ) + (y ξ ) )
The AO-eqns (6.48) with (6.61) are quasilinear and coupled, and can be solved iter-
atively. Unweighted AO grids (with ϕ(ξ, η) = 1) are generally smooth and unfolded,
being near to orthogonality and having near-uniform areas. The smoothness of the
grids is rather unexpected, since the generating equations are not elliptic.
A variation on the AO-functional is the ‘AO-squared functional’, which is given in
unweighted form by

2
I AO-squared = (g 11 g 22 ) dξ dη. (6.62)
R
The Euler-Lagrange equations whose solutions minimize this functional generally
produce very good grids.

6.4.6 Other orthogonality functionals

Orthogonality two functional
This functional is given by
2
(g 12 )
I O,2 = dξ dη. (6.63)

R g 11 g 22
Since
(g 12 ) 2 (r ξ · r η ) 2 r ξ r η 2
= = · ,
g 11 g 22 (r ξ · r ξ )(r η · r η ) |r ξ | |r η |