Page 179 - Basic Structured Grid Generation

168 Basic Structured Grid Generation

where g 12 = (x ξ x η +y ξ y η ). Like the previous functionals, this takes only non-negative
values. It would take a minimum value of zero only for a completely orthogonal grid
(if one exists), with g 12 = 0everywhere.

Exercise 4. Show that the Euler-Lagrange equations may be expressed as
(g 12 r η ) ξ + (g 12 r ξ ) η = 0 (6.55)
and also in the form (6.48), with
2
(x η ) x η y η (4x ξ x η + 2y ξ y η ) (x ξ y η + x η y ξ )
A 11 = , A 12 = ,
(y η ) 2 (x ξ y η + x η y ξ ) (4y ξ y η + 2x ξ x η )
x η y η
2
(x ξ ) x ξ y ξ 0
A 22 = 2 , S = . (6.56)
x ξ y ξ (y ξ ) 0
If the resulting minimum (or lower bound) for I O is greater than zero, then clearly
we are as close as we can get to an orthogonal grid in a ‘least-squares’ sense. With no
weight function involved, the ‘metric’ g 12 maybesaidto be equidistributed. However,
in practice it is found that even equidistribution may fail to produce a satisfactory grid,
particularly in non-convex physical domains where a perfectly orthogonal grid may
fail to exist, and the functional (6.54) on its own is not recommended.

6.4.5 Combination of functionals

A weighted linear combination of length-, area-, and orthogonality-functional may be
formulated to achieve a compromise between the various properties controlled by these
functionals separately. So we may consider
I = w L I L + w A I A + w O I O , (6.57)

where the weights w L , w A , w O , are non-negative constants satisfying w L +w A +w O =
1. These weights can be used in numerical experiments to obtain the combination of
functionals that produces the most satisfactory grid. For example, the choice w L = 0.1,
w A = 0.9, w O = 0 of length and area functionals generally gives smooth, unfolded
grids, this combination usually overcoming the separate limitations of lack of smooth-
ness in the area-functional and a tendency for folding in the length-functional. However,
in certain geometries the resulting grid may suffer from severe skewness and/or a fail-
ure to converge. In such cases there is a need to experiment so as to ‘tune’ the weight
parameters to obtain a satisfactory grid.

The area-orthogonality-functional
The choice of weight parameters w L = 0, w A = 0.5, w O = 0.5 of area and orthog-
onality functionals leads to a robust automatic grid generator known as the Area-
Orthogonality(AO)-functional. A weight function ϕ(ξ, η) can also be employed, so
that we get a functional
1 1 g (g 12 ) 2
I AO = + dξ dη. (6.58)
2 R 2 ϕ ϕ