Page 173 - Basic Structured Grid Generation

162 Basic Structured Grid Generation

a uniformly-spaced set of points in ξ-space (the computational space 0 ξ 1) maps
to a non-uniformly spaced set of points in χ-space (the parametric space 0 χ 1),
which generate a suitable set of points r(χ) on the curve. Here we show that map-
pings χ(ξ) may be obtained as the solutions of Euler-Lagrange equations for some
variational problems.
An appropriate functional incorporating a weight function ϕ(r) in physical space
suggested by eqn (6.14) would be
1 1 g 11
I = dξ,
2 0 [ϕ(r)] 2
2
2
2
where g 11 = (x ξ ) + (y ξ ) + (z ξ ) . We may regard r and hence ϕ as a function of the
parameter χ, and consider the slightly more general functional
1 1 H(g 11 )
I = dξ, (6.29)
2 0 [ϕ(χ)] 2
for some function H. By eqn (2.54), we can write
2
1 1 H(˜g 11 [χ ξ ] )
I = dξ. (6.30)
2 0 [ϕ(χ)] 2
2
2
2
where ˜g 11 = (x χ ) + (y χ ) + (z χ ) , which may be regarded as a function of χ.
Thus the integrand is not explicitly dependent on ξ, and we can call on eqn (6.4) to
write the Euler-Lagrange equation as

d H 1 ∂ 2
− χ ξ H(˜g 11 [χ ξ ] ) = 0,
2
dξ ϕ 2 ϕ ∂χ ξ
from which we obtain

2
2

d H − 2(χ ξ ) ˜g 11 H (˜g 11 [χ ξ ] )
= 0. (6.31)
dξ ϕ 2
In the particular case where H = g 11 , this equation reduces to

2
d ˜ g 11 [χ ξ ] − 2 ˜g 11 [χ ξ ] 2 d ˜ g 11 [χ ξ ] 2
=− = 0, (6.32)
dξ ϕ 2 dξ ϕ 2
which yields
2
˜ g 11 [χ ξ ] g 11
= = const. (6.33)
ϕ 2 ϕ 2
The end conditions are of course χ(0) = 0, χ(1) = 1.
√
For the special case ϕ = 1, we obtain g 11 = const. and hence also g 11 = const.
By eqn (2.47), equally spaced points on the ξ interval will produce a set of points on
the curve with equal arc-length between points.
On the accompanying disk the subdirectory Book/one.d.gds contains a file curve.SOR.f
which solves eqn (6.32) in the case of a plane parabolic curve defined by

x = χ
. (6.34)
y = χ(1 − χ)