Page 51 - Basic Structured Grid Generation

40 Basic Structured Grid Generation

Here we put
dr dr
2 2 2
(ds) = · (dχ) =˜g 11 (dχ) ,
dχ dχ

so that ds/dχ = ˜ g 11 . Clearly we have
2 2
dr dr dr dr dχ dχ

g 11 = · = · =˜g 11 . (2.54)
dξ dξ dχ dχ dξ dξ
If we wanted grid points on C to be evenly distributed in the sense that the length
of the curve between neighbouring points was always the same, we would require that
χ i+1
˜ g 11 dχ = const. = L/n, (2.55)
χ i
where L is the total length of the curve. This equation is an example of an equidistri-
bution principle; these principles in general are prescriptions for controlling the density
of grid points.
Equation (2.55) suggests that the mapping ξ → χ should satisfy the differential
equation
dχ L
, (2.56)
=
dξ ˜ g 11
or, in approximate form,
χ i+1 − χ i L
, (2.57)
=
ξ i+1 − ξ i ˜ g 11

with ˜ g 11 evaluated, say, at the mid-point χ 1 of the corresponding χ interval. Here
i+ 2
the ξ i values are evenly distributed, so that (ξ i+1 − ξ i ) = 1/n. Thus the spacing of

points on the χ interval is proportional to 1/ ˜g 11 .
Equation (2.57) is satisfactory for numerical work except for cases where ˜g 11
becomes too small, or zero. A better choice might be based on
dχ c
(2.58)
dξ 1 +˜g 11
=
instead of eqn (2.56), for some constant c,or
dχ c
, (2.59)
=
dξ 2
1 + α ˜g 11
where extra flexibility in controlling grid density is provided by the parameter α.
The optimal spacing of grid points may also be influenced by the curvature κ of C.
To obtain higher grid-point density in regions of high curvature, the equation
dχ c
= (2.60)
dξ (1 + β |κ|) 1 + α ˜g 11
2
2
may be used.
Thus the grid generation procedure may be represented in general by
dχ
= cϕ(χ), χ(0) = 0, (2.61)
dξ