Page 175 - Basic Structured Grid Generation

164 Basic Structured Grid Generation

For example, in the case considered here, from eqns (6.34) and (6.35), we would
have

˜ g 11 1 2
= (2 − 4χ i + 4χ ). (6.37)
i
ϕ 2 ϕ 2
i i
The presence of the user-specified weight function ϕ implies that the grid can be
generated adaptively, so that grid generation can be coupled dynamically to the solution
of the hosted equations. For example, the gradient of field variables can be used as a
weight function to ensure that the grid-density increases where there are sharp changes
in these variables.
The program on the file employs SOR, and takes constant ϕ = 1. Having obtained the
solution χ(ξ),we can findthe setof χ points corresponding to a uniformly spaced set
of ξ points, and then from eqns (6.34) calculate the co-ordinates of the corresponding
points on the curve. According to the observations above for constant ϕ these points
should be equally spaced (in terms of arc-length) along the curve.
It may be pointed out that for this case (ϕ = 1) eqns (6.32) admit an analytic solution
which gives χ implicitly in terms of ξ, and in principle this may be compared with
the numerical one to assess accuracy.

6.4 Two-dimensional grids

In the variational approach grid properties such as grid-node spacing, cell area, and
orthogonality can be controlled by minimizing an appropriate functional. In fact
these properties can be controlled simultaneously by choosing a functional which is
a weighted combination of specific functionals which control individual properties.
The appropriate choice of weight functions may, however, not be clear-cut, since the
question of which grid generator is best is not as readily answered in two and three-
dimensional problems as it is in one dimension. But variational methods allow intuition
to be used in selecting functionals appropriate to a particular problem.
Here we shall be mainly concerned with ‘length’ (L) functionals, ‘area’ (A) func-
tionals, and ‘orthogonality’ (O) functionals, and various combinations of these. These
functionals will be expressed in terms of the transformation x = x(ξ, η), y = y(ξ, η)
from a computational domain to a two-dimensional physical domain (or its inverse),
with a uniform grid in computational space transforming to a curvilinear co-ordinate
grid in physical space. By eqn (1.42), the length of elements of grid lines in physical
space, given a fixed uniform grid in computational space, is governed by the terms g 11
and g 22 of the covariant metric tensor, while the angle between intersecting grid lines
is related to g 12 . In a completely orthogonal grid g 12 would be zero everywhere. The
area of a grid cell, using eqns (1.44), (1.159), and (1.160), depends on the local value
√
of the Jacobian g of the transformation. So here we discuss variational problems
√
where the functionals depend on g 11 , g 22 g 12 ,and g. The associated Euler-Lagrange
equations then provide the corresponding differential model of grid generation. (On
the floppy disk included with this book we present a number of numerical examples
with various geometries and varying degrees of complexity to show how to solve the
Euler-Lagrange equations associated with certain functionals, and it is hoped that this