Page 129 - Basic Structured Grid Generation

118 Basic Structured Grid Generation

of R, this suggests that they will also vary monotonically along the grid-lines, and
that folding will not occur. In fact it can be proved theoretically that the curvilinear
co-ordinate system generated in R is non-degenerate.
The partial differential equations for the corresponding inverse problem may be
obtained directly from the two-dimensional version of eqn (1.114):
2
∂ y k
∂y k
2 i
(∇ x ) =−g ij .
i
∂x i ∂x ∂x j
Using eqn (5.1), we obtain
2
∂ y k
g ij = 0, k = 1, 2. (5.2)
i
∂x ∂x j
2
1
Hence, substituting x = ξ, x = η, y 1 = x, y 2 = y, and using eqn (1.163), we have
2
2
2
∂ x ∂ x ∂ x
g 22 − 2g 12 + g 11 = 0,
∂ξ 2 ∂ξ∂η ∂η 2
2
2
2
∂ y ∂ y ∂ y
g 22 − 2g 12 + g 11 = 0, (5.3)
∂ξ 2 ∂ξ∂η ∂η 2
for x and y as functions of ξ, η, with g 11 ,g 12 ,and g 22 given by eqn (1.158). These
equations, which here we call the Winslow equations, Winslow (1967), are also elliptic.
However, although eqns (5.1) are linear and uncoupled in ξ and η, eqns (5.3) are in
general non-linear and coupled in x and y through the coefficients g ij .
An intrinsic property of elliptic grid generators is that the grids generated are always
smooth. This means that discontinuities in slope at the boundaries are not propagated
into the interior of the physical domain. In fact, as will be shown in Chapter 6, the
grid generator based on Laplace’s eqns (5.1) produces a grid which minimizes (for all
admissible functions ξ(x, y), η(x, y)) the integral

2
2
I = (|∇ξ| +|∇η| ) dx dy (5.4)
R
evaluated over the physical domain R. Given that the grid in the computational plane
will be associated with equal increments in ξ and η, we might expect |∇ξ| and |∇η|
to give measures of grid-point density in the physical plane, with large gradients in ξ
and η where points are closely clustered together. Minimizing the integral I may then
be regarded as equivalent to generating the (in some sense) smoothest possible grid.
Since we have a choice of models of grid generation for any given problem, the
possibility of selecting the pair of Laplace equations
2
2
∂ x ∂ x
+ = 0,
∂ξ 2 ∂η 2
2
2
∂ y ∂ y
+ = 0, (5.5)
∂ξ 2 ∂η 2
instead of the more complex system (5.3) suggests itself. However, although this model
may generate satisfactory grids in some cases, in general the non-vanishing of the
Jacobian of the transformation from the computational to the physical domain cannot
be guaranteed, and folding (or overlapping) of grid lines may occur.