Page 91 - Basic Structured Grid Generation

80 Basic Structured Grid Generation

In general we seek to avoid mappings of a sheet in the physical region into a square
in the computational region which would involve folding the sheet, resulting in a map
which is not one–one. A simple example of such a transformation is the mapping of
the rectangle 0 x 1, −1 y 1 into the square 0 ξ 1, 0 η 1, with

2
ξ = x, η = y , (4.12)
which folds the sheet in the xy-plane along the x-axis. The Jacobian

∂ξ ∂ξ

∂x ∂y

∂η ∂η

∂x ∂y

of the transformation vanishes when y = 0 (along the x-axis, where the fold is located),
and this is the mathematical cause of the difficulty. Hence, in order to generate good
grids, we seek to avoid mappings where the Jacobian is zero (or infinite, in which case
the Jacobian of the inverse mapping would be zero) at points inside the physical region.
Similar considerations apply in three dimensions, where we seek transformations
between a physical domain in which there is a cartesian co-ordinate system Oxyz and
a unit cube 0 ξ 1, 0 η 1, 0 ς 1 in computational space with ξ, η, ς
as co-ordinates. The transformations can be imagined to deform a sponge-like object
occupying the physical domain into the unit cube, or vice versa, with corresponding
transformations of co-ordinate curves and co-ordinate surfaces. Again, the Jacobians
of the transformations are required to be non-zero and finite.
Once we have recognized that the boundary curves (or boundary surfaces in three
dimensions) in the physical space can be regarded as co-ordinate curves which map onto
the sides of a square in computational space (or co-ordinate surfaces which map onto
the faces of a cube in three dimensions), certain simple methods of interpolating a set
of grid points may suggest themselves. These algebraic methods based on interpolation
are extensively used in computational fluid dynamics, exploiting their advantages of
ease of computation (compared with differential models involving the solution of partial
differential equations) and the capability of direct control over grid node location. On
the other hand, algebraic methods may not generate smooth grids; in particular, they
tend to preserve the features of boundaries, and any discontinuities in the slope of
boundary curves will generally propagate into the interior region. A common use of
algebraic methods is to generate a first attempt at a grid, which may then be used as a
starting point in the iterative implementation of differential models of grid generation,
to be considered in Chapter 5.
In the following section we review some basic methods of interpolation.

4.2 Unidirectional interpolation

4.2.1 Polynomial interpolation

A need for interpolation may arise when dealing with a boundary of complex shape. For
example, consider a plane curve representing an airfoil section, for which measurement