Page 127 - Basic Structured Grid Generation

5

Differential models for grid

generation

5.1 The direct and inverse problems

In Section 4.1 a number of examples were given in two dimensions of the way in which
grids could be generated in a non-rectangular physical domain by utilizing a transfor-
mation between the domain and a rectangular (or square) domain in computational
space, whereby the rectangular cartesian co-ordinates in computational space become
curvilinear co-ordinates in physical space. As curvilinear co-ordinates, moreover, they
are boundary-conforming, so that the physical boundaries become co-ordinate curves
on which one of the curvilinear co-ordinates is constant. In three dimensions, in the
same way, boundary surfaces of the physical domain become co-ordinate surfaces,
again with one of the three curvilinear co-ordinates constant. The principal advantage
of such transformations is naturally that physical boundary conditions expressed in
terms of the new curvilinear co-ordinates are simplified and thus easier to incorporate
in numerical work. This advantage comes at the cost of increasing the complexity of
the partial differential equations to be solved.
Thus the ‘direct problem’ in two dimensions, given a physical domain R in the plane
Oxy bounded by four segments of curves, is to determine two functions ξ(x, y), η(x, y)
for x and y within the domain, such that ξ is constant (say ξ = 0 and 1, although
other choices might be convenient) on two opposite boundaries and η is constant
(0 and 1 again, say) on the other two boundaries. Furthermore, on each boundary
where ξ is constant, η must increase monotonically (from 0 to 1) so as to make
η(x, y) continuous with a one–one relationship between points (x, y) and η on that
boundary curve; similarly on the other two boundaries, where η is constant and ξ
varies monotonically. The details of precisely how ξ and η vary monotonically on
the boundaries remain at our disposal. A structured grid can then be generated in
the physical domain by selecting a network of ξ and η co-ordinate curves. The same
considerations apply to a doubly-connected physical domain (Fig. 5.1) once we have
made a branch-cut, except that the cut introduces two artificial boundaries on which
any boundary conditions must coincide, since they correspond to the same points in
physical space.
Analysis of the accuracy of the numerical solution of the hosted partial differential
equations based on a grid indicates as one would expect that grid spacing needs to be
small to obtain accurate resolution where gradients in the solution are large. Moreover,