Page 10 - Basic Structured Grid Generation

Preface

Over the past two decades, efficient methods of grid generation, together with the
power of modern digital computers, have been the key to the development of numer-
ical finite-difference (as well as finite-volume and finite-element) solutions of linear
and non-linear partial differential equations in regions with boundaries of complex
shape. Although much of this development has been directed toward fluid mechanics
problems, the techniques are equally applicable to other fields of physics and engi-
neering where field solutions are important. Structured grid generation is, broadly
speaking, concerned with the construction of co-ordinate systems which provide co-
ordinate curves (in two dimensions) and co-ordinate surfaces (in three dimensions)
that remain coincident with the boundaries of the solution domain in a given problem.
Grid points then arise in the interior of the solution domain at the intersection of these
curves or surfaces, the grid cells, lying between pairs of intersecting adjacent curves
or surfaces, being generally four-sided figures in two dimensions and small volumes
with six curved faces in three dimensions.
It is very helpful to have a good grasp of the underlying mathematics, which is
principally to be found in the areas of differential geometry (of what is now a fairly
old-fashioned variety) and tensor analysis. We have tried to present a reasonably self-
contained account of what is required from these subjects in Chapters 1 to 3. It is
hoped that these chapters may also serve as a helpful source of background reference
equations.
The following two chapters contain an introduction to the basic techniques (mainly
in two dimensions) of structured grid generation, involving algebraic methods and dif-
ferential models. Again, in an attempt to be reasonably inclusive, we have given a
brief account of the most commonly-used numerical analysis techniques for interpo-
lation and for solving algebraic equations. The differential models considered cover
elliptic and hyperbolic partial differential equations, with particular reference to the
use of forcing functions for the control of grid-density in the solution domain. For
solution domains with complex geometries, various techniques are used in practice,
including the multi-block method, in which a complex solution domain is split up
into simpler sub-domains. Grids may then be generated in each sub-domain (using the
sort of methods we have presented), and a matching routine, which reassembles the
sub-domains and matches the individual grids at the boundaries of the sub-domains, is
used. We show a simple matching routine at the end of Chapter 5.
A number of variational approaches (preceded by a short introduction to variational
methods in general) are presented in Chapter 6, showing how grid properties such