Thompson, Warsi, Mastin - Numerical Grid Generation

PREFACENumerical grid generation has now become a fairly common tool for use in thenumerical solution of partial differential equations on arbitrarily shaped regions. This isespecially true in computational fluid dynamics, from whence has come much of the impetusfor the development of this technique, but the procedures are equally applicable to allphysical problems that involve field solutions. Numerically generated grids have providedthe key to removing the problem of boundary shape from finite difference methods, andthese grids also can serve for the construction of finite element meshes.

With such grids allnumerical algorithms, finite difference or finite element, are implemented on a square grid ina rectangular computational region regardless of the shape and configuration of the physicalregion. (Finite volume methods are effectively a type of conservative finite differencemethod on these grids.)In this text, grid generation and the use thereof in numerical solutions of partialequations are both discussed. The intent was to provide the necessary basic information,from both the standpoint of mathematical background and from that of codingimplementation, for numerical solutions of partial differential equations to be constructed ongeneral regions. Since these numerical solutions are ultimately constructed on a square gridin a rectangular computational region, any solution algorithm that can treat equations withvariable coefficients is basically applicable, and therefore discussion of specific algorithms isleft to classical texts on the numerical solution of partial differential equations.The area of numerical grid generation is relatively young in practice, although its rootsin mathematics are old.

This somewhat eclectic area involves the engineer’s feel for physicalbehavior, the mathematician’s understanding of functional behavior, and a lot ofimagination, with perhaps a little help from Urania. The physics of the problem at hand mustultimately direct the grid points to congregate so that a functional relationship on thesepoints can represent the physical solution with sufficient accuracy. The mathematics controlsthe points by sensing the gradients in the evolving physical solution, evaluating the accuracyof the discrete representation of that solution, communicating the needs of the physics to thepoints, and by providing mutual communication among the points as they respond to thephysics.Numerical grid generation can be thought of as a procedure for the orderly distributionof observers, or sampling stations, over a physical field in such a way that efficientcommunication among the observers is possible and that all physical phenomena on theentire continuous field may be represented with sufficient accuracy by this finite collectionof observations.

The structure of an intersecting net of families of coordinate lines allows theobservers to be readily identified in relation to each other, and results in much more simplecoding than would the use of a triangular structure or a random distribution of points. Thegrid generation system provides some influence of each observer on the others, so that if onemoves to get into a better position for observation of the solution, its neighbors will follow tosome extent in order to maintain smooth coverage of the field.Another way to think of the grid is as the structure on which the numerical solution isbuilt.

As the design of the lightest structure requires consideration of the load distribution, sothe most economical distribution of grid points requires that the grid be influenced by boththe geometric configuration and by the physical solution being done thereon.

In any case,since resources are limited in any numerical solution, it is the function of the numerical gridgeneration to make the best use of the number of points that are available, and thus to makethe grid points an active part of the numerical solution.This is a rapidly developing area, being now only about ten years old, and thus is stillin search of new ideas. Therefore no book on the subject at this time could possibly beconsidered to be definitive. However, enough material has now accumulated in the literature,and enough basic concepts have emerged, that a fundamental text is now needed to meet theneeds of the rapidly expanding circle of interest in the area. It is with the knowledge of boththese needs and these limitations that this text has been written.

Some of the techniquesdiscussed will undoubtedly be superceded by better ideas, but the fundamental conceptsshould serve for understanding, and hopefully also for some inspiration, of new directions.The only background assumed of the student is a senior-level understanding of numericalanalysis and partial differential equations. Concepts from differential geometry and tensoranalysis are introduced and explained as needed.Numerical grid generation draws on various areas of mathematics, and emphasisthroughout is placed on the development of the relations involved, as well as on thetechniques of application. This text is intended to provide the student with the understandingof both the mathematical background and the application techniques necessary to generategrids and to develop codes based on numerically generated grids for the numerical solutionof partial differential equations on regions of arbitrary shape.The writing of this text has been a cooperative effort over the last two years, spurredon by the institution of a graduate course in numerical grid generation, as well as an annualshort course, at Mississippi State.

The students in both of these courses have contributedsignificantly in revising the text as it evolved. The last appendix is the result of a classassignment prepared by Col. Hyun Jin Kim, graduate student in the computational fluiddynamics program, who also compiled the index. Our colleage, Dr. Helen V.McConnaughey of Mathematics contributed significantly through continual discussions andwrote most of Chapter IV.We are indebted to a large number of former students and fellow researchers aroundthe world for the development of the ideas that have crystallized into numerical gridgeneration.

The complete debt can be acknowledged only through mention of thebibliographies contained in the several surveys cited herein. A list here would either be toolong to note the strongest influences or too short to acknowledge all the significant ones. Wemust, however, acknowledge the many long and fruitful discussions with Peter Eiseman ofColumbia University.0f vital importance is the support that has been provided for the research from whichthe developments discussed in this book have emerged, including NASA; the researchoffices of the Air Force, Army, and Navy; the National Science Foundation, and variousindustrial concerns.

The interest and contributions of a number of contract monitors has beenessential over the years. We are especially appreciative of Bud Bobbitt and Jerry South ofNASA Langley Research Center, who provided the initial support for an unknown with anidea.Particular debts are owed to W. H. Chu for an idea in the Journal of ComputationalPhysics in 1971, and to Frank Thames who put the idea into a dissertation.In the preparation of the text we had the conscientious and untiring efforts of two mostable secretaries, Rita Curry and Susan Triplett, who typed on in good spirits through a yearof numerous revisions and frustrations as the text evolved.Finally, we were particularly fortunate to have the services of Yeon Seok Chae,graduate student in the computational fluid dynamics program and illustrator par excellence,who did all the figures with understanding of the intended meaning as well as artisticcompetence.

His meticulous efforts were extensions of our thoughts.Joe F. ThompsonZ. U. A. WarsiC. Wayne MastinMississippi State, MississippiJanuary 1985I. INTRODUCTIONThe numerical solution of partial differential equations requires some discretization ofthe field into a collection of points or elemental volumes (cells).

The differential equationsare approximated by a set of algebraic equations on this collection, and this system ofalgebraic equations is then solved to produce a set of discrete values which approximates thesolution of the partial differential system over the field. The discretization of the fieldrequires some organization for the solution thereon to be efficient, i.e., it must be possible toreadily identify the points or cells neighboring the computation site. Furthermore, thediscretization must conform to the boundaries of the region in such a way that boundaryconditions can be accurately represented. This organization is provided by a coordinatesystem, and the need for alignment with the boundary is reflected in the routine choice ofcartesian coordinates for rectangular regions, cylindrical coordinates for circular regions,etc., to the extent of the handbook’s resources.The current interest in numerically-generated, boundary-conforming coordinatesystems arises from this need for organization of the discretization of the field for generalregions, i.e., to provide computationally for arbitrary regions what is available in thehandbook for simple regions.