Preface to Volume 3

This volume appears for the first time in a separate form. Though part of it has been updated from the second volume of the fourth edition, in the main it is an entirely new work. Its objective is to separate the fluid mechanics formulations and applications from those of solid mechanics and thus perhaps to reach a different interest group. Though the introduction to the finite element method contained in the first volume (the basis) is general, in it we have used, in the main, examples of elastic solids. Only a few applications to areas such as heat conduction, porous media flow and potential field problems have been presented. The reason for this is that all such problems are self-adjoint and that for such self-adjoint problems Galerkin procedures are optimal. For convection dominatedproblems the Galerkin process is no longer optimal and it is here that most of the fluid mechanics problems lie. The present volume is devoted entirely to fluid mechanics and uses in the main the methods introduced in Volume 1. However, it then enlarges these to deal with the non-self-adjoint problems of convection which are essential to fluid mechanics problems. It is our intention that the present volume could be used by investigators familiar with the finite element method in general terms and introduce them to the subject of fluid mechanics. It can thus in many ways stand alone. However, many of the general finite element procedures available in Volume 1 may not be familiar to a reader introduced to the finite element method through different texts and therefore we recommend that this volume be used in conjunction with Volume 1 to which we make frequent reference. In fluid mechanics several difficulties arise. ( I ) The first is that of dealing with incompressible or almost incompressible situations. These, as we already know, present special difficulties in formulation even in solids. (2) Second and even more important is the difficulty introduced by the convection which requires rather specialized treatment and stabilization. Here particularly in the field of compressible high-speed gas flow many alternative finite element approaches are possible and often different algorithms for different ranges of flow have been suggested. Although slow creeping flows may well be dealt with by procedures almost identical to those of solid mechanics, the high-speed range of supersonic and hypersonic flow may require a very particular treatment. In this text we shall generally use only one algorithm the so-called characteristic basedsplit (CBS),introduced a few years ago by the authors. It turns out that

xiv

Preface t o Volume 3

this algorithm is applicable to all ranges of flow and indeed gives results which are at least equal to those of specialized methods. We shall therefore stress its development and give details of its use in the third chapter dealing with discretization. We hope that the book will be useful in introducing the reader to the complex subject of fluid mechanics and its many facets. Further we hope it will also be of use to the experienced practitioner of computational fluid dynamics (CFD) who may find the new presentation of interest and practical application.

Acknowledgements The authors would like to thank Professor Peter Bettess for largely contributing the chapter on waves (Chapter 8) in which he has made so many achievementst and to Dr. Pablo Ortiz who, with the first author, was the first to apply the CBS algorithm to shallow-water equations. Our gratitude also goes to Professor Eugenio Oiiate for adding the section on free surface flows in the incompressible flow chapter (Chapter 5) documenting the success and usefulness of the procedure in ship hydrodynamics. Thanks are also due to Professor J. Tinsley Oden for the short note describing the discontinuous Galerkin method and to Professor Ramon Codina whose participation in recent research work has been extensive. Thanks are also due to Drs Joanna Szmelter and Jie Wu who both contributed in the early developments leading to the final form of the CBS algorithm. The establishment of finite elements in CFD applications to high-speed convectiondominated flows was first accomplished at Swansea by the research team working closely with Professor Ken Morgan. His former students include Professor Rainald Lohner and Professor Jaime Peraire as well as many others to whom frequent reference is made. We are very grateful to Professor Nigel Weatherili and Dr. Oubay Hassan who have contributed several of the diagrams and colour plates and, in particular, the cover of the book. The recent work on the CBS algorithm has been accomplished by the first author with substantial support from NASA (Grant NAGW/2127, Ames Control Number 90-144). Here the support, encouragement and help given by Dr. Kajal K. Gupta is most gratefully acknowledged. Finally the first author (O.C. Zienkiewicz) is extremely grateful to Dr. Perumal Nithiarasu who worked with him for several years developing the CBS algorithm and who has given to him very much help in achieving the present volume. OCZ and RLT

t As already mentioned in the acknowledgement of Volume 1, both Peter and Jackie Bettess have helped us by writing a general subject index for Volumes 1 and 3. $Complete source code for all programs in the three volumes may be obtained at no cost from the publisher’s web page: http://www.bh.com/companions/fem