Springer.Computational.Engineering.Introdution.to.Numerical.Methods.Apr.2006: Springer.Computational.Engineering.Introdution.to.Numerical.Methods.Apr.2006.pdf

Springer.Computational.Engineering.Introdution.to.Numerical.Methods.Apr.2006

Professor Dr. rer. nat. Michael Schäfer
Chair of Numerical Methods in Mechanical Engineering
Technische Universität Darmstadt
Petersenstr. 30
64287 Darmstadt
Germany
schaefer@fnb.tu-darmstadt.de
Solutions to the exercises:
www.fnb.tu-darmstadt.de/ceinm/ or www.springer.com/3-540-30686-2
The book is the English edition of the German book: Numerik im Maschinenbau
Library of Congress Control Number: 2005938889
ISBN-10 3-540-30685-4 Springer Berlin Heidelberg New York
ISBN-13 978-3-540-30685-6 Springer Berlin Heidelberg New York
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication
of this publication or parts thereof is permitted only under the provisions of the German Copyright
Law of September 9, 1965, in its current version, and permission for use must always be obtained from
Springer. Violations are liable for prosecution under the German Copyright Law.
Springer is a part of Springer Science+Business Media
springer.com
© Springer-Verlag Berlin Heidelberg 2006
Printed in Germany
The use of general descriptive names, registered names, trademarks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant
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Cover Design: Frido Steinen-Broo, EStudio Calamar, Spain
Production: LE-TEX Jelonek, Schmidt&Vöckler GbR, Leipzig
Printed on acid-free paper 7/3100/YL 5 4 3 2 1 0

Preface
Due to the enormous progress in computer technology and numerical methods
that have been achieved in recent years, the use of numerical simulation methods
in industry gains more and more importance. In particular, this applies
to all engineering disciplines. Numerical computations in many cases offer a
cost effective and, therefore, very attractive possibility for the investigation
and optimization of products and processes.
Besides the need for developers of corresponding software, there is a strong
– and still rapidly growing – demand for qualified specialists who are able to
efficiently apply numerical simulation tools to complex industrial problems.
The successful and efficient application of such tools requires certain basic
knowledge about the underlying numerical methodologies and their possibilities
with respect to specific applications. The major concern of this book is
the impartation of this knowledge in a comprehensive way.
The text gives a practice oriented introduction in modern numerical methods
as they typically are applied in engineering disciplines like mechanical,
chemical, or civil engineering. In corresponding applications the by far most
frequent tasks are related to problems from heat transfer, structural mechanics,
and fluid mechanics, which, therefore, constitute a thematical focus of the
text.
The topic must be seen as a strongly interdisciplinary field in which aspects
of numerical mathematics, natural sciences, computer science, and the corresponding
engineering area are simultaneously important. As a consequence,
usually the necessary information is distributed in different textbooks from
the individual disciplines. In the present text the subject matter is presented
in a comprehensive multidisciplinary way, where aspects from the different
fields are treated insofar as it is necessary for general understanding.
Following this concept, the text covers the basics of modeling, discretization,
and solution algorithms, whereas an attempt is always made to establish
the relationships to the engineering relevant application areas mentioned
above. Overarching aspects of the different numerical techniques are emphasized
and questions related to accuracy, efficiency, and cost effectiveness, which
VI Preface
are most relevant for the practical application, are discussed. The following
subjects are addressed in detail:
Modelling: simple field problems, heat transfer, structural mechanics, fluid
mechanics.
Discretization: connection to CAD, numerical grids, finite-volume methods,
finite-element methods, time discretization, properties of discrete systems.
Solution algorithms: linear systems, non-linear systems, coupling of variables,
adaptivity, multi-grid methods, parallelization.
Special applications: finite-element methods for elasto-mechanical problems,
finite-volume methods for incompressible flows, simulation of turbulent
flows.
The topics are presented in an introductory manner, such that besides basic
mathematical standard knowledge in analysis and linear algebra no further
prerequisites are necessary. For possible continuative studies hints for corresponding
literature with reference to the respective chapter are given.
Important aspects are illustrated by means of application examples. Many
exemplary computations done “by hand” help to follow and understand the
numerical methods. The exercises for each chapter give the possibility of reviewing
the essentials of the methods. Solutions are provided on the web page
www.fnb.tu-darmstadt.de/ceinm/. The book is suitable either for self-study or
as an accompanying textbook for corresponding lectures. It can be useful for
students of engineering disciplines, but also for computational engineers in
industrial practice. Many of the methods presented are integrated in the flow
simulation code FASTEST, which is available from the author.
The text evolved on the basis of several lecture notes for different courses
at the Department of Numerical Methods in Mechanical Engineering at Darmstadt
University of Technology. It closely follows the German book Numerik
im Maschinenbau (Springer, 1999) by the author, but includes several modifications
and extensions.
The author would like to thank all members of the department who have
supported the preparation of the manuscript. Special thanks are addressed to
Patrick Bontoux and the MSNM-GP group of CNRS at Marseille for the warm
hospitality at the institute during several visits which helped a lot in completing
the text in time. Sincere thanks are given to Rekik Alehegn Mekonnen
for proofreading the English text. Last but not least the author would like to
thank the Springer-Verlag for the very pleasant cooperation.
Darmstadt
Spring 2006 Michael Sch¨afer

Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Usefulness of Numerical Investigations . . . . . . . . . . . . . . . . . . . . . 1
1.2 Development of Numerical Methods. . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Characterization of Numerical Methods . . . . . . . . . . . . . . . . . . . . 6
2 Modeling of Continuum Mechanical Problems . . . . . . . . . . . . . 11
2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Basic Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 Moment of Momentum Conservation . . . . . . . . . . . . . . . . . 19
2.2.4 Energy Conservation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.5 Material Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Scalar Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Simple Field Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.2 Heat Transfer Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Structural Mechanics Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.1 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.2 Bars and Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.3 Disks and Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.4 Linear Thermo-Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.5 Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Fluid Mechanical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.1 Incompressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.2 Inviscid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6 Coupled Fluid-Solid Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.6.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6.2 Examples of applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Exercises for Chap. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
VIII Contents
3 Discretization of Problem Domain . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 Description of Problem Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Numerical Grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.1 Grid Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.2 Grid Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Generation of Structured Grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.1 Algebraic Grid Generation. . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.2 Elliptic Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Generation of Unstructured Grids . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.1 Advancing Front Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4.2 Delaunay Triangulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Exercises for Chap. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Finite-Volume Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1 General Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Approximation of Surface and Volume Integrals . . . . . . . . . . . . . 81
4.3 Discretization of Convective Fluxes . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.1 Central Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.2 Upwind Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3.3 Flux-Blending Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4 Discretization of Diffusive Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5 Non-Cartesian Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.6 Discrete Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.7 Treatment of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 95
4.8 Algebraic System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.9 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Exercises for Chap. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5 Finite-Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.1 Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Finite-Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3 One-Dimensional Linear Elements . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3.2 Global and Local View. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.4 Practical Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4.1 Assembling of Equation Systems . . . . . . . . . . . . . . . . . . . . 118
5.4.2 Computation of Element Contributions . . . . . . . . . . . . . . 120
5.4.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.5 One-Dimensional Cubic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.5.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.5.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.6 Two-Dimensional Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.6.1 Variable Transformation for Triangular Elements . . . . . . 129
5.6.2 Linear Triangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.6.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Contents IX
5.6.4 Bilinear Parallelogram Elements . . . . . . . . . . . . . . . . . . . . . 138
5.6.5 Other Two-Dimensional Elements . . . . . . . . . . . . . . . . . . . 140
5.7 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Exercises for Chap. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.1 Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 Explicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.3 Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Exercises for Chap. 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7 Solution of Algebraic Systems of Equations . . . . . . . . . . . . . . . . 167
7.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.1.1 Direct Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.1.2 Basic Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.1.3 ILU Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.1.4 Convergence of Iterative Methods . . . . . . . . . . . . . . . . . . . 174
7.1.5 Conjugate Gradient Methods . . . . . . . . . . . . . . . . . . . . . . . 176
7.1.6 Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.1.7 Comparison of Solution Methods . . . . . . . . . . . . . . . . . . . . 179
7.2 Non-Linear and Coupled Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Exercises for Chap. 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8 Properties of Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.1 Properties of Discretization Methods . . . . . . . . . . . . . . . . . . . . . . . 187
8.1.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.1.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.1.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.1.4 Conservativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.1.5 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.2 Estimation of Discretization Error . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.3 Influence of Numerical Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.4 Cost Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Exercises for Chap. 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9 Finite-Element Methods in Structural Mechanics . . . . . . . . . . 209
9.1 Structure of Equation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
9.2 Finite-Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
9.3 Examples of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Exercises for Chap. 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
X Contents
10 Finite-Volume Methods for Incompressible Flows . . . . . . . . . . 223
10.1 Structure of Equation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
10.2 Finite-Volume Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
10.3 Solution Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
10.3.1 Pressure-Correction Methods . . . . . . . . . . . . . . . . . . . . . . . 231
10.3.2 Pressure-Velocity Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 235
10.3.3 Under-Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.3.4 Pressure-Correction Variants . . . . . . . . . . . . . . . . . . . . . . . . 244
10.4 Treatment of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 247
10.5 Example of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Exercises for Chap. 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
11 Computation of Turbulent Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 259
11.1 Characterization of Computational Methods . . . . . . . . . . . . . . . . 259
11.2 Statistical Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
11.2.1 The k-ε Turbulence Model . . . . . . . . . . . . . . . . . . . . . . . . . . 263
11.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
11.2.3 Discretization and Solution Methods . . . . . . . . . . . . . . . . . 270
11.3 Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
11.4 Comparison of Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
12 Acceleration of Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
12.1 Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
12.1.1 Refinement Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
12.1.2 Error Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
12.2 Multi-Grid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
12.2.1 Principle of Multi-Grid Method . . . . . . . . . . . . . . . . . . . . . 282
12.2.2 Two-Grid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
12.2.3 Grid Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
12.2.4 Multigrid Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
12.2.5 Examples of Computations . . . . . . . . . . . . . . . . . . . . . . . . . 290
12.3 Parallelization of Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
12.3.1 Parallel Computer Systems . . . . . . . . . . . . . . . . . . . . . . . . . 296
12.3.2 Parallelization Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
12.3.3 Efficieny Considerations and Example Computations . . . 302
Exercises for Chap. 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

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