Computational engineering: introduction to numerical methods

326 Pages·2006·2.033 MB·English

by  Michael Schäfer

MichaelSchäfer
ComputationalEngineering–IntroductiontoNumericalMethods
Michael Schäfer
Computational Engineering –
Introduction to
Numerical Methods
With204Figures
123
ProfessorDr.rer.nat.MichaelSchäfer
ChairofNumericalMethodsinMechanicalEngineering
TechnischeUniversitätDarmstadt
Petersenstr.30
64287Darmstadt
Germany
[email protected]
Solutionstotheexercises:
www.fnb.tu-darmstadt.de/ceinm/orwww.springer.com/3-540-30686-2
ThebookistheEnglisheditionoftheGermanbook:NumerikimMaschinenbau
LibraryofCongressControlNumber:2005938889
ISBN-10 3-540-30685-4 SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-30685-6 SpringerBerlinHeidelbergNewYork
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Preface
Duetotheenormousprogressincomputertechnologyandnumericalmethods
thathavebeenachievedinrecentyears,theuseofnumericalsimulationmeth-
ods 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.
Besidestheneedfordevelopersofcorrespondingsoftware,thereisastrong
– 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 possibil-
ities with respect to specific applications. The major concern of this book is
the impartation of this knowledge in a comprehensive way.
Thetextgivesapracticeorientedintroductioninmodernnumericalmeth-
ods 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 mechan-
ics,andfluidmechanics,which,therefore,constituteathematicalfocusofthe
text.
Thetopicmustbeseenasastronglyinterdisciplinaryfieldinwhichaspects
of numerical mathematics, natural sciences, computer science, and the corre-
sponding 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, discretiza-
tion, and solution algorithms, whereas an attempt is always made to estab-
lish the relationships to the engineering relevant application areas mentioned
above. Overarching aspects of the different numerical techniques are empha-
sizedandquestionsrelatedtoaccuracy,efficiency,andcosteffectiveness,which
VI Preface
are most relevant for the practical application, are discussed. The following
subjects are addressed in detail:
Modelling:simplefieldproblems,heattransfer,structuralmechanics,fluid
mechanics.
Discretization: connection to CAD, numerical grids, finite-volume meth-
ods,finite-elementmethods,timediscretization,propertiesofdiscretesys-
tems.
Solution algorithms: linear systems, non-linear systems, coupling of vari-
ables, adaptivity, multi-grid methods, parallelization.
Special applications: finite-element methods for elasto-mechanical prob-
lems, finite-volume methods for incompressible flows, simulation of turbu-
lent 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 corre-
sponding literature with reference to the respective chapter are given.
Importantaspectsareillustratedbymeansofapplicationexamples.Many
exemplary computations done “by hand” help to follow and understand the
numerical methods. The exercises for each chapter give the possibility of re-
viewingtheessentialsofthemethods.Solutionsareprovidedonthewebpage
www.fnb.tu-darmstadt.de/ceinm/.Thebookissuitableeitherforself-studyor
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
attheDepartmentofNumericalMethodsinMechanicalEngineeringatDarm-
stadt University of Technology. It closely follows the German book Numerik
im Maschinenbau (Springer, 1999) by the author, but includes several modi-
fications and extensions.
The author would like to thank all members of the department who have
supportedthepreparationofthemanuscript.Specialthanksareaddressedto
PatrickBontouxandtheMSNM-GPgroupofCNRSatMarseilleforthewarm
hospitality at the institute during several visits which helped a lot in com-
pletingthetextintime.SincerethanksaregiventoRekikAlehegnMekonnen
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 Scha¨fer
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
1
Introduction
Inthisintroductorychapterweelucidatethevalueofusingnumericalmethods
in engineering applications. Also, a brief overview of the historical develop-
ment of computers is given, which, of course, are a major prerequisite for
the successful and efficient use of numerical simulation techniques for solving
complex practical problems.
1.1 Usefulness of Numerical Investigations
The functionality or efficiency of technical systems is always determined by
certain properties. An ample knowledge of these properties is frequently the
key to understanding the systems or a starting point for their optimization.
Numerous examples from various engineering branches could be given for
this. A few examples, which are listed in Table 1.1, may be sufficient for the
motivation.
Table1.1.Examplesforthecorrelationofpropertieswithfunctionality
and efficiency of technical systems
Property Functionality/Efficiency
Aerodynamics of vehicles Fuel consumption
Statics of bridges Carrying capacity
Crash behavior of vehicles Chances of passenger survival
Pressure drop in vacuum cleaners Sucking performance
Pressure distribution in brake pipes Braking effect
Pollutants in exhaust gases Environmental burden
Deformation of antennas Pointing accuracy
Temperature distributions in ovens Quality of baked products