Michael Schäfer Computational Engineering – Introduction to Numerical Methods Michael Schäfer Computational Engineering – Introduction to Numerical Methods With 204 Figures 123 Professor Dr. rer. nat. Michael Schäf er 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: N u merik im M aschinenbau Library of Congress C ontrol 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 cop yright. 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Typesetting: Digital data supplied by author Cover Design: Frido Steinen-Broo, EStudio Calamar, Spain Production: LE-T E XJelonek,Schmidt&VöcklerGbR,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 meth- 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. 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 possibil- ities 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 meth- 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, 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 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- sized 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 meth- ods, finite-element methods, time discretization, properties of discrete sys- 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. 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 re- viewing 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 Darm- 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 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 com- pleting 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 UsefulnessofNumericalInvestigations 1 1.2 Developmentof NumericalMethods 4 1.3 CharacterizationofNumericalMethods 6 2 Modeling of Continuum Mechanical Problems 11 2.1 Kinematics 11 2.2 BasicConservationEquations 15 2.2.1 MassConservation 16 2.2.2 MomentumConservation 18 2.2.3 Moment of Momentum Conservation . . . . . . . . . . . . . . . . . 19 2.2.4 EnergyConservation 19 2.2.5 Material Laws 20 2.3 Scalar Problems 20 2.3.1 SimpleFieldProblems 21 2.3.2 HeatTransferProblems 23 2.4 StructuralMechanicsProblems 26 2.4.1 LinearElasticity 27 2.4.2 BarsandBeams 30 2.4.3 Disks andPlates 35 2.4.4 LinearThermo-Elasticity 39 2.4.5 Hyperelasticity 40 2.5 FluidMechanicalProblems 42 2.5.1 Incompressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.2 InviscidFlows 45 2.6 CoupledFluid-SolidProblems 46 2.6.1 Modeling 47 2.6.2 Examples ofapplications 49 ExercisesforChap.2 56 VIII Contents 3 Discretization of Problem Domain 57 3.1 Description of Problem Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 NumericalGrids 60 3.2.1 GridTypes 61 3.2.2 GridStructure 62 3.3 GenerationofStructuredGrids 66 3.3.1 AlgebraicGridGeneration 67 3.3.2 Elliptic Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4 GenerationofUnstructuredGrids 71 3.4.1 AdvancingFrontMethods 72 3.4.2 DelaunayTriangulations 74 ExercisesforChap.3 76 4 Finite-Volume Methods 77 4.1 GeneralMethodology 77 4.2 Approximation of SurfaceandVolumeIntegrals 81 4.3 Discretization of Convective Fluxes . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.1 CentralDifferences 85 4.3.2 UpwindTechniques 86 4.3.3 Flux-BlendingTechnique 88 4.4 Discretization of Diffusive Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.5 Non-CartesianGrids 91 4.6 DiscreteTransportEquation 94 4.7 Treatment of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 95 4.8 AlgebraicSystemofEquations 97 4.9 NumericalExample 100 ExercisesforChap.4 103 5 Finite-Element Methods 107 5.1 GalerkinMethod 107 5.2 Finite-ElementDiscretization 110 5.3 One-DimensionalLinearElements 112 5.3.1 Discretization 112 5.3.2 GlobalandLocalView 115 5.4 Practical Realization 118 5.4.1 Assembling of Equation Systems . . . . . . . . . . . . . . . . . . . . 118 5.4.2 Computation ofElementContributions 120 5.4.3 NumericalExample 121 5.5 One-DimensionalCubicElements 123 5.5.1 Discretization 123 5.5.2 NumericalExample 126 5.6 Two-DimensionalElements 128 5.6.1 Variable Transformation for Triangular Elements . . . . . . 129 5.6.2 LinearTriangularElements 131 5.6.3 NumericalExample 132 Contents IX 5.6.4 Bilinear Parallelogram Elements . . . . . . . . . . . . . . . . . . . . . 138 5.6.5 Other Two-DimensionalElements 140 5.7 NumericalIntegration 143 ExercisesforChap.5 146 6 Time Discretization 149 6.1 Basics 149 6.2 ExplicitMethods 154 6.3 ImplicitMethods 157 6.4 NumericalExample 161 ExercisesforChap.6 165 7 Solution of Algebraic Systems of Equations 167 7.1 Linear Systems 167 7.1.1 DirectSolutionMethods 168 7.1.2 Basic Iterative Methods 169 7.1.3 ILUMethods 171 7.1.4 Convergence of Iterative Methods . . . . . . . . . . . . . . . . . . . 174 7.1.5 ConjugateGradientMethods 176 7.1.6 Preconditioning 178 7.1.7 Comparisonof Solution Methods 179 7.2 Non-Linearand CoupledSystems 182 ExercisesforChap.7 184 8 Properties of Numerical Methods 187 8.1 Properties of DiscretizationMethods 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 EstimationofDiscretization Error 199 8.3 Influence of Numerical Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 8.4 CostEffectiveness 206 ExercisesforChap.8 206 9 Finite-Element Methods in Structural Mechanics 209 9.1 StructureofEquationSystem 209 9.2 Finite-ElementDiscretization 211 9.3 Examplesof Applications 215 ExercisesforChap.9 221 XContents 10 Finite-Volume Methods for Incompressible Flows 223 10.1 StructureofEquation System 223 10.2 Finite-VolumeDiscretization 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 Exampleof Application 251 ExercisesforChap.10 258 11 Computation of Turbulent Flows 259 11.1 Characterization of Computational Methods 259 11.2 StatisticalTurbulenceModeling 261 11.2.1 The k-ε TurbulenceModel 263 11.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 11.2.3 Discretizationand SolutionMethods 270 11.3 Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 11.4 ComparisonofApproaches 275 12 Acceleration of Computations 277 12.1 Adaptivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 12.1.1 RefinementStrategies 278 12.1.2 ErrorIndicators 280 12.2 Multi-GridMethods 281 12.2.1 Principleof Multi-GridMethod 282 12.2.2 Two-Grid Method 284 12.2.3 Grid Transfers 287 12.2.4 MultigridCycles 288 12.2.5 ExamplesofComputations 290 12.3 Parallelization of Computations 295 12.3.1 Parallel Computer Systems 296 12.3.2 ParallelizationStrategies 297 12.3.3 Efficieny Considerations and Example Computations . . . 302 ExercisesforChap.12 306 List of Symbols 307 References 313 Index 317 1 Introduction In this introductory chapter we elucidate the value of using numerical methods 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. Table 1.1. Examples for the correlation of properties with functionality 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 [...]... summary, it can be stated that related to the application of numerical methods for engineering problems, the following areas are of particular importance: Mathematical modelling of continuum mechanical processes Development and analysis of numerical algorithms Implementation of numerical methods into computer codes Adaption and application of numerical methods to concrete problems Validation, verification,... Parallel computers 105 IBM SP 104 Vector supercomputers Fujitsu NWT 3 10 Cray X-MP 102 101 100 1970 CDC 7600 1980 1990 2000 2010 Fig 1.1 Developments in computer technology (bottom) and numerical methods (top) 1.3 Characterization of Numerical Methods To illustrate the different aspects that play a role when employing numerical simulation techniques for the solution of engineering problems, the general... from engineering science, natural sciences, numerical mathematics, and computer science (see Fig 1.5) are involved An important prerequisite for the successful and efficient use of 10 1 Introduction Engineering science Numerical mathematics c E Numerical ' simulation T Computer science Physics Chemistry Fig 1.5 Interdisciplinarity of numerical simulation of engineering problems numerical simulation methods. .. models, Let us turn to the question of what possibilities are available for obtaining knowledge on the properties of systems, since here, compared to alternative investigation methods, the great potential of numerical methods can be seen In general, the following approaches can be distinguished: theoretical methods, experimental investigations, numerical simulations Theoretical methods, i.e., analytical... not address this topic further here 1.3 Characterization of Numerical Methods 9 After the results are available in an interpretable form, it is essential to inspect them with respect to their quality During all prior steps, errors are inevitably introduced, and it is necessary to get clarity about their quantity (e.g., reference experiments for model error, systematic computations for numerical errors)... investigations are performed 4 1 Introduction by means of numerical methods on computers The advantages of numerical simulations compared to purely experimental investigations are quite obvious: Numerical results often can be obtained faster and at lower costs Parameter variations on the computer usually are easily realizable (e.g., aerodynamics of different car bodies) A numerical simulation often gives... model The continuous problem that result from the modeling – usually systems of differential or integral equations derived in the framework of continuum mechanics – must then be suitably approximated by a discrete problem, i.e., the unknown quantities to be computed have to be represented by a finite 1.3 Characterization of Numerical Methods Engineering problem ? Experimental data Math models Differential... continuously increasing demand for qualified specialists, who are able to apply numerical methods in an efficient way for complex industrial problems An important aspect here is that the possibilities and also the limitations of numerical methods and the corresponding computer software for the respective application area are properly assessed 6 1 Introduction 105 Speed-up 104 Multigrid 103 PCG SOR 2 10 101 100... realistically, for which as near to exact and detailed measuring data are indispensable Thus, both areas, numerics and experiments, must be further developed and ideally used in a complementary way to achieve optimal solutions for the different requirements 1.2 Development of Numerical Methods The possibility of obtaining approximative solutions via the application of finite-difference methods to the partial differential... technology have had a crucial influence on the possibilities of numerical simulation methods, but also the continuous further development of the numerical algorithms has contributed significantly to this This becomes apparent when one contrasts the developments in both areas in recent years as indicated in Fig 1.1 The improved 1.2 Development of Numerical Methods 5 Table 1.2 Development of computing power and . Michael Schäfer Computational Engineering – Introduction to Numerical Methods Michael Schäfer Computational Engineering – Introduction to Numerical Methods With 204 Figures 123 Professor. NWT IBM SP Vector supercomputers Parallel computers Fig. 1.1. Developments in computer technology (bottom) and numerical methods (top) 1.3 Characterization of Numerical Methods To illustrate. 2006 Michael Sch¨afer Contents 1 Introduction 1 1.1 UsefulnessofNumericalInvestigations 1 1.2 Developmentof NumericalMethods 4 1.3 CharacterizationofNumericalMethods 6 2 Modeling of Continuum