The Finite Element Method for Fluid Dynamics

8th Edition - November 20, 2024
Authors: R. L. Taylor, P. Nithiarasu
Language: English
Hardback ISBN:
9 7 8 - 0 - 3 2 3 - 9 5 8 8 6 - 8
eBook ISBN:
9 7 8 - 0 - 3 2 3 - 9 5 8 8 7 - 5

The Finite Element Method for Fluid Dynamics provides a comprehensive introduction to the application of the finite element method in fluid dynamics. The book begins with a useful… Read more

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The Finite Element Method for Fluid Dynamics provides a comprehensive introduction to the application of the finite element method in fluid dynamics. The book begins with a useful summary of all relevant partial differential equations, progressing to the discussion of convection stabilization procedures, steady and transient state equations, and numerical solution of fluid dynamic equations.

In this expanded eighth edition, the book starts by explaining the character-based split (CBS) scheme, followed by an exploration of various other methods, including SUPG/PSPG, space-time, and VMS methods. Emphasising the fundamental knowledge, mathematical, and analytical tools necessary for successful implementation of computational fluid dynamics (CFD), The Finite Element Method for Fluid Dynamics stands as the authoritative introduction of choice for graduate level students, researchers, and professional engineers.

Title of Book
Cover image
Title page
Table of Contents
Copyright
Dedication
List of figures
List of tables
Preface
Chapter 1: The equations of fluid dynamics
1.1. General remarks and classification of fluid dynamics problems
1.2. The governing equations of fluid dynamics
1.2.1. Stresses in fluids
1.2.2. Constitutive relations for fluids
1.2.3. Mass conservation
1.2.4. Momentum conservation: dynamic equilibrium
1.2.5. Energy conservation and equation of state
1.2.6. Boundary conditions
1.2.7. Navier–Stokes and Euler equations
1.3. Inviscid, incompressible flow
1.3.1. Velocity potential solution
1.4. Incompressible (or nearly incompressible) flows
1.5. Concluding remarks
Chapter 2: The finite element approximation
2.1. Introduction
2.2. Numerical solutions: weak forms, weighted residual and finite element approximation
2.2.1. Strong and weak forms
2.2.2. A finite volume approximation
2.3. Concluding remarks
Chapter 3: Convection dominated problems – finite element approximations to the convection–diffusion–reaction equation
3.1. Introduction
3.2. The steady-state problem in one dimension
3.2.1. General remarks
3.2.2. Petrov–Galerkin methods for upwinding in one dimension
3.2.3. Balancing diffusion in one dimension
3.2.4. A variational principle in one dimension
3.2.5. Galerkin least-squares approximation (GLS) in one dimension
3.2.6. Subgrid scale (SGS) approximation
3.2.7. The finite increment calculus (FIC) for stabilising the convective–diffusion equation in one dimension
3.2.8. Higher-order approximations
3.3. The steady-state problem in two (or three) dimensions
3.3.1. General remarks
3.3.2. Streamline (upwind) Petrov–Galerkin weighting (SUPG)
3.3.3. Galerkin least squares (GLS) and finite increment calculus (FIC) in multi-dimensional problems
3.4. Steady state – concluding remarks
3.5. Transients – introductory remarks
3.5.1. Mathematical background
3.5.2. Possible discretization procedures
3.6. Characteristic-based methods
3.6.1. Mesh updating and interpolation methods
3.6.2. Characteristic–Galerkin procedures
3.6.3. A simple explicit characteristic–Galerkin procedure
3.6.4. Boundary conditions for convection dominated problems
3.7. Taylor–Galerkin procedures for scalar variables
3.8. Steady-state condition
3.9. Non-linear waves and shocks
3.10. Boundary conditions for pure convection problems
3.11. Weak Dirichlet conditions for convection–diffusion problems
3.12. Summary and concluding remarks
Chapter 4: Fractional step methods: the characteristic-based split (CBS) algorithm for compressible and incompressible flows
4.1. Introduction
4.2. Non-dimensional form of the governing equations
4.3. Characteristic-based split (CBS) algorithm
4.3.1. The split – general remarks
4.3.2. The split – temporal discretization
4.3.3. Spatial discretization and solution procedure
4.3.4. Mass diagonalization (lumping)
4.4. Explicit, semi-implicit and nearly implicit forms
4.4.1. Fully explicit form
4.4.2. Semi-implicit form
4.4.3. Quasi-(nearly) implicit form
4.4.4. Evaluation of time step limits. Local and global time steps
4.5. Artificial compressibility and dual time stepping
4.5.1. Artificial compressibility for steady problems
4.5.2. Artificial compressibility in transient problems (dual time stepping)
4.6. ‘Circumvention’ of the Babuška–Brezzi (BB) restrictions
4.7. A single-step version
4.8. Splitting error
4.8.1. Elimination of first-order pressure error
4.9. Boundary conditions
4.9.1. Fictitious boundaries
4.9.2. Real boundaries
4.9.3. Application of real boundary conditions in the discretization using the CBS algorithm
4.10. The performance of two- and single-step algorithms on an inviscid problem
4.11. Performance of dual time stepping to remove pressure error
4.12. Concluding remarks
Chapter 5: Incompressible Newtonian laminar flows
5.1. Introduction and the basic equations
5.2. CBS algorithms for incompressible flow
5.2.1. The fully explicit artificial compressibility-based matrix-free form
5.2.2. The semi-implicit form
5.2.3. Quasi-implicit solution
5.3. Slow flows – mixed and penalty formulations
5.3.1. Analogy with incompressible elasticity
5.3.2. Mixed and penalty discretization
5.4. Concluding remarks
Chapter 6: Incompressible non-Newtonian flows
6.1. Introduction
6.2. Non-Newtonian flows – metal and polymer forming
6.2.1. Non-Newtonian flows including viscoplasticity and plasticity
6.2.2. Steady-state problems of forming
6.2.3. Transient problems with changing boundaries
6.2.4. Elastic springback and viscoelastic fluids
6.3. Viscoelastic flows
6.3.1. Conservation equations
6.4. Direct displacement approach to transient metal forming
6.5. Concluding remarks
Chapter 7: Free surface flows
7.1. Introduction
7.2. Free surface flows – solution methods
7.3. Lagrangian method
7.4. Eulerian methods
7.4.1. Mesh updating or regeneration methods
7.4.2. Hydrostatic adjustment
7.4.3. Numerical examples using mesh regeneration methods
7.5. Arbitrary Lagrangian–Eulerian (ALE) method
7.5.1. ALE implementation
7.6. Concluding remarks
Chapter 8: Buoyancy driven flows
8.1. Introduction
8.2. Buoyancy driven flows – changes to equations
8.3. Finite element solution
8.4. Flow in an enclosure and channel
8.5. Concluding remarks
Chapter 9: Compressible high-speed gas flow
9.1. Introduction
9.2. The governing equations
9.3. Boundary conditions – subsonic and supersonic flow
9.3.1. Euler equation
9.3.2. Navier–Stokes equations
9.4. Numerical approximations
9.5. Shock capture
9.5.1. Second derivative based methods
9.5.2. Residual based methods
9.6. Variable smoothing
9.7. Some preliminary examples for the Euler equation
9.8. Three-dimensional inviscid examples in steady state
9.8.1. The recasting of element formulation in an edge form
9.8.2. Multigrid approaches
9.8.3. Parallel computation
9.9. Viscous flows
9.10. Boundary layer–inviscid Euler solution coupling
9.11. Concluding remarks
Chapter 10: Adaptive mesh refinement
10.1. Introduction
10.2. The h-refinement processes
10.2.1. Mesh enrichment
10.2.2. Remeshing
10.3. Second gradient (curvature) based refinement
10.3.1. Local patch interpolation. Superconvergent values
10.3.2. Estimation of second derivatives at nodes
10.3.3. Element elongation
10.4. First derivative (gradient) based refinement
10.5. Choice of variables
10.6. Adaptive remeshing for compressible flows
10.6.1. Solution of the flow pattern around a complete aircraft
10.7. Viscous compressible flow problems
10.8. Concluding remarks
Chapter 11: Turbulent flows
11.1. Introduction
11.1.1. Time averaging
11.1.2. Relationship between κ, ϵ and νT
11.2. Treatment of incompressible turbulent flows
11.2.1. Reynolds-averaged Navier–Stokes
11.2.2. One-equation models
11.2.3. Two-equation models
11.2.4. Non-dimensional form of the governing equations
11.3. Shortest distance to a solid wall
11.4. Solution procedure for turbulent flow equations
11.5. Treatment of compressible flows
11.5.1. Mass-weighted (Favre) time averaging
11.6. Large eddy simulation
11.7. Detached eddy simulation and monotonically integrated LES
11.8. Direct numerical simulation
11.9. Concluding remarks
Chapter 12: Flow and heat transport in porous media
12.1. Introduction
12.2. A generalised porous medium flow approach
12.2.1. Dimensionless scales
12.3. Discretization procedure
12.3.1. Semi- and quasi-implicit forms
12.4. Forced convection
12.5. Natural convection
12.5.1. Constant porosity medium
12.6. Concluding remarks
Chapter 13: Shallow-water problems
13.1. Introduction
13.2. The basis of the shallow-water equations
13.3. Numerical approximation
13.4. Examples of application
13.4.1. Transient one-dimensional problems – a performance assessment
13.4.2. Two-dimensional periodic tidal motions
13.4.3. Tsunami waves
13.4.4. Steady-state solutions
13.5. Drying areas
13.6. Shallow-water transport
13.7. Concluding remarks
Chapter 14: Long and medium waves
14.1. Introduction and equations
14.2. Waves in closed domains – finite element models
14.3. Difficulties in modelling surface waves
14.4. Bed friction and other effects
14.5. The short-wave problem
14.6. Waves in unbounded domains (exterior surface wave problems)
14.6.1. Background to wave problems
14.6.2. Wave diffraction
14.6.3. Incident waves, domain integrals and nodal values
14.7. Unbounded problems
14.8. Local non-reflecting boundary conditions
14.8.1. Sponge Layers, Perfectly Matched Layers or PMLs
14.9. Infinite elements
14.10. Mapped periodic (unconjugated) infinite elements
14.10.1. Introducing the wave component
14.11. Ellipsoidal-type infinite elements of Burnett and Holford
14.12. Wave envelope (or conjugated) infinite elements
14.13. Accuracy of infinite elements
14.13.1. Other applications
14.14. Trefftz-type infinite elements
14.15. Convection and wave refraction
14.16. Transient problems
14.17. Linking to exterior solutions (or DtN mapping)
14.17.1. Linking to boundary integrals
14.17.2. Linking to series solutions
14.18. Three-dimensional effects in surface waves
14.18.1. Large-amplitude water waves
14.18.2. Cnoidal and solitary waves
14.18.3. Stokes waves
14.19. Concluding remarks
Chapter 15: Short waves
15.1. Introduction
15.2. Background
15.3. Errors in wave modelling
15.4. Short wave modelling
15.5. Transient solution of electromagnetic scattering problems
15.6. Finite elements incorporating wave shapes
15.6.1. Shape functions using products of polynomials and waves
15.6.2. Shape functions using sums of polynomials and waves
15.6.3. The discontinuous enrichment method
15.6.4. Ultra-weak formulation
15.6.5. Trefftz-type finite elements for waves
15.7. Refraction
15.7.1. Wave speed refraction
15.7.2. Refraction caused by flows
15.8. Spectral finite elements for waves
15.9. Discontinuous Galerkin finite elements (DGFE)
15.10. Concluding remarks
Chapter 16: Fluid–structure interaction
16.1. Introduction
16.2. One-dimensional fluid–structure interaction
16.2.1. Equations
16.2.2. Characteristic analysis
16.2.3. Boundary conditions
16.2.4. Solution method
16.2.5. Some results
16.3. Multidimensional problems
16.3.1. Equations and discretization
16.3.2. Segregated approach
16.3.3. Mesh moving procedures
16.4. Concluding remarks
Chapter 17: Biofluid dynamics – blood flow
17.1. Introduction
17.2. Flow in human arterial system
17.2.1. Heart
17.2.2. Reflections
17.2.3. Aortic valve
17.2.4. Vessel branching
17.2.5. Terminal vessels
17.2.6. Numerical solution
17.3. Image based subject-specific flow modelling
17.3.1. Image segmentation
17.3.2. Geometrical potential force
17.3.3. Numerical solution, initial and boundary conditions
17.3.4. Domain discretisation
17.3.5. Flow solution
17.4. Concluding remarks
Chapter 18: Data-driven methods
18.1. Introduction
18.2. Purely data driven machine learning models
18.3. Data-driven models with offline finite element calculations
18.4. Direct data-driven finite element calculations
18.5. Concluding remarks
Chapter 19: Computer implementation of the CBS algorithm
19.1. Introduction
19.1.1. Python users
19.2. The data input module
19.2.1. Mesh data – nodal coordinates and connectivity
19.2.2. Boundary data
19.2.3. Other necessary data and flags
19.2.4. Preliminary subroutines and checks
19.3. Solution module
19.3.1. Time step
19.3.2. Shock capture
19.3.3. CBS algorithm. Steps
19.3.4. Boundary conditions
19.3.5. Solution of simultaneous equations – semi-implicit form
19.3.6. Different forms of energy equation
19.3.7. Convergence to steady state
19.4. Output module
Appendix A: Self-adjoint differential equations
Appendix B: Non-conservative form of Navier–Stokes equations
Appendix C: Computing drag force and stream function
C.1. Drag calculation
C.2. Stream function
Appendix D: Integration formulae
D.1. Linear triangles
D.2. Linear tetrahedron
Appendix E: Convection–diffusion equations: vector-valued variables
E.1. The Taylor–Galerkin method used for vector-valued variables
E.2. Two-step predictor–corrector methods. Two-step Taylor–Galerkin operation
E.2.1. Multiple wave speeds
Appendix F: Edge-based finite element formulation
Appendix G: Multigrid method
Appendix H: Boundary layer–inviscid flow coupling
Appendix I: Mass-weighted averaged turbulence transport equations
I.1. Spalart–Allmaras turbulence model
I.2. κ−ω turbulence model
Appendix J: Introduction to neural networks
J.1. Neural networks for regression
J.1.1. Forward pass
J.1.2. Backpropagation and automatic gradient
Index

R. L. Taylor

Professor R.L. Taylor has more than 60 years of experience in the modelling and simulation of structures and solid continua including eighteen years in industry. He is Professor of the Graduate School and the Emeritus T.Y. and Margaret Lin Professor of Engineering at the University of California, Berkeley and also Corporate Fellow at Dassault Systèmes Americas Corp. in Johnston, Rhode Island. In 1991 he was elected to membership in the US National Academy of Engineering in recognition of his educational and research contributions to the field of computational mechanics. Professor Taylor is a Fellow of the US Association for Computational Mechanics – USACM (1996) and a Fellow of the International Association of Computational Mechanics – IACM (1998). He has received numerous awards including the Berkeley Citation, the highest honour awarded by the University of California, Berkeley, the USACM John von Neumann Medal, the IACM Gauss–Newton Congress Medal and a Dr.-Ingenieur ehrenhalber awarded by the Technical University of Hannover, Germany. Professor Taylor has written several computer programs for finite element analysis of structural and non-structural systems, one of which, FEAP, is used world-wide in education and research environments. A personal version, FEAPpv, available on GitHub, is incorporated into this book.

Affiliations and expertise

Emeritus Professor of Engineering, University of California, Berkeley, USA

P. Nithiarasu

P. Nithiarasu is Professor at Zienkiewicz Institute for Modelling, Data and AI and Associate Dean for Research, Innovation and Impact, Faculty of Science and Engineering, Swansea University. Previously he has served as the Head of Zienkiewicz Centre for Computational Engineering, Deputy Head of College of Engineering and Dean of Academic Leadership. He was awarded the Zienkiewicz silver medal from the ICE London in 2002, the ECCOMAS Young Investigator award in 2004, and the prestigious EPSRC Advanced Fellowship in 2006.

Affiliations and expertise

Professor, Zienkiewicz Institute for Modelling, Data and AI and Associate Dean for Research, Innovation and Impact, Faculty of Science and Engineering, Swansea University

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The Finite Element Method for Fluid Dynamics

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R. L. Taylor

P. Nithiarasu