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Donea J., Huerta A. Finite Element Methods for Flow Problems

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Donea J., Huerta A. Finite Element Methods for Flow Problems
Chichester: John Wiley & Sons Ltd, 2003. — 358 p. — ISBN 0471496669.
In recent years there have been significant developments in the development of stable and accurate finite element procedures for the numerical approximation of a wide range of fluid mechanics problems. Taking an engineering rather than a mathematical bias, this valuable reference resource details the fundamentals of stabilised finite element methods for the analysis of steady and time-dependent fluid dynamics problems. Organised into six chapters, this text combines theoretical aspects and practical applications and offers coverage of the latest research in several areas of computational fluid dynamics.
Introduction and preliminaries
Finite elements in fluid dynamics
Subjects covered
Kinematical descriptions of the flow field
Lagrangian and Eulerian descriptions
ALE description of motion
The fundamental ALE equation
Time derivative of integrals over moving volumes
The basic conservation equations
Mass equation
Momentum equation
Internal energy equation
Total energy equation
ALE form of the conservation equations
Closure of the initial boundary value problem
Basic ingredients of the finite element method
Mathematical preliminaries
Trial solutions and weighting functions
Compact integral forms
Strong and weak forms of a boundary value problem
Finite element spatial discretization
Steady transport problems
Problem statement
Strong form
Weak form
Galerkin approximation
Piecewise linear approximation in 1D
Analysis of the discrete equation
Piecewise quadratic approximation in ID
Analysis of the discrete equations
Early Petrov—Galerkin methods
Upwind approximation of the convective term
First finite elements of upwind type
The concept of balancing diffusion
Stabilization techniques
The SUPG method
The Galerkin/Least-squares method
The stabilization parameter
Other stabilization techniques and new trends
Finite increment calculus
Bubble Junctions and wavelet approximations
The variational multiscale method
Applications and solved exercises
Construction of a bubble function method
One-dimensional transport
Convection—diffusion across a source term
Convection—diffusion skew to the mesh
Convection—diffusion-reaction in 2D
The Hemker problem
Unsteady convective transport
Problem statement
The method of characteristics
The concept of characteristic lines
Properties of the linear convection equation
Methods based on the characteristics
Classical time and space discretization techniques
Time discretization
Galerkin spatial discretization
Stability and accuracy analysis
Analysis of stability by Fourier techniques
Analysis of classical time-stepping schemes
The modified equation method
Taylor—Galerkin Methods
The need for higher-order time schemes
Third-order explicit Taylor—Galerkin method
Fourth-order explicit leap-frog method
Two-step explicit Taylor—Galerkin methods
An introduction to monotonicity-preserving schemes
Least-squares-based spatial discretization
Least-squares approach for the 6 family of methods
Taylor least-squares method
The discontinuous Galerkin method
Space—time formulations
Time-discontinuous Galerkin formulation
Time-discontinuous least-squares formulation
Applications and solved exercises
Propagation of a cosine profile
Travelling wave package
The rotating cone problem
Propagation of a steep front Compressible Flow Problems
Compressible Flow Problems
Nonlinear hyperbolic equations
Scalar equations
Weak solutions and entropy condition
Time and space discretization
The Euler equations
Strong form of the conservation equations
The quasi-linear form of the Euler equations
Basic properties of the Euler equations
Boundary conditions
Spatial discretization techniques
Galerkin formulation
Upwind-type discretizations
Numerical treatment of shocks
Early artificial diffusion methods
High-resolution methods
Nearly incompressible flows
Fluid-structure interaction
Acoustic approximation
Nonlinear transient dynamic problems
llustrative examples
Solved exercises
One-step Taylor—Galerkin solution of Burgers' equation
The shock tube problem
Unsteady convection—diffusion problems
Problem statement
Time discretization procedures
Classical methods
Fractional-step methods
High-order time-stepping schemes
Spatial discretization procedures
Galerkin formulation of the semi-discrete scheme
Galerkin formulation of 0 family methods
Galerkin formulation of explicit Pade schemes
Galerkin formulation of implicit multistage schemes
Stabilization of the semi-discrete scheme
Stabilization of multistage schemes
Stabilized space-time formulations
Solved exercises
Convection-diffusion of a Gaussian hill
Transient rotating pulse
Steady rotating pulse problem
Nonlinear propagation of a step
Burgers' equation in 1D
Two-dimensional Burgers' equation
Appendix Least-squares in transient/relaxation problems
Viscous incompressible flows
Basic concepts
Strain rate and spin tensors
The stress tensor in a Newtonian fluid
The Navier—Stokes equations
Main issues in incompressible flow problems
Trial solutions and weighting functions
Stationary Stokes problem
Formulation in terms of Cauchy stress
Formulation in terms of velocity and pressure
Galerkin formulation
Matrix problem
Solvability condition and solution procedure
The LBB compatibility condition
Some popular velocity—pressure couples
Stabilization of the Stokes problem
Penalty method
Steady Navier—Stokes problem
Weak form and Galerkin formulation
Matrix problem
Unsteady Navier—Stokes equations
Weak formulation and spatial discretization
Stabilized finite element formulation
Time discretization by fractional-step methods
Applications and solved exercices
Stokes flow with analytical solution
Cavity flow problem
Plane jet simulation
Natural convection in a square cavity
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