Li Ben Q. Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer

Springer, 2006. — 586 p. — (Computational Fluid and Solid Mechanics). — ISBN10: 1-85233-988-8, ISBN13: 978-1-85233-988-3The discontinuous finite element method (also known as the discontinuous Galerkin method) embodies the advantages of both finite element and finite difference methods. It can be used in convection-dominant applications while maintaining geometric flexibility and higher local approximations throught the use of higher-order elements. Element-by element connection propagates the effect of boundary conditions and the local formulation obviates the need for global matrix assembly. All of this adds up to a method which is not unduly memory-intensive and uniquely useful for working with computational dynamics, heat transfer and fluid flow calculations. Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer offers its readers a systematic and practical introduction to the discontinuous finite element method. It moves from a brief review of the fundamental laws and equations governing thermal and fluid systems, through a discussion of different approaches to the formulation of discontinuous finite element solutions for boundary and initial value problems, to their applicaton in a variety of thermal-system and fluid-related problems, including: heat conduction problems; convection-dominant problems; compressible and incompressible flows; external radiation problems; internal radiation and radiative transfer; free- and moving-boundary problems; micro- and nanoscale heat transfer and fluid flow; thermal fluid flow under the influence of applied magnetic fields. Mesh generation and adaptivity, parellelization algorithms and a priori and a posteriori error analysis are also introduced and explained, rounding out a comprehensive review of the subject. Each chapter features worked examples and exercises illustrating situations ranging from simple benchmarks to practical engineering questions. This textbook is written to form the foundations of senior undergraduate and graduate learning and also provides scientists, applied mathematicians and research engineers with a thorough treatment of basic concepts, specific techniques and methods for the use of discontinuous Galerkin methods in computational fluid dynamics and heat transfer applicationsConservation Laws for a Continuum Medium.
Conservation of Mass.
Conservation of Momentum.
Conservation of Energy.
Constitutive Relations.
Governing Equations in Terms of Primitive Variables.
Vector Form.
Component Form in Cartesian Coordinates.
Component Form in Cylindrical Coordinates.Species Transport Equations.
Governing Equations in Translating and Rotating Frames of Reference.
Boundary and Initial Conditions.
General Boundary Conditions.
Free Boundary Conditions.
Moving Interface Conditions.
Phase Change Conditions.
Governing Equations for Flows Through Porous Media.
Governing Equations in Conservation Form.
Exercises.Discontinuous Finite Element Procedures.
The Concept of Discontinuous Finite Elements.
Weakly Imposed Cross-element Continuity.
Numerical Boundary Fluxes for Discontinuity.
Boundary Constraint Minimization.
Treatment of Discontinuity for Non-conservative Systems.
Transient Problems.
Discontinuous Finite Element Formulation.
Integral Formulation.
Time Integration.
Solution Procedures.
Advantages and Disadvantages of Discontinuous Finite Element.
Formulations.
Advantages.
Disadvantages.
Examples.
Exercises.Shape Functions and Elemental Calculations.
Shape Functions.
D Shape Functions.
D Shape Functions.
Triangular Elements.
Quadrilateral Elements.
D Shape Functions.
Tetrahedral Elements.
Hexahedral Elements.
Construction of Special Elements.
Non-standard Elements.
Construction of Element Shape Functions by.
Node Collapsing.
Spectral Elements.
Hierarchical Shape Functions.
D Hierarchical Correction.
Canonical Square and Cubic Elements.
Triangular and Tetrahedral Elements.
Obtaining Hierarchical Elements Through Coordinate.
Transformations.
Orthogonal Mass Matrix Construction.
Interpolation Error Analysis.
Hilbert Space and Various Error Measures.
Interpolation Error Analysis for 1-D Elements.
Interpolation Error Analysis for 2-D/3-D Elements.
Numerical Integration.
D Numerical Integration.
D and 3-D Numerical Integration.
Integration for Triangular and Tetrahedral Elements.
Elemental Calculations.
Domain Calculations.
Boundary Calculations.
Exercises.Conduction Heat Transfer and Potential Flows.
D Steady State Heat Conduction.
Steady State Heat Conduction in Multidimensions.
D Transient Heat Conduction.
Alternating Upwinding Scheme.
Central Fluxes.
Unified Representation.
Numerical Implementation.
Runge Kutta Time Integration.
Computational Procedures.
Transient Heat Conduction in Multidimensions.
Potential Flows and Flows in Porous Media.
Selection of Numerical Fluxes.
Stability for Steady StateProblems.
Stability and Numerical Fluxes.
Discontinuous and Mixed Finite Element.
Formulations.
Stability for Time Dependent Problems.
Numerical Fluxes for Transient Problems.
Stability Analysis Using Matrix.
Fourier Analysis.
Exercises.Convection-dominated Problems.
Pure Convection Problems.
D Pure Convection.
Method of Characteristics.
Discontinuous Finite Element Formulation.
Pure Convection in Multidimensions.
Stability Analysis L.
Stability – Integral Analysis L.
Stability – Discretized Analysis.
Fourier Analysis.
Steady State Convection-diffusion.
D Problem.
Origin of Oscillatory Stability.
Steady Convection-Diffusion in Multidimensions.
Transient Convection-diffusion.
Multidimensional Problem.
Stability Analysis L.
Stability – Integral Analysis L.
Stability – Discretized Analysis.
Fourier Analysis.
Nonlinear Problems.
D Inviscid Burgers’ Equation.
Basic Considerations.
Discontinuous Finite Element Formulation.
Multidimensional Inviscid Burgers’ Equation.
Discontinuous Finite Element Formulation.
Characteristic Decomposition.
Higher Order Approximations and TVD Formulations.
Concept of Total Variation Diminishing.
Flux Limiters.
Slope Limiters.
TVD-Runge Kutta Schemes.
Viscous Burgers’ Equations.
D Burgers’ Equation.
D Viscous Burgers’ Equation.
Exercises.Incompressible Flows.
Primitive Variable Approach.
Fractional Step (Projection) Approach.
Vorticity and Stream Function Approach.
Coupled Flow and Heat Transfer.
Exercises.Compressible Fluid Flows.
D Compressible Flows.
Governing Equations.
Basic Properties of the Euler Equations.
The Rankine–Hugoniot Conditions.
D Riemann Solver – Exact Solution.
D Riemann Solver – Approximate Solution.
Discontinuous Finite Element Formulation.
Low Order (Finite Volume) Approximations.
The Godunov Scheme.
The Roe Scheme.
High Order TVD Approximations.
Numerical Examples.
Multidimensional Inviscid Compressible Flows.
Governing Equations.
Basic Properties of the Split 3-D Euler Equations.
Discontinuous Finite Element Formulation.
Multidimensional Compressible Viscous Flows.
ALE Formulation.
ALE Kinematic Description.
Conservation of Mass.
Conservation of Momentum.
Conservation of Energy.
Summary of ALE Equations.
Constitutive Relations.
ALE Description of Compressible Flows.
Discontinuous Finite Element Formulation.
Exercises.External Radiative Heat Transfer.
Integral Equation for Surface Radiation Exchanges.
Governing Equation.
Kernel Functions.
Discontinuous Galerkin Finite Element Formulation.
Shadowing Algorithms.
Shadowing Algorithm for 2-D Geometry.
Shadowing Algorithm for Axisymmetric Configurations.
Shadowing Algorithm for 3-D Geometry.
Coupling with Other Heat Transfer Calculations.
Direct Coupling.
Iterative Coupling.
Numerical Examples.
Exercises.Radiative Transfer In Participating Media.
Governing Equation and Boundary Conditions.
Radiative Transfer Equation.
Boundary Conditions.
Approximation Methods.
The Discrete Ordinate Method.
The Spherical Harmonics Method.
Discontinuous Finite Element Formulation.
Numerical Implementation.
D Calculations.
D Calculations.
Integration of the Source Term.
The Emitting Contribution.
The Scattering Contribution.
Radiation In Systems of Axisymmetry.
Governing Equation in Cylindrical Coordinates.
Volume Integration.
Surface Integration Over * p.
Integration Over * M.
Mapping.
Treatment of the Emitting and Scattering Term.
Use of RTE for External Radiation Calculations.
Coupling of the Discontinuous Method with Other Methods.
Constant Element Approximation.
Exercises.Free and Moving Boundary Problems.
Free and Moving Boundaries.
Basic Relations for a Curved Surface.
Description of a Surface.
Differential and Integral Relations for Curved Surfaces.
Physical Constraints at a Moving Boundary.
Kinematic Conditions at a Moving Boundary.
Stress Condition at a Moving Interface.
Thermal Conditions at a Moving Interface.
Moving Grids vs. Fixed Grids for Numerical Solutions.
Moving Grid Methods.
Moving Boundaries Between Fluids.
Moving Phase Boundaries.
Solid Liquid Phase Transition.
Liquid Vapor Phase Transition.
Fixed Grid methods.
Volume of Fluid Method.
Structured Mesh.
Unstructured Mesh.
The Marker-and-Cell Method.
The Level Set Method.
Fixed Grid Methods for Phase Change Problems.
Flow-based Methods.
Enthalpy-based Methods.
Phase Field Modeling of Moving Boundaries.
Basic Ideas of Phase Field Models.
Governing Equations for Interfacial Phenomena.
Discontinuous Finite Element Formulation.
Phase Field Modeling of Microstructure Evolution.
Governing Equations.
Discontinuous Formulation.
Numerical Examples.
Flow and Orientation Effects on Microstructure Evolution.
Flow Effects on Microstructure Evolution.
Microstructure Evolution During Polycystalline.
Solidification.
Exercises.Micro and Nano Scale Fluid Flow and Heat Transfer.
Micro Scale Heat Conduction.
Two-temperature Equations.
Phonon Scattering Equation.
Phonon Radiative Transfer Equation.
Discontinuous Finite Element Formulation.
Micro and Nano Fluid Flow and Heat Transfer.
The Boltzmann Transport Equation and Numerical Solution.
The Boltzmann Integral-Differential Equation.
Numerical Solution of the Boltzmann Transport Equation.
The Boltzman BGK Equation and Numerical Solution.
The Boltzmann BGK Equation.
Discontinuous Finite Element Formulation.
The Lattice Boltzman Equation and Numerical Solution.
Derivation of the Lattice Boltzmann Equation.
Boundary Conditions.
Bounce Back Boundary Conditions (No Slip).
Symmetric Boundary Conditions (Free Slip).
Inflow and Outflow Boundary Conditions (No Gradient).
Force Field Conditions.
Moving Wall Conditions.
Discontinuous Finite Element Formulation.
Exercises.Fluid Flow and Heat Transfer in Electromagnetic Fields.
Maxwell Equations and Boundary Conditions.
Maxwell Equations.
Boundary Conditions.
Interface Boundary Condition.
Perfect Conducting Surface.
Impedance Condition.
Sommerfield Radiation Condition.
Symmetry Boundary Condition.
Maxwell Stresses and Energy Sources.
Discontinuous Formulation of the Maxwell Equations.
Solution in Time Domain.
Solution in Frequency Domain.
Solution in Other Forms.
Electroosmotic Flows.
Governing Equations.
Discontinuous Finite Element Formulation.
Microwave Heating.
Electrically Deformed Free Surfaces.
Compressible flows in Magnetic Fields.
Exercises.