Table Of ContentComputational Fluid and Solid Mechanics
SeriesEditor:
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Advisors:
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AvailableVolumes
D.Chapelle,K.J.Bathe
TheFiniteElementAnalysisofShells–Fundamentals
2003
D.Drikakis,W.Rider
High-ResolutionMethodsforIncompressibleandLow-SpeedFlows
2005
M.Kojic,K.J.Bathe
InelasticAnalysisofSolidsandStructures
2005
E.N.Dvorkin,M.B.Goldschmit
NonlinearContinua
2005
B.Q.Li
DiscontinuousFiniteElementsinFluidDynamicsandHeatTransfer
2006
Ben Q. Li
Discontinuous
Finite Elements in
Fluid Dynamics and
Heat Transfer
With167Figures
123
Author
BenQ.Li,PhD
ProfessorofMechanicalEngineering
SchoolofMechanicalandMaterialsEngineering
WashingtonStateUniversity
Pullman,Washington,USA
BritishLibraryCataloguinginPublicationData
Li,BenQ.
Discontinuousfiniteelementsinfluiddynamicsandheat
transfer.-(Computationalfluidandsolidmechanics)
1.Fluiddynamics-Mathematicalmodels2.Heat-
Transmission-Mathematicalmodels3.Finiteelementmethod
4.Galerkinmethods
I.Title
532’00151825
ISBN-10:1852339888
LibraryofCongressControlNumber:2005935664
ComputationalFluidandSolidMechanicsSeriesISSN1860-482X
ISBN-10: 1-85233-988-8 e-ISBN 1-84628-205-5 Printedonacid-freepaper
ISBN-13: 978-1-85233-988-3
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To my children: Thomas, Katherine and Lauren
Preface
Over the past several years, significant advances have been made in developing the
discontinuous Galerkin finite element method for applications in fluid flow and
heat transfer. Certain unique features of the method have made it attractive as an
alternative for other popular methods such as finite volume and finite elements in
thermal fluids engineering analyses.
This book is written as an introductory textbook on the discontinuous finite
element method for senior undergraduate and graduate students in the area of
thermal science and fluid dynamics. It also can be used as a reference book for
researchers and engineers who intend to use the method for research in
computational fluid dynamics and heat transfer. A good portion of this book has
been used in a course for computational fluid dynamics and heat transfer for senior
undergraduate and first year graduate students. It also has been used by some
graduate students for self-study of the basics of discontinuous finite elements.
This monograph assumes that readers have a basic understanding of
thermodynamics, fluid mechanics and heat transfer and some background in
numerical analysis. Knowledge of continuous finite elements is not necessary but
will be helpful. The book covers the application of the method for the simulation of
both macroscopic and micro/nanoscale fluid flow and heat transfer phenomena.
Background information on the subjects that are not covered in standard textbooks
is also presented. Examples of different levels of difficulty are given, which help
readers understand the concept and capability of the discontinuous finite element
method and the computational procedures involved in the use of the method.
Chapter 1 of the book presents a brief review of fundamental laws and
mathematical equations for thermal and fluid systems including both
incompressible and compressible fluids and for generic boundary and initial
conditions.
In Chapter 2, different approaches to formulate discontinuous finite element
solutions for boundary and initial value problems are discussed. The numerical
procedures for the discontinuous finite element formulation, elemental calculations
and element-by-element solution are discussed in detail through simple, elementary
and illustrative examples. The advantages and disadvantages of the discontinuous
finite element formulations are also given.
viii Preface
Chapter 3 is concerned with the development of shape functions and elemental
calculations for discontinuous finite elements. The Lagrangian basis functions,
hierarchical shape functions, spectral elements and special elements are discussed.
A majority of discontinuous finite element formulations use unstructured meshes
made of triangular/tetrahedral elements. Construction of these elements from the
Lagrangian interpolation functions or from coordinate transformations of the
existing finite elements is presented. Numerical integration and elemental
calculations are also given.
Starting with Chapter 4, we discuss the application of the discontinuous finite
element methods for the solution of thermal and fluids problems. Chapter 4 deals
with the heat conduction problems. Heat conduction is the first mode of heat
transfer and the simple mathematical form of the governing equation serves as a
good entry point for a numerical analysis of thermal problems. We present the
detailed discontinuous formulations for both steady state and transient heat
conduction problems. The stability analysis and selection of numerical fluxes for
discontinuous finite element solution of diffusive systems are discussed.
Convection-dominant problems are discussed in Chapter 5, which covers pure
convection, diffusion-convection, and inviscid and viscous nonlinear convection.
The stability analysis and selection of various convection fluxes are also discussed.
Both steady state and transient problems are considered. A good portion of the
discussion is devoted to the numerical stability analysis and control of numerical
oscillations.
Incompressible flow problems are discussed in Chapter 6, where the
discontinuous finite element formulations for both isothermal and non-isothermal
systems are given. The formulations are further used in the later chapters.
Chapter 7 is concerned about computational compressible flows using the
discontinuous finite element method. The numerical procedure for both Euler and
Navier–Stokes equations is presented. The use of various numerical fluxes, flux
limiter and slope limiters in both 1-D and multidimensions is also discussed.
Chapter 8 discusses the discontinuous Galerkin boundary element method for
the numerical solution of external radiation problems. Most numerical books on
thermal and fluid flow analysis either have no or give very little coverage of the
topic of radiation heat transfer. We present the discontinuous concept and its
numerical implementation with the selection of kernel functions for external
radiation calculations. Shadowing algorithms are discussed for detecting the
internal blockages in 2-D, axisymmetric and 3-D enclosures.
Internal radiation occurs in many high temperature processes and is governed
by the radiative transfer equation. The solution of the radiative transfer equation
governing internal radiation is discussed in Chapter 9. This type of equation is
difficult to solve using continuous finite elements but is almost ideal for
discontinuous finite element computations. Detailed procedures for the numerical
solution of the internal radiation problems are given. The analytical formulae for
typical elements used for the discontinuous finite element formulation of 1-D, 2-D
and 3-D simulations are given. Numerical examples include both simple pure
internal radiation systems and complex thermal systems in which multiple heat
transfer modes occur.
Preface ix
Chapter 10 discusses the use of the discontinuous finite element method for the
solution of free and moving boundary problems. Both moving and fixed grid
methods are discussed and the discontinuous finite element based algorithms for
the solution of these problems are given. The concepts of the methods for moving
boundary problems such as the volume of fluid method, the marker-and-cell
method and the level set method are discussed. Incorporation of these fixed grid
methods and moving grid methods into discontinuous finite element solvers using
both structured and unstructured meshes are presented. The discontinuous finite
element formulation of the phase field model, which has emerged as a powerful
tool for modeling moving boundary problems at local scales, is also given, along
with the 1-D, 2-D and 3-D examples of the evolution of very complex moving
boundaries in phase change moving boundary problems.
The use of the discontinuous finite elements for the simulation of microscale
and nanoscale heat transfer and fluid flow problems is discussed in Chapter 11.
Some of the recently developed models describing the microscale heat transfer
phenomena have mathematical forms that are particularly suited for the
discontinuous finite element formulations. The numerical solution of non-Fourier
heat transfer equations and the lattice Boltzmann equation is also given.
Chapter 12 deals with the discontinuous finite element solution of the thermal
and fluid flow problems under the influence of applied electromagnetic fields. The
basic theory of electromagnetism is presented. Numerical examples are given on
the discontinuous finite element simulation of electroosmotive flows in
microchannels, microwave heating and electrically-induced droplet deformation.
This book is printed in shades of grey. Color versions of some of the figures in
this book can be downloaded in a pdf from springer.com. Computer codes used for
some of the calculations may also be downloaded from the same web site.
Both the theory and applications of the discontinuous finite element method are
still evolving and writing a book on this particular subject proved to be a major
task. It was impossible to accomplish this task without assistance from various
sources. I am most grateful to those whose contributions have made this
monograph possible. I am indebted to Professors C.-W. Shu at Brown University
and P. Castillo at University of Puerto Rico for helpful discussions on the
mathematical theory of the discontinuous Galerkin finite element method. My
appreciation also goes to Professor K. J. Bathe at Massachusetts Institute of
Technology for sharing some of his latest work on the mixed finite element and the
ALE methods and for stimulating comments. Professor H.-M. Yin of Department
of Pure and Applied Mathematics, Washington State University, provided
constructive comments on the basic theory of error analyses. I wish to thank my
current and former graduate students, in particular, Drs. X. Ai, Y. Shu, B. Xu and
X. Cui, who have helped in checking the examples and the exercises. I am also
grateful to Ms. K. Faunce for her assistance in preparing the manuscript and
formatting the final layout of the book, and to Messrs. A. Doyle and O. Jackson of
Springer-Verlag and Ms. S. Moosdorf of LE-TeX for their continuous support.
Ben Q. Li
January, 2005
Contents
1 Introduction.......................................................................................................1
1.1 Conservation Laws for a Continuum Medium...........................................1
1.1.1 Conservation of Mass.....................................................................1
1.1.2 Conservation of Momentum...........................................................2
1.1.3 Conservation of Energy..................................................................2
1.1.4 Constitutive Relations.....................................................................3
1.2 Governing Equations in Terms of Primitive Variables..............................5
1.2.1 Vector Form....................................................................................5
1.2.2 Component Form in Cartesian Coordinates....................................5
1.2.3 Component Form in Cylindrical Coordinates.................................6
1.2.4 Summary........................................................................................8
1.3 Species Transport Equations......................................................................8
1.4 Governing Equations in Translating and Rotating Frames
of Reference...............................................................................................9
1.5 Boundary and Initial Conditions................................................................9
1.5.1 General Boundary Conditions......................................................10
1.5.2 Free Boundary Conditions............................................................11
1.5.3 Moving Interface Conditions........................................................12
1.5.4 Phase Change Conditions.............................................................12
1.6 Governing Equations for Flows Through Porous Media.........................13
1.7 Governing Equations in Conservation Form...........................................15
Exercises...........................................................................................................17
References.........................................................................................................18
2 Discontinuous Finite Element Procedures....................................................21
2.1 The Concept of Discontinuous Finite Elements.......................................22
2.1.1 Weakly Imposed Cross-element Continuity.................................23
2.1.2 Numerical Boundary Fluxes for Discontinuity.............................25
2.1.3 Boundary Constraint Minimization..............................................26
2.1.4 Treatment of Discontinuity for Non-conservative Systems..........27
2.1.5 Transient Problems.......................................................................28
2.2 Discontinuous Finite Element Formulation.............................................29
xii Contents
2.2.1 Integral Formulation.....................................................................29
2.2.2 Time Integration...........................................................................30
2.3 Solution Procedures.................................................................................31
2.4 Advantages and Disadvantages of Discontinuous Finite Element
Formulations............................................................................................32
2.4.1 Advantages...................................................................................32
2.4.2 Disadvantages...............................................................................33
2.5 Examples..................................................................................................34
Exercises...........................................................................................................42
References.........................................................................................................42
3 Shape Functions and Elemental Calculations..............................................45
3.1 Shape Functions.......................................................................................46
3.1.1 1-D Shape Functions....................................................................46
3.1.2 2-D Shape Functions....................................................................50
3.1.2.1 Triangular Elements.......................................................50
3.1.2.2 Quadrilateral Elements...................................................54
3.1.3 3-D Shape Functions....................................................................58
3.1.3.1 Tetrahedral Elements.....................................................58
3.1.3.2 Hexahedral Elements.....................................................60
3.2 Construction of Special Elements............................................................65
3.2.1 Non-standard Elements.................................................................65
3.2.2 Construction of Element Shape Functions by
Node Collapsing...........................................................................66
3.2.3 Spectral Elements.........................................................................67
3.3 Hierarchical Shape Functions..................................................................69
3.3.1 1-D Hierarchical Correction.........................................................69
3.3.2 Canonical Square and Cubic Elements.........................................71
3.3.3 Triangular and Tetrahedral Elements...........................................73
3.3.4 Obtaining Hierarchical Elements Through Coordinate
Transformations............................................................................77
3.3.5 Orthogonal Mass Matrix Construction.........................................78
3.4 Interpolation Error Analysis....................................................................80
3.4.1 Hilbert Space and Various Error Measures..................................80
3.4.2 Interpolation Error Analysis for 1-D Elements.............................84
3.4.3 Interpolation Error Analysis for 2-D/3-D Elements.....................87
3.5 Numerical Integration..............................................................................88
3.5.1 1-D Numerical Integration............................................................88
3.5.2 2-D and 3-D Numerical Integration..............................................90
3.5.3 Integration for Triangular and Tetrahedral Elements...................91
3.6 Elemental Calculations............................................................................93
3.6.1 Domain Calculations....................................................................93
3.6.2 Boundary Calculations.................................................................98
Exercises.........................................................................................................102
References.......................................................................................................103
