Table Of ContentINTRODUCTION TO FINITE ELEMENT,
BOUNDARY ELEMENT, AND Pepper INTRODUCTION TO FINITE
Kassab
MESHLESS METHODS
Divo ELEMENT, BOUNDARY
With Applications to Heat Transfer
and Fluid Flow ELEMENT, AND MESHLESS
Darel W. Pepper, Alain J. Kassab, and Eduardo A. Divo METHODS
When students once master the concepts of the fnite element
method (and meshing), it’s not long before they begin to look at other
numerical techniques and applications, especially the boundary Darrell W. Pepper Alain J. Kassab
element and meshless methods (since a mesh is not required).
The expert authors of this book provide a simple explanation of
these three powerful numerical schemes and show how they all
fall under the umbrella of the more universal method of weighted
residuals.
The book is structured in four sections. The frst introductory
section provides the method of weighted residuals development of
fnite differences, fnite volume, fnite element, boundary element,
and meshless methods along with 1D examples of each method.
The following three sections of the book present a more detailed
development of the fnite element method, then progress through the
boundary element method, and end with meshless methods. Each
section serves as a stand-alone description, but it is apparent how
each conveniently leads to the other techniques. It is recommended
Eduardo A. Divo
that the reader begin with the fnite element method, as this serves
as the primary basis for defning the method of weighted residuals.
Computer fles in MathCad, MATLAB, MAPLE and FORTRAN are
available from the fbm.centecorp.com website, along with example
data fles.
With Applications to
Heat Transfer and
Fluid Flow
Two Park Avenue
New York, NY 10016, USA
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INTRODUCTION TO FINITE ELEMENT,
BOUNDARY ELEMENT, AND MESHLESS METHODS
AN INTRODUCTION TO
FINITE ELEMENT,
BOUNDARY ELEMENT,
AND MESHLESS METHODS
With Applications to
Heat Transfer and
Fluid Flow
Darrell W. Pepper
University of Nevada Las Vegas
Alain J. Kassab
University of Central Florida
Eduardo A. Divo
Embry-Riddle Aeronautical University
© 2014, American Society of Mechanical Engineers (ASME), 2 Park Avenue, New York, NY 10016, USA
(www.asme.org)
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DeDication
To the students and masters of these elegant numerical methods, as well as future numerical
methods yet to come.
Table of ConTenTs
Preface ix
Overview xi
the Method of Weighted Residuals (MWR) xi
MWR example Problem: FDM, FVM, FeM, BeM and MM xiv
Finite Difference Method (FDM) – collocation MWR with Local
P olynomial trial Functions xv
Finite Volume Method – Subdomain MWR with Local Polynomial
trial Functions xviii
Finite element Method – Galerkin MWR with Local Polynomial
trial Functions xxi
Boundary element Method – collocation MWR of Boundary
integral equation xxv
Meshless Method – collocation MWR with Global Radial-Basis
F unction (RBF) trial Functions xxviii
References xxxiii
appendix a
Derivation of the 1D Fundamental Solution for T˝ + T = –δ(x – xi) xxxii
appendix B-MatLaB xxxv
appendix c-MaPLe xlix
PART I THE FINITE ELEMENT METHOD 1
Chapter 1 Introduction 3
Chapter 2 Governing Equations 5
2.1 Mass conservation 5
2.2 navier-Stokes 5
2.3 energy conservation 5
2.4 Mass transport 6
2.5 Boundary conditions 6
Chapter 3 The Finite Element Method 7
3.1 error in Finite element approximation 8
3.2 one-Dimensional elements 8
3.2.1 Linear element 8
3.2.2 Quadratic and Higher order elements 9
vi ■ table of contents
3.3 two-Dimensional elements 10
3.3.1 triangular elements 10
3.3.2 Quadrilateral elements 12
3.3.3 isoparametric elements 13
3.4 three-Dimensional elements 17
3.5 Quadrature 18
3.6 Reduced integration 20
3.7 time Dependence 21
3.7.1 the q Method 21
3.7.2 Mass Lumping 22
3.8 Petrov-Galerkin Method 23
3.9 taylor-Galerkin Method 25
Chapter 4 Mesh Generation 27
4.1 Mesh Generation Guidelines 27
4.2 Bandwidth 29
4.3 adaptation 30
4.3.1 Mesh Regeneration 31
4.3.2 element Subdivision 32
4.3.3 adaptation Rules 33
4.3.4 Mesh adaptation example 34
Chapter 5 Fluid Flow Applications 37
5.1 constant-Density Flows 38
5.1.1 Mixed Formulation 38
5.1.2 Fractional Step Method 42
5.1.3 Penalty Function Formulation 43
5.1.4 calculation of Pressure 44
5.1.5 open Boundaries 44
5.2 Free Surface Flows 45
5.3 Flows in Rotating Systems 46
5.4 isothermal Flow Past a circular cylinder 47
5.5 turbulent Flow 48
5.5.1 Large eddy Simulation (LeS) 51
5.5.2 Subgrid-Scale (SGS) Modeling 54
5.6 compressible Flow 55
5.6.1 Supersonic Flow impinging on a cylinder 57
5.6.2 transonic Flow through a Rectangular nozzle 58
Chapter 6 List of Commercial Codes 61
Chapter 7 Conclusion 65
References 66
aPPenDiX a 71
Symbols 71
Subscripts 73
Superscripts 73
aPPenDiX B 75
B.1 Matrix equations and Solution Method 76
B.2 temporal evolution of the Semi-implicit Scheme 76
B.2.1 Momentum 76
B.2.2 continuity 77
B.2.3 energy 78
table of contents ■ vii
B.2.4 t urbulent Kinetic energy and Specific
Dissipation Rate (k-w) 78
B.2.5 Matrix Formulation 79
References 80
PART II THE BOUNDARY ELEMENT METHOD 81
Chapter 1 Introduction 83
Chapter 2 BEM Fundamentals 85
2.1 a Familiar example: Green’s third identity for
Potential Problems 85
2.2 the 2D Heat conduction Problem 87
2.3 G enerating the integral equation: Weighting Function and
Green’s Second identity 88
2.4 a nalytical Solution: Green’s Function Method and the
auxiliary Problem 90
2.5 n umerical Solution: the BeM and the Boundary integral
equation 93
appendix a Derivation of the Green’s Function for the 2D
Problem in a Square 106
appendix B D erivation of the Green’s Free Space (Fundamental)
Solution to the Laplace equation 107
Chapter 3 Numerical Implementation of the BEM 109
3.1 two-Dimensional Boundary elements 109
3.2 three-Dimensional Boundary elements 115
3.3 adaptive Quadrature in 3D 119
3.4 numerical Solution of the BeM equations 121
a ppendix a conjugate Gradient and GMReS MatHcaD
Pseudo-codes 123
Chapter 4 Steady Heat Conduction with Variable Heat Conductivity 129
4.1 nonlinear thermal conductivity 129
4.2 anisotropic Heat conductivity 131
4.3 non-Homogenous thermal conductivity 133
Chapter 5 Heat Conduction in Media with Energy Generation 139
5.1 Special Form of Generation Leading to contour integrals 139
5.2 Use of Particular Solutions 141
5.3 the Dual Reciprocity Boundary element Method 142
Chapter 6 A pplications of the BEM to Heat Transfer and
Inverse Problems 149
6.1 axi-Symmetric Problems 149
6.2 Heat conduction in thin Plates and extended Surfaces 151
6.3 conjugate Heat transfer 154
6.4 Large-Scale Heat transfer 157
6.5 non-Homogeneous Heat conduction: Generalized Bie 162
6.6 inverse Problems applications of the BeM 166
Chapter 7 Conclusion 173
References 173
viii ■ table of contents
PART III THE MESHLESS METHOD 179
Chapter 1 Introduction and Background 181
Chapter 2 Radial-Basis Function (RBF) Interpolation 183
Chapter 3 The Localized Collocation Meshless Method (LCMM) Framework 187
Chapter 4 The Moving Least-Squares (MLS) Smoothing Scheme 193
Chapter 5 The Finite-Differencing Enhanced LCMM 195
Chapter 6 Upwinding Schemes 199
6.1 one-Dimensional LcMM Upwinding test 200
6.2 t wo-Dimensional LcMM Upwinding test for
an inclined Wave 203
6.3 t wo-Dimensional LcMM Upwinding test for
a turning Wave 205
Chapter 7 Automatic Point Distribution 207
Chapter 8 Parallelization 209
Chapter 9 Applications 211
9.1 incompressible Fluid Flow and conjugate Heat transfer 211
9.1.1 Decaying Vortex Flow 215
9.1.2 Lid-Driven Flow in a Square cavity 218
9.1.3 air Jet into a Square cavity 220
9.1.4 conjugate Heat transfer between Parallel Plates 221
9.1.5 c onjugate Heat transfer Flow over a
Rectangular obstruction 223
9.1.6 conjugate Film-cooling Heat transfer 225
9.1.7 Flow over a cylinder 227
9.1.8 Steady Blood Flow through a Femoral Bypass 229
9.1.9 Pulsatile Blood Flow through a Femoral Bypass 233
9.2 natural convection 235
9.2.1 Buoyancy-Driven Flow in a Square cavity 236
9.2.2 B uoyancy-Driven Flow of Liquid aluminum in a
Rectangular cavity 238
9.3 turbulent Fluid Flows 239
9.3.1 turbulent Flow over a Flat Plate 241
9.3.2 turbulent Flow over a Backward-Facing Step 242
9.4 compressible Fluid Flows 243
9.4.1 Subsonic and Supersonic Smooth expanding Diffuser 245
9.4.2 characteristic nozzle Flow 247
9.4.3 Subsonic and Supersonic Flow Past an airfoil 248
9.4.4 turbulent Wake Flow 251
9.5 two-Phase Flow 252
9.5.1 Dam-Breaking test of two-Phase Flow Formulation 253
9.6 Solid Mechanics and thermo-elasticity 254
9.6.1 cantilever Beam under constant Distributed Load 256
9.6.2 c ortical Bone with Fixation element under
Bending Moment 256
9.7 Porous Media Flow and Poro-elasticity 258
9.7.1 Rectangular Poro-elastic Medium 260
9.7.2 air Flow coupled with Poro-elastic Balloon 260
9.7.3 coupled tracheo-Bronchial Poro-elastic Lung 262
9.7.4 Groundwater Flow through a Poro-elastic Levee 263
Chapter 10 Conclusions 265
References 266
PReFace
This book stems from our experiences in teaching numerical methods to both engineering
students and experienced, practicing engineers in industry. The emphasis in this book deals
with finite element, boundary element, and meshless methods. Much of the material comes
from courses we have conducted over many years at our institutions, including AIAA home
study and ASME short courses presented over several decades, as well as from the sug-
gestions and recommendations of our colleagues and students. There are numerous books
on applied numerical methods, many of them being finite element and boundary element
textbooks available in the literature today. However, there are very few books dealing with
meshless methods, especially those showing how nearly all of these numerical schemes
originate from the fundamental principles of the method of weighted residuals. We find that
when students once master the concepts of the finite element method (and meshing), it’s
not long before they begin to look at more advanced numerical techniques and applications,
especially the boundary element and meshless methods (since a mesh is not required). Our
intent in this book is to provide a simple explanation of these three powerful numerical
schemes, and to show how they all fall under the umbrella of the more universal method of
weighted residuals approach.
The book is divided into three sections, beginning with the finite element method,
then progressing through the boundary element method, and finally ending with the mesh-
less method. Each section serves as a stand-alone description, but it is apparent to see how
each conveniently leads to the other techniques. We recommend that the reader begin with
the finite element method, as this serves as the primary basis for defining the method of
weighted residuals.
We begin by introducing the basic fundamentals of the finite element method using
simple examples. Particular attention is given to the development of the discrete set of al-
gebraic equations, beginning with simple one-dimensional problems that can be solved by
inspection, and continuing to two- and three-dimensional elements. Once these principles
are grasped, we then introduce the concept of boundary elements, and the relative ease with
which one reduces the dimensionality of a problem (a great relief when solving large prob-
lems, or problems with infinite domain boundaries). The boundary element technique is a
natural extension of the finite element method, and becomes greatly appreciated by users.
While the method has some limitations regarding the wide range of applications afforded
by the finite element technique, it is still a very popular and useful method. It is finding use
in crack growth and related applications dealing with structural mechanics, and couples
nicely with finite element meshes.
The more recent introduction of meshless methods is rapidly becoming a method now
being used by practitioners of both finite element and boundary element methods. The
method is simple to grasp, and simple to implement. The power of the method is becom-
ing more appreciated with time. The meshless method has been shown to yield solutions
with accuracies comparable to finite element methods employing an extensive number of
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