The finite element method : basic concepts and applications with MATLAB, MAPLE, and COMSOL

628 Pages·2017·5.844 MB·English

by  Heinrich, Juan C., Pepper, Darrell W

The Finite
Element Method
Basic Concepts and Applications with MATLAB®, MAPLE, and COMSOL
THIRD EDITION
Series in Computational and Physical Processes
in Mechanics and Thermal Sciences
Series Editors
W. J. Minkowycz
Mechanical and Industrial Engineering
University of Illinois at Chicago
Chicago, Illinois
E. M. Sparrow
Mechanical Engineering
University of Minnesota
Minneapolis, Minnesota

The Finite Element Method: Basic Concepts and Applications, Darrell W. Pepper and
Juan C. Heinrich
Numerical Simulation of Heat Exchangers: Advances in Numerical Heat Transfer,
Volume V, edited by W. J. Minkowycz, E. M. Sparrow, J.P Abraham,
and J. M. Gorman
Nanoparticle Heat Transfer and Fluid Flow, Volume IV, edited by W. J. Minkowycz,
E. M. Sparrow, and J. P. Abraham
Advances in Numerical Heat Transfer, Volume III, edited by W. J. Minkowycz,
and E. M. Sparrow
Advances in Numerical Heat Transfer, Volume II, edited by W. J. Minkowycz,
and E. M. Sparrow
Advances in Numerical Heat Transfer, Volume I, edited by W. J. Minkowycz
Series in Computational and Physical Processes
in Mechanics and Thermal Sciences
The Finite
Series Editors
W. J. Minkowycz
Mechanical and Industrial Engineering Element Method
University of Illinois at Chicago
Chicago, Illinois
E. M. Sparrow
Mechanical Engineering
University of Minnesota Basic Concepts and Applications with MATLAB®, MAPLE, and COMSOL
Minneapolis, Minnesota

The Finite Element Method: Basic Concepts and Applications, Darrell W. Pepper and THIRD EDITION
Juan C. Heinrich
Numerical Simulation of Heat Exchangers: Advances in Numerical Heat Transfer,
Volume V, edited by W. J. Minkowycz, E. M. Sparrow, J.P Abraham,
and J. M. Gorman
Nanoparticle Heat Transfer and Fluid Flow, Volume IV, edited by W. J. Minkowycz,
E. M. Sparrow, and J. P. Abraham
Advances in Numerical Heat Transfer, Volume III, edited by W. J. Minkowycz,
and E. M. Sparrow
Advances in Numerical Heat Transfer, Volume II, edited by W. J. Minkowycz,
and E. M. Sparrow
Advances in Numerical Heat Transfer, Volume I, edited by W. J. Minkowycz Darrell W. Pepper • Juan C. Heinrich
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks
does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion
of MATLAB® software or related products does not constitute endorsement or sponsorship by The
MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.
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To our parents:
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Carlos and Ruby Heinrich

Contents
Preface, xiii
Authors, xvii
CHAPTER 1 ◾ Introduction 1
1.1 BACKGROUND 1
1.2 SHORT HISTORY 2
1.3 ORIENTATION 4
1.4 CLOSURE 6
REFERENCES 7
CHAPTER 2 ◾ Method of Weighted Residuals and Galerkin
Approximations 11
2.1 BACKGROUND 11
2.2 CLASSICAL SOLUTIONS 12
2.3 THE “WEAK” STATEMENT 14
2.4 CLOSURE 26
EXERCISES 26
REFERENCES 29
CHAPTER 3 ◾ Finite Element Method in One Dimension 31
3.1 BACKGROUND 31
3.2 SHAPE FUNCTIONS 31
3.2.1 Linear Elements 32
3.2.2 Quadratic Elements 35
3.2.3 Cubic Elements 37
vii
viii ◾ Contents
3.3 STEADY CONDUCTION EQUATION 39
3.3.1 Galerkin Formulation 39
3.3.2 Variable Conduction and Boundary Convection 45
3.4 AXISYMMETRIC HEAT CONDUCTION 55
3.5 NATURAL COORDINATE SYSTEM 60
3.6 TIME DEPENDENCE 77
3.6.1 Spatial Discretization 79
3.6.2 Time Discretization 81
3.7 MATRIX FORMULATION 90
3.8 SOLUTION METHODS 93
3.9 CLOSURE 102
PROBLEMS 103
REFERENCES 109
CHAPTER 4 ◾ Two-Dimensional Triangular Element 111
4.1 BACKGROUND 111
4.2 THE MESH 112
4.3 SHAPE FUNCTIONS 116
4.3.1 Linear Shape Functions 116
4.3.2 Quadratic Shape Functions 122
4.4 AREA COORDINATES 123
4.5 NUMERICAL INTEGRATION 133
4.6 CONDUCTION IN A TRIANGULAR ELEMENT 138
4.7 STEADY-STATE CONDUCTION WITH BOUNDARY
CONVECTION 143
4.8 THE AXISYMMETRIC CONDUCTION EQUATION 147
4.9 THE QUADRATIC TRIANGULAR ELEMENT 150
4.10 TIME-DEPENDENT DIFFUSION EQUATION 166
4.11 BANDWIDTH 176
4.12 MASS LUMPING 181
4.13 CLOSURE 183
Contents ◾ ix
EXERCISES 183
REFERENCES 191
CHAPTER 5 ◾ Two-Dimensional Quadrilateral Element 193
5.1 BACKGROUND 193
5.2 ELEMENT MESH 193
5.3 SHAPE FUNCTIONS 195
5.3.1 Bilinear Rectangular Element 195
5.3.2 Quadratic Rectangular Element 197
5.4 NATURAL COORDINATE SYSTEM 200
5.5 NUMERICAL INTEGRATION USING GAUSSIAN
QUADRATURES 211
5.6 STEADY-STATE CONDUCTION EQUATION 215
5.7 STEADY-STATE CONDUCTION WITH BOUNDARY
CONVECTION 226
5.8 THE QUADRATIC QUADRILATERAL ELEMENT 239
5.9 TIME-DEPENDENT DIFFUSION 253
5.10 COMPUTER PROGRAM EXERCISES 254
5.11 CLOSURE 257
EXERCISES 258
REFERENCES 262
CHAPTER 6 ◾ Isoparametric Two-Dimensional Elements 263
6.1 BACKGROUND 263
6.2 NATURAL COORDINATE SYSTEM 264
6.3 SHAPE FUNCTIONS 266
6.3.1 Bilinear Quadrilateral 266
6.3.2 Eight-Noded Quadratic Quadrilateral 268
6.3.3 Linear Triangle 269
6.3.4 Quadratic Triangle 269
6.3.5 Directional Cosines 270