Textbook
A First Course in Finite ElementsMay 2007, ©2007

Table of Contents
Preface xi
1 Introduction 1
1.1 Background 1
1.2 Applications of Finite elements 7
References 9
2 Direct Approach for Discrete Systems 11
2.1 Describing the Behavior of a Single Bar Element 11
2.2 Equations for a System 15
2.2.1 Equations for Assembly 18
2.2.2 Boundary Conditions and System Solution 20
2.3 Applications to Other Linear Systems 24
2.4 TwoDimensional Truss Systems 27
2.5 Transformation Law 30
2.6 ThreeDimensional Truss Systems 35
References 36
Problems 37
3 Strong andWeak Forms for OneDimensional Problems 41
3.1 The Strong Form in OneDimensional Problems 42
3.1.1 The Strong Form for an Axially Loaded Elastic Bar 42
3.1.2 The Strong Form for Heat Conduction in One Dimension 44
3.1.3 Diffusion in One Dimension 46
3.2 TheWeak Form in One Dimension 47
3.3 Continuity 50
3.4 The Equivalence Between theWeak and Strong Forms 51
3.5 OneDimensional Stress Analysis with Arbitrary Boundary Conditions 58
3.5.1 Strong Form for OneDimensional Stress Analysis 58
3.5.2 Weak Form for OneDimensional Stress Analysis 59
3.6 OneDimensional Heat Conduction with Arbitrary Boundary Conditions 60
3.6.1 Strong Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions 60
3.6.2 Weak Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions 61
3.7 TwoPoint Boundary Value Problem with Generalized Boundary Conditions 62
3.7.1 Strong Form for TwoPoint Boundary Value Problems with Generalized Boundary Conditions 62
3.7.2 Weak Form for TwoPoint Boundary Value Problems with Generalized Boundary Conditions 63
3.8 Advection–Diffusion 64
3.8.1 Strong Form of Advection–Diffusion Equation 65
3.8.2 Weak Form of Advection–Diffusion Equation 66
3.9 Minimum Potential Energy 67
3.10 Integrability 71
References 72
Problems 72
4 Approximation of Trial Solutions,Weight Functions and Gauss Quadrature for OneDimensional Problems 77
4.1 TwoNode Linear Element 79
4.2 Quadratic OneDimensional Element 81
4.3 Direct Construction of Shape Functions in One Dimension 82
4.4 Approximation of theWeight Functions 84
4.5 Global Approximation and Continuity 84
4.6 Gauss Quadrature 85
Reference 90
Problems 90
5 Finite Element Formulation for OneDimensional Problems 93
5.1 Development of Discrete Equation: Simple Case 93
5.2 Element Matrices for TwoNode Element 97
5.3 Application to Heat Conduction and Diffusion Problems 99
5.4 Development of Discrete Equations for Arbitrary Boundary Conditions 105
5.5 TwoPoint Boundary Value Problem with Generalized Boundary Conditions 111
5.6 Convergence of the FEM 113
5.6.1 Convergence by Numerical Experiments 115
5.6.2 Convergence by Analysis 118
5.7 FEM for Advection–Diffusion Equation 120
References 122
Problems 123
6 Strong andWeak Forms for Multidimensional Scalar Field Problems 131
6.1 Divergence Theorem and Green’s Formula 133
6.2 Strong Form 139
6.3 Weak Form 142
6.4 The Equivalence BetweenWeak and Strong Forms 144
6.5 Generalization to ThreeDimensional Problems 145
6.6 Strong andWeak Forms of Scalar SteadyState Advection–Diffusion in Two Dimensions 146
References 148
Problems 148
7 Approximations of Trial Solutions,Weight Functions and Gauss Quadrature for Multidimensional Problems 151
7.1 Completeness and Continuity 152
7.2 ThreeNode Triangular Element 154
7.2.1 Global Approximation and Continuity 157
7.2.2 Higher Order Triangular Elements 159
7.2.3 Derivatives of Shape Functions for the ThreeNode Triangular Element 160
7.3 FourNode Rectangular Elements 161
7.4 FourNode Quadrilateral Element 164
7.4.1 Continuity of Isoparametric Elements 166
7.4.2 Derivatives of Isoparametric Shape Functions 166
7.5 Higher Order Quadrilateral Elements 168
7.6 Triangular Coordinates 172
7.6.1 Linear Triangular Element 172
7.6.2 Isoparametric Triangular Elements 174
7.6.3 Cubic Element 175
7.6.4 Triangular Elements by Collapsing Quadrilateral Elements 176
7.7 Completeness of Isoparametric Elements 177
7.8 Gauss Quadrature in Two Dimensions 178
7.8.1 Integration Over Quadrilateral Elements 179
7.8.2 Integration Over Triangular Elements 180
7.9 ThreeDimensional Elements 181
7.9.1 Hexahedral Elements 181
7.9.2 Tetrahedral Elements 183
References 185
Problems 186
8 Finite Element Formulation for Multidimensional Scalar Field Problems 189
8.1 Finite Element Formulation for TwoDimensional Heat Conduction Problems 189
8.2 Verification and Validation 201
8.3 Advection–Diffusion Equation 207
References 209
Problems 209
9 Finite Element Formulation for Vector Field Problems – Linear Elasticity 215
9.1 Linear Elasticity 215
9.1.1 Kinematics 217
9.1.2 Stress and Traction 219
9.1.3 Equilibrium 220
9.1.4 Constitutive Equation 222
9.2 Strong andWeak Forms 223
9.3 Finite Element Discretization 225
9.4 ThreeNode Triangular Element 228
9.4.1 Element Body Force Matrix 229
9.4.2 Boundary Force Matrix 230
9.5 Generalization of Boundary Conditions 231
9.6 Discussion 239
9.7 Linear Elasticity Equations in Three Dimensions 240
Problems 241
10 Finite Element Formulation for Beams 249
10.1 Governing Equations of the Beam 249
10.1.1 Kinematics of Beam 249
10.1.2 Stress–Strain Law 252
10.1.3 Equilibrium 253
10.1.4 Boundary Conditions 254
10.2 Strong Form toWeak Form 255
10.2.1 Weak Form to Strong Form 257
10.3 Finite Element Discretization 258
10.3.1 Trial Solution andWeight Function Approximations 258
10.3.2 Discrete Equations 260
10.4 Theorem of Minimum Potential Energy 261
10.5 Remarks on Shell Elements 265
Reference 269
Problems 269
11 Commercial Finite Element Program ABAQUS Tutorials 275
11.1 Introduction 275
11.1.1 SteadyState Heat Flow Example 275
11.2 Preliminaries 275
11.3 Creating a Part 276
11.4 Creating a Material Definition 278
11.5 Defining and Assigning Section Properties 279
11.6 Assembling the Model 280
11.7 Configuring the Analysis 280
11.8 Applying a Boundary Condition and a Load to the Model 280
11.9 Meshing the Model 282
11.10 Creating and Submitting an Analysis Job 284
11.11 Viewing the Analysis Results 284
11.12 Solving the Problem Using Quadrilaterals 284
11.13 Refining the Mesh 285
11.13.1 Bending of a Short Cantilever Beam 287
11.14 Copying the Model 287
11.15 Modifying the Material Definition 287
11.16 Configuring the Analysis 287
11.17 Applying a Boundary Condition and a Load to the Model 288
11.18 Meshing the Model 289
11.19 Creating and Submitting an Analysis Job 290
11.20 Viewing the Analysis Results 290
11.20.1 Plate with a Hole in Tension 290
11.21 Creating a New Model 292
11.22 Creating a Part 292
11.23 Creating a Material Definition 293
11.24 Defining and Assigning Section Properties 294
11.25 Assembling the Model 295
11.26 Configuring the Analysis 295
11.27 Applying a Boundary Condition and a Load to the Model 295
11.28 Meshing the Model 297
11.29 Creating and Submitting an Analysis Job 298
11.30 Viewing the Analysis Results 299
11.31 Refining the Mesh 299
Appendix 303
A.1 Rotation of Coordinate System in Three Dimensions 303
A.2 Scalar Product Theorem 304
A.3 Taylor’s Formula with Remainder and the Mean Value Theorem 304
A.4 Green’s Theorem 305
A.5 Point Force (Source) 307
A.6 Static Condensation 308
A.7 Solution Methods 309
Direct Solvers 310
Iterative Solvers 310
Conditioning 311
References 312
Problem 312
Index 313
Author Information
Dr. Fish has 20 years of experience (both industry and academia) in the field of multiscale computational engineering, which bridges the gap between modeling, simulation and design of products based on multiscale principles. Dr. Fish has published over one hundred journal articles and book chapters. Two of his papers, one on development of multilevel solution techniques for large scale systems presented at the 1995 ASME International Computers in Engineering Conference and the second one, on fatigue crack growth in aging aircraft presented at the 1993 Structures, Structural Dynamics, and Materials Conference have won the Best Paper Awards. Dr. Fish is a recipient of 2005 USACM Computational Structural Mechanics Award given “in recognition of outstanding and sustained contributions to the broad field of Computational Structural Mechanics”. He is editor of the International Journal for Multiscale Computational Engineering.
Ted Belytschko, Department of Mechanical Engineering2145 North Sheridan Road, Northwestern University, Evanston, IL 60208311
Ted Belytschko's main interests lie in the development of computational methods for engineering problems. He has developed explicit finite element methods that are widely used in crashworthiness analysis and virtual prototyping. He is also interested in engineering education, and he chaired the committee that developed the "Engineering First Program" at Northwestern. He obtained his B.S. and Ph.D. at Illinois Institute of Technology in 1965 and 1968, respectively. He has been at Northwestern since 1977 where he is currently Walter P. Murphy Professor and McCormick Distinguished Professor of Computational Mechanics. He is coauthor of the book NONLINEAR FINITE ELEMENTS FOR CONTINUA AND STRUCTURES with W.K.Liu and B. Moran (published by Wiley and in the third printing) and he has edited more than 10 other books. In January 2004, he was listed as the 4th most cited researcher in engineering. He is past Chairman of the Engineering Mechanics Division of the ASCE, the Applied Mechanics Division of ASME, past President of USACM, and a member of the National Academy of Engineering (elected in 1992) and the American Academy of Arts and Sciences (elected in 2002). He is the editor of Numerical Methods in Engineering.
The Wiley Advantage
 The book comes with a copy of ABAQUS Student Edition finite element software which retails for $99. (www.abaqus.com)
 ABAQUS is the most widely used commercial FE software globally and is used in most engineering departments in industry and universities. (Abaqus will be strongly marketing the book)
 It takes a generic approach and so can be used by students from various disciplines in both engineering and science. Finite Elements are a mandatory course on most undergraduate engineering courses.
 A practical course for both lecturers constructing and planning a finite element module, and for students using the text in private study. It includes quizzes and a solution manual.
 The accompanying website includes ABAQUS Student Edition, Matlab data and programs, the solutions manual and instructor resources.
 Accompanied by a book companion website housing supplementary material that can be found at http://www.wileyeurope.com/college/Fish
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