First Course in Finite Elements

Preface
This book is written to be an undergraduate and introductory graduate level textbook, depending on whether the more advanced topics appearing at the end of each chapter are covered.

Without the advanced topics, the book is of a level readily comprehensible by junior and senior undergraduate students in science and engineering.With theadvanced topics included, thebookcan serve as the textbook for the first course in finite elements at the graduate level. The text material evolved from over 50 years of combined teaching experience by the authors of graduate and undergraduate finite element courses.

The book focuses on the formulation and application of the finite element method. It differs from other elementary finite element textbooks in the following three aspects:
1. It is introductory andself-contained.Only a modest background in mathematics and physics is needed,all of which is covered in engineering and science curricula in the firsttwo years. Furthermore,many of the specific topics in mathematics, such as matrix algebra, some topics in differential equations, and mechanics and physics, such as conservation laws and constitutive equations, are reviewed prior to their application.
2. It is generic. While most introductory finite element textbooks are application specific, e.g. focusing on linear elasticity, the finite element method in this book is formulated as a general purpose numerical procedure for solving engineering problems governed by partial differential equations. The methodology for obtaining weak forms for the governing equations, a crucial step in the development and understanding of finite elements, is carefully developed. Consequently, students from various engineering and science disciplines will benefit equally from the exposition of the subject.
3. It isa hands-on experience.Thebookintegrates finite element theory, finite element code development and the application of commercial software package. Finite element code development is introduced throughMATLAB exercises and aMATLAB program, whereas ABAQUS is used for demonstrating the use of commercial finite element software.
The material in the book can be covered in a single semester and a meaningful course can be constructed from a subset of the chapters in this book for a one-quarter course. The course material is organized in three chronological units of about one month each: (1) finite elements for one-dimensional problems; (2) finite elements for scalar field problems in two dimensions and (3)finite elements for vectorfield problems in two dimensions and beams. In each case, the weak form is developed, shape functions are described and these ingredients are synthesized to obtain the finite element equations. Moreover, in a web-base chapter, the application of general purpose finite element software using ABAQUS is given for linear heat conduction and elasticity.
Each chapter contains a comprehensive set of homework problems, some of which require programming with MATLAB. Each book comes with an accompanying ABAQUS Student Edition CD, and MATLAB finite element programs can be downloaded from the accompanying website hosted by John Wiley&Sons:www.wileyeurope/college/Fish. Atutorial for theABAQUSexample problems, written by ABAQUS staff, is also included in the book.

Depending on the interests and background of the students, three tracks have been developed:
1. Broad Science and Engineering (SciEng) track
2. Advanced (Advanced) track
3. Structural Mechanics (StrucMech) track
The SciEng track is intended for a broad audience of students in science and engineering. It is aimed at presenting FEM as a versatile tool for solving engineering design problems and as a tool for scientific discovery. Students who have successfully completed this track should be able to appreciate and apply the finite element method for the types of problems described in the book, but more importantly, the SciEng track equips them with a set of skills that will allow them to understand and develop the method for a variety of problems that have not been explicitly addressed in the book. This is our recommended track.

The Advanced track is intended for graduate students as well as undergraduate students with a strong focus on applied mathematics, who are less concerned with specialized applications, such as beams and trusses, but rather with a more detailed exposition of the method. Although detailed convergence proofs in multidimensions are left out, the Advanced track is an excellent stepping stone for students interested in a comprehensive mathematical analysis of the method.
The StrucMech track is intended for students in Civil, Mechanical and Aerospace Engineering whose main interests are in structural and solid mechanics. Specialized topics, such as trusses, beams and energy based principles, are emphasized in this track, while sections dealing with topics other than solid mechanics in multidimensions are classified as optional.

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 Two-Dimensional Truss Systems 27
2.5 Transformation Law 30
2.6 Three-Dimensional Truss Systems 35
References 36
Problems 37
3 Strong andWeak Forms for One-Dimensional Problems 41
3.1 The Strong Form in One-Dimensional 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 One-Dimensional Stress Analysis with Arbitrary Boundary Conditions 58
3.5.1 Strong Form for One-Dimensional Stress Analysis 58
3.5.2 Weak Form for One-Dimensional Stress Analysis 59

3.6 One-Dimensional 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 Two-Point Boundary Value Problem with
Generalized Boundary Conditions 62
3.7.1 Strong Form for Two-Point Boundary Value Problems
with Generalized Boundary Conditions 62
3.7.2 Weak Form for Two-Point 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 One-Dimensional Problems 77
4.1 Two-Node Linear Element 79
4.2 Quadratic One-Dimensional 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 One-Dimensional Problems 93
5.1 Development of Discrete Equation: Simple Case 93
5.2 Element Matrices for Two-Node 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 Two-Point 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 Three-Dimensional Problems 145
6.6 Strong andWeak Forms of Scalar Steady-State
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 Three-Node 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
Three-Node Triangular Element 160
7.3 Four-Node Rectangular Elements 161
7.4 Four-Node 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 Three-Dimensional 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 Two-Dimensional
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 Three-Node 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 Steady-State 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

Total 344 pages 6.3 mb
Download