A First Course in Finite Elements

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 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