Basic Theory of Finite Element Method, livre ebook

This book is an introduction to the basic theory of the finite element method. In addition, it formulates a scheme for finite element computation of elasticity problems and provides typical computer programs for implementing the finite element method. The basic concepts and key techniques involved in finite element computation are presented to enable readers to quickly grasp the knowledge system of the finite element method. Main contents include: (1) introduction of finite element method; (2) fundamentals of elasticity mechanics; (3) weak form of equivalent integration; (4) elements and shape functions; (5) isoparametric element and numerical integration; (6) finite element computation scheme of elasticity problems; (7) solutions of linear algebraic equations; (8) error estimation and adaptive analysis; (9) programs of finite element method. This book is intended for undergraduate and postgraduate students majoring in engineering mechanics, civil engineering, and mining engineering. It provides also reference for researchers in related fields. To facilitate teaching and learning, this book provides selected homework exercises and programming projects with well-designed solutions. Online course and the source code of the related programs are also available.

Preface..................................................... V

About the Author............................................ VII

CHAPTER 1

Introduction of Finite ElementMethod............................. 1

1.1 Development Process of Finite ElementMethod ................. 1

1.2 Computation Procedure of Finite ElementMethod ............... 3

1.3 Main Contents of the Book................................. 5

1.4 Exercises...............................................6

CHAPTER 2

Fundamentals of Elasticity Mechanics.............................. 7

2.1 Displacements........................................... 8

2.2 Strains................................................ 9

2.3 Stresses................................................9

2.4 GeometricEquations...................................... 11

2.5 Constitutive Equations.................................... 12

2.6 Equilibrium Equations.................................... 14

2.6.1 Three-Dimensional Problems.......................... 14

2.6.2 Two-Dimensional Plane Stress and StrainProblems......... 14

2.6.3 Two-Dimensional Axisymmetric Problems................ 15

2.7 Boundary Conditions..................................... 15

2.8Exercises............................................... 16

CHAPTER 3

Weak Form of Equivalent Integration.............................. 17

3.1 Weak Form of Equivalent Integration forDifferential Equations ..... 17

3.2 Weak Form of One-Dimensional ElasticityProblems .............. 17

3.3 Finite Element Computation Based on WeakForm ............... 20

3.3.1 Galerkin Method................................... 20

3.3.2 Finite Element Computation.......................... 24

3.4 Global Assembly from One-DimensionalElements................ 26

3.5 Treatments on Boundary Conditions.......................... 28

3.6Exercises............................................... 32

CHAPTER 4

Elements and Shape Functions................................... 33

4.1 One-Dimensional Lagrange Element.......................... 33

4.1.1 Linear Element with Two Nodes....................... 33

4.1.2 Higher-Order Lagrange Element....................... 34

4.1.3 Quadratic Lagrange Element.......................... 36

4.2 Two-Dimensional Triangle Element........................... 37

4.2.1 Triangle with ThreeNodes............................ 37

4.2.2 Higher-Order Triangle Element........................ 40

4.2.3 Quadratic Triangle Element........................... 43

4.2.4 Cubic Triangle Element.............................. 43

4.3 Two-Dimensional Rectangle Element......................... 43

4.3.1 Linear Rectangle Element with Four Nodes............... 43

4.3.2 Higher-Order Rectangle Element....................... 46

4.3.3 Quadratic Rectangle Element......................... 47

4.4 Three-Dimensional TetrahedronElement....................... 48

4.4.1 Linear Tetrahedron Element with FourNodes ............. 48

4.4.2 Higher-Order Tetrahedron Element..................... 51

4.4.3 Quadratic TetrahedronElement........................ 51

4.4.4 Cubic Tetrahedron Element........................... 52

4.5 Three-Dimensional Hexahedron Element....................... 52

4.5.1 Hexahedron with Eight Nodes......................... 52

4.5.2 Higher-Order Hexahedron Element..................... 54

4.5.3 Quadratic Hexahedron Element........................ 55

4.6Exercises............................................... 56

CHAPTER 5

Isoparametric Element and NumericalIntegration..................... 59

5.1 Isoparametric Element.................................... 59

5.1.1 One-Dimensional Isoparametric LagrangeElement .......... 59

5.1.2 Two-Dimensional Isoparametric TriangleElement .......... 60

5.1.3 Two-Dimensional Isoparametric RectangleElement ......... 63

5.1.4 Three-Dimensional IsoparametricTetrahedron Element ...... 64

5.1.5 Three-Dimensional IsoparametricHexahedron Element ...... 67

5.1.6 Requirements of Isoparametric Element.................. 69

5.2 Numerical Integration..................................... 70

5.2.1 One-Dimensional Integration for LagrangeElement ......... 70

5.2.2 Two-Dimensional Integration for TriangleElement ......... 73

5.2.3 Two-Dimensional Integration for RectangleElement ........ 74

5.2.4 Three-Dimensional Integration forTetrahedron Element ..... 75

5.2.5 Three-Dimensional Integration forHexahedron Element...... 76

5.2.6 Required Order of Numerical Integration................. 78

5.3Exercises............................................... 78

X Contents

CHAPTER 6

Finite Element Computation Scheme of ElasticityProblems ............. 81

6.1 Weak Form for General Elasticity Problems.................... 81

6.2 Finite Element Method for SolvingElasticity Problems ............ 84

6.3 Global Assembly from High-DimensionalElements ............... 86

6.4 Treatments on Boundary Conditions.......................... 93

6.5Exercises............................................... 98

CHAPTER 7

Solutions of Linear Algebraic Equations............................ 101

7.1 LU Decomposition Method................................. 101

7.2Exercises............................................... 106

CHAPTER 8

Error Estimation and AdaptiveAnalysis............................ 107

8.1 Error Estimation of Finite ElementSolutions ................... 107

8.1.1 Error of Finite Element Solutions...................... 107

8.1.2 Superconvergent Patch Recovery Method................. 108

8.2 Adaptive Finite Element Method............................ 110

8.2.1 Categories of Adaptive Finite ElementMethod ............ 110

8.2.2 h-Version Adaptive Finite Element Method............... 111

8.2.3 hp-Version Adaptive Finite Element Method.............. 112

8.3Exercises............................................... 114

CHAPTER 9

Programs of Finite Element Method............................... 115

9.1 One-Dimensional Program of Beam Deformation................ 115

9.1.1 Main Program..................................... 116

9.1.2 Numerical Example................................. 118

9.1.3 Interactive Interface................................. 118

9.2 Two-Dimensional Program of Plane StrainProblem .............. 128

9.2.1 Main Program..................................... 128

9.2.2 Numerical Example................................. 132

9.2.3 Interactive Interface................................. 134

9.3 Three-Dimensional Program of SolidCompression................ 145

9.3.1 Main Program..................................... 145

9.3.2 Numerical Example................................. 148

9.3.3 Interactive Interface................................. 151

9.4Exercises............................................... 162

Contents XI

Appendix A. Keyword Index ....................................165

Appendix B. Matrix Calculation.................................. 169

B.1 Definition.............................................. 169

B.2 Matrix Addition or Subtraction............................. 170

B.3 Transpose ..............................................171

B.4 Transpose of a Product................................... 172

B.5 Inverse................................................ 172

B.6 Symmetric Matrices...................................... 172

B.7 Partitioning ............................................172

Appendix C. Summary of Elements and ShapeFunctions ............... 175

C.1 One-Dimensional Lagrange Element.......................... 175

C.2 Two-Dimensional Triangle Element.......................... 175

C.3 Two-Dimensional Rectangle Element......................... 176

C.4 Three-Dimensional Tetrahedron Element...................... 177

C.5 Three-Dimensional Hexahedron Element...................... 178

Appendix D. Gaussian Integration Points andWeights ................. 179

Appendix E. Exercise Solutions.................................. 189

References.................................................. 207