A First Course in Finite Elements - Jacob Fish

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Résumé :

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 and Weak 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 The Weak Form in One Dimension 47

3.3 Continuity 50

3.4 The Equivalence Between the Weak 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 the Weight 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 and Weak 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 Between Weak and Strong Forms 144

6.5 Generalization to Three-Dimensional Problems 145

6.6 Strong and Weak 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 T...

Biographie:
instructions for such use are given in the preface. The course material is organized in three chronological units of one month each: 1) the finite element formulation for one-dimensional problems, 2) the finite element formulation for scalar field problems in two dimensions and 3) finite element programming and application to scalar field problems...

Sommaire:
The text material evolved from over 50 years of combined teaching experience it deals with a formulation and application of the finite element method. A meaningful course can be constructed from a subset of the chapters in this book for a quarter course...