Finite elements. An Introduction

Start Here......Page 1
CONTENTS......Page 5
PREFACE......Page 7
1.1 Orientation......Page 9
1.2 The Statement of the Model Problem......Page 10
1.3.1 A symmetric variational formulation......Page 11
1.4 Galerkin Approximations......Page 14
1.5 Finite Element Basis Functions......Page 16
1.6 Finite Element Calculations......Page 20
1.7 Interpretation of the Approximate Solution......Page 24
1.8 Accuracy of the Finite Element Approximation......Page 26
2.2.1 Linear second-order problems......Page 29
2.2.2 Physical origins of two-point problems......Page 30
2.3 Variational Formulation of Two-Point Boundary-Value Problems......Page 35
2.4 Galerkin Approximations......Page 38
2.5 Minimization of Energy Functionals......Page 39
2.6 Finite Element Interpolation......Page 41
2.7.1 Partitioning n and selection of shape functions......Page 45
2.7.2 Calculation of element matrices and equations......Page 46
2.7.3 Element assembly......Page 47
2.7.4 Boundary conditions......Page 48
2.7.5 Error estimates......Page 51
3.1 Computer Implementation of the Finite Element Method......Page 52
3.2 Description of CODE1......Page 53
3.3 Element Calculations......Page 54
3.3.2 Shape-function routine-SHAPE......Page 56
3.3.3 Element routine-ELEM......Page 57
3.3.4 Material-property routine-GETMAT......Page 58
3.4.3 Nodal-point coordinate definition: routine RNODE......Page 59
3.4.4 Element data definition: routine-RELEM......Page 60
3.5.2 Formulation of K and F: routine-FORMKF......Page 61
3.5.4 Boundary conditions: routine-APLYBC......Page 62
3.5.7 Reduction of K to triangular form: routine TRI......Page 63
3.6 Postprocessing......Page 64
3.6.2 Evaluation of\tfinite element and exact solutions: routine EVAL......Page 65
3.6.4 Calculation of exact solution: routine-EXACT......Page 66
3.8.1 Topics in code development......Page 67
3.8.2 Numerical experiments......Page 70
3.8.3 Applications using CODE1......Page 71
4.1 Introduction......Page 74
4.2.1 Some preliminaries......Page 75
4.2.2 Physical principles......Page 77
4.2.3 Statement of the boundary-value problem......Page 79
4.3 Variational Boundary-Value Problems......Page 81
4.4.1 Discretization......Page 83
4.4.2 Piecewise-linear interpolation on triangles......Page 85
4.4.3 Other triangular elements......Page 86
4.4.4 Rectangular elements......Page 88
4.4.5 Interpolation error......Page 89
4.5.1 Approximation of two-dimensional boundary-value problems......Page 90
4.5.2 An example......Page 93
5.2 Element Transformations......Page 96
5.2.1 The master element......Page 97
5.2.2 Construction of the transformations Te......Page 98
5.3 Finite Element Calculations......Page 102
5.3.1 Master-element calculations......Page 103
5.3.2 Computational aspects......Page 105
5.4.1 Quadrilateral elements......Page 106
5.4.2 Triangular elements......Page 108
5.5.2 Data storage in CODE2......Page 113
5.5.3 Preprocessor routines......Page 114
5.5.4 Processor routines......Page 116
5.5.5 Postprocessor......Page 119
6.2 Three-Dimensional Problems......Page 120
6.3.1 Two-point boundary-value problems......Page 123
6.3.2 Finite element approximations......Page 125
6.3.3 A two-dimensional problem......Page 126
6.4.1 A one-dimensional system......Page 129
6.4.2 Plane-stress problems......Page 130
6.5 Time-Dependent Problems......Page 132
INDEX......Page 135