Preface..................................................... V
About the Author............................................ VII
CHAPTER 1
Introduction of Finite Element Method............................. 1
1.1 Development Process of Finite Element Method ................. 1
1.2 Computation Procedure of Finite Element Method ............... 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 Strain Problems......... 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 for Differential Equations ..... 17
3.2 Weak Form of One-Dimensional Elasticity Problems .............. 17
3.3 Finite Element Computation Based on Weak Form ............... 20
3.3.1 Galerkin Method................................... 20
3.3.2 Finite Element Computation.......................... 24
3.4 Global Assembly from One-Dimensional Elements................ 26
3.5 Treatments on Boundary Conditions.......................... 28
3.6 Exercises............................................... 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 Three Nodes............................ 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 Tetrahedron Element....................... 48
4.4.1 Linear Tetrahedron Element with Four Nodes ............. 48
4.4.2 Higher-Order Tetrahedron Element..................... 51
4.4.3 Quadratic Tetrahedron Element........................ 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
4.6 Exercises............................................... 56
CHAPTER 5
Isoparametric Element and Numerical Integration..................... 59
5.1 Isoparametric Element.................................... 59
5.1.1 One-Dimensional Isoparametric Lagrange Element .......... 59
5.1.2 Two-Dimensional Isoparametric Triangle Element .......... 60
5.1.3 Two-Dimensional Isoparametric Rectangle Element ......... 63
5.1.4 Three-Dimensional Isoparametric Tetrahedron Element ...... 64
5.1.5 Three-Dimensional Isoparametric Hexahedron Element ...... 67
5.1.6 Requirements of Isoparametric Element.................. 69
5.2 Numerical Integration..................................... 70
5.2.1 One-Dimensional Integration for Lagrange Element ......... 70
5.2.2 Two-Dimensional Integration for Triangle Element ......... 73
5.2.3 Two-Dimensional Integration for Rectangle Element ........ 74
5.2.4 Three-Dimensional Integration for Tetrahedron Element ..... 75
5.2.5 Three-Dimensional Integration for Hexahedron Element...... 76
5.2.6 Required Order of Numerical Integration................. 78
5.3 Exercises............................................... 78
CHAPTER 6
Finite Element Computation Scheme of Elasticity Problems ............. 81
6.1 Weak Form for General Elasticity Problems.................... 81
6.2 Finite Element Method for Solving Elasticity Problems ............ 84
6.3 Global Assembly from High-Dimensional Elements ............... 86
6.4 Treatments on Boundary Conditions.......................... 93
6.5 Exercises............................................... 98
CHAPTER 7
Solutions of Linear Algebraic Equations............................ 101
7.1 LU Decomposition Method................................. 101
7.2 Exercises............................................... 106
CHAPTER 8
Error Estimation and Adaptive Analysis............................ 107
8.1 Error Estimation of Finite Element Solutions ................... 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 Element Method ............ 110
8.2.2 h-Version Adaptive Finite Element Method............... 111
8.2.3 hp-Version Adaptive Finite Element Method.............. 112
8.3 Exercises............................................... 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 Strain Problem .............. 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 Solid Compression................ 145
9.3.1 Main Program..................................... 145
9.3.2 Numerical Example................................. 148
9.3.3 Interactive Interface................................. 151
9.4 Exercises............................................... 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 Shape Functions ............... 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 and Weights ................. 179
Appendix E. Exercise Solutions.................................. 189
References.................................................. 207
