Basic Theory of Finite Element Method

Contents 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 4 1.4 Exercises 5 Chapter 2 Fundamentals of Elasticity Mechanics 6 2.1 Displacements 7 2.2 Strains 8 2.3 Stresses 8 2.4 Geometric Equations 9 2.5 Constitutive Equations 10 2.6 Equilibrium Equations 12 2.6.1 Three-Dimensional Problems 12 2.6.2 Two-Dimensional Plane Stress and Strain Problems 13 2.6.3 Two-Dimensional Axisymmetric Problems 13 2.7 Boundary Conditions 13 2.8 Exercises 14 Chapter 3 Weak Form of Equivalent Integration 15 3.1 Weak Form of Equivalent Integration for Differential Equations 15 3.2 Weak Form of One-Dimensional Elasticity Problems 15 3.3 Finite Element Computation Based on Weak Form 17 3.3.1 Galerkin Method 17 3.3.2 Finite Element Computation 20 3.4 Global Assembly from One-Dimensional Elements 23 3.5 Treatments on Boundary Conditions 24 3.6 Exercises 28 Chapter 4 Elements and Shape Functions 29 4.1 One-Dimensional Lagrange Element 29 4.1.1 Linear Element with Two Nodes 29 4.1.2 Higher-Order Lagrange Element 29 4.1.3 Quadratic Lagrange Element 31 4.2 Two-Dimensional Triangle Element 32 4.2.1 Triangle with Three Nodes 32 4.2.2 Higher-Order Triangle Element 35 4.2.3 Quadratic Triangle Element 37 4.2.4 Cubic Triangle Element 37 4.3 Two-Dimensional Rectangle Element 37 4.3.1 Linear Rectangle Element with Four Nodes 37 4.3.2 Higher-Order Rectangle Element 39 4.3.3 Quadratic Rectangle Element 40 4.4 Three-Dimensional Tetrahedron Element 41 4.4.1 Linear Tetrahedron Element with Four Nodes 41 4.4.2 Higher-Order Tetrahedron Element 44 4.4.3 Quadratic Tetrahedron Element 44 4.4.4 Cubic Tetrahedron Element 44 4.5 Three-Dimensional Hexahedron Element 45 4.5.1 Hexahedron with Eight Nodes 45 4.5.2 Higher-Order Hexahedron Element 46 4.5.3 Quadratic Hexahedron Element 47 4.6 Exercises 48 Chapter 5 Isoparametric Element and Numerical Integration 50 5.1 Isoparametric Element 50 5.1.1 One-Dimensional Isoparametric Lagrange Element 50 5.1.2 Two-Dimensional Isoparametric Triangle Element 51 5.1.3 Two-Dimensional Isoparametric Rectangle Element 53 5.1.4 Three-Dimensional Isoparametric Tetrahedron Element 54 5.1.5 Three-Dimensional Isoparametric Hexahedron Element 56 5.1.6 Requirements of Isoparametric Element 58 5.2 Numerical Integration 59 5.2.1 One-Dimensional Integration for Lagrange Element 59 5.2.2 Two-Dimensional Integration for Triangle Element 62 5.2.3 Two-Dimensional Integration for Rectangle Element 63 5.2.4 Three-Dimensional Integration for Tetrahedron Element 64 5.2.5 Three-Dimensional Integration for Hexahedron Element 65 5.2.6 Required Order of Numerical Integration 66 5.3 Exercises 67 Chapter 6 Finite Element Computation Scheme of Elasticity Problems 68 6.1 Weak Form for General Elasticity Problems 68 6.2 Finite Element Method for Solving Elasticity Problems 71 6.3 Global Assembly from High-Dimensional Elements 72 6.4 Treatments on Boundary Conditions 78 6.5 Exercises 82 Chapter 7 Solutions of Linear Algebraic Equations 83 7.1 LU Decomposition Method 83 7.2 Exercises 87 Chapter 8 Error Estimation and Adaptive Analysis 88 8.1 Error Estimation of Finite Element Solutions 88 8.1.1 Error of Finite Element Solutions 88 8.1.2 Superconvergent Patch Recovery Method 88 8.2 Adaptive Finite Element Method 90 8.2.1 Categories of Adaptive Finite Element Method 90 8.2.2 h-Version Adaptive Finite Element Method 91 8.2.3 hp-Version Adaptive Finite Element Method 92 8.3 Exercises 94 Chapter 9 Programs of Finite Element Method 95 9.1 One-Dimensional Program of Beam Deformation 95 9.1.1 Main Program 96 9.1.2 Numerical Example 98 9.1.3 Interactive Interface 98 9.2 Two-Dimensional Program of Plane Strain Problem 108 9.2.1 Main Program 108 9.2.2 Numerical Example 112 9.2.3 Interactive Interface 114 9.3 Three-Dimensional Program of Solid Compression 124 9.3.1 Main Program 124 9.3.2 Numerical Example 128 9.3.3 Interactive Interface 131 9.4 Exercises 141 Appendix A. Keyword Index 143 Appendix B. Matrix Calculation 145 B.1 Definition 145 B.2 Matrix Addition or Subtraction 146 B.3 Transpose 146 B.4 Transpose of a Product 147 B.5 Inverse 147 B.6 Symmetric Matrices 148 B.7 Partitioning 148 Appendix C. Summary of Elements and Shape Functions 150 C.1 One-Dimensional Lagrange Element 150 C.2 Two-Dimensional Triangle Element 150 C.3 Two-Dimensional Rectangle Element 151 C.4 Three-Dimensional Tetrahedron Element 151 C.5 Three-Dimensional Hexahedron Element 152 Appendix D. Gaussian Integration Points and Weights 154 Appendix E. Exercise Solutions 158 References 174