The Trefftz finite and boundary element method Machine generated contents note: Chapter 1 Introduction 1 11 Finite element technique 1 12 Shape functions and element stiffness matrix 7 13 Brief historical background 15 14 Basic relationships in engineering problems 16 15 Modified variational principles 21 16 The concept of T complete solution 23 17 Comparison of T elements with conventional finite elements 24 18 Comparison of T elements with boundary elements 35 References 38 Chapter 2 Potential Problems 43 21 Introduction 43 22 Statement of the problem 44 23 T complete functions 45 24 Assumed fields 46 25 Generation of element matrix equation 49 26 Rank condition 53 27 Special purpose functions 54 28 Sensitivity to mesh distortion 57 29 Orthotropic case 57 210 The Helmholtz equation 60 211 HT element with boundary traction frame 62 212 Frameless T elements 67 References 71 Chapter 3 Linear Elastostatics 73 31 Introduction 73 32 Linear theory of elasticity 73 33 Assumed fields in plane elasticity 79 34 T complete functions 81 35 Variational formulations 84 36 Element stiffness equation 86 37 Special purpose elements 88 38 p extension approach 96 39 Three dimensional elasticity 97 310 Numerical examples 98 References 101 Chapter 4 Thin Plates 103 41 Introduction 103 42 Thin plate theory 103 43 Assumed field 107 44 T complete functions and particular solutions 109 45 Variational formulations for plate bending 111 46 Generation of element stiffness matrix 113 47 p method elements 115 48 Special purpose functions 116 49 Extension to thin plates on elastic foundation 119 410 Two alternative plate bending p elements 122 411 Numerical examples and assessment 126 References 136 Chapter 5 Thick Plates 139 51 Introduction 139 52 Basic equations for Reissner Mindlin plate theory 140 53 Assumed fields and particular solution 141 54 Variational formulation for HT thick plate elements 149 55 Implementation of the new family of HT elements 151 56 A 12 DOF quadrilateral element free of shear locking 153 57 Extension to thick plates on elastic foundation 157 58 Sensitivity to mesh distortion 160 59 Numerical assessment 160 References 170 Chapter 6 Transient Heat Conduction 173 61 Introduction 173 62 Elements of heat conduction 174 63 Time step formula 174 64 Element matrix formulations 175 65 T complete functions and particular solutions 179 66 Numerical examples 179 References 184 Chapter 7 Geometrically Nonlinear Analysis of Plate Bending Problems 187 71 Introduction 187 72 Basic equations of nonlinear thin plate bending 187 73 Assumed fields and Trefftz functions 191 74 Particular solutions 194 75 Modified Variational principle 195 76 Element matrix 197 77 Iterative scheme 199 78 Extension to post buckling thin plates on elastic foundation 200 79 Geometrically nonlinear analysis of thick plates 202 710 Numerical examples 210 References 216 Chapter 8 Elastoplasticity 219 81 Introduction 219 82 Time discretization 219 83 Basic relations 220 84 Assumed fields 223 85 Constraints on the approximation functions 224 86 Finite element equilibrium and compatibility equations 225 87 Finite element equations 226 88 Finite element governing system 229 References 230 Chapter 9 Dynamics of Plate Bending Problems 231 91 Introduction 231 92 Basic equations 231 93 Time stepping formulation 232 94 Numerical examples 235 References 238 Chapter 10 Trefftz Boundary Element Method 239 101 Introduction 239 102 Potential problems 239 103 Plane elasticity 250 104 Thin plate bending 256 105 Moderately thick plates 260 References 265 Appendix A Numerical Integration 267 Appendix B Matrix Algebra and Calculus 271