Although the Trefftz finite element method (FEM) has become a powerful computational tool in the analysis of plane elasticity, thin and thick plate bending, Poisson’s equation, heat conduction, and piezoelectric materials, there are few books that offer a comprehensive computer programming treatment of the subject. Collecting results scattered in the literature,
MATLAB® and C Programming for Trefftz Finite Element Methods provides the detailed MATLAB® and C programming processes in applications of the Trefftz FEM to potential and elastic problems. The book begins with an introduction to the hybrid-Trefftz (HT) FEM that covers basic concepts and general element formulations of the method. It then concentrates on both the essentials and subroutines of MATLAB and C programming. The next few chapters present applications of T-elements to potential problems and linear plane elasticity, discuss how to solve body force in elasticity through radial basis functions, and examine how special purpose functions can be constructed. The final chapter explores advanced topics, such as the construction of Trefftz p -elements, dimensionless transformation, and an alternative formulation to HT FEM.
Unifying the computer programming aspects of the Trefftz FEM, this book will stimulate the development and application of this novel method in many facets of practical engineering.
Author(s): Qing-Hua Qin, Hui Wang
Edition: 1
Publisher: Taylor & Francis
Year: 2008
Language: English
Pages: 451
MATLAB® and C Programming for Trefftz Finite Element Methods......Page 3
Preface......Page 5
Contents......Page 7
1.1 Historical background......Page 13
1.2.1 Basic field equations and boundary conditions......Page 15
1.2.2 Assumed fields......Page 16
1.2.3 Element stiffness 1.2.3 Element stiffness......Page 20
1.3 Variational principles......Page 21
1.4 Concept of the T-complete solution......Page 22
1.5.1 Assumed element fields......Page 24
1.5.2 Variational functionals......Page 26
1.5.3 Assessment of the two techniques......Page 27
1.5.3.1 Geometry-induced singularities and stress concentrations......Page 28
1.6 Comparison of T-elements with boundary elements......Page 29
1.6.1 Boundary elements......Page 31
1.6.3 Assessment of the two numerical models......Page 32
References......Page 35
2.2.1 Array and variable......Page 40
2.2.3 Built-in variables......Page 41
2.3.2 Matrix indexing......Page 42
2.3.3 Common array and matrix operations......Page 43
2.3.4 Hierarchy of operations......Page 46
2.4.1 Relational and logical operators......Page 47
2.4.3 The switch construct......Page 48
2.4.5 The while loop......Page 49
2.4.6 Jump commands for loop control......Page 50
2.5.1 M-file function structure......Page 51
2.5.3 Executing an m-file function......Page 53
2.6.1 Open and close a file......Page 54
2.6.2 Input manipulation......Page 55
2.6.3 Output manipulation......Page 57
2.7 Vectorization programming with MATLAB......Page 58
2.8 Common built-in MATLAB functions......Page 59
References......Page 61
3.1.1 Data types......Page 63
3.1.2 Variable declaration......Page 64
3.1.3 Operators......Page 66
3.2.1 if-else structure......Page 67
3.2.2 switch-case structure......Page 68
3.2.3 for loop......Page 69
3.2.4 while loop......Page 70
3.3 Advanced array and pointer action......Page 71
3.3.2 Pointers......Page 72
3.3.4 Initialisation of array and storage management......Page 73
3.4.1 Types of functions......Page 74
3.4.2 Function call and parameter transfer......Page 76
3.5.1 Open and close a file......Page 78
3.6 Create and execute C codes in visual C++ platform......Page 79
3.6.2 Creating a C source file......Page 80
3.6.3 Compile, build and execute a C program......Page 82
3.7 Common library functions and related head files......Page 83
References......Page 85
4.2.1 Input of data......Page 87
4.2.2 MATLAB codes for input of data......Page 91
4.2.3 C codes for input of data......Page 94
4.2.4 Output of results......Page 97
4.2.5 MATLAB codes for output of results......Page 98
4.2.6 C codes for output of results......Page 100
4.3 Numerical integration over element edges......Page 102
4.3.1 MATLAB codes......Page 103
4.3.2 C codes......Page 104
4.4 Shape functions along element edge......Page 106
4.4.1 MATLAB codes......Page 107
4.4.2 C codes......Page 109
4.5 Assembly of elements......Page 110
4.5.2 C codes......Page 111
4.6 Introduction of essential boundary conditions......Page 112
4.6.1 MATLAB codes......Page 114
4.6.2 C codes......Page 115
4.7 Solution of global stiffness equation......Page 116
4.7.2 C codes......Page 118
References......Page 121
5.1 Introduction......Page 122
5.2 Basic equations of potential problems......Page 123
5.3.1 Non-conforming intra-element field......Page 124
5.3.2 Auxiliary conforming frame field......Page 125
5.3.3 Modified variational principle......Page 126
5.3.4 Recovery of rigid-body motion......Page 128
5.4 T-complete functions......Page 129
5.5.2 Computation of H matrix......Page 131
5.5.3 Computation of G matrix......Page 132
5.6 Computation of equivalent nodal load......Page 133
5.8 MATLAB programming for potential problems......Page 134
5.9 C computer programming......Page 150
5.10 Numerical examples......Page 180
References......Page 194
6.2 Linear theory of elasticity......Page 196
6.2.2 Strain-displacement relations......Page 197
6.2.3 Constitutive relations (stress-strain relations)......Page 198
6.2.4 Boundary conditions......Page 199
6.3.1 Non-conforming intra-element field......Page 200
6.3.2 Auxiliary conforming frame field......Page 201
6.3.3 Modified variational functional......Page 203
6.4 T-complete functions......Page 205
6.5.1 Geometric characteristics of boundary edges......Page 208
6.5.2 Computation of matrix H......Page 209
6.6 Evaluation of equivalent nodal loads......Page 210
6.7 MATLAB functions for plane elastic problems......Page 212
6.8 C computer programming......Page 223
6.9 Numerical examples......Page 243
References......Page 253
7.1 Introduction......Page 255
7.2.1 Basics of radial basis functions......Page 256
7.2.2 RBF approximation......Page 257
7.2.3 Stability and convergence of RBF approximation......Page 259
7.3.1 Basic equations for Poisson’s problems......Page 261
7.3.2 Basic equations for plane stress/strain problems......Page 263
7.4.1 Assumed fields......Page 264
7.4.2 Variational functional......Page 265
7.5 Particular solutions in terms of RBFs......Page 266
7.5.1 Particular solutions for Poisson’s equation......Page 267
7.5.2 Particular solutions for plane stress/strain equations......Page 268
7.6.1 Forming equivalent nodal flux......Page 273
7.7.1 Two-dimensional Poisson’s problems......Page 274
7.7.2 Plane stress/strain problems......Page 285
7.8.1 Two-dimensional Poisson’s problems......Page 299
7.8.2 Plane stress/strain problems......Page 317
7.9.1 Poisson’s problems......Page 338
7.9.2 Plane stress/strain problems......Page 342
References......Page 348
8.2 Basic concept of special Trefftz functions......Page 350
8.3.1 Trefftz-complete solutions for circular hole elements......Page 353
8.4.1 Special Trefftz solutions for circular hole elements......Page 354
8.5.1 Data preparation......Page 361
8.6.1 Potential problems......Page 362
8.6.2 Elastic problems......Page 373
8.7.1 Potential problems......Page 386
8.7.2 Elastic problems......Page 401
8.8.1 Potential problems......Page 417
8.8.2 Plane elastic problems......Page 420
References......Page 425
9.2 Construction of Trefftz elements......Page 426
9.3 Dimensionless transformation......Page 427
9.3.1 Dimensionless transformation in regular HT element for plane potential problems......Page 428
9.3.3 Dimensionless transformation in regular element for plane elastic problems......Page 432
9.3.4 Dimensionless transformation in hole element for plane elastic problems......Page 433
9.4 smoothNodalstressevaluation- techniques......Page 434
9.6 Sparse matrix generation and solving procedure......Page 436
9.7 An alternative formulation to HT-FEM......Page 438
References......Page 441
Appendix A: Format of input data......Page 443
Appendix B: Glossary of variables......Page 444
Appendix C: Glossary of subroutines......Page 448
Appendix D: Plane displacement and stress transformations......Page 450
