MATLAB and C Programming for Trefftz Finite Element Methods

by Qin; Qing-Hua

• ISBN13: 9781420072754
• ISBN10: 1420072757
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-07-21
• Publisher: CRC Press

Summary

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 basisfunctions, 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.

Table of Contents

Introduction to Trefftz finite element method	p. 1
Historical background	p. 1
Trefftz finite element procedure	p. 3
Basic field equations and boundary conditions	p. 3
Assumed fields	p. 4
Element stiffness equation	p. 8
Variational principles	p. 9
Concept of the T-complete solution	p. 10
Comparison of Trefftz FEM and conventional FEM	p. 12
Assumed element fields	p. 12
Variational functionals	p. 14
Assessment of the two techniques	p. 15
Comparison of T-elements with boundary elements	p. 17
Boundary elements	p. 19
T-elements	p. 20
Assessment of the two numerical models	p. 20
References	p. 23
Foundation of Matlab programming	p. 29
Introduction	p. 29
Basic data types in Matlab	p. 29
Array and variable	p. 29
Types of variables	p. 30
Built-in variables	p. 30
Matrix manipulations	p. 31
Initialising matrix variable	p. 31
Matrix indexing	p. 31
Common array and matrix operations	p. 32
Hierarchy of operations	p. 35
Control structures	p. 36
Relational and logical operators	p. 36
The if construct	p. 37
The switch construct	p. 37
The for loop	p. 38
The while loop	p. 38
Jump commands for loop control	p. 39
M-file functions	p. 40
M-file function structure	p. 40
Global and local variables	p. 42
Executing an m-file function	p. 42
I/O file manipulation	p. 43
Open and close a file	p. 43
Input manipulation	p. 44
Output manipulation	p. 46
Vectorization programming with Matlab	p. 47
Common built-in Matlab functions	p. 48
References	p. 50
C programming	p. 53
Data types, variable declaration and operators	p. 53
Data types	p. 53
Variable declaration	p. 54
Operators	p. 56
Control structures	p. 57
if-else structure	p. 57
switch-case structure	p. 58
for loop	p. 59
while loop	p. 60
Jump control commands	p. 61
Advanced array and pointer action	p. 61
Arrays	p. 62
Pointers	p. 62
Pointers and arrays	p. 63
Initialisation of array and storage management	p. 63
Functions and parameter transfer	p. 64
Types of functions	p. 64
Function call and parameter transfer	p. 66
File manipulation	p. 68
Open and close a file	p. 68
Input data from a file	p. 69
Output data to a file	p. 69
Create and execute C codes in visual C++ platform	p. 69
Creating a project	p. 70
Creating a C source file	p. 70
Compile, build and execute a C program	p. 72
Output result	p. 73
Common library functions and related head files	p. 73
References	p. 75
Commonly used subroutines	p. 77
Introduction	p. 77
Input and output	p. 77
Input of data	p. 77
Matlab codes for input of data	p. 81
C codes for input of data	p. 84
Output of results	p. 87
Matlab codes for output of results	p. 88
C codes for output of results	p. 90
Numerical integration over element edges	p. 92
Matlab codes	p. 93
C codes	p. 94
Shape functions along element edge	p. 96
Matlab codes	p. 97
C codes	p. 99
Assembly of elements	p. 100
Matlab codes	p. 101
C codes	p. 101
Introduction of essential boundary conditions	p. 102
Matlab codes	p. 104
C codes	p. 105
Solution of global stiffness equation	p. 106
Matlab codes	p. 108
C codes	p. 108
References	p. 111
Potential problems	p. 113
Introduction	p. 113
Basic equations of potential problems	p. 114
Trefftz FE formulation	p. 115
Non-conforming intra-element field	p. 115
Auxiliary conforming frame field	p. 116
Modified variational principle	p. 117
Recovery of rigid-body motion	p. 119
T-complete functions	p. 120
Computation of H and G matrix	p. 122
Geometric characteristics of boundary edges	p. 122
Computation of H matrix	p. 122
Computation of G matrix	p. 123
Computation of equivalent nodal load	p. 124
Program structure for HT-FEM	p. 125
Matlab programming for potential problems	p. 125
C computer programming	p. 141
Numerical examples	p. 171
References	p. 185
Plane stress/strain problems	p. 187
Introduction	p. 187
Linear theory of elasticity	p. 187
Equilibrium equations	p. 188
Strain-displacement relations	p. 188
Constitutive relations (stress-strain relations)	p. 189
Boundary conditions	p. 190
Governing equations in terms of displacements	p. 191
Trefftz finite element formulation	p. 191
Non-conforming intra-element field	p. 191
Auxiliary conforming frame field	p. 192
Modified variational functional	p. 194
Recovery of rigid-body motion	p. 196
T-complete functions	p. 196
Computation of H and G matrix	p. 199
Geometric characteristics of boundary edges	p. 199
Computation of matrix H	p. 200
Computation of matrix G	p. 201
Evaluation of equivalent nodal loads	p. 201
Matlab functions for plane elastic problems	p. 203
C computer programming	p. 214
Numerical examples	p. 234
References	p. 244
Treatment of inhomogeneous terms using RBF approximation	p. 247
Introduction	p. 247
Radial basis functions	p. 248
Basics of radial basis functions	p. 248
RBF approximation	p. 249
Stability and convergence of RBF approximation	p. 251
Non-homogeneous problems	p. 253
Basic equations for Poisson's problems	p. 253
Basic equations for plane stress/strain problems	p. 255
Solution procedure of HT-FEM for non-homogenous problems	p. 256
Assumed fields	p. 256
Variational functional	p. 257
Discussion	p. 258
Particular solutions in terms of RBFs	p. 258
Particular solutions for Poisson's equation	p. 259
Particular solutions for plane stress/strain equations	p. 260
Modification of the program structure	p. 265
Forming equivalent nodal flux	p. 265
Introducing nodal potential condition	p. 266
Matlab functions for particular solutions	p. 266
Two-dimensional Poisson's problems	p. 266
Plane stress/strain problems	p. 277
C programming	p. 291
Two-dimensional Poisson's problems	p. 291
Plane stress/strain problems	p. 309
Numerical examples	p. 330
Poisson's problems	p. 330
Plane stress/strain problems	p. 334
References	p. 340
Special purpose T-elements	p. 343
Introduction	p. 343
Basic concept of special Trefftz functions	p. 343
Special purpose elements for potential problems	p. 346
Trefftz-complete solutions for circular hole elements	p. 346
Special purpose elements for linear elastic problems	p. 347
Special Trefftz solutions for circular hole elements	p. 347
Programming implementation	p. 354
Data preparation	p. 354
Special Trefftz functions	p. 355
Output quantities	p. 355
Matlab functions for special T-elements	p. 355
Potential problems	p. 355
Elastic problems	p. 366
C programming for special T-elements	p. 379
Potential problems	p. 379
Elastic problems	p. 394
Test examples	p. 410
Potential problems	p. 410
Plane elastic problems	p. 413
References	p. 418
Advanced topics for further programming development	p. 419
Introduction	p. 419
Construction of Trefftz elements	p. 419
Dimensionless transformation	p. 420
Dimensionless transformation in regular HT element for plane potential problems	p. 421
Dimensionless transformation in special HT element for plane potential problems	p. 425
Dimensionless transformation in regular element for plane elastic problems	p. 425
Dimensionless transformation in hole element for plane elastic problems	p. 426
Nodal stress evaluation-smooth techniques	p. 427
Generating intra-element points for outputting field results	p. 429
Sparse matrix generation and solving procedure	p. 429
An alternative formulation to HT-FEM	p. 431
References	p. 434
Format of input data	p. 437
Glossary of variables	p. 439
Glossary of subroutines	p. 443
Plane displacement and stress transformations	p. 445
Index	p. 447
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