1st Edition
MATLAB and C Programming for Trefftz Finite Element Methods
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.
Historical background
Trefftz FE procedure
Variational principles
Concept of the T-complete solution
Comparison of Trefftz FEM and conventional FEM
Comparison of T-elements with boundary elements
Foundation of MATLAB Programming
Introduction
Basic data types in MATLAB
Matrix manipulations
Control structures
M-file functions
I/O file manipulation
Vectorization programming with MATLAB
Common built-in MATLAB functions
C Programming
Data types, variable declaration, and operators
Control structures
Advanced array and pointer action
Functions and parameter transfer
File manipulation
Create and execute C codes in visual C++ platform
Common library functions and related head files
Commonly Used Subroutines
Introduction
Input and output
Numerical integration over element edges
Shape functions along element edge
Assembly of elements
Introduction of essential boundary conditions
Solution of global stiffness equation
Potential Problems
Introduction
Basic equations of potential problems
Trefftz FE formulation
T-complete functions
Computation of H and G matrix
Computation of equivalent nodal load
Program structure for HT FEM
MATLAB programming for potential problems
C computer programming
Numerical examples
Plane Stress/Strain Problems
Introduction
Linear theory of elasticity
Trefftz FE formulation
T-complete functions
Computation of H and G matrix
Evaluation of equivalent nodal loads
MATLAB functions for plane elastic problems
C computer programming
Numerical examples
Treatment of Inhomogeneous Terms Using RBF Approximation
Introduction
Radial basis functions (RBFs)
Non-homogeneous problems
Solution procedure of HT FEM for non-homogeneous problems
Particular solutions in terms of RBFs
Modification of the program structure
MATLAB functions for particular solutions
C programming
Numerical examples
Special Purpose T-Elements
Introduction
Basic concept of special Trefftz functions
Special purpose elements for potential problems
Special purpose elements for linear elastic problems
Programming implementation
MATLAB functions for special T-elements
C programming for special T-elements
Test examples
Advanced Topics for Further Programming Development
Introduction
Construction of Trefftz elements
Dimensionless transformation
Nodal stress evaluation-smooth techniques
Generating intra-element points for outputting field results
Sparse matrix generation and solving procedure
An alternative formulation to HT FEM
Appendix A: Format of Input Data
Appendix B: Glossary of Variables
Appendix C: Glossary of Subroutines
Appendix D: Plane Displacement and Stress Transformations
Index
References appear at the end of each chapter.
Biography
Hui Wang, Qing-Hua Qin
"The book begins with an introduction to the hybrid-Tretz (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 Tretz p-elements, dimensionless transformation, and an alternative formulation to HT FEM. Unifying the computer programming aspects of the Tretz FEM, this book will stimulate the development and application of this novel method in many facets of practical engineering."
—Zentralblatt MATH
