To fully comprehend and quantify any physical phenomenon, such as structural or fluid behaviour, thermal transfer, wave propagation, and biological cell growth, mathematics is required. Partial differential equations are used to describe the majority of these processes (PDEs). Numerical algorithms have been developed over the last few decades to allow a computer to solve these PDEs, and one of the most popular today is the finite element method. In two or three space variables, the FEM is a general numerical approach for solving partial differential equations (i.e., some boundary value problems). The FEM breaks a big system into smaller, simpler sections called finite elements in order to solve a problem. The classic domains of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential are typical issue areas of interest.
This book provides comprehensive insights into the field of finite elements. It talks extensively about the finite element analysis of harmonic problems and their other applications and uses. It deals with structural analysis, continuum mechanics problems, finite element meshes and biharmonic problems. It also focuses on harmonic problems, elliptic boundary value problems and variational formulation. It is compiled in such a manner, that it will provide in-depth knowledge about the theory and practice of the subject. Coherent flow of topics, student-friendly Language and extensive use of example make this book an invaluable source of knowledge. For all those who are interested in this topic, this book can prove to be an essential guide.
Preface
1. Introduction and Structural Analysis
1. Introduction
2. Computer Programming
3. Structural Analysis
4. Case Study: Bending of a Tapered Beam
Conclusion
References
2. Continuum Mechanics Problems
1. Introduction
2. Continuum Mechanics Equations
3. Classification of Partial Differential Equations
4. Methods for Solving Harmonic and Biharmonic Equations
Conclusion
References
3. Finite Element Analysis of Harmonic Problems
1. Introduction
2. Derivation of the Element Stiffness Matrix
3. Assembly of the Overall Stiffness Matrix
4. Comparison with the Finite Difference Approach
5. Variational Formulation
6. Boundary Conditions
7. Solution of the Linear Equations
8. Convergence of Finite Element Methods
9. A Computer Program for Harmonic Problems
Conclusion
References
4. Finite Element Meshes
1. Introduction
2. Choice of Mesh
3. Mesh Data in Numerical Form
4. Generation of Mesh Data
5. Mesh Modification
Conclusion
References
5. Some Harmonic Problems
1. Introduction
2. Case Study: Downstream Viscous Flow in a Rectangular
Channel
3. Case Study: Torsion of Prismatic Bars
Conclusion
References
6. Finite Element Analysis of Biharmonic Problems
1. Introduction
2. Derivation of the Element Stiffness Matrix
3. Assembly of the Overall Stiffness Matrix
4. Boundary Conditions
5. A Computer Program for Problems of the Biharmonic Plane
Strain or Plane Stress Type
Conclusion
References
7. Some Biharmonic Problems
1. Introduction
2. Case Study: Plane Strain Compression
3. Case Study: Stresses in Concentric Cylinders
4. Case Study: Stress Concentration near a Hole in a Flat Plate
Conclusion
References
8. Further Applications
1. Introduction
2. Axi-symmetric Problems
3. Higher-order Elements
4. Three-dimensional Problems
5. Biharmonic Problems Involving Incompressible Materials
6. Plate and Shell Problems
7. Isoparametric Elements
8. Nonlinear Problems
Conclusion
References
9. FEA of Elliptic Boundary Value Problems
1. Introduction
2. Galerkin Approximation
3. Existence and uniqueness of Galerkin Approximations
Conclusion
References
10. Variational Formulation of Boundary Value Problems
1. Introduction
2. Formulation of Variational Boundary-Value problem
3. Coercive Bilinear Forms
4. Existence and Uniqueness of Solution
Conclusion
References
11. Various Types of Finite Element Methods
1. Introduction
2. Stiffness matrix
Conclusion
References
Index
Saundra Peltier is a Ph.D. holder in Fluid Mechanics & Engineering from Public Research University, La Pampa. She has served as a Guest Lecturer for various universities across America and Europe. Her areas of expertise cover multidisciplinary areas: from design and optimisation of energy systems to fundamental problems focused on modelling and simulation. Peltier is author of more than 100 scientific papers and member of editorial board and scientific committee of a number of journals and conferences. She has been also a principal investigator in several projects for universities and companies.
