Finite volume methods can be compared and contrasted with the finite difference methods, which approximate derivatives using nodal values, or finite element methods, which create local approximations of a solution using local data, and construct a global approximation by stitching them together. In contrast a finite volume method evaluates exact expressions for the average value of the solution over some volume, and uses this data to construct approximations of the solution within cells. The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem.
This book elucidates the concepts and innovative models around prospective developments with respect to finite volume methods. It talks in detail about the various concepts and theories related to finite element assembly, applications in one-dimension, quadratic elements, spectral elements and legendre polynomials. It also deals with two dimensions, viscous flow and spectral element methods in three dimensions. The aim of this book is to equip students with the fundamental techniques and methods used in this area. Through this book, we attempt to further enlighten the readers about the new concepts in this field. Coherent flow of topics, student-friendly language and extensive use of examples make this book an invaluable source of knowledge. There is no doubt that the book will have recognition by scientists, students, researchers, operatives, and technologists.
1. The Finite Element Method in One Dimension 1
1. Introduction 1
2. Steady diffusion with linear elements 2
3. Linear element interpolation 2
4. Galerkin projection 5
5. Formulation of a linear algebraic system 5
6. FIux at the Dirichlet end 8
7. Galerkin finite element equations via the Dirac delta function 13
8. Relation to the finite difference method 19
8. Finite element assembly 28
9. Assembly of a linear system 29
10. Thomas algorithm for a tridiagonal system 32
11. Convection (Robin or mixed) boundary condition 34
12. Variational formulation and weighted residuals 34
13. Helmholtz equation 38
14. Element nodes and global nodes 39
15. conclusion 39
Reference 41
2. Further Applications in One Dimension 44
1. Introduction 44
2. Unsteady diffusion 45
3. Galerkin projection 48
4. Integrating ODEs 51
5. Forward Euler method 55
6. Numerical stability 56
7. Crank-Nicolson integration 58
8. Convection 59
9. Numerical dispersion due to spatial discretization 68
10. Quadratic elements 69
11. Steady linear convection-diffusion 70
12. Beam bending 92
13. Euler-Bernoulli beam 93
14. Finite element methods for beam bending 95
15. FEM Equations 95
16. Shape Function Construction 95
17. Element Matrices 99
18. Hermitian elements 100
19. One-element cantilever beam 102
20. Beam buckling 107
21. conclusion 109
References 110
3. High-order and Spectral Elements in One Dimension 112
1. Introduction 112
2. Element nodal sets 113
3. Lagrange interpolation 114
4. Evenly spaced nodes 117
5. Element matrices 135
6. Change of element nodal sets 136
7. Spectral interpolation 138
8. Lobatto nodal base 148
9. Discretization code 150
10. Legendre polynomials 150
11. Conclusion 183
References 184
4. The Finite Element Method in Two Dimensions 187
1 Introduction 187
2. Three noded linear triangular element 188
3. Four noded linear rectangular element 189
4. Two-variable 3-noded linear triangular element 190
5. Structural mechanics applications in 2 dimension 203
6. Conclusion 231
Reference 232
5. Quadratic and Spectral Elements in Two Dimensions 234
1. Introduction 234
2. Six-node triangular elements 235
3. Isoparametric interpolation and element matrices 237
4. L-shaped domain 241
5. Laplace and Poisson equations 244
6. Eigenvalues of the Laplacian operator 247
7. The Lebesgue constant 248
8. Incomplete biorthogonality 251
9. Implementation of the modal expansion 253
10. Properties of the modal expansion 257
11. Surface elements 259
12. High-order quadrilateral elements 259
13. Grid nodes via tensor-product expansions 264
14. conclusion 264
References 265
6. Viscous Flow 267
1. Introduction 267
2. The importance of steady-state stability analysis 276
3. Transient and dynamic stability analysis 277
4. Conclusion 277
References 278
7. Finite and Spectral Element Methods in Three Dimensions 280
1. Introduction 280
2. Parametric representation 292
3. Domain discretization into four-node tetrahedral 295
4. conclusion. 308
References 309
Wan Altman obtained his Ph.D. in Theoretical and Applied Mechanics from Public Science University, Santa Catarina. He has edited numerous articles, journals, and research papers on analytical mechanics, computational mechanics and electromechanics. He is a Research Professor at Community College, Veracruz. His research interest focuses on engineering mathematics, mechanics, industrial design, computer aided design, finite element analysis and implementation of risk management processes. Altman holds prominent memberships of renowned associations and has been an active participant at international conferences and forums.
