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The Finite Element Method: Its Basis and Fundamentals
- 7th Edition - August 22, 2013
- Authors: O. C. Zienkiewicz, R. L. Taylor
- Language: English
- Hardback ISBN:9 7 8 - 1 - 8 5 6 1 7 - 6 3 3 - 0
- eBook ISBN:9 7 8 - 0 - 0 8 - 0 9 5 1 3 5 - 5
The Finite Element Method: Its Basis and Fundamentals offers a complete introduction to the basis of the finite element method, covering fundamental theory and worked examples… Read more
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Request a sales quoteThe Finite Element Method: Its Basis and Fundamentals offers a complete introduction to the basis of the finite element method, covering fundamental theory and worked examples in the detail required for readers to apply the knowledge to their own engineering problems and understand more advanced applications.
This edition sees a significant rearrangement of the book’s content to enable clearer development of the finite element method, with major new chapters and sections added to cover:
- Weak forms
- Variational forms
- Multi-dimensional field problems
- Automatic mesh generation
- Plate bending and shells
- Developments in meshless techniques
Focusing on the core knowledge, mathematical and analytical tools needed for successful application, The Finite Element Method: Its Basis and Fundamentals is the authoritative resource of choice for graduate level students, researchers and professional engineers involved in finite element-based engineering analysis.
- A proven keystone reference in the library of any engineer needing to understand and apply the finite element method in design and development
- Founded by an influential pioneer in the field and updated in this seventh edition by an author team incorporating academic authority and industrial simulation experience
- Features reworked and reordered contents for clearer development of the theory, plus new chapters and sections on mesh generation, plate bending, shells, weak forms and variational forms
Mechanical, Civil and Electrical Engineers, applied mathematicians and computer aided engineering software developers
Author Biography
Dedication
List of Figures
List of Tables
Preface
Chapter 1. The Standard Discrete System and Origins of the Finite Element Method
Abstract
1.1 Introduction
1.2 The structural element and the structural system
1.3 Assembly and analysis of a structure
1.4 The boundary conditions
1.5 Electrical and fluid networks
1.6 The general pattern
1.7 The standard discrete system
1.8 Transformation of coordinates
1.9 Problems
References
Chapter 2. Problems in Linear Elasticity and Fields
Abstract
2.1 Introduction
2.2 Elasticity equations
2.3 General quasi-harmonic equation
2.4 Concluding remarks
2.5 Problems
References
Chapter 3. Weak Forms and Finite Element Approximation: 1-D Problems
Abstract
3.1 Weak forms
3.2 One-dimensional form of elasticity
3.3 Approximation to integral and weak forms: The weighted residual (Galerkin) method
3.4 Finite element solution
3.5 Isoparametric form
3.6 Hierarchical interpolation
3.7 Axisymmetric one-dimensional problem
3.8 Transient problems
3.9 Weak form for one-dimensional quasi-harmonic equation
3.10 Concluding remarks
3.11 Problems
References
Chapter 4. Variational Forms and Finite Element Approximation: 1-D Problems
Abstract
4.1 Variational principles
4.2 “Natural” variational principles and their relation to governing differential equations
4.3 Establishment of natural variational principles for linear, self-adjoint differential equations
4.4 Maximum, minimum, or a saddle point?
4.5 Constrained variational principles
4.6 Constrained variational principles: Penalty function and perturbed Lagrangian methods
4.7 Least squares approximations
4.8 Concluding remarks: Finite difference and boundary methods
4.9 Problems
References
Chapter 5. Field Problems: A Multidimensional Finite Element Method
Abstract
5.1 Field problems: Quasi-harmonic equation
5.2 Partial discretization: Transient problems
5.3 Numerical examples: An assessment of accuracy
5.4 Problems
References
Chapter 6. Shape Functions, Derivatives, and Integration
Abstract
6.1 Introduction
6.2 Two-dimensional shape functions
6.3 Three-dimensional shape functions
6.4 Other simple three-dimensional elements
6.5 Mapping: Parametric forms
6.6 Order of convergence for mapped elements
6.7 Computation of global derivatives
6.8 Numerical integration
6.9 Shape functions by degeneration
6.10 Generation of finite element meshes by mapping
6.11 Computational advantage of numerically integrated finite elements
6.12 Problems
References
Chapter 7. Elasticity: Two- and Three-Dimensional Finite Elements
Abstract
7.1 Introduction
7.2 Elasticity problems: Weak form for equilibrium
7.3 Finite element approximation by the Galerkin method
7.4 Boundary conditions
7.5 Numerical integration and alternate forms
7.6 Infinite domains and infinite elements
7.7 Singular elements by mapping: Use in fracture mechanics, etc.
7.8 Reporting results: Displacements, strains, and stresses
7.9 Discretization error and convergence rate
7.10 Minimization of total potential energy
7.11 Finite element solution process
7.12 Numerical examples
7.13 Concluding remarks
7.14 Problems
References
Chapter 8. The Patch Test, Reduced Integration, and Nonconforming Elements
Abstract
8.1 Introduction
8.2 Convergence requirements
8.3 The simple patch test (Tests A and B): A necessary condition for convergence
8.4 Generalized patch test (Test C) and the single-element test
8.5 The generality of a numerical patch test
8.6 Higher order patch tests
8.7 Application of the patch test to plane elasticity elements with “standard” and “reduced” quadrature
8.8 Application of the patch test to an incompatible element
8.9 Higher order patch test: Assessment of robustness
8.10 Concluding remarks
8.11 Problems
References
Chapter 9. Mixed Formulation and Constraints: Complete Field Methods
Abstract
9.1 Introduction
9.2 Mixed form discretization: General remarks
9.3 Stability of mixed approximation: The patch test
9.4 Two-field mixed formulation in elasticity
9.5 Three-field mixed formulations in elasticity
9.6 Complementary forms with direct constraint
9.7 Concluding remarks: Mixed formulation or a test of element “robustness”
9.8 Problems
References
Chapter 10. Incompressible Problems, Mixed Methods, and Other Procedures of Solution
Abstract
10.1 Introduction
10.2 Deviatoric stress and strain, pressure, and volume change
10.3 Two-field incompressible elasticity (u-p form)
10.4 Three-field nearly incompressible elasticity (u-p- form)
10.5 Reduced and selective integration and its equivalence to penalized mixed problems
10.6 A simple iterative solution process for mixed problems: Uzawa method
10.7 Stabilized methods for some mixed elements failing the incompressibility patch test
10.8 Concluding remarks
10.9 Problems
References
Chapter 11. Multidomain Mixed Approximations
Abstract
11.1 Introduction
11.2 Linking of two or more subdomains by Lagrange multipliers
11.3 Linking of two or more subdomains by perturbed Lagrangian and penalty methods
11.4 Problems
References
Chapter 12. The Time Dimension: Semi-Discretization of Field and Dynamic Problems
Abstract
12.1 Introduction
12.2 Direct formulation of time-dependent problems with spatial finite element subdivision
12.3 Analytical solution procedures: General classification
12.4 Free response: Eigenvalues for second-order problems and dynamic vibration
12.5 Free response: Eigenvalues for first-order problems and heat conduction, etc.
12.6 Free response: Damped dynamic eigenvalues
12.7 Forced periodic response
12.8 Transient response by analytical procedures
12.9 Symmetry and repeatability
12.10 Problems
References
Chapter 13. Plate Bending Approximation: Thin and Thick Plates
Abstract
13.1 Introduction
13.2 Governing equations
13.3 General plate theory
13.4 The patch test: An analytical requirement
13.5 A nonconforming three-node triangular element
13.6 Numerical example for thin plates
13.7 Thick plates
13.8 Irreducible formulation: Reduced integration
13.9 Mixed formulation for thick plates
13.10 Elements with rotational bubble or enhanced modes
13.11 Linked interpolation: An improvement of accuracy
13.12 Discrete “exact” thin plate limit
13.13 Limitations of plate theory
13.14 Concluding remarks
13.15 Problems
References
Chapter 14. Shells as a Special Case of Three-Dimensional Analysis
Abstract
14.1 Introduction
14.2 Shell element with displacement and rotation parameters
14.3 Special case of axisymmetric thick shells
14.4 Special case of thick plates
14.5 Convergence
14.6 Some shell examples
14.7 Concluding remarks
14.8 Problems
References
Chapter 15. Errors, Recovery Processes, and Error Estimates
Abstract
15.1 Definition of errors
15.2 Superconvergence and optimal sampling points
15.3 Recovery of gradients and stresses
15.4 Superconvergent patch recovery (SPR)
15.5 Recovery by equilibration of patches (REP)
15.6 Error estimates by recovery
15.7 Residual-based methods
15.8 Asymptotic behavior and robustness of error estimator: The Babuška patch test
15.9 Error bounds and error estimates for quantities of interest
15.10 Which errors should concern us?
15.11 Problems
References
Chapter 16. Adaptive Finite Element Refinement
Abstract
16.1 Introduction
16.2 Adaptive h-refinement
16.3 p-refinement and hp-refinement
16.4 Concluding remarks
16.5 Problems
References
Chapter 17. Automatic Mesh Generation
Abstract
17.1 Introduction
17.2 Geometrical representation of the domain
17.3 Two-dimensional mesh generation: Advancing front method
17.4 Surface mesh generation
17.5 Three-dimensional mesh generation: Delaunay triangulation
17.6 Concluding remarks
References
Chapter 18. Computer Procedures for Finite Element Analysis
Abstract
18.1 Introduction
18.2 Pre-processing module: Mesh creation
18.3 Solution module
18.4 Post-processor module
18.5 User modules
References
Appendix A: Matrix Algebra
Definition of a matrix
Matrix addition or subtraction
Transpose of a matrix
Inverse of a matrix
A sum of products
Transpose of a product
Symmetric matrices
Partitioning
The standard eigenvalue problem
The generalized eigenvalue problem
Appendix B: Some Vector Algebra
Addition and subtraction
“Scalar” products
Length of vector
Direction cosines
“Vector” or cross-product
Elements of area and volume
Appendix C: Tensor-Indicial Notation in the Approximation of Elasticity Problems
Introduction
Indicial notation: Summation convention
Derivatives and tensorial relations
Coordinate transformation
Equilibrium and energy
Elastic constitutive equations
Finite element approximation
Relation between indicial and matrix notation
References
Appendix D: Solution of Simultaneous Linear Algebraic Equations
Direct solution
Iterative solution
References
Appendix E: Triangle and Tetrahedron Integrals
Triangles
Tetrahedron
Appendix F: Integration by Parts in Two or Three Dimensions (Green’s Theorem)
Appendix G: Solutions Exact at Nodes
References
Appendix H: Matrix Diagonalization or Lumping
References
Author Index
Subject Index
- No. of pages: 756
- Language: English
- Edition: 7
- Published: August 22, 2013
- Imprint: Butterworth-Heinemann
- Hardback ISBN: 9781856176330
- eBook ISBN: 9780080951355
OZ
O. C. Zienkiewicz
O. C. Zienkiewicz was one of the early pioneers of the finite element method and is internationally recognized as a leading figure in its development and wide-ranging application. He was awarded numerous honorary degrees, medals and awards over his career, including the Royal Medal of the Royal Society and Commander of the British Empire (CBE). He was a founding author of The Finite Element Method books and developed them through six editions over 40 years up to his death in 2009. Previous positions held by O.C. Zienkiewicz include UNESCO Professor of Numerical Methods in Engineering at the International Centre for Numerical Methods in Engineering, Barcelona, Director of the Institute for Numerical Methods in Engineering at the University of Wales, Swansea, U.K.
Affiliations and expertise
Swansea University, Swansea, WalesRT
R. L. Taylor
R.L Taylor is a Professor of the Graduate School at the Department of Civil and Environmental Engineering, University of California, Berkeley, USA. Awarded the Daniel C. Drucker Medal by the American Society of Mechanical Engineering in 2005, the Gauss-Newton Award and Congress Medal by the International Association for Computational Mechanics in 2002, and the Von Neumann Medal by the US Association for Computational Mechanics in 1999.
Affiliations and expertise
Emeritus Professor of Engineering, University of California, Berkeley, USARead The Finite Element Method: Its Basis and Fundamentals on ScienceDirect