# The Finite Element Method: Its Basis and Fundamentals

## 6th Edition

**Authors:**Olek Zienkiewicz Robert Taylor J.Z. Zhu

**eBook ISBN:**9780080472775

**Imprint:**Butterworth-Heinemann

**Published Date:**18th April 2005

**Page Count:**752

## Description

The Sixth Edition of this influential best-selling book delivers the most up-to-date and comprehensive text and reference yet on the basis of the finite element method (FEM) for all engineers and mathematicians. Since the appearance of the first edition 38 years ago, *The Finite Element Method* provides arguably the most authoritative introductory text to the method, covering the latest developments and approaches in this dynamic subject, and is amply supplemented by exercises, worked solutions and computer algorithms.

• The classic FEM text, written by the subject's leading authors

• Enhancements include more worked examples and exercises

• With a new chapter on automatic mesh generation and added materials on shape function development and the use of higher order elements in solving elasticity and field problems

Active research has shaped *The Finite Element Method* into the pre-eminent tool for the modelling of physical systems. It maintains the comprehensive style of earlier editions, while presenting the systematic development for the solution of problems modelled by linear differential equations.

Together with the second and third self-contained volumes (0750663219 and 0750663227), *The Finite Element Method* Set (0750664312) provides a formidable resource covering the theory and the application of FEM, including the basis of the method, its application to advanced solid and structural mechanics and to computational fluid dynamics.

## Key Features

- The classic introduction to the finite element method, by two of the subject's leading authors
- Any professional or student of engineering involved in understanding the computational modelling of physical systems will inevitably use the techniques in this key text

## Readership

Senior students, researchers and practicing engineers in mechanical, automotive, aeronautical and civil engineering. Key topic for applied mathematicians and engineering software developers.

## Table of Contents

**Chapter 1: The standard discrete system and origins of the finite element method**
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

**Chapter 2: A direct physical approach to problems in elasticity: plane stress**
2.1 Introduction
2.2 Direct formulation of finite element characteristics
2.3 Generalization to the whole region ¨C internal nodal force concept abandoned
2.4 Displacement approach as a Minimization of total potential energy
2.5 Convergence criteria
2.6 Discretization error and convergence rate
2.7 Displacement functions with discontinuity between elements ¨C non-conforming elements and the patch test
2.8 Finite element solution process
2.9 Numerical examples
2.10 Concluding remarks
2.11 Problems

**Chapter 3: Generalization of finite element concepts**
3.1 Introduction
3.2 Integral or ¡®weak¡¯ statements equivalent to the differential equations
3.3 Approximation to integral formulations: the weighted residual-Galerkin method
3.4 Virtual work as the ¡®weak form¡¯ of equilibrium equations for analysis of solids or fluids
3.5 Partial discretization
3.6 Convergence
3.7 What are ¡®variational principles¡¯?
3.8 ¡®Natural¡¯ variational principles and their relation to governing differential equations
3.9 Establishment of natural variational principles for linear, self-adjoint, differential equations
3.10 Maximum, minimum, or a saddle point?
3.11 Constrained variational principles. Lagrange multipliers
3.12 Constrained variational principles. Penalty function and perturbed lagrangian methods
3.13 Least squares approximations
3.14 Concluding remarks ¨C finite difference and boundary methods
3.15 Problems

**Chapter 4: Element shape functions**
4.1 Introduction
4.2 Standard and hierarchical concepts
4.3 Rectangular elements ¨C some preliminary considerations
4.4 Completeness of polynomials
4.5 Rectangular elements ¨C Lagrange family
4.6 Rectangular elements ¨C ¡®serendipity¡¯ family
4.7 Triangular element family
4.8 Line elements
4.9 Rectangular prisms ¨C Lagrange family
4.10 Rectangular prisms ¨C ¡®serendipity¡¯ family
4.11 Tetrahedral elements
4.12 Other simple three-dimensional elements
4.13 Hierarchic polynomials in one dimension
4.14 Two- and three-dimensional, hierarchical elements of the ¡®rectangle¡¯ or ¡®brick¡¯ type
4.15 Triangle and tetrahedron family
4.16 Improvement of conditioning with hierarchical forms
4.17 Global and local finite element approximation
4.18 Elimination of internal parameters before assembly ¨C substructures
4.19 Concluding remarks
4.20 Problems

**Chapter 5: Mapped elements and numerical integration**
5.1 Introduction
5.2 Use of ¡®shape functions¡¯ in the establishment of coordinate transformations
5.3 Geometrical conformity of elements
5.4 Variation of the unknown function within distorted, curvilinear elements. Continuity requirements
Contents ix
5.5 Evaluation of element matrices. Transformation in ¦Î, ¦Â, ¦Æ coordinates
5.6 Evaluation of element matrices. Transformation in area and volume coordinates
5.7 Order of convergence for mapped elements
5.8 Shape functions by degeneration
5.9 Numerical integration ¨C rectangular (2D) or brick regions (3D)
5.10 Numerical integration ¨C triangular or tetrahedral regions
5.11 Generation of finite element meshes by mapping. Blending functions
5.12 Required order of numerical integration
5.13 Meshes by blending functions
5.14 Infinite domains and infinite elements
5.15 Singular elements by mapping ¨C use in fracture mechanics, etc.
5.16 Computational advantage of numerically integrated finite elements
5.17 Problems

**Chapter 6: Linear elasticity**
6.1 Introduction
6.2 Governing equations
6.3 Finite element approximation
6.4 Reporting of results: displacements, strains and stresses
6.5 Numerical examples
6.6 Problems

**Chapter 7: Field problems**
7.1 Introduction
7.2 General quasi-harmonic equation
7.3 Finite element solution process
7.4 Partial discretization ¨C transient problems
7.5 Numerical examples ¨C an assessment of accuracy
7.6 Concluding remarks
7.7 Problems

**Chapter 8: Automatic mesh generation**
8.1 Introduction
8.2 Two-dimensional mesh generation ¨C advancing front method
8.3 Surface mesh generation
8.4 Three-dimensional mesh generation ¨C Delaunay triangulation
8.5 Concluding remarks
8.6 Problems

**Chapter 9: The patch test and reduced integration**
9.1 Introduction
9.2 Convergence requirements
9.3 The simple patch test (tests A and B) ¨C a necessary condition for convergence
9.4 Generalized patch test (test C) and the single-element test
9.5 The generality of a numerical patch test
9.6 Higher order patch tests
9.7 Application of the patch test to plane elasticity elements with ¡®standard¡¯ and ¡®reduced¡¯ quadrature
9.8 Application of the patch test to an incompatible element
9.9 Higher order patch test ¨C assessment of robustness
9.10 Conclusion
9.11 Problems

**Chapter 10: Mixed formulation and constraints**
10.1 Introduction
10.2 Discretization of mixed forms ¨C some general remarks
10.3 Stability of mixed approximation. The patch test
10.4 Two-field mixed formulation in elasticity
10.5 Three-field mixed formulations in elasticity
10.6 Complementary forms with direct constraint
10.7 Concluding remarks ¨C mixed formulation or a test of element ¡®robustness¡¯
10.8 Problems

**Chapter 11: Incompressible problems, mixed methods and other procedures of solution**
11.1 Introduction
11.2 Deviatoric stress and strain, pressure and volume change
11.3 Two-field incompressible elasticity (u¨Cp form)
11.4 Three-field nearly incompressible elasticity (u¨Cp¨C¦Åv form)
11.5 Reduced and selective integration and its equivalence to penalized mixed problems
11.6 A simple iterative solution process for mixed problems: Uzawa method
11.7 Stabilized methods for some mixed elements failing the incompressibility patch test
11.8 Concluding remarks
11.9 Exercises

**Chapter 12 Multidomain mixed approximations ¨C domain decomposition and ¡®frame¡¯ methods**
12.1 Introduction
12.2 Linking of two or more subdomains by Lagrange multipliers
12.3 Linking of two or more subdomains by perturbed lagrangian and penalty methods
12.4 Interface displacement ¡®frame¡¯
12.5 Linking of boundary (or Trefftz)-type solution by the ¡®frame¡¯ of specified displacements
12.6 Subdomains with ¡®standard¡¯ elements and global functions
12.7 Concluding remarks
12.8 Problems

**Chapter 13: Errors, recovery processes and error estimates**
13.1 Definition of errors
13.2 Superconvergence and optimal sampling points
13.3 Recovery of gradients and stresses
13.4 Superconvergent patch recovery ¨C SPR
13.5 Recovery by equilibration of patches ¨C REP
13.6 Error estimates by recovery
13.7 Residual-based methods
13.8 Asymptotic behaviour and robustness of error estimators ¨C the Babu¡¦ska patch test
13.9 Bounds on quantities of interest
13.10 Which errors should concern us?
13.11 Problems

**Chapter 14: Adaptive finite element refinement**
14.1 Introduction
14.2 Adaptive h-refinement
14.3 p-refinement and hp-refinement
14.4 Concluding remarks
14.5 Problems

** Chapter 15: Point-based and partition of unity approximations**
15.1 Introduction
15.2 Function approximation
15.3 Moving least squares approximations ¨C restoration of continuity of approximation
15.4 Hierarchical enhancement of moving least squares expansions
15.5 Point collocation ¨C finite point methods
15.6 Galerkin weighting and finite volume methods
15.7 Use of hierarchic and special functions based on standard finite elements satisfying the partition of unity requirement
15.8 Closure
15.9 Problems

**Chapter 16: Semi-discretization and analytical solution**
16.1 Introduction
16.2 Direct formulation of time-dependent problems with spatial finite element subdivision
16.3 General classification
16.4 Free response ¨C eigenvalues for second-order problems and dynamic vibration
16.5 Free response ¨C eigenvalues for first-order problems and heat conduction, etc.
16.6 Free response ¨C damped dynamic eigenvalues
16.7 Forced periodic response
16.8 Transient response by analytical procedures
16.9 Symmetry and repeatability
16.10 Problems

**Chapter 17: Discrete approximation in time**
17.1 Introduction
17.2 Simple time-step algorithms for the first-order equation
17.3 General single-step algorithms for first and second order equations
17.4 Stability of general algorithms
17.5 Multistep recurrence algorithms
17.6 Some remarks on general performance of numerical algorithms
17.7 Time discontinuous Galerkin approximation
17.8 Concluding remarks
17.9 Problems

** Chapter 18: Coupled systems**
18.1 Coupled problems ¨C definition and classification
18.2 Fluid¨Cstructure interaction (Class I problem)
18.3 Soil¨Cpore fluid interaction (Class II problems)
18.4 Partitioned single-phase systems ¨C implicit¨Cexplicit partitions (Class I problems)
18.5 Staggered solution processes
18.6 Concluding remarks

** Chapter 19: Computer procedures for finite element analysis**
19.1 Introduction
19.2 Pre-processing module: mesh creation
19.3 Solution module
19.4 Post-processor module
19.5 User modules

Appendix A: Matrix algebra Appendix B: Tensor-indicial notation in elasticity Appendix C: Solution of linear algebraic equations Appendix D: Integration formulae for a triangle Appendix E: Integration formulae for a tetrahedron Appendix F: Some vector algebra Appendix G: Integration by parts Appendix H: Solutions exact at nodes Appendix I: Matrix diagonalization or lumping

## Details

- No. of pages:
- 752

- Language:
- English

- Copyright:
- © Butterworth-Heinemann 2005

- Published:
- 18th April 2005

- Imprint:
- Butterworth-Heinemann

- eBook ISBN:
- 9780080472775

## About the Author

### Olek 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.

### Affiliations and Expertise

Finite element method pioneer and former UNESCO Professor of Numerical Methods in Engineering, Barcelona, Spain

### Robert Taylor

R. L. Taylor is Emeritus Professor of Engineering and Professor in the Graduate School, Department of Civil and Environmental Engineering at the University of California, Berkeley.

### Affiliations and Expertise

Emeritus Professor of Engineering, University of California, Berkeley, USA.

### J.Z. Zhu

J. Z. Zhu is a Senior Scientist at ProCAST, ESI Group, USA.

### Affiliations and Expertise

Senior Scientist at ProCast Inc., ESI-Group North America, USA

## Reviews

It is very difficult to write a book which covers the entire finite element field. ..The authors have made a splendid attempt at a very difficult task. The books remain a tremendous bargain...and are an invaluable guide to the entire field of finite elements. If you are serious about working on finite elements you cannot do without this book. Mathematics Today, August 2001. "...the publication of the first edition was an epoch making event...it is written by...the greatest theorist of the subject. If you are serious about finite elements, this is a book that you simply cannot afford to be without." International Journal of Numerical Methods in Engineering. "..the pre-eminent reference work on finite element analysis." Applied Mechanical Review "...a very good book...presentation is first class...will be of great assistance to all engineers and scientists interested in the method...a very commendable piece of work." Journal of the British Society for Strain Measurement