The Finite Element Method: Its Basis and Fundamentals

The Finite Element Method: Its Basis and Fundamentals

7th Edition - August 22, 2013

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  • Authors: Olek C Zienkiewicz, R. L. Taylor
  • Hardcover ISBN: 9781856176330
  • eBook ISBN: 9780080951355

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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 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.

Key Features

  • 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

Table of Contents

  • Author Biography


    List of Figures

    List of Tables


    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. Problems in Linear Elasticity and Fields


    2.1 Introduction

    2.2 Elasticity equations

    2.3 General quasi-harmonic equation

    2.4 Concluding remarks

    2.5 Problems


    Chapter 3. Weak Forms and Finite Element Approximation: 1-D Problems


    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


    Chapter 4. Variational Forms and Finite Element Approximation: 1-D Problems


    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


    Chapter 5. Field Problems: A Multidimensional Finite Element Method


    5.1 Field problems: Quasi-harmonic equation

    5.2 Partial discretization: Transient problems

    5.3 Numerical examples: An assessment of accuracy

    5.4 Problems


    Chapter 6. Shape Functions, Derivatives, and Integration


    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


    Chapter 7. Elasticity: Two- and Three-Dimensional Finite Elements


    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


    Chapter 8. The Patch Test, Reduced Integration, and Nonconforming Elements


    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


    Chapter 9. Mixed Formulation and Constraints: Complete Field Methods


    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


    Chapter 10. Incompressible Problems, Mixed Methods, and Other Procedures of Solution


    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-image 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


    Chapter 11. Multidomain Mixed Approximations


    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


    Chapter 12. The Time Dimension: Semi-Discretization of Field and Dynamic Problems


    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


    Chapter 13. Plate Bending Approximation: Thin and Thick Plates


    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


    Chapter 14. Shells as a Special Case of Three-Dimensional Analysis


    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


    Chapter 15. Errors, Recovery Processes, and Error Estimates


    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


    Chapter 16. Adaptive Finite Element Refinement


    16.1 Introduction

    16.2 Adaptive h-refinement

    16.3 p-refinement and hp-refinement

    16.4 Concluding remarks

    16.5 Problems


    Chapter 17. Automatic Mesh Generation


    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


    Chapter 18. Computer Procedures for Finite Element Analysis


    18.1 Introduction

    18.2 Pre-processing module: Mesh creation

    18.3 Solution module

    18.4 Post-processor module

    18.5 User modules


    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


    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


    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


    Appendix D: Solution of Simultaneous Linear Algebraic Equations

    Direct solution

    Iterative solution


    Appendix E: Triangle and Tetrahedron Integrals



    Appendix F: Integration by Parts in Two or Three Dimensions (Green’s Theorem)

    Appendix G: Solutions Exact at Nodes


    Appendix H: Matrix Diagonalization or Lumping


    Author Index

    Subject Index

Product details

  • No. of pages: 756
  • Language: English
  • Copyright: © Butterworth-Heinemann 2013
  • Published: August 22, 2013
  • Imprint: Butterworth-Heinemann
  • Hardcover ISBN: 9781856176330
  • eBook ISBN: 9780080951355

About the Authors

Olek 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

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

R. L. Taylor

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

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