The Finite Element Method: Its Basis and Fundamentals

The Finite Element Method: Its Basis and Fundamentals

6th Edition - April 18, 2005

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  • Authors: O. C. Zienkiewicz, R. L. Taylor, J.Z. Zhu
  • eBook ISBN: 9780080472775

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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 problemsActive 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


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

Product details

  • No. of pages: 752
  • Language: English
  • Copyright: © Butterworth-Heinemann 2005
  • Published: April 18, 2005
  • Imprint: Butterworth-Heinemann
  • eBook ISBN: 9780080472775

About the Authors

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

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.

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

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