The Finite Element Method: Its Basis and Fundamentals - 6th Edition - ISBN: 9780750663205, 9780080472775

The Finite Element Method: Its Basis and Fundamentals

6th Edition

Authors: Olek Zienkiewicz Robert Taylor J.Z. Zhu
eBook ISBN: 9780080472775
Paperback ISBN: 9781493302888
Imprint: Butterworth-Heinemann
Published Date: 18th April 2005
Page Count: 752
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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 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


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


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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


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 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