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Skip to main content# The Finite Element Method: Its Basis and Fundamentals

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Request a sales quote**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### Olek C Zienkiewicz

### Robert L. Taylor

### J.Z. Zhu

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6th Edition - April 18, 2005

Authors: Olek C Zienkiewicz, Robert L. Taylor, J.Z. Zhu

Language: EnglisheBook ISBN:

9 7 8 - 0 - 0 8 - 0 4 7 2 7 7 - 5

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… Read more

LIMITED OFFER

Immediately download your ebook while waiting for your print delivery. No promo code is needed.

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.

• 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

Together with the second and third self-contained volumes (0750663219 and 0750663227),

- 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

14.1 Introduction

14.2 Adaptive h-refinement

14.3 p-refinement and hp-refinement

14.4 Concluding remarks

14.5 Problems

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

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

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

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

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

- No. of pages: 752
- Language: English
- Edition: 6
- Published: April 18, 2005
- Imprint: Butterworth-Heinemann
- eBook ISBN: 9780080472775

OZ

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, SpainRT

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.JZ

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