The Finite Element Method

Its Basis and Fundamentals

8th Edition - November 21, 2024
Authors: O. C. Zienkiewicz, R. L. Taylor, S. Govindjee
Language: English
eBook ISBN:
9 7 8 - 0 - 4 4 3 - 1 6 0 4 5 - 5

The Finite Element Method: Its Basis and Fundamentals, Eighth Edition offers a complete introduction to the basis of the finite element method, covering fundamental theory an… Read more

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The Finite Element Method: Its Basis and Fundamentals, Eighth Edition offers a complete introduction to the basis of the finite element method, covering fundamental theory and worked examples in a kind of detail required for readers to apply the knowledge to their own engineering problems and understand more advanced applications. This edition includes a significant addition of content addressing coupling problems, including: Finite element analysis formulations for coupled problems; Details of algorithms for solving coupled problems; Examples showing how algorithms can be used to solve for piezoelectricity and poroelasticity problems.

Focusing on the core knowledge, mathematical and analytical tools needed for successful application, this book is the authoritative resource of choice for graduate level students, researchers and professional engineers involved in finite element-based engineering analysis.

Title of Book
Cover image
Title page
Table of Contents
Copyright
Dedication
List of figures
List of tables
Biography
Preface
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. 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. Concluding remarks
1.10. Problems
2: Problems in linear fields
2.1. Introduction
2.2. General quasi-harmonic equation
2.2.1. Governing equations: flux and continuity
2.2.2. Boundary conditions
2.2.3. Initial condition
2.2.4. Constitutive behaviour
2.2.5. Irreducible form
2.2.6. Anisotropic and isotropic forms for k: transformations
2.3. Governing equations: cylindrical coordinates
2.4. Concluding remarks
2.5. Problems
3: Weak forms and approximate solutions: 1-D field problems
3.1. Weak forms
3.2. One dimensional weak form of field problem
3.2.1. Cylindrical coordinate form
3.3. Approximation to integral and weak forms: the weighted residual-Galerkin method
3.3.1. Galerkin solution of quasi-harmonic equation
3.4. Concluding remarks
3.5. Problems
4: Finite element approximate solutions: 1-D field problems
4.1. Finite element solution
4.1.1. Boundary flux computation
4.2. Requirements for finite element approximations
4.3. Isoparametric form
4.3.1. Integrals on the parent element: numerical integration
4.4. Transient problems
4.4.1. Discrete time methods
4.4.2. ‘Spurious’ oscillations
4.5. Concluding remarks
4.6. Problems
5: Finite element forms exact at nodes
5.1. Introduction
5.1.1. Adjoint forms
5.2. Solutions exact at nodes
5.2.1. Cylindrical problem
5.2.2. Convection-diffusion equation
5.3. Concluding remarks
5.4. Problems
6: Variational forms and finite element approximation: 1-D problems
6.1. Variational principles
6.2. ‘Natural’ variational principles and their relation to governing differential equations
6.2.1. Euler equations
6.3. Establishment of natural variational principles for linear, self-adjoint differential equations
6.4. Maximum, minimum, or a saddle point?
6.5. Constrained variational principles
6.5.1. Lagrange multipliers
6.5.2. Identification of Lagrange multipliers. Forced boundary conditions and modified variational principles
6.6. Constrained variational principles. Penalty function and perturbed Lagrangian methods
6.6.1. Penalty functions
6.6.2. Perturbed Lagrangian
6.7. Least squares approximations
6.8. Concluding remarks – finite difference and boundary methods
6.9. Problems
7: Field problems: a multi-dimensional finite element method
7.1. Field problems: quasi-harmonic equation
7.1.1. Irreducible form
7.1.2. Finite element discretization
7.1.3. Shape functions for triangle, tetrahedron and rectangle
7.1.4. Quadrature for rectangular elements
7.2. Numerical examples – an assessment of accuracy
7.2.1. Torsion of prismatic bars
7.2.2. Transient heat conduction
7.2.3. Anisotropic seepage
7.2.4. Electrostatic and magnetostatic problems
7.2.5. Lubrication problems
7.2.6. Irrotational and free surface flows
7.3. Problems
8: Shape functions, derivatives and integration
8.1. Introduction
8.2. Lagrange interpolation
8.2.1. One dimensional quadratic element
8.3. Two dimensional shape functions
8.3.1. Shape functions for triangles
8.3.2. Shape functions for quadrilaterals
8.4. Three dimensional shape functions
8.4.1. Tetrahedral elements
8.4.2. Hexahedral elements – brick family
8.5. Other simple three-dimensional elements
8.5.1. ‘Serendipity’ quadratic
8.6. Mapping: parametric forms
8.7. Order of convergence for mapped elements
8.8. Computation of global derivatives
8.8.1. Placement of element coordinates
8.9. Numerical integration
8.9.1. Quadrilateral elements
8.9.2. Brick elements
8.9.3. Triangular elements
8.9.4. Tetrahedral elements
8.9.5. Required order of numerical integration
8.9.6. Matrix singularity due to numerical integration
8.9.7. Computational advantage of numerically integrated finite elements
8.10. Shape functions by degeneration
8.10.1. Higher order degenerate elements
8.11. Problems
9: Problems in linear elasticity
9.1. Introduction
9.2. Elasticity equations
9.2.1. Displacement function
9.2.2. Strain matrix
9.2.3. Stress matrix
9.2.4. Equilibrium equations
9.2.5. Boundary conditions
9.2.6. Initial conditions
9.2.7. Transformation of stress and strain
9.2.8. Stress-strain relations – elasticity matrix
9.3. Concluding remarks
9.4. Problems
10: Elasticity: two and three dimensional finite elements
10.1. Introduction
10.2. Elasticity problems: weak form for equilibrium
10.2.1. Displacement method: irreducible form
10.3. Finite element approximation by Galerkin method
10.3.1. Quasi-static problems
10.3.2. Strains: two-dimensional case
10.3.3. Element stresses
10.4. Boundary conditions
10.4.1. Displacement boundary conditions
10.4.2. Traction boundary conditions
10.5. Numerical integration and alternate forms
10.6. Infinite domains and infinite elements
10.6.1. The mapping function
10.7. Singular elements by mapping – use in fracture mechanics, etc.
10.8. Reporting results: displacements, strains and stresses
10.9. Discretization error and convergence rate
10.10. Minimization of total potential energy
10.10.1. Bound on strain energy in a displacement formulation
10.10.2. Direct minimization
10.11. Finite element solution process
10.12. Numerical examples
10.13. Concluding remarks
10.14. Problems
11: The patch test, reduced integration, and non-conforming elements
11.1. Introduction
11.2. Convergence requirements
11.3. The patch test – a necessary condition for convergence
11.4. The generality of a numerical patch test
11.5. Higher order patch tests
11.6. Application of the patch test to plane elasticity elements with ‘standard’ and ‘reduced’ quadrature
11.7. Application of the patch test to an incompatible element
11.8. Higher order patch test – assessment of robustness
11.9. Concluding remarks
11.10. Problems
12: Mixed formulation and constraints – complete field methods
12.1. Introduction
12.2. Mixed form discretization – general remarks
12.3. Stability of mixed approximation. The patch test
12.3.1. Solvability requirement
12.3.2. Locking
12.3.3. The mixed patch test
12.4. Two-field mixed formulation in elasticity
12.4.1. General
12.4.2. The u–σ mixed form
12.4.3. Stability of two-field approximation in elasticity (u–σ)
12.5. Three-field mixed formulations in elasticity
12.5.1. The u–σ–ε mixed form
12.5.2. Stability condition of three-field approximation (u–σ–ε)
12.5.3. The u–σ–εen form. Enhanced strain formulation
12.6. Complementary forms with direct constraint
12.6.1. General forms
12.6.2. Solution using auxiliary functions
12.7. Concluding remarks – mixed formulation or a test of element ‘robustness’
12.8. Problems
13: Incompressible problems, mixed methods and other procedures of solution
13.1. Introduction
13.2. Deviatoric stress and strain, pressure and volume change
13.3. Two-field incompressible elasticity (u–p form)
13.4. Three-field nearly incompressible elasticity (u–p–εv form)
13.4.1. The B-bar method for nearly incompressible problems
13.5. Reduced and selective integration and its equivalence to penalized mixed problems
13.6. A simple iterative solution process for mixed problems: Uzawa method
13.6.1. General
13.6.2. Iterative solution for incompressible elasticity
13.7. Stabilized methods for some mixed elements failing the incompressibility patch test
13.7.1. Galerkin least squares method
13.7.2. Direct pressure stabilization
13.7.3. Numerical comparisons
13.8. Concluding remarks
13.9. Problems
14: Transient analysis of solids
14.1. Transient problems
14.2. Discrete time methods
14.2.1. Newmark method
14.2.2. HHT algorithm
14.2.3. First order form
14.3. Semi-discrete method
14.4. Stability, damping and period behaviour
14.4.1. HHT and (β,γ)=(1/4,1/2) Newmark
14.4.2. First order form algorithm
14.5. Concluding remarks
14.6. Problems
15: Finite element solutions: multi-physics problems
15.1. Introduction
15.2. Linear thermoelasticity
15.2.1. Isotropic case
15.2.2. Variational equations
15.2.3. Finite element solution
15.2.4. Equation structure quasi-static case
15.2.5. Transient integration
15.2.6. Operator splitting
15.2.7. Application of operator splitting to thermoelasticity
15.3. Linear chemoelasticity
15.3.1. Theory
15.3.2. Variational equations
15.3.3. Finite element solution
15.3.4. Time integration
15.4. Linear poroelasticity
15.4.1. Theory
15.4.2. Variational equations
15.4.3. Finite element solution
15.5. Piezoelectricity
15.5.1. Governing equations
15.5.2. Variational equations
15.5.3. Finite element solution
15.6. Concluding remarks
16: Multi-physics battery simulation
16.1. Introduction
16.2. Chemoelasticity for energy storage – batteries
16.2.1. Battery material model
16.2.2. Variational equations
16.2.3. Nonlinear equation solving
16.3. Concluding remarks
17: Multidomain mixed approximations
17.1. Introduction
17.2. Linking of two or more subdomains by Lagrange multipliers
17.2.1. Linking subdomains for quasi-harmonic equations
17.2.2. Treatment for forced boundary conditions
17.2.3. Mortar and dual mortar methods
17.2.4. Linking subdomains for elasticity equations
17.3. Linking of two or more subdomains by perturbed Lagrangian and penalty methods
17.3.1. Nitsche method and discontinuous Galerkin approximation
17.4. Problems
18: The time dimension – semi-discretization of field and dynamic problems
18.1. Introduction
18.2. Direct formulation of time-dependent problems with spatial finite element subdivision
18.2.1. The ‘quasi-harmonic’ equation with first and second time derivative
18.2.2. Dynamic behaviour of elastic structures with linear damping
18.2.3. ‘Mass’ or ‘damping’ matrices for some typical elements
18.2.4. Mass ‘lumping’ or diagonalization
18.3. Analytical solution procedures: general classification
18.4. Free response – eigenvalues for second-order problems and dynamic vibration
18.4.1. Free dynamic vibration – real eigenvalues
18.4.2. Determination of eigenvalues
18.4.3. Free vibration with a singular K matrix
18.4.4. Reduction of the eigenvalue system
18.4.5. Examples
18.5. Free response – eigenvalues for first-order problems and heat conduction, etc.
18.6. Free response – damped dynamic eigenvalues
18.7. Forced periodic response
18.8. Transient response by analytical procedures
18.8.1. General
18.8.2. Frequency response procedures
18.8.3. Modal decomposition analysis
18.8.4. Damping and participation of modes
18.9. Symmetry and repeatability
18.10. Problems
19: Plate bending approximation: thin and thick plates
19.1. Introduction
19.2. Governing equations
19.2.1. One dimensional theory: cylindrical bending
19.2.2. Weak form for cylindrical bending
19.2.3. Finite element approximation
19.2.4. Exact nodal solution for thick plate
19.3. General plate theory
19.3.1. The boundary conditions
19.3.2. The irreducible, thin plate approximation
19.3.3. Finite element approximation
19.3.4. Continuity requirement for shape functions (C1 continuity)
19.4. The patch test – an analytical requirement
19.5. A non-conforming 9-node triangular element
19.6. Numerical example for thin plates
19.7. Thick plates
19.8. Irreducible formulation – reduced integration
19.9. Mixed formulation for thick plates
19.9.1. The approximation
19.9.2. Continuity requirements
19.9.3. Equivalence of mixed forms with discontinuous Q interpolation and reduced (selective) integration
19.9.4. Why elements fail: patch test for thick plates
19.9.5. Design of some useful elements
19.10. Elements with rotational bubble or enhanced modes
19.11. Linked interpolation – an improvement of accuracy
19.11.1. Linking function for linear triangles and quadrilaterals
19.11.2. Linked interpolation for quadratic elements
19.12. Discrete ‘exact’ thin plate limit
19.13. Limitations of plate theory
19.14. Concluding remarks
19.15. Problems
20: Shells as a special case of three-dimensional analysis
20.1. Introduction
20.2. Shell element with displacement and rotation parameters
20.2.1. Geometric definition of an element
20.2.2. Displacement field
20.2.3. Definition of strains and stresses
20.2.4. Element properties and necessary transformations
20.2.5. Some remarks on stress representation
20.3. Special case of axisymmetric thick shells
20.4. Special case of thick plates
20.5. Convergence
20.6. Some shell examples
20.7. Concluding remarks
20.8. Problems
21: Computer procedures for finite element analysis
21.1. Introduction
21.2. Pre-processing module: mesh creation
21.2.1. Element library
21.3. Solution module
21.4. Post-processor module
21.5. User modules
A: Matrix algebra
B: Some vector algebra
C: Mathematical properties, error estimates and convergence
Canonical 1-D steady state problem
Inner products and norms
Properties of the canonical problem
Stability of the exact and finite element solutions
Orthogonality of error
Best approximation: energy norm
Stability of the finite element error
Error of the interpolant
An L2 norm estimate on the FE error
D: Tensor-indicial notation in elasticity problems
E: Solution of simultaneous linear algebraic equations
F: Integration by parts – Green's theorem
G: Triangle and tetrahedron integrals
H: Matrix diagonalization or lumping
I: Recovery process
Author index
Subject index

O. C. Zienkiewicz

Professor O.C. Zienkiewicz, CBE, FRS, FREng died on 2 January 2009. Prior to his death he was Professor Emeritus at the Civil and Computational Engineering Centre, University of Wales Swansea and previously was Director of the Institute for Numerical Methods in Engineering at the University of Wales Swansea, UK. He also held the UNESCO Chair of Numerical Methods in Engineering at the Technical University of Catalunya, Barcelona, Spain. He was the head of the Civil Engineering Department at the University of Wales Swansea between 1961 and 1989. During this period he established that department as one of the primary centres of finite element research. In 1968 he became the Founder Editor of the International Journal for Numerical Methods in Engineering which still remains today the major journal in this field. The recipient of 27 honorary degrees and many medals, Professor Zienkiewicz was a member of five academies – an honour he received for his many contributions to the fundamental developments of the finite element method. In 1978, he became a Fellow of the Royal Society and the Royal Academy of Engineering. This was followed by his election as a foreign member to the US National Academy of Engineering (1981), the Polish Academy of Science (1985), the Chinese Academy of Sciences (1998), and the National Academy of Science, Italy (Academia dei Lincei) (1999). He published the first edition of this book in 1967 and it remained the only book on the subject until 1971.

Affiliations and expertise

Swansea University, Swansea, Wales

R. L. Taylor

Professor R.L. Taylor has more than 60 years of experience in the modelling and simulation of structures and solid continua including eighteen years in industry. He is Professor of the Graduate School and the Emeritus T.Y. and Margaret Lin Professor of Engineering at the University of California, Berkeley and also Corporate Fellow at Dassault Systèmes Americas Corp. in Johnston, Rhode Island. In 1991 he was elected to membership in the US National Academy of Engineering in recognition of his educational and research contributions to the field of computational mechanics. Professor Taylor is a Fellow of the US Association for Computational Mechanics – USACM (1996) and a Fellow of the International Association of Computational Mechanics – IACM (1998). He has received numerous awards including the Berkeley Citation, the highest honour awarded by the University of California, Berkeley, the USACM John von Neumann Medal, the IACM Gauss–Newton Congress Medal and a Dr.-Ingenieur ehrenhalber awarded by the Technical University of Hannover, Germany. Professor Taylor has written several computer programs for finite element analysis of structural and non-structural systems, one of which, FEAP, is used world-wide in education and research environments. A personal version, FEAPpv, available on GitHub, is incorporated into this book.

Affiliations and expertise

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

S. Govindjee

S. Govindjee is the Horace, Dorothy, and Katherine Johnson Endowed Professor in Engineering and a Distinguish Professor of Civil and Environmental Engineering at the University of California, Berkeley (1993–2006, 2008–present). His main interests are in theoretical and computational mechanics with an emphasis on micro-mechanics, shape memory alloys, and elastomers. Prior to joining Berkeley he worked as an engineer at the Lawrence Livermore National Laboratory (1991–1993) in Livermore, California. He was also Professor of Mechanics at ETH Zürich (2006–2008) in Zürich, Switzerland. Dr. Govindjee serves as a consultant to several governmental agencies and private corporations. He is an active member of the American Society of Mechanical Engineers (ASME), the US Association for Computational Mechanics (USACM), and the International Association for Computational Mechanics (IACM) and has served or serves on the editorial boards of several major journals. He is also a registered Professional Mechanical Engineer in the state of California. He is the author of four texts books on engineering mechanics. Noteworthy honours include a National Science Foundation Career Award, the inaugural 1998 Zienkiewicz Prize and Medal, an Alexander von Humboldt Foundation Fellowship 1999, a Berkeley Chancellor’s Professorship 2006–2011, and a guest Professorship at ETH Zürich 2008–2013. In 2015 he was named a Fellow of the US Association for Computational Mechanics. In 2018 he received a Humboldt-Forschungspreis (Alexander von Humboldt Research Award).

Affiliations and expertise

Professor of Engineering, University of California, Berkeley, USA

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The Finite Element Method

Its Basis and Fundamentals

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O. C. Zienkiewicz

R. L. Taylor

S. Govindjee