THE FINITE ELEMENT METHOD: ITS BASIS AND FUNDAMENTALS
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By O. C. Zienkiewicz, UNESCO Professor of Numerical Methods in Engineering, International Centre for Numerical Methods in Engineering, Barcelona, Spain R. L. Taylor, Professor in the Graduate School, Department of Civil and Environmental Engineering, University of California at Berkeley, USA J.Z. Zhu, Senior Scientist at ProCast Inc., ESI-Group North America, USA
Description 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, plus a companion website
with a solutions manual and downloadable algorithms – 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.
Audience
Senior students, researchers and practicing engineers in mechanical, automotive, aeronautical and civil engineering. Key topic for applied mathematicians and engineering software developers.
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 |I, |A, |? 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 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|?v 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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