This book is dedicated to our wives Helen, Mary Lou and Song and our families for their support and patience during the preparation of this book, and also to all of our students and colleagues who over the years have contributed to our knowledge of the finite element method. In particular we would like to mention Professor Eugenio Oniate and his group at CIMNE for their help, encouragement and support during the preparation process.

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 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 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 - 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 -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 Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches

 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 Vitual 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, differentaal 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 - finite difference and boundary methods

 3.15 Problems

4 Standard' and 'hierarchical' element shape functions: some general families of Co continuity

 4.1 Introduction

 4.2 Standard and hierarchical concepts

 4.3 Rectangular elements - some preliminary considerations

 4.4 Completeness of polynomials

 4.5 Rectangular elements - Lagrange family

 4.6 Rectangular dements - 'serendipity' family

 4.7 Triangular element family

 4.8 Line elements

 4.9 Rectangular prisms - Lagrange family

 4.10 Rectangular prisms - 'serendipity' family

 4.11 Tetrahedral dements

 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 - substructures

 4.19 Concluding remarks

 4.20 Problems

5 Mapped elements and numerical integration - 'infinite' and 'singularity elements'

 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

 5.5 Evaluation of element matrices. Transformation in ε, η, ζ coordinates

 5.6 Evaluation of element matrices. Transformation in area and volumecoordinates

 5.7 Order of convergence for mapped elements

 5.8 Shape functions by degeneration

 5.9 Numerical integration - one dimensional

 5.10 Numerical integration - rectangular (2D) or brick regions (3D)

 5.11 Numerical integration - triangular or tetrahedral regions

 5.12 Required order of numerical integration

 5.13 Generation of finite element meshes by mapping. Blending functions

 5.14 Infinite domains and infinite elements

 5.15 Singular elements by mapping - use in fracture mechanics, etc.

 5.16 Computational advantage of numerically integrated finite elements

 5.17 Problems

6 Problems in 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

7 Field problems - heat conduction, electric and magnetic potential and fluid flow

 7.1 Introduction

 7.2 General quasi-harmonic equation

 7.3 Finite element solution process

 7.4 Partial discretization - transient problems

 7.5 Numerical examples - an assessment of accuracy

 7.6 Concluding remarks

 7.7 Problems

8 Automatic mesh generation

 8.1 Introduction

 8.2 Two-dimensional mesh generation - advancing front method

 8.3 Surface mesh generation

 8.4 Three-dimensional mesh generation - Delaunay triangulation

 8.5 Concluding remarks

 8.6 Problems

9 The patch test, reduced integration, and non-conforming elements

 9.1 Introduction

 9.2 Convergence requirements

 9.3 The simple patch test (tests A and B) - 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 dements with 'standard' and 'reduced' quadrature

 9.8 Application of the patch test to an incompatible element

 9.9 Higher order patch test - assessment of robustness

 9.10 Concluding remarks

 9.11 Problems

10 Mixed formulation and constraints - complete field methods

 10.1 Introduction

 10.2 Discretization of mixed forms - some general remarks

 10.3 Stability of mixed approximation. The patch test

 10.4 Two-fidd mixed formulation in elasticity

 10.5 Three-field mixed formulations in elasticity

 10.6 Complementary forms with direct constraint

 10.7 Concluding remarks - mixed formulation or a test of element 'robustness'

 10.8 Problems

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 (up form)

 11.4 Three-field nearly incompressible elasticity (u-p-~o 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 Problems

12 Multidomain mixed approximations - 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

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 -, SPR

 13.5 Recovery by equilibration of patches - REP

 13.6 Error estimates by recovery

 13.7 Residual-based methods

 13.8 Asymptotic behaviour and robustness of error estimators - the Babuska patch test

 13.9 Bounds on quantities of interest

 13.10 Which errors should concern us?

 13.11 Problems

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

15 Point-based and partition of unity approximations. Extended finite element methods

 15.1 Introduction

 15.2 Function approximation

 15.3 Moving least squares approximations - restoration of continuity of approximation

 15.4 Hierarchical enhancement of moving least squares expansions

 15.5 Point collocation - 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 Concluding remarks

 15.9 Problems

16 The time dimension - semi-discretization of field and dynamic problems and analytical solution procedures

 16.1 Introduction

 16.2 Direct formulation of time-dependent problems with spatial finite element subdivision

 16.3 General classification

 16.4 Free response - eigenvalues for second-order problems and dynamic vibration

 16.5 Free response - eigenvalues for first-order problems and heat conduction, etc.

 16.6 Free response - damped dynamic eigenvalues

 16.7 Forced periodic response

 16.8 Transient response by analytical procedures

 16.9 Symmetry and repeatability

 16.10 Problems

17 The time dimension - 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

18 Coupled systems

 18.1 Coupled problems - definition and classification

 18.2 Fluid-structure interaction (Class I problems)

 18.3 Soil-pore fluid interaction (Class II problems)

 18.4 Partitioned single-phase systems - implicit--explicit partitions(Class I problems)

 18.5 Staggered solution processes

 18.6 Concluding remarks

19 Computer procedures for finite dement 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 the approximation of elasticity problems

Appendix C: Solution of simultaneous linear algebraic equations

Appendix D: Some integration formulae for a triangle

Appendix E: Some integration formulae for a tetrahedron

Appendix F: Some vector algebra

Appendix G: Integration by parts in two or three dimensions (Green's theorem)

Appendix H: Solutions exact at nodes

Appendix I: Matrix diagonalization or lumping

Author index

Subject index