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 nonconforming 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 residualGalerkin 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, selfadjoint, 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 threedimensional elements

4.13 Hierarchic polynomials in one dimension

4.14 Two and threedimensional, 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

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 quasiharmonic 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 Twodimensional mesh generation ¨C advancing front method

8.3 Surface mesh generation

8.4 Threedimensional 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 singleelement 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 Twofield mixed formulation in elasticity

10.5 Threefield 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 Twofield incompressible elasticity (u¨Cp form)

11.4 Threefield 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 Residualbased 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 hrefinement

14.3 prefinement and hprefinement

14.4 Concluding remarks

14.5 Problems


Chapter 15: Pointbased 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: Semidiscretization and analytical solution

16.1 Introduction

16.2 Direct formulation of timedependent problems with spatial finite element subdivision

16.3 General classification

16.4 Free response ¨C eigenvalues for secondorder problems and dynamic vibration

16.5 Free response ¨C eigenvalues for firstorder 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 timestep algorithms for the firstorder equation

17.3 General singlestep 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 singlephase 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 Preprocessing module: mesh creation

19.3 Solution module

19.4 Postprocessor module

19.5 User modules

Appendix A: Matrix algebra

Appendix B: Tensorindicial 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
