The Finite Element Method: Its Basis and Fundamentals

Author Biography

Dedication

List of Figures

List of Tables

Preface

Chapter 1. The Standard Discrete System and Origins of the Finite Element Method

Abstract

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

References

Chapter 2. Problems in Linear Elasticity and Fields

Abstract

2.1 Introduction

2.2 Elasticity equations

2.3 General quasi-harmonic equation

2.4 Concluding remarks

2.5 Problems

References

Chapter 3. Weak Forms and Finite Element Approximation: 1-D Problems

Abstract

3.1 Weak forms

3.2 One-dimensional form of elasticity

3.3 Approximation to integral and weak forms: The weighted residual (Galerkin) method

3.4 Finite element solution

3.5 Isoparametric form

3.6 Hierarchical interpolation

3.7 Axisymmetric one-dimensional problem

3.8 Transient problems

3.9 Weak form for one-dimensional quasi-harmonic equation

3.10 Concluding remarks

3.11 Problems

References

Chapter 4. Variational Forms and Finite Element Approximation: 1-D Problems

Abstract

4.1 Variational principles

4.2 “Natural” variational principles and their relation to governing differential equations

4.3 Establishment of natural variational principles for linear, self-adjoint differential equations

4.4 Maximum, minimum, or a saddle point?

4.5 Constrained variational principles

4.6 Constrained variational principles: Penalty function and perturbed Lagrangian methods

4.7 Least squares approximations

4.8 Concluding remarks: Finite difference and boundary methods

4.9 Problems

References

Chapter 5. Field Problems: A Multidimensional Finite Element Method

Abstract

5.1 Field problems: Quasi-harmonic equation

5.2 Partial discretization: Transient problems

5.3 Numerical examples: An assessment of accuracy

5.4 Problems

References

Chapter 6. Shape Functions, Derivatives, and Integration

Abstract

6.1 Introduction

6.2 Two-dimensional shape functions

6.3 Three-dimensional shape functions

6.4 Other simple three-dimensional elements

6.5 Mapping: Parametric forms

6.6 Order of convergence for mapped elements

6.7 Computation of global derivatives

6.8 Numerical integration

6.9 Shape functions by degeneration

6.10 Generation of finite element meshes by mapping

6.11 Computational advantage of numerically integrated finite elements

6.12 Problems

References

Chapter 7. Elasticity: Two- and Three-Dimensional Finite Elements

Abstract

7.1 Introduction

7.2 Elasticity problems: Weak form for equilibrium

7.3 Finite element approximation by the Galerkin method

7.4 Boundary conditions

7.5 Numerical integration and alternate forms

7.6 Infinite domains and infinite elements

7.7 Singular elements by mapping: Use in fracture mechanics, etc.

7.8 Reporting results: Displacements, strains, and stresses

7.9 Discretization error and convergence rate

7.10 Minimization of total potential energy

7.11 Finite element solution process

7.12 Numerical examples

7.13 Concluding remarks

7.14 Problems

References

Chapter 8. The Patch Test, Reduced Integration, and Nonconforming Elements

Abstract

8.1 Introduction

8.2 Convergence requirements

8.3 The simple patch test (Tests A and B): A necessary condition for convergence

8.4 Generalized patch test (Test C) and the single-element test

8.5 The generality of a numerical patch test

8.6 Higher order patch tests

8.7 Application of the patch test to plane elasticity elements with “standard” and “reduced” quadrature

8.8 Application of the patch test to an incompatible element

8.9 Higher order patch test: Assessment of robustness

8.10 Concluding remarks

8.11 Problems

References

Chapter 9. Mixed Formulation and Constraints: Complete Field Methods

Abstract

9.1 Introduction

9.2 Mixed form discretization: General remarks

9.3 Stability of mixed approximation: The patch test

9.4 Two-field mixed formulation in elasticity

9.5 Three-field mixed formulations in elasticity

9.6 Complementary forms with direct constraint

9.7 Concluding remarks: Mixed formulation or a test of element “robustness”

9.8 Problems

References

Chapter 10. Incompressible Problems, Mixed Methods, and Other Procedures of Solution

Abstract

10.1 Introduction

10.2 Deviatoric stress and strain, pressure, and volume change

10.3 Two-field incompressible elasticity (u-p form)

10.4 Three-field nearly incompressible elasticity (u-p- form)

10.5 Reduced and selective integration and its equivalence to penalized mixed problems

10.6 A simple iterative solution process for mixed problems: Uzawa method

10.7 Stabilized methods for some mixed elements failing the incompressibility patch test

10.8 Concluding remarks

10.9 Problems

References

Chapter 11. Multidomain Mixed Approximations

Abstract

11.1 Introduction

11.2 Linking of two or more subdomains by Lagrange multipliers

11.3 Linking of two or more subdomains by perturbed Lagrangian and penalty methods

11.4 Problems

References

Chapter 12. The Time Dimension: Semi-Discretization of Field and Dynamic Problems

Abstract

12.1 Introduction

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

12.3 Analytical solution procedures: General classification

12.4 Free response: Eigenvalues for second-order problems and dynamic vibration

12.5 Free response: Eigenvalues for first-order problems and heat conduction, etc.

12.6 Free response: Damped dynamic eigenvalues

12.7 Forced periodic response

12.8 Transient response by analytical procedures

12.9 Symmetry and repeatability

12.10 Problems

References

Chapter 13. Plate Bending Approximation: Thin and Thick Plates

Abstract

13.1 Introduction

13.2 Governing equations

13.3 General plate theory

13.4 The patch test: An analytical requirement

13.5 A nonconforming three-node triangular element

13.6 Numerical example for thin plates

13.7 Thick plates

13.8 Irreducible formulation: Reduced integration

13.9 Mixed formulation for thick plates

13.10 Elements with rotational bubble or enhanced modes

13.11 Linked interpolation: An improvement of accuracy

13.12 Discrete “exact” thin plate limit

13.13 Limitations of plate theory

13.14 Concluding remarks

13.15 Problems

References

Chapter 14. Shells as a Special Case of Three-Dimensional Analysis

Abstract

14.1 Introduction

14.2 Shell element with displacement and rotation parameters

14.3 Special case of axisymmetric thick shells

14.4 Special case of thick plates

14.5 Convergence

14.6 Some shell examples

14.7 Concluding remarks

14.8 Problems

References

Chapter 15. Errors, Recovery Processes, and Error Estimates

Abstract

15.1 Definition of errors

15.2 Superconvergence and optimal sampling points

15.3 Recovery of gradients and stresses

15.4 Superconvergent patch recovery (SPR)

15.5 Recovery by equilibration of patches (REP)

15.6 Error estimates by recovery

15.7 Residual-based methods

15.8 Asymptotic behavior and robustness of error estimator: The Babuška patch test

15.9 Error bounds and error estimates for quantities of interest

15.10 Which errors should concern us?

15.11 Problems

References

Chapter 16. Adaptive Finite Element Refinement

Abstract

16.1 Introduction

16.2 Adaptive h-refinement

16.3 p-refinement and hp-refinement

16.4 Concluding remarks

16.5 Problems

References

Chapter 17. Automatic Mesh Generation

Abstract

17.1 Introduction

17.2 Geometrical representation of the domain

17.3 Two-dimensional mesh generation: Advancing front method

17.4 Surface mesh generation

17.5 Three-dimensional mesh generation: Delaunay triangulation

17.6 Concluding remarks

References

Chapter 18. Computer Procedures for Finite Element Analysis

Abstract

18.1 Introduction

18.2 Pre-processing module: Mesh creation

18.3 Solution module

18.4 Post-processor module

18.5 User modules

References

Appendix A: Matrix Algebra

Definition of a matrix

Matrix addition or subtraction

Transpose of a matrix

Inverse of a matrix

A sum of products

Transpose of a product

Symmetric matrices

Partitioning

The standard eigenvalue problem

The generalized eigenvalue problem

Appendix B: Some Vector Algebra

Addition and subtraction

“Scalar” products

Length of vector

Direction cosines

“Vector” or cross-product

Elements of area and volume

Appendix C: Tensor-Indicial Notation in the Approximation of Elasticity Problems

Introduction

Indicial notation: Summation convention

Derivatives and tensorial relations

Coordinate transformation

Equilibrium and energy

Elastic constitutive equations

Finite element approximation

Relation between indicial and matrix notation

References

Appendix D: Solution of Simultaneous Linear Algebraic Equations

Direct solution

Iterative solution

References

Appendix E: Triangle and Tetrahedron Integrals

Triangles

Tetrahedron

Appendix F: Integration by Parts in Two or Three Dimensions (Green’s Theorem)

Appendix G: Solutions Exact at Nodes

References

Appendix H: Matrix Diagonalization or Lumping

References

Author Index

Subject Index