Finite Element Analysis with Error Estimators

Preface

Features of the text and accompanying resources

Notation

Chapter 1: Introduction

1.1 Finite element methods

1.2 Capabilities of FEA

1.3 Outline of finite element procedures

1.4 Assembly into the system equations

1.5 Error section-titles

1.6 Exercises

Chapter 2: Mathematical preliminaries

2.1 Introduction

2.2 Linear spaces and norms

2.3 Sobolev norms *

2.4 Dual problem, self-adjointness

2.5 Weighted residuals

2.6 Boundary condition terms

2.7 Adding more unknowns

2.8 Numerical integration

2.9 Integration by parts

2.10 Finite element model problem

2.11 Continuous nodal flux recovery

2.12 A one-dimensional example error analysis

2.13 General boundary condition choices

2.14 General matrix partitions

2.15 Elliptic boundary value problems

2.16 Initial value problems

2.17 Eigen-problems

2.18 Equivalent forms *

2.19 Exercises

Chapter 3: Element interpolation and local coordinates

3.1 Introduction

3.2 Linear interpolation

3.3 Quadratic interpolation

3.4 Lagrange interpolation

3.5 Hermitian interpolation

3.6 Hierarchical interpolation

3.7 Space-time interpolations*

3.8 Nodally exact interpolations *

3.9 Interpolation error *

3.10 Gradient estimates *

3.11 Exercises

Chapter 4: One-dimensional integration

4.1 Introduction

4.2 Local coordinate Jacobian

4.3 Exact polynomial integration *

4.4 Numerical integration

4.5 Variable Jacobians

4.6 Exercises

Chapter 5: Error estimates for elliptic problems

5.1 Introduction

5.2 Error estimates

5.3 Hierarchical error indicator

5.4 Flux balancing error estimates

5.5 Element adaptivity

5.6 H-adaptivity

5.7 P-adaptivity

5.8 HP-adaptivity

5.9 Exercises

Chapter 6: Super-convergent patch recovery

6.1 Patch implementation database

6.2 SCP nodal flux averaging

6.3 Computing the SCP element error estimates

6.4 Hessian matrix *

6.5 Exercises

Chapter 7: Variational methods

7.1 Introduction

7.2 Structural mechanics

7.3 Finite element analysis

7.4 Continuous elastic bar

7.5 Thermal loads on a bar *

7.6 Reaction flux recovery for an element

7.7 Heat transfer in a rod

7.8 Element validation *

7.9 Euler’s equations of variational calculus *

7.10 Exercises

Chapter 8: Cylindrical analysis problems

8.1 Introduction

8.2 Heat conduction in a cylinder

8.3 Cylindrical stress analysis

8.4 Exercises

Chapter 9: General interpolation

9.1 Introduction

9.2 Unit coordinate interpolation

9.3 Natural coordinates

9.4 Isoparametric and subparametric elements

9.5 Hierarchical interpolation

9.6 Differential geometry *

9.7 Mass properties *

9.8 Interpolation error *

9.9 Element distortion*

9.10 Space-time interpolation *

9.11 Exercises

Chapter 10: Integration methods

10.1 Introduction

10.2 Unit coordinate integration

10.3 Simplex coordinate integration

10.4 Numerical integration

10.4.1 Unit coordinate quadrature

10.4.2 Natural coordinate quadrature

10.5 Typical source distribution integrals *

10.6 Minimal, optimal, reduced and selected integration

10.7 Exercises

Chapter 11: Scalar fields

11.1 Introduction

11.2 Variational formulation

11.3 Element and boundary matrices

11.4 Linear triangular element

11.5 Linear triangle applications

11.5.1 Internal source

11.6 Bilinear rectangles *

11.7 General 2-d elements

11.8 Numerically integrated arrays

11.9 Strong diagonal gradient SCP test case

11.10 Orthotropic conduction

11.11 Axisymmetric conductions

11.12 Torsion

11.13 Introduction to linear flows

11.14 Potential flow

11.15 Axisymmetric plasma equilibria

11.16 Slider bearing lubrication

11.17 Transient scalar fields

11.18 Exercises

Chapter 12: Vector fields

12.1 Introduction

12.2 Displacement based stress analysis summary

12.3 Planar models

12.4 Matrices for the constant strain triangle (CST)

12.5 Stress and strain transformations *

12.6 Axisymmetric solid stress *

12.7 General solid stress *

12.8 Anisotropic materials *

12.9 Circular hole in an infinite plate

12.10 Dynamics of solids

12.11 Exercises

Index