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1 Finite Element Basis Functions
1.1 Representing a One-Dimensional Field
1.2 Linear Basis Functions
1.3 Basis Functions as Weighting Functions
1.4 Quadratic Basis Functions
1.5 Two- and Three-Dimensional Elements
1.6 Higher Order Continuity
1.7 Triangular Elements
1.8 Curvilinear Coordinate Systems
1.9 CMISS Examples

2 Steady-State Heat Conduction
2.1 One-Dimensional Steady-State Heat Conduction
2.1.1 Integral equation
2.1.2 Integration by parts
2.1.3 Finite element approximation
2.1.4 Element integrals
2.1.5 Assembly
2.1.6 Boundary conditions
2.1.7 Solution
2.1.8 Fluxes
2.2 An x-Dependent Source Term
2.3 The Galerkin Weight Function Revisited
2.4 Two and Three-Dimensional Steady-State Heat Conduction
2.5 Basis Functions – Element Discretisation
2.6 Integration
2.7 Assemble Global Equations
2.8 Gaussian Quadrature
2.9 CMISS Examples

3 The Boundary Element Method
3.1 Introduction
3.2 The Dirac-Delta Function and Fundamental Solutions
3.2.1 Dirac-Delta function
3.2.2 Fundamental solutions
3.3 The Two-Dimensional Boundary Element Method
3.4 Numerical Solution Procedures for the Boundary Integral Equation
3.5 Numerical Evaluation of Coefficient Integrals
3.6 The Three-Dimensional Boundary Element Method
3.7 A Comparison of the FE and BE Methods
3.8 More on Numerical Integration
3.8.1 Logarithmic quadrature and other special schemes
3.8.2 Special solutions
3.9 The Boundary Element Method Applied to other Elliptic PDEs
3.10 Solution of Matrix Equations
3.11 Coupling the FE and BE techniques
3.12 Other BEM techniques
3.13 Symmetry
3.14 Axisymmetric Problems
3.15 Infinite Regions
3.16 Appendix: Common Fundamental Solutions
3.17 CMISS Examples

4 Transient Heat Conduction
4.1 Introduction
4.2 Finite Differences
4.2.1 Explicit Transient Finite Differences
4.2.2 Von Neumann Stability Analysis
4.2.3 Higher Order Approximations
4.3 The Transient Advection-Diffusion Equation
4.4 Mass lumping
4.5 CMISS Examples

5 Linear Elasticity
5.1 Introduction
5.2 Truss Elements
5.3 Beam Elements
5.4 Plane Stress Elements
5.4.1 Notes on calculating nodal loads
5.5 Three-Dimensional Elasticity
5.6 Linear Elasticity with Boundary Elements
5.7 Fundamental Solutions
5.8 Boundary Integral Equation
5.9 Body Forces (and Domain Integrals in General)
5.10 CMISS Examples

6 Modal Analysis
6.1 Introduction
6.2 Free Vibration Modes
6.3 An Analytic Example
6.4 Proportional Damping
6.5 CMISS Examples

7 Domain Integrals in the BEM
7.1 Achieving a Boundary Integral Formulation
7.2 Removing Domain Integrals due to Inhomogeneous Terms
7.2.1 The Galerkin Vector technique
7.2.2 The Monte Carlo method
7.2.3 Complementary Function-Particular Integral method
7.3 Domain Integrals Involving the Dependent Variable
7.3.1 The Perturbation Boundary Element Method
7.3.2 The Multiple Reciprocity Method
7.3.3 The Dual Reciprocity Boundary Element Method

8 The BEM for Parabolic PDES
8.1 Time-Stepping Methods
8.1.1 Coupled Finite Difference – Boundary Element Method
8.1.2 Direct Time-Integration Method
8.2 Laplace Transform Method
8.3 The DR-BEM For Transient Problems
8.4 The MRM for Transient Problems

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