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The presence of the electrodes is taken into account via appropriate boundary conditions
which will appear as modifications to the equation for normal current, 2.6, on Γ. Electrode
models are discussed in section 2.3.3. Equations 2.2 and 2.6 comprise the forward
problem in EIT and are used to find the voltage distribution within the medium. Analytic
methods using series approximations [76][33] have been used to solve the forward problem
for simplified models such as a single contrast in a circular medium. However, solving the
forward problem for models with arbitrary geometries requires numerical techniques such
as the finite element method. With such methods continuum problems of 2.2 and 2.6 are
converted into discrete algebraic problems that can be solved with a computer.
2.3 Finite Element Method
The Finite Element Method (FEM) is a numerical analysis technique for obtaining approximate
solutions to a wide variety of engineering problems. It was originally developed as
a tool for aircraft design but has since been extended to many other fields including the
modeling of electromagnetic and electrostatic fields [106]. Due to its ability to model arbitrary
geometries and various boundary conditions the Finite Element Method is the most
common method currently used for the numerical solution of EIT problems [92][100]. The
following paragraphs borrow heavily from [75].
In a continuum problem of any dimension the field variable, such as the electric potential
in EIT, is defined over an infinite number of values because it is a function of the infinite
number of points in the body. The finite element method first discretizes the medium under
analysis into a finite number of elements collectively called a finite element mesh. Within
each element the field variable is approximated by simple functions that are defined only
within the individual element. The approximating functions (sometimes called interpolation
or shape functions) are defined in terms of the values of the field variables at specified
points on the element called nodes. Most EIT work uses linear shape functions in which all
nodes lie on the element boundaries where adjacent elements are connected. Higher order
shape functions will have interior nodes. In summary, the finite element method reduces
a continuum problem of infinite dimension to a discrete problem of finite dimension in
which the nodal values of the field variable and the interpolation functions for the elements
completely define the behaviour of the field variable within the elements and the individual
elements collectively define the behaviour of the field over the entire medium.
There are three different methods typically used to formulate Finite Element problems:
1. Direct approach. The Direct approach is so called because of its origins in the direct
stiffness method of structural analysis. The method is limited, however it is the most
intuitive way to understand the finite element method.
2. Variational approach. Element properties obtained by the direct approach can also
be determined by the Variational approach which relies on the calculus of variations
and involves extremizing a functional such as the potential energy. The variational
approach is necessary to extend the finite element method to a class of problems that
cannot be handled by direct methods. For example problems involving elements with
non-constant conductivity, for problems using higher order interpolation functions and
for element shapes other than triangles and tetrahedrons.
3. Method of Weighted Residuals (MWR). The most versatile approach to deriving element
properties is the Method of Weighted Residuals. The weighted residuals approach
begins with the governing equations of the problem and proceeds without
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