Notation

Two types of notation are used:

Indicial notation: Equations of continuum mechanics are usually written in this form.
Matrix notation: Used for equations pertinent to the finite element implementation.

Index Notation

Components of tensors and matrices are given explicitly. A vector, which is a first order tensor, is denoted in indicial notation by $x_{i}$ . The range of the index is the dimension of the vector.

To avoid confusion with nodal values, coordinates will be written as $x$ , $y$ or $z$ rather than using subscripts. Similarly, for a vector such as the velocity $v_{i}$ , numerical subscripts are avoided so as to avoid confusion with node numbers. So, $x_{1} = x, x_{2} = y, x_{3} = z$ and $v_{1} = v_{x}, v_{2} = v_{y}$ and $v_{3} = v_{2}$ .

Indices repeated twice in a list are summed. Indices which refer to components of tensors are always written in lower case. Nodal indices are always indicated by upper case Latin letters. For instance, $v_{i I}$ is the i-component of the velocity vector at node I. Upper case indices repeated twice are summed over their range.

A second order tensor is indicated by two subscripts. For example, $E_{i j}$ is a second order tensor whose components are $E_{x x}, E_{x y}$ , ...

Matrix Notation

Matrix notation is used in the implementation of finite element models. For instance, equation(1)

r^{2} = x_{i} \cdot x_{i} = x_{1} \cdot x_{1} + x_{2} \cdot x_{2} + x_{3} \cdot x_{3}

is written in matrix notation as:(2)

r^{2} = x^{T} x

All vectors such as the velocity vector $ν$ will be denoted by lower case letters. Rectangular matrices will be denoted by upper case letters.