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
. The range of the index is the dimension of the vector.
To avoid confusion with nodal values, coordinates will be written as
,
or
rather than using subscripts. Similarly, for a vector
such as the velocity
, numerical subscripts are avoided so as to avoid
confusion with node numbers. So,
and
and
.
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,
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,
is a second order tensor whose components are
, ...
Matrix Notation
Matrix notation is used in the implementation of finite element models. For instance,
equation
(1)
is written in matrix notation as:
(2)
All vectors such as the velocity vector
will be denoted by lower case letters. Rectangular matrices will be denoted by upper
case letters.