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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWG4baaaa@39D4@ , y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWG4baaaa@39D4@ or z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWG4baaaa@39D4@ 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 x i = x 1 x 1 + x 2 x 2 + x 3 x 3
is written in matrix notation as:(2)
r 2 = x T x

All vectors such as the velocity vector ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqaH9oGBaaa@3A8F@ will be denoted by lower case letters. Rectangular matrices will be denoted by upper case letters.