These programs and sample data files generate the semi-analytical stiffness
matrix of the elements indicated. The results should lead to the same
stiffness matrix that would have been generated by the numerical integration 
rule indicated, but in a fraction of the CPU time. The subroutines were 
developed using Computer Algebra Systems (CAS).

The formulations are for straight-sided elements, so when appropriate, 
the data reads only the corner coordinates, with any intermediate and 
internal nodes computed automatically.

The 4-node quadrilateral formulation is based on the paper "Stiffness matrix 
of the four-node quadrilateral element in closed form." by D.V. Griffiths, 
Int J Numer Methods Eng, vol.37, no.6, pp.1027-1038, (1994).
 
Both of the 8-node quadrilateral formulations (plane strain and axisymmetry)
use an extrapolation of the methodology used for the 4-node quadrilateral.

The plane strain coding was developed primarily by J-P. Cardoso and described
in "Generation of finite element matrices using computer algebra" 
by J-P. Cardoso, Masters Thesis, School of Engineering, 
Univeristy of Manchester (1994)

The axisymmetric coding was developed primarily by Iris lozada and described 
"Semi-analytical integration of the elastic stiffness matrix of an axisymmetric 
8-noded finite element" by I.J. Lozada, D.V. Griffiths and M. Cerrolaza 
To appear in Numer Meth Part Diff Eqns (2009)


The element node numbering in all cases is the same as that indicated in 
Appendix B of the text, "Programming the finite element method" by 
I.M. Smith and D.V. Griffiths, 4th ed., Wiley, 2004