The problem to solve
by Nasser M. Abbasi (oct 2009)
In this solution, I start directly by solving for the vector field {u,v} and starting from the general degrees of freedom, and from it by matrix inversion, find the shape function matrix N (in terms of nodal degrees of freedom). This involves inversting a 6 by 6 matrix. But Ok, I am using a computer. By hand, I would use the method I showed in the analytical note part of this assignment which involves inverting only a 3 by 3 matrix.
Start by defining the u and v trial functions (linear polynomials in x and y)
set up the u=X a equation
Now find the shape functions. Start by expression nodal unknowns in terms of nodal coordinates
Write the u=A a equation
Find 
from the above by matrix inversion
Now find the B matrix from the above N matrix by multiplying the N matrix by the following differetial operartors matrix
Now find B = oper * N
Factor the determinant term from the above to the outside.
But area of triangle is
Hence B matrix becomes
