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