# computational tool for higher order Lagrangian interpolation for finite element

In finite element, I can calculate the Lagrangian interpolation shape functions for each degree of freedom in an element, from the the number of nodal degrees of freedom and the number of nodes required in the element, if I know the data points ($x$,$y$ coordinates) required to generate linear algebraic conditions, which is in number of total degrees of freedom in the element.

In my problem, for a 2D element I have 6 degree of freedom in each node, namely $\phi, \phi _x, \phi _y, \phi _{xx}, \phi _{yy}$ and $\phi _{xy}$. As the name indicate, the five degrees of freedom are derivative of the $\phi$ degree of freedom.

If I want to calculate, shape functions for degree of freedom in a quadrilateral of four nodes, I would select a incomplete polynomial of degree five and $24$ terms. But the complexity will increase, if the number of nodes are more.

I would like to know that is there any programming library available which can be used for generation of these shape functions for higher nodes?

• do you know how to compute the dual basis of $P_1(T)$ where $T =$ simplex of $R^d$, as in this problem: scicomp.stackexchange.com/questions/25719/… – user177196 Dec 5 '16 at 15:24
• @user177196 Sorry, I'm very beginner in computational science – user294664 Dec 5 '16 at 15:51
• For what kind of problem do you want to do this? – P. G. Dec 5 '16 at 20:14
• @P.G., trying to solve non-linear analysis problem by optimising implicit constitutive relation involving Airy's stress potential. These are the stress degree of freedom given in the problem, derived from Airy's stress potential – user294664 Dec 6 '16 at 5:57
• @user294664 ok, not what I expected. But anyway, I looked up Airy's stress function. To me it looks a bit like the equations for plate bending. If you have four nodes, you can't have Lagrangian shape functions with 24 terms. You should look into hermitian shape fuctions for this problem. – P. G. Dec 6 '16 at 11:45