Two-dimensional Shape Functions

Shape functions are selected to fit as exact as possible the Finite Element Solution. If this solution is a combination of polynomial functions of n^th order, these functions should include a complete polynomial of equal order.

That is, a complete polynomial of n^th order can be written as:

with:     the number of terms.

More specifically:

polynomial order n	number of terms p
Constant:
Linear:
Quadratic:

For example, in the case of a lineal polynomial:

f(x,y) = α₁ + α₂x + α₃y

can only fit polynomial functions of p^th order if they content a polynomial function

for any polynomial function of pth order it is enough to use p-1 integration points.

References

1. Pascal's triangle