# 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 pth order, these functions should include a complete polynomial of equal order.

That is, a complete polynomial of pth order can be written as:

$f(x,y)=\sum_{i=1}^p \alpha_i x^j y^k \qquad j+k \le p$

can only fit polynomial functions of pth 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