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 nth order, these functions should include a complete polynomial of equal order.
That is, a complete polynomial of nth order can be written as:
with: the number of terms.
|polynomial order n||number of terms p|
A quick way to easily obtain the terms of a complete polynomial is by using the Pascal's triangle:
|order n||new polynomial terms||number of terms p|
For example, in the case of a lineal polynomial:
- f(x,y) = α1 + α2x + α3y
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.