Two-dimensional Shape Functions
From KratosWiki
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) = α_{1} + α_{2}x + α_{3}y
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.