# Two-dimensional Shape Functions

(Difference between revisions)
 Revision as of 09:47, 4 November 2009 (view source)JMora (Talk | contribs)← Older edit Revision as of 09:49, 4 November 2009 (view source)JMora (Talk | contribs) Newer edit → Line 1: Line 1: − 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. + 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 pth order can be written as: + That is, a complete polynomial of nth order can be written as: + + + :$f(x,y)=\sum_{i=1}^p \alpha_i x^j y^k \qquad j+k \le n$ + + + with $p=frac{(n+1)(n+2)}{2}$ − :$f(x,y)=\sum_{i=1}^p \alpha_i x^j y^k \qquad j+k \le p$ can only fit polynomial functions of pth order if they content a polynomial function can only fit polynomial functions of pth order if they content a polynomial function

## Revision as of 09:49, 4 November 2009

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:

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

with p = frac(n + 1)(n + 2)2

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