# Two-dimensional Shape Functions

From KratosWiki

(Difference between revisions)

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 | + | 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<sup>th</sup> order, these functions should include a complete polynomial of equal order. |

− | That is, a complete polynomial of | + | That is, a complete polynomial of n<sup>th</sup> order can be written as: |

+ | |||

+ | |||

+ | :<math>f(x,y)=\sum_{i=1}^p \alpha_i x^j y^k \qquad j+k \le n</math> | ||

+ | |||

+ | |||

+ | with <math>p=frac{(n+1)(n+2)}{2}</math> | ||

− | |||

can only fit polynomial functions of p<sup>th</sup> order if they content a polynomial function | can only fit polynomial functions of p<sup>th</sup> 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 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 *p* = *f**r**a**c*(*n* + 1)(*n* + 2)2

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.