Two-dimensional Shape Functions

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Shape functions are selected to fit as exact as possible the Finite Element Solution. If this solution is a combination of polynomial functions of  p<sup>th</sup> order, these functions should include a complete polynomial of equal order.  
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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 p<sup>th</sup> order can be written as:
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That is, a complete polynomial of n<sup>th</sup> order can be written as:
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:<math>f(x,y)=\sum_{i=1}^p \alpha_i x^j y^k \qquad j+k \le n</math>
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with <math>p=frac{(n+1)(n+2)}{2}</math>
  
  
:<math>f(x,y)=\sum_{i=1}^p \alpha_i x^j y^k \qquad j+k \le p</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 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
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