Two-dimensional Shape Functions

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with: &nbsp; <math>\qquad p=\frac{(n+1)(n+2)}{2}</math> &nbsp; the number of terms.
 
with: &nbsp; <math>\qquad p=\frac{(n+1)(n+2)}{2}</math> &nbsp; the number of terms.
  
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More specifically:
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{|
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! polynomial order '''n''' | number of terms '''p''' | <math>f(x,y) \,</math>
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|-
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| Constant: <math>0 \,</math> | <math>1 \,</math> | <math>\alpha \,</math>
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|-
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| Linear: <math>1 \,</math> | <math>3 \,</math> | <math>\alpha_1+\alpha_2 x + \alpha_3 y \,</math>
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|-
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| Quadratic: <math>2 \,</math> | <math>6 \,</math> | <math>\alpha_1+\alpha_2 x + \alpha_3 + \alpha_4 x y +\alpha_5 x^2 + \alpha_6 y^2\,</math>
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|}
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For example, in the case of a lineal polynomial:
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:<math>f(x,y)=\alpha_1+\alpha_2 x + \alpha_3 y</math>
  
  

Revision as of 10:17, 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:   \qquad p=\frac{(n+1)(n+2)}{2}   the number of terms.


More specifically:


number of terms p | f(x,y) \,
1 \, | \alpha \,
3 \, | \alpha_1+\alpha_2 x + \alpha_3 y \,
6 \, | \alpha_1+\alpha_2 x + \alpha_3 + \alpha_4 x y +\alpha_5 x^2 + \alpha_6 y^2\,



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.




References

  1. Pascal's triangle
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