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 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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