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:

polynomial order n number of terms p f(x,y) \,
Constant: 0 \, 1 \, \alpha \,
Linear: 1 \, 3 \, \alpha_1+\alpha_2 x + \alpha_3 y \,
Quadratic: 2 \, 6 \, \alpha_1+\alpha_2 x + \alpha_3 + \alpha_4 x y +\alpha_5 x^2 + \alpha_6 y^2\,

A quick way to easily obtain the terms of a complete polynomial is by using the Pascal's triangle:

order n new polynomial terms number of terms p
  1 \, 1
Linear x \qquad y \, 3
Quadratic x^2 \qquad 2 x y  \qquad y^2\, 6
Cubic x^3 \qquad 3 x^2 y  \qquad 3 x y^2 \qquad y^3\, 10
Quartic x^4 \qquad 4 x^3 y  \qquad 6 x^2 y^2 \qquad 4 x y^3 \qquad y^4\, 15

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.


  1. Pascal's triangle
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