Two-dimensional Shape Functions
From KratosWiki
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+ | A quick way to easily obtain the terms of a complete polynomial is by using the '''Pascal's triangle''': | ||
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+ | {| border="1" cellpadding="5" cellspacing="0" class="wikitable" style="margin:auto; background:white;" | ||
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+ | | || <math>1 \,</math> | ||
+ | |- align="center" | ||
+ | | Linear || <math>x \qquad y \,</math> | ||
+ | |- align="center" | ||
+ | | Quadratic || <math>x^2 \qquad 2 x y \qquad y^2\,</math> | ||
+ | |- align="center" | ||
+ | | Cubic || <math>x^3 \qquad 3 x^2 y \qquad 3 x y^2 \qquad y^3\,</math> | ||
+ | |- align="center" | ||
+ | | Quartic || <math>x^4 \qquad 4 x^3 y \qquad 6 x^2 y^2 \qquad 4 x y^3 \qquad y^4\,</math> | ||
+ | |} | ||
Revision as of 10:28, 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: the number of terms.
More specifically:
polynomial order n | number of terms p | |
---|---|---|
Constant: | ||
Linear: | ||
Quadratic: |
A quick way to easily obtain the terms of a complete polynomial is by using the Pascal's triangle:
Linear | |
Quadratic | |
Cubic | |
Quartic |
For example, in the case of a lineal polynomial:
- f(x,y) = α_{1} + α_{2}x + α_{3}y
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.