Two-dimensional Shape Functions
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with: <math>\qquad p=\frac{(n+1)(n+2)}{2}</math> the number of terms. | with: <math>\qquad p=\frac{(n+1)(n+2)}{2}</math> the number of terms. | ||
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+ | More specifically: | ||
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+ | |||
+ | {| | ||
+ | ! polynomial order '''n''' | number of terms '''p''' | <math>f(x,y) \,</math> | ||
+ | |- | ||
+ | | Constant: <math>0 \,</math> | <math>1 \,</math> | <math>\alpha \,</math> | ||
+ | |- | ||
+ | | Linear: <math>1 \,</math> | <math>3 \,</math> | <math>\alpha_1+\alpha_2 x + \alpha_3 y \,</math> | ||
+ | |- | ||
+ | | 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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+ | For example, in the case of a lineal polynomial: | ||
+ | |||
+ | |||
+ | :<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 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:
number of terms p | |
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| |
| |
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.