Two-dimensional Shape Functions

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Revision as of 10:19, 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:

$f(x,y)=\sum_{i=1}^p \alpha_i x^j y^k \qquad j+k \le n$

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: $2 \,$	$6 \,$	$\alpha_1+\alpha_2 x + \alpha_3 + \alpha_4 x y +\alpha_5 x^2 + \alpha_6 y^2\,$

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.

References

Pascal's triangle

@@ Line 14: / Line 14: @@
 {| border="1" cellpadding="5" cellspacing="0" class="wikitable" style="margin:auto; background:white;"
-! polynomial order '''n''' !!  number of terms '''p''' !! <math>f(x,y) \,</math>
+! polynomial order ''n'' !!  number of terms ''p'' !! <math>f(x,y) \,</math>
 |-
 | Constant: <math>0 \,</math> || <math>1 \,</math> || <math>\alpha \,</math>

Two-dimensional Shape Functions

Revision as of 10:19, 4 November 2009

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