Two-dimensional Shape Functions

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Revision as of 10:31, 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: $2 \,$	$6 \,$	$\alpha_1+\alpha_2 x + \alpha_3 + \alpha_4 x y +\alpha_5 x^2 + \alpha_6 y^2\,$

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) = 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 30: / Line 30: @@
 ! order ''n'' !! new polynomial terms !! number of terms ''p''
 |- align="center"
-| &nbsp; || <math>1 \,</math> || 1
+| &nbsp; || <math>1 \,</math> || <math>1 \,</math>
 |- align="center"
-| Linear || <math>x \qquad y \,</math> || 3
+| Linear || <math>x \qquad y \,</math> || <math>3 \,</math>
 |- align="center"
-| Quadratic  || <math>x^2 \qquad 2 x y  \qquad y^2\,</math>  || 6
+| Quadratic  || <math>x^2 \qquad 2 x y  \qquad y^2\,</math>  || <math>6 \,</math>
 |- align="center"
-| Cubic || <math>x^3 \qquad 3 x^2 y  \qquad 3 x y^2 \qquad y^3\,</math>  || 10
+| Cubic || <math>x^3 \qquad 3 x^2 y  \qquad 3 x y^2 \qquad y^3\,</math>  || <math>10 \,</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>  || 15
+| Quartic || <math>x^4 \qquad 4 x^3 y  \qquad 6 x^2 y^2 \qquad 4 x y^3 \qquad y^4\,</math>  || <math>15 \,</math>
 |}

Two-dimensional Shape Functions

Revision as of 10:31, 4 November 2009

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