Two-dimensional Shape Functions

From KratosWiki
(Difference between revisions)
Jump to: navigation, search
Line 13: Line 13:
  
  
{|
+
{| 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>
+
| 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>
+
| 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>
+
| 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>
 
|}
 
|}
  

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 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\,



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.




References

  1. Pascal's triangle
Personal tools
Categories