Two-dimensional Shape Functions
From KratosWiki
(Difference between revisions)
Line 30: | Line 30: | ||
! order ''n'' !! new polynomial terms !! number of terms ''p'' | ! order ''n'' !! new polynomial terms !! number of terms ''p'' | ||
|- align="center" | |- align="center" | ||
− | | || <math>1 \,</math> || 1 | + | | || <math>1 \,</math> || <math>1 \,</math> |
|- align="center" | |- align="center" | ||
− | | Linear || <math>x \qquad y \,</math> || 3 | + | | Linear || <math>x \qquad y \,</math> || <math>3 \,</math> |
|- align="center" | |- 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" | |- 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" | |- 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> |
|} | |} | ||
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:
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:
order n | new polynomial terms | number of terms p |
---|---|---|
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.