
Finite Element Integral
Hey guys, I'm currently completing a Finite Element course and there is a tricky integral that we use to evaluate forces in a 3point triangle. I was hoping someone would know how to explain the general result,
$\displaystyle \int_A L_1^p L_2^q L_3^r dA = \frac{2A p! q! r!}{(p+q+r+2)!} $
Where, p, q and r are constants. A represents the total area of a 3 sided triangle and,
$\displaystyle L_1 = \frac{ A_1}{A} $
$\displaystyle L_2 = \frac{ A_2}{A} $
$\displaystyle L_3 = \frac{ A_3}{A} $
$\displaystyle A_1, A_2, $ and $\displaystyle A_3 $ represent the divided areas of the triangle when we are looking at a point. For example, say we have a 3sided triangle and point P is smack dab in the middle. If we draw lines from the 3 points that connect the triangle to the point of interest, the triangle will be divided into 3 areas. Those represent $\displaystyle A_1, A_2, $ and $\displaystyle A_3 $.
In essense the Ls represent the area coordinates of our point of interest.
I'm thinking there may be some geometry involved here?! It actually doesn't matter because all we need is the general result for our computations, but I would like to know! hehe.
Cheers
Allan