Question regarding normalization condition

Asalih3d

Hello guys!

I am trying to make sense of the statement that the integration of the kernel over the volume should equal 1.

I am using the Wendland kernel in 2D and the function and constant is taken from DualSPHysics wiki. So I have two questions right now:

1. My Normalization approach is correct?
2. Do I ever use this rule in computation? I need to have W(q) with units for this to make sense




I hope somebody could help me out?

Kind regards

Asalih3d

Okay what I've found out until now is that, forget my pdf. The normalizasation is given by the coefficent infront, alpha_D, which ensures this integral becomes 1. Something else happens at boundaries/free surfaces though, since there are not enough particles, but that is another topic.

I've found an example for a kernel in spherical coordinate system here:

https://physics.stackexchange.com/questions/138700/kernel-normalization-in-smoothed-particle-hydrodynamcs



And this integral ends up giving 1 (unitless) as it should for any choice of h. I've tried applying it to the Wendland kernel and then only using the first two limits, 0 to h and 0 to pi, but no success - if anybody could point me in the right direction I would be very happy.