Wendland.awen and Wendland.bwen
Hello
I understand that awen comes from 7/4pi = 0.557 in the 2D case, but I cannot figure out where bwen comes from?
My understanding is that the "-" comes because of "odd^3" will always produce odd, and that the extra 1/h comes from gradient estimation. Could anyone kindly explain it in bit more detail / show me briefly the math to derive -2.7852?
Kind regards
Comments
Does anybody know or some literature where it is clearly explained?
Kind regards
Does no one know where -2.7852 comes from?
Okay, I finally realized how. Say we have:
Then it is easy to prove "0.557", since:
But how to prove -2.7852?
The trick is to realize, that the derivative of W w.r.t. q, which is needed for the gradient calculation is given as:
Where we spot that the factor 5 is present. If we multiply on 0.557 from before:
5*7/4pi = 2.78521150411
Which is equal to 2.78! Now we only need to find out where the minus sign comes from. One can simply rewrite the parentheses as such:
And now with -5, we have -2.78.
So this is how it came to be.
I am unsure as to why one would do this, because why not just take +- 5/8 and remove the factors completely? Perhaps for some historical reasons, @Alex might be able to explain, but I see other SPH codes uses a similar approach
EDIT: The extra "h" in the denominator comes from when one makes the math to get the gradient, then a factor 1/h is produced.
Kind regards
Looking at the derivative calculation in DualSPHysics (fac) I get a bit worried, because I do not see the divide by 8 any where, and that it uses 1-0.5q, instead of 2-q, which I cannot figure out why.
In the return statement I also see that "dr (relative displacement between point 1 and point 2 in one direction)" is missing, but further in the code for In/Out I see how this function is used:
So that part is correct. What I cannot get to match is:
Kind regards