Hi Koen,
I'm afraid since I'm away from Houdini I can't tell you how those gather screen shots compare to mine. I did realize I made a nasty mistake in my writeup above, in case you were following that and not the paper. The first step in calculating the gathering term is to subtract the goal field from the density, *not* densityblur from the density. Sorry about that.
Anyway, if I can make a couple of suggestions in general on “how to implement a Siggraph paper”. Feel free to ignore any or all.
1) If you can spare the $9.99, go to
encore.siggraph.org and download the video of the presentation the authors gave at the original Siggraph if it's available (this one is). It's almost always worth it. This paper is not too impenetrable on its own, but I always like getting the authors' take on what's important, tricky, etc.
2) Simplify things as much as possible when trying to implement at first. In this case that means 2D simulations, do the driving force term first, worry about gather later, etc. Sounds like you're already on this path.
3) Try to recreate their examples if possible. In this case that's pretty simple. You can extrude some text and do an Isooffset and recreate either the single-character transitions from the paper, or the squiggly line into the “SIGGRAPH” letters. You probably found this via Google, but movies from the paper are
here [cs.huji.ac.il].
So, along those lines, I'd encourage you to get it working it with the Gas Blur first, then switch the driving force to use an SDF later. That just removes one variable that might trip you up in recreating the examples. You can make it more controllable once yo get the effect working overall.
In general, the main requirement on the driving force is that the pressure projection stage can cancel out all velocities when the density field matches the goal field. That's the “rest state” they talk about in the paper. They achieve this in their formula by having the blurred density and goal fields cancel each other out, so you're left with just the gradient of the blurred field. The pressure projection should completely cancel the force out when it is the gradient of a smooth potential field. If you can meet this requirement with your smoothed SDF gradient, great, but I'd still do that as a second step once you already have it working like in the paper.
As for the gather, you should expect some negative values. The term is essentially the result of diffusing the error between the current state and the goal over time. In some case that means more density needs to be added to the current state, in some cases it means some should be removed. If you look at the figures in the paper where they compare the results with and without the gathering term, you'll see not only is there more density in the target position, there's less in the areas immediately surrounding it.
From my recollection of implementing this, with a 2D, unit square simulation, you ought to be able to come very close to matching that first figure (the single letter transition) with just the driving force at a resolution of about 128x128. If you try to transition to another goal state (say, another letter) from there, you'll just start to lose too much density to numerical dissipation. And none of the examples with a constantly moving target (like the bouncing smily face) will work without the gathering term.
I'll be back in front of my computer with Houdini on it in a few days and should be more helpful.
Good luck,
John