Grain size distribution in Voronoi Tessellation

[jplauzie]

Hi,

I was wondering if there's already been any discussion/thoughts about whether it would be possible to control the distribution in Voronoi grains? In addition to the average grain size, it would be nice to also control the spread via something akin to a standard deviation.

From my initial look, it seems like it would be possible in principle, although there are some challenges I'm not sure about. In particular I'm not sure if it's compatible with a moving window frame, where new grains would need to be generated locally?

[ilateur]

Hi,

During Voronoi tessellation, the domain is split into tiles. Each Voronoi grain has a center and the number of centers per tile is Poisson distributed. This only has a single parameter lambda (related to the grainsize), which is both the mean and the variance of the distribution. In principle one could take a different approach to distributing grains, but a Poisson distribution seems like the most natural choice. A hypothetical moving window (which does not exist in mumax+ but is not written off either) would not interfere with this change I think.

Hopefully this is clear :)

[jplauzie]

That makes sense. When you say that Poisson is a natural choice, is there a physics reason for it to be Poisson? I don't know that much about grain growth, but I assume there is probably some underlying reason.

When I was looking, it seems like one could extend the Voronoi tessellation to a Laguerre tessellation, which is essentially just the current process, but with random weights added to each grain (and for weights=0, it reduces back down to Voronoi). This would allow one to control the variance separately from the mean. However, there is no easy conversion from the weights to a standard deviation, so what people seem to do is to iterate the weights to match the distribution they want.

But this iteration process seems to be basically a global process, I couldn't find a good way to introduce new grains locally, which was a bit sad. I was hoping there was a clean way that was consistent with current/future mumax+ that could be implemented in a PR, but perhaps that isn't possible without compromises? It isn't a critical feature for me, it just always seemed strange one couldn't control the mean and the variance separately.

[ilateur]

I think the reasoning is that crystal grains tend to grow out from a single nucleation point, until they hit a different crystal grain. A logical way to randomly distribute nucleation points on a surface is an independent nonnegative number of times, similar to distributing chocolate pieces on a cookie or pieces of chewing gum on the ground. The Wikipedia page of the Poisson distribution even lists "locations of defects and dislocations in materials" as an example. ;)

I hadn't heard of the Laguerre tessellation before, but I guess it's easily compatible with the reasoning above if you suppose not every crystal grain orientation grows equally as fast or starts growing at the same time. It's sad (and a bit strange) that there is no way to predict the resulting standard deviation beforehand though. Otherwise it also sounds easily compatible with the existing code structure, where the distance calculation of the cell to the closest centers can be modified by the weight of each center.

In any case, the only thing Voronoi tessellation does right now, is return an array of region indices. You can use any kind of Python function you'd like for this, so a different iterative tessellation procedure is certainly possible. :)