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I'm new in fipy and I would like to know if its possible to compute grain boundaries diffusion using the Fisher model with the following set of equations : 

dc/dt = D*(d2c/dx2 + d2c/dy2)

dcb/dt = Db*(d2cb/dy2) + (2D/d)*(dc/dx)|x=d/2

with the condition at the border between grain boundary and bulk : 

cb(y,t) = c(d/2, y, t)

where c is the concentration in the bulk, cb is the concentration in the grain boundary, D is the diffusivity in the bulk and Db in the grain boundary. And d corresponds to the grain boundary width.

Thank's.

Best regards,

Adrien

alexander.polomodov
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ShottaPanda
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  • The equations as written aren't very clear. Can you define the solution domain more clearly and also in what part of the solution domain each equation is applied? Is the solution in the grain boundary 1D? – wd15 Nov 27 '17 at 16:34
  • Hi, thank you for your answer ! Firstly, I would like to create a 2D mesh where I consider two different grains and one grain boundary. The first equation describes the diffusion inside the grains and the second in the grain boundary. The second term of the RHS in the second equation describes the leakage from the grain boundary to the grains. – ShottaPanda Dec 01 '17 at 13:51
  • But, my real aim is to compute the diffusion of one or severals elements through a polycristal, maybe using a voronoi tessellation. But I don't know if it's possible in fipy to detect voronoi cells and faces to distinguish grains (voronoi cells) and grain boundaries (faces) and compute the diffusion inside this structure. With a diffusion coefficient higher in faces and lower in voronoi cells. – ShottaPanda Dec 01 '17 at 13:58
  • The voronoi faces would need to have some volume. FiPy calculates diffusion between fipy cells, across fipy faces. The diffusivity coefficient is most accurately defined on those faces, but then there is no diffusivity in the cells, per se. To do this, you'd need to have fipy cells defining grain regions and fipy cells defining grain boundaries. You could use voronoi tessellation to generate the initial microstructure, but then you'd need to subdivide that space. FiPy provides no tools to do that step. – jeguyer Feb 16 '19 at 13:48

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