Anisotropic Diffusion

Much of the research we are doing on modelling diffusion applies to membranes other than skin, hence the use of geometries in some of the other pictures on these pages that are not truely representative of the enormous changes in scale in the stratum corneum between corneocyte and lipid. For example even considering the 2-d brick-and-mortar representation there are large differences in the expected behaviour of the diffusion through the geometry. Examples of the differences of both the aspect ratio of the corneocytes (length/height) and the lipid volume fraction (lipid width/corneocyte height) are shown to the right where in the top figure an idealised shape is consideed where the more realistic skin geometry is shown below.

With such fine geometries it is important for reasons of efficiency to consider how finely they need to be meshed. When using local refinement the grid adaptivity is almost always centred around the edges of the corneocytes, as shown in the picture below left. For the diffusion cases being covered in this work there are no sharp features needing resolving along the course of the path from diffusion in to diffusion out, and the only singularities are at the edges. This means that it is possible to use long stretched elements along the length of the corneocytes away from the corneocyte junctions, as shown in the picture below right.

Interesting questions are then posed as to what is the correct mesh for these cases, how do we detect where we can but long thin elements in, and how can we drive the mesh generation to give us the structure we want. Much of our work on anisotropic meshes thus far has focused on how good the indicators combining solution and mesh quality are for these cases.

In generalising these meshes to 3-d more complex patterns are seen, as in the pictures below, along with how partial staggering affects the generated stretched geometries.

Institute for Computational and Systems Science, School of Computing, University of Leeds