Tauriello G., Koumoutsakos P., SIAM J. Sci. Comput., 35(6), B1285–B1303, 2013

Reaction-diffusion systems on the surface and the interior of complex domains are potent models of growth in living organisms. The simulation of these models requires numerical methods capable of handling large deformations and the accurate coupling of the evolution of substances in the lumen and on the surface of the deforming geometries. Here, we develop a novel computational method to handle such problems by combining a remeshed particle method with a phase field method. Remeshed particle methods are well suited to discretizing deforming geometries, while the phase field method is used to impose boundary conditions that effectuate the coupling of substances evolving in their lumen and on their surfaces. We demonstrate that this hybrid method enables for the first time the accurate coupling of reaction-diffusion on a deformable surface and its interior. The method is validated on benchmark problems and the effect of lumen diffusion to a pattern forming reaction-diffusion system on a deforming surface is discussed.

http://dx.doi.org/10.1137/130906441 (pdf)