Realistic pattern formations on surfaces by adding arbitrary roughness

We are interested in generating surfaces with arbitrary roughness and forming patterns on the surfaces. Two methods are applied to construct rough surfaces. In the first method, some superposition of wave functions with random frequencies and angles of propagation are used to get periodic rough surfaces with analytic parametric equations. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace-Beltrami operator and in the generation of pattern formation. Numerical experiments show that the patterns become irregular as the amplitude and frequency of the rough surface increase. For the sake of easy generalization to closed manifolds, we propose a second construction method for rough surfaces, which uses random nodal values and discretized heat filters. We provide numerical evidence that both surface {construction methods} yield comparable patterns to those {observed} in real-life animals.Comment: 22 pages, 16 figure

Turing patterns in a 3D morpho-chemical bulk-surface reaction-diffusion system for battery modeling

In this paper we introduce a bulk-surface reaction-diffusion (BSRD) model in three space dimensions that extends the DIB morphochemical model to account for the electrolyte contribution in the application, in order to study structure formation during discharge-charge processes in batteries. Here we propose to approximate the model by the Bulk-Surface Virtual Element Method on a tailor-made mesh that proves to be competitive with fast bespoke methods for PDEs on Cartesian grids. We present a selection of numerical simulations that accurately match the classical morphologies found in experiments. Finally, we compare the Turing patterns obtained by the coupled 3D BS-DIB model with those obtained with the original 2D version.Comment: 25 pages, 11 figures, 1 tabl