A modified Allen-Cahn model for pattern synthesis on surfaces

TL;DR

This paper introduces a modified Allen-Cahn model for pattern synthesis on curved surfaces, enabling controlled generation of spots, inverted spots, or stripes through a new reaction term and an efficient numerical scheme.

Contribution

It extends the Allen-Cahn model to curved surfaces with a novel reaction term, improving pattern control and computational efficiency.

Findings

The new model can generate diverse surface patterns.

Pattern type depends on the proposed reaction term.

The operator splitting scheme effectively discretizes the model.

Abstract

We propose an extension of the Allen-Cahn model for pattern synthesis on two dimensional curved surfaces. This model is based on a single PDE and it offers improved ability of controlling the type of generated surface patterns via the chosen reaction-diffusion coefficient, thus, obtaining patterns in form of spots, inverted spots, or stripes. We investigate the dependence of the type of the obtained pattern on the new proposed reaction term. An efficient operator splitting scheme is used to discretize the model on a surface. Experiments on surfaces with varying initial conditions illustrate a variety of patterns.