A wide variety of problems in soft condensed matter physics require the solution of a nonlinear diffusion equation on a sphere. Examples are the phase separation of lipids in giant vesicles, the time evolution of liquid crystalline order, and the distribution of defects that determine the shapes of vesicles. The goal of this project is to develop a simple and effective scheme for the numerical description of nonlinear diffusion on a sphere. The first specific problem to be considered is the solution of the Cahn-Hilliard equation on the unit sphere. The investigators solve the Cahn-Hilliard and related equations numerically, using real space discretization. Numerical simulations are sensitive to details of the underlying lattice. Because it is not possible to construct a regular lattice on a sphere (if the number of points >20), they plan to use a random lattice, obtained by placing points randomly on the surface of the sphere. In devising a strategy towards this goal, three fascinating topics emerge: Voronoi tessellation of the sphere to identity the nearest neighbors of each lattice site; stable and efficient scheme to approximate differential operators; and regularizing the grid.
This project provides a new algorithm to identify nearest neighbors among random points on a sphere. This is useful for solving a wide variety of problems, ranging from dominance regions of military bases around the world and the locating of fuel depots to modeling the behavior of vesicles and related biological objects. In addition, it gives insights into numerical stability associated with the approximants of differential operators on random lattices, and regularization strategies for improving random grids. It enables the effective numerical solution of nonlinear diffusion equations on a sphere, and thereby makes possible numerical descriptions of a wide variety of problems in soft condensed matter physics. An important aspect of the project is to get students and postdocs from underrepresented groups actively involved in this exciting research.
