A level set approach to anisotropic flows with curvature regularization

Modeling and simulation of faceting effects on surfaces are topics of growing importance in modern nanotechnology. Such effects pose various theoretical and computational challenges, since they are caused by non-convex surface energies, which lead to ill-posed evolution equations for the surfaces. In order to overcome the ill-posedness, regularization of the energy by a curvature-dependent term has become a standard approach, which seems to be related to the actual physics, too. The use of curvature-dependent energies yields higher order partial differential equations for surface variables, whose numerical solution is a very challenging task.In this paper, we investigate the numerical simulation of anisotropic growth with curvature-dependent energy by level set methods, which yield flexible and robust surface representations. We consider the two dominating growth modes, namely attachment–detachment kinetics and surface diffusion. The level set formulations are given in terms of metric gradient flows, which are discretized by finite element methods in space and in a semi-implicit way as local variational problems in time. Finally, the constructed level set methods are applied to the simulation of faceting of embedded surfaces and thin films.