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.