In this paper we present a few numerical simulations of a nonsymmetric anisotropic evolution by mean curvature which leads to the so-called fattening of the interface. The numerical simulations are based on a diffused interface approximation via a bistable reaction-diffusion equation which is then discretized by means of finite elements in space and forward differences in time. An adaptive strategy, together with the use of a dynamic mesh, is used to take advantage of the local behaviour of the equation at hand. However, this particular choice of anisotropy, with a large ratio between the maximal and minimal surface energy, seems to be highly critical for the type of approximation presented here, leading to very expensive computations in terms of CPU time.