  Based on the thermodynamic variation to the free energy functional, we
propose a sharp-interface model for simulating solid-state dewetting of thin
films on rigid curved substrates in two dimensions. This model describes the
interface evolution which occurs through surface diffusion-controlled mass
transport and contact point migration along the curved substrate. Furthermore,
the surface energy anisotropy is easily included into the model, and the
contact point migration is explicitly described by the relaxed contact angle
boundary condition. We implement the mathematical model by a semi-implicit
parametric finite element method to study several interesting phenomena, such
as "small" particle migration on curved substrates and templated solid-state
dewetting on a pre-patterned substrate. Based on ample numerical simulations,
we demonstrate that, the migration velocity of a "small" solid particle is
proportional to the substrate curvature gradient $\hat{\kappa}'$ and inversely
proportional to the square root of the area of the particle $\sqrt{A}$, and it
decreases when the isotropic Young angle $\theta_i$ increases. In addition, we
also observe four periodic categories of dewetting on a pre-patterned
sinusoidal substrate. Our approach can provide a convenient and powerful tool
to exploring how to produce well-organized nanoparticles by making use of
template-assisted solid-state dewetting.
