3. Surface Evolution

The simulation of the topography changing processes requires a method capable to describe geometric deformations over time. To observe the final profile it is primarily important to track the surface ${\mathcal{S}}({t})$ over time. The initial surface ${\mathcal{S}}({t}=0)$ can be extracted from the initial geometry ( ${\mathcal{S}}=\partial{\mathcal{M}}$ ). During topography simulations the surface velocities ${V}({\vec{x}})$ in normal direction are calculated at the surface points ${\vec{x}}\in{\mathcal{S}}({t})$ . To obtain the final profile after the complete process time, ${t}_{\text{process}}$ , the following problem must be solved:

• Given: ${\mathcal{S}}({t}=0)$ and ${V}({\vec{x}})$ for all ${\vec{x}}\in{\mathcal{S}}({t})$ and ${t}\in\left[0,{t}_{\text{process}}\right]$ ,
• Required: ${\mathcal{S}}({t}_{\text{process}})$ .

This chapter first gives an overview of different numerical methods for tracking moving boundaries. The pros and cons of these techniques are briefly discussed. Finally the focus is placed on the LS method which is probably the most popular method used in modern topography simulators, especially for three dimensions.