3.1.3 The Level Set Method

Another technique to describe moving boundaries over time is the LS method [86]. The surface ${\mathcal {S}}$ is described as the zero LS of a continuous function ${\Phi}({\vec{x}},{t})$ defined on the entire simulation domain

$${\mathcal{S}}({t})=\lbrace{{\vec{x}}:{\Phi}({\vec{x}},{t})=0}\rbrace.$$ (3.1)

Such a function can be obtained by a signed distance transform [53,77]

$${\Phi}({\vec{x}},{t}=0):= \begin{cases}-{\displaystyle\min_{{\vec... ...al{S}}({t}=0)}}\lVert{\vec{x}}-{\vec{x}}'\rVert \par & \text{else}. \end{cases}$$ (3.2)

Using the implicit representation of the surface (3.1) the time evolution of the surface driven by a scalar velocity field ${V}({\vec{x}})$ can be described by the LS equation

$$\frac{\partial{\Phi}}{\partial{t}}+{V}({\vec{x}})\lVert\nabla{\Phi}\rVert=0.$$ (3.3)

The velocity field can be obtained from the velocities given on the surface by extrapolation, as will be described later in Section 3.2.4.

The LS method is able to describe the position of the surface with sub-grid accuracy. Furthermore, this method allows an accurate calculation of geometric variables, such as the surface normal or the curvature. Similarly to cell-based methods, the implicit description of the surface position allows handling of topographic changes without special consideration.

After successful demonstrations for the applicability of the LS method to topography simulation [4,5,6,108], this technique has become the most popular technique to track a surface over time, especially in three dimensions. The LS method is used by many academic groups for two-dimensional [8,112,113] or three-dimensional topography simulation [46,65,96,98]. The topography simulators earlier developed at the Institute for Microelectronics at the Vienna University of Technology, as ELSA [40] and Topo3D [111], also use the LS method for surface evolution. Furthermore, the newer commercial topography simulators such as the two- and three-dimensional Victory process simulator by Silvaco [43,126], the two-dimensional Sentaurus Topography simulator by Synopsys [127], or PLENTE by Process-Evolution [16,17] are also based on the LS method.