Accurate representation of moving geometrical structures are very important in the simulation of many physical processes. Examples of such processes are the motion of interfaces in multiphase and free surface flows, high frequency wave front propagation and the dynamical evolution of phase transitions.
The segment projection method is a new computational technique for the representation of moving interfaces. The representation of an interface is in this method comprised by a union of overlapping patches. When the interfaces are curves in the plane, the patches are curve segments, chosen such that they can be represented as functions of one coordinate variable. For surfaces in $\R^3$, the segment functions are functions of two coordinate variables. These segment functions are discretized on Eulerian grids and their motions are given by partial differential equations.
Considering surfaces, we present evolution under motion of mean curvature. This implementation is still under development. The two-dimensional version of the method has been implemented and used in different applications. One application is simulation of immiscible incompressible multiphase flows, where the interfaces separating two different fluids are tracked and topological changes in the form of merging bubbles are allowed.
In the area of geometrical optics, our method has been applied to the tracking of wave fronts. Problems where wave fronts develop cusps, corners and caustics are difficult to handle by PDE calculations in the xy-plane, due to the loss of information that arises in these complicated situations. By defining the segments also as functions of one extra variable (the direction of propagation) and viewing the wave fronts as projections onto the xy-plane, this loss of information can be avoided.
We have also applied our method to simulations of epitaxial growth, i.e. growth of thin films in the production of semi-conductors. Considering the interface physics, diffusion in the tangential direction along boundaries for growth areas is of importance. Due to the explicit Eulerian discretization of the interfaces in the segment projection method, separate diffusion equations can be defined and approximated on the segments.
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