Phase Transitions , Curve Evolution , and the Control ofSemiconductor

Perhaps the most common approach to solving PDEs in engineering applications is through use of nite-element approximations. In free-boundary problems this usually requires adaptive meshing that tracks the surface motion. These meshing schemes may not be able to handle shock formation or topological changes without arcane logic or user intervention. Inconvenient in simulation, this can cause grave diiculties in real-time control. Thus alternatives are needed. We nd these alternatives under development in the surprisingly related areas of phase transition physics and computer vision. We present (i) A mathematical representation of free surface motion that is particularly well-suited to real-time implementation, (ii) a technique for estimating an isotropic and homogeneous normal velocity based on a simple measurement and (iii) application to a semiconductor etching problem.