Evolution, Implementation, and Application of Level Set and Fast Marching Methods for Advancing Fronts

A variety of numerical techniques are available for tracking moving interfaces. In this review, we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws. This link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws. We discuss the evolution of these techniques, the fundamental numerical approximations, involved, implementation details, and applications. In particular, we review some work on three aspects of materials sciences: semiconductor process simulations, seismic processing, and optimal structural topology design.