Takeshi Ohtsuka: A level set method for geometric evolution of spirals and its application

In 1951 Burton, Cabrera and Frank proposed a theory of crystal growth with aid of screw dislocations. According to their theory, screw dislocations causes a helical structure in lattice of atoms, and provides a spiral-shaped steps (discontinuity) in the crystal height. Atoms bond with the crystal structure at the step, and then the crystal surface grows with evolution of spiral steps. The dynamics of the spiral steps includes some geometrical problems. Velocity of steps are given as an eikonal-curvature equation, which is given as not only isotropic, but also an anisotropic evolution reflecting the anisotropic surface energy density of the crystal. Moreover, there are possibly several spiral centers on the growing crystal surface, which causes singularities on the steps by collision. Therefore, implicit description of spirals can be of an attractive option for the spiral crystal growth. In this talk, we introduce a simple level set formulation for evolving spirals with a single auxiliary function and a pre-determined function reflecting the helical structure of the crystal lattice. Moreover, we introduce some mathematical and numerical results on the evolution of spirals or crystal surface, and some challenges to a singular anisotropic problems.