A level set approach for evolving spirals is introduced to handle merging spiral steps. For this purpose, the level set method is extended to describe curves by an auxiliary surface and a surface defined by a pre-determined multivalued function, like as a Riemann surface. Since the level set equation is a degenerate parabolic type, its solution is considered in the viscosity sense. The comparison principle or the existence and uniqueness of viscosity solution globally-in-time are explained as the results of the mathematical analysis. This method can be applied to compute the growth rate of a crystal surface that is evolving via spiral steps. As an application of this, the growth rate of a crystal surface with several screw dislocations is investigated numerically. We improved the estimate of the surface growth rate compared to that reported by Burton et al. (Philos. Trans. R. Soc. London A, 243(1951), 299–358).

In epithelial cell sheets, forces generated by individual cells cause deformation, such as elongation, bending, and twisting, resulting in the formation of three-dimensional organs and tissues. Mechanistically analyzing the deformation behavior from the cellular level to the tissue level is necessary to elucidate the morphogenesis mechanism. In this paper, we discuss a vertex model that mechanistically considers the behavior of individual cells and introduce the possibilities for conducting research using this model.