A Level Set Approach Reflecting Sheet Structure with Single Auxiliary Function for Evolving Spirals on Crystal Surfaces

We introduce a new level set method to simulate motion of spirals in a crystal surface governed by an eikonal–curvature flow equation. Our formulation allows collision of several spirals and different strength (different modulus of Burgers vectors) of screw dislocation centers. We represent a set of spirals by a level set of a single auxiliary function $$u$$u minus a pre-determined multi-valued sheet structure function $$\theta $$θ, which reflects the strength of spirals (screw dislocation centers). The level set equation used in our method for $$u - \theta $$u-θ is the same as that of the eikonal–curvature flow equation. The multi-valued nature of the sheet structure function is only invoked when preparing the initial auxiliary function, which is nontrivial, and in the final step when extracting information such as the height of the spiral steps. Our simulation enables us not only to reproduce all speculations on spirals in a classical paper by Burton et al. (Philos Trans R Soc Lond Ser A Math Phys Sci 243, 299–358, 1951) but also to find several new phenomena.

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