Phase Field Topology Constraints
This paper presents a morphological approach to extract topologically critical regions in phase field models. There are a few studies regarding topological properties of phase fields. One line of work related to our problem addresses constrained phase field evolution. This approach is based on modifying the optimization problem to limit connectedness of the interface. However, this approach results in a complex optimization problem, and it provides nonlocal control. We adapted a non-simple point concept from digital topology to local regions using structuring masks. These regions can be used to constrain the evolution locally. Besides this approach is flexible as it allows the design of structuring elements. Such a study to define topological structures specific to phase field dynamics has not been done to our knowledge.
- 5.Eckhardt, U., Latecki, L.: Digital Topology. In Current Topics in Pattern Recognition Research, Research Trends, Council of Scientific Information, Vilayil Gardens, Trivandrum (1995)Google Scholar
- 7.Gunther, D.: Topological analysis of discrete scalar data. Ph.D. thesis, Max-Planck-Institut Informatik (2012)Google Scholar
- 8.Han, X., Xu, C., Prince, J.: A topology preserving level set method for geometric deformable models. IEEE Trans. Pattern Anal. Mach. Intell. 25(6), 755–768 (2003)Google Scholar
- 12.Le Guyader, C., Vese, L.A.: Self-repelling snakes for topology-preserving segmentation models. IEEE Trans. Image Process. 17(5), 767–779 (2008)Google Scholar
- 14.Melin, E.: Connectedness and continuity in digital spaces with the Khalimsky topology (2003)Google Scholar
- 18.Wojtowytsch, S.: Phase-field models for thin elastic structures: Willmore’s energy and topological constraints. Ph.D. thesis, Durham University (2017)Google Scholar