Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Multiphase volume-preserving interface motions via localized signed distance vector scheme

Pages: 969 - 988, Volume 8, Issue 5, October 2015      doi:10.3934/dcdss.2015.8.969

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Rhudaina Z. Mohammad - Graduate School of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa, 920-1192, Japan (email)
Karel Švadlenka - Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa, 920-1192, Japan (email)

Abstract: We develop a signed distance vector approach for approximating volume-preserving mean curvature motions of interfaces separating multiple phases -- a variant of the BMO (Bence-Merriman-Osher) thresholding dynamics. We adopt a variational method employing the idea of a vector-type discrete Morse flow, which allows us to easily treat volume constraint via penalization without having to change the threshold value. Moreover, employing a vector-valued analogue of the signed distance function, the scheme is designed to allow subgrid accuracy on uniform grids without adaptive refinement; thereby, alleviating the well-known BMO time and grid restrictions. Finally, we present numerical tests and examples.

Keywords:  Multiphase mean curvature flow, vector-valued signed distance, volume preservation, constrained variational method, BMO thresholding.
Mathematics Subject Classification:  Primary: 53C44, 76T30; Secondary: 35K55.

Received: December 2013;      Revised: August 2014;      Available Online: July 2015.