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Threshold Dynamics for Networks with Arbitrary Surface Tensions
Author(s) -
Esedoḡ Lu Selim,
Otto Felix
Publication year - 2015
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21527
Subject(s) - mathematics , surface (topology) , isotropy , convergence (economics) , curvature , motion (physics) , range (aeronautics) , phase (matter) , mean curvature , class (philosophy) , mathematical analysis , statistical physics , geometry , classical mechanics , physics , computer science , materials science , quantum mechanics , artificial intelligence , economics , composite material , economic growth
We present and study a new algorithm for simulating the N ‐phase mean curvature motion for an arbitrary set of (isotropic)N ( N − 1 ) 2surface tensions. The departure point is the threshold dynamics algorithm of Merriman, Bence, and Osher for the two‐phase case. A new energetic interpretation of this algorithm allows us to extend it in a natural way to the case of N phases, for arbitrary surface tensions and arbitrary (isotropic) mobilities. For a large class of surface tensions, the algorithm is shown to be consistent in the sense that at every time step, it decreases an energy functional that is an approximation (in the sense of Gamma convergence) of the interfacial energy. A broad range of numerical tests shows good convergence properties. An important application is the motion of grain boundaries in polycrystalline materials: It is also established that the above‐mentioned large class of surface tensions contains the Read‐Shockley surface tensions, both in the two‐dimensional and three‐dimensional settings.© 2015 Wiley Periodicals, Inc.