Abstract
We provide a new convergence proof of the celebrated Merriman–Bence–Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedoḡlu and Otto and De Giorgi’s general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation.
Highlights
The thresholding scheme is a highly efficient computational scheme for multiphase mean curvature flow (MCF) which was originally introduced by Merriman, Bence, and Osher [27, 28]
The main motivation for MCF comes from metallurgy where it models the slow relaxation of grain boundaries in polycrystals [31]
The effective surface tension σi j (ν) and the mobility μi j (ν) of a grain boundary depend on the mismatch between the lattices of the two adjacent grains i and j and on the relative orientation of the grain boundary, given by its normal vector ν
Summary
The thresholding scheme is a highly efficient computational scheme for multiphase mean curvature flow (MCF) which was originally introduced by Merriman, Bence, and Osher [27, 28]. The effective surface tension σi j (ν) and the mobility μi j (ν) of a grain boundary depend on the mismatch between the lattices of the two adjacent grains i and j and on the relative orientation of the grain boundary, given by its normal vector ν. It is well known that for small mismatch angles, the dependence on the normal can be neglected [32]. Where Vi j and Hi j denote the normal velocity and mean curvature of the grain boundary i j = ∂ i ∩ ∂ j , respectively. These equations are coupled by the Herring angle condition
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
