A Generalized Fast Marching Method for Dislocation Dynamics

In this paper, we consider a generalized fast marching method (GFMM) as a numerical method to compute dislocation dynamics. The dynamics of a dislocation hypersurface in $\mathbb{R}^N$ (with $N=2$ for physical applications) is given by its normal velocity which is a nonlocal function of the whole shape of the hypersurface itself. For this dynamics, we show a convergence result of the GFMM as the mesh size goes to zero. We also provide some numerical simulations in dimension $N=2$.

[1]  Nicolas Forcadel.  Comparison Principle for a Generalized Fast Marching Method , 2009, SIAM J. Numer. Anal..

[2]  James A. Sethian,et al.  The Fast Construction of Extension Velocities in Level Set Methods , 1999 .

[3]  P. Lions,et al.  Two approximations of solutions of Hamilton-Jacobi equations , 1984 .

[4]  David L. Chopp,et al.  Another Look at Velocity Extensions in the Level Set Method , 2009, SIAM J. Sci. Comput..

[5]  S. Osher,et al.  A PDE-Based Fast Local Level Set Method 1 , 1998 .

[6]  M. Falcone,et al.  Convergence of a Generalized Fast Marching Method for a non-convex eikonal equation , 2008 .

[7]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid , 2012 .

[8]  M. Aurada,et al.  Convergence of adaptive BEM for some mixed boundary value problem , 2012, Applied numerical mathematics : transactions of IMACS.

[9]  Maurizio Falcone,et al.  Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign , 2008, SIAM J. Numer. Anal..

[10]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[11]  D. Rodney,et al.  Phase eld methods and dislocations , 2001 .

[12]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[13]  Guy Barles,et al.  Global Existence Results and Uniqueness for Dislocation Equations , 2008, SIAM J. Math. Anal..

[14]  A. Vladimirsky.  Static PDEs for time-dependent control problems , 2006 .

[15]  P. Cardaliaguet,et al.  Existence and uniqueness for dislocation dynamics with nonnegative velocity , 2005 .

[16]  E. Rouy,et al.  Convergence of a first order scheme for a non-local Eikonal equation , 2006 .

[17]  J. Tsitsiklis,et al.  Efficient algorithms for globally optimal trajectories , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[18]  Régis Monneau,et al.  Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution , 2006 .

[19]  S. Osher,et al.  Regular Article: A PDE-Based Fast Local Level Set Method , 1999 .

[20]  Alex M. Andrew,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition) , 2000 .