Title: A numerical study for the homogenisation of one-dimensional models describing the motion of dislocations

Authors: M-A. Ghorbel, P. Hoch, R. Monneau

Addresses: CERMICS, Ecole Nationale des Ponts et Chaussees, 6 and 8, Avenue Blaise Pascal, cite Descartes, Champs-sur-Marne, 77455 Marne-la-Vallee Cedex 2, France. ' CEA/DAM Ile de France, BP 12, 91680 Bruyeres Le Chatel, France. ' CERMICS, Ecole Nationale des Ponts et Chaussees, 6 and 8, Avenue Blaise Pascal, cite Descartes, Champs-sur-Marne, 77455 Marne-la-Vallee Cedex 2, France

Abstract: In this paper we are interested in the collective motion of dislocations defects in crystals. Mathematically, we study the homogenisation of a non-local Hamilton-Jacobi equation. We prove some qualitative properties on the effective Hamiltonian and we provide a numerical scheme which is proved to be monotone under some suitable CFL conditions. Using this scheme, we compute numerically the effective Hamiltonian. Furthermore, we provide numerical computations of the effective Hamiltonian for several models corresponding to the dynamics of dislocations where no theoretical analysis is available.

Keywords: continuous viscosity solution; dislocations dynamics; eikonal equation; effective Hamiltonian; finite difference scheme; Hamilton-Jacobi equation; non-local equation; numerical homogenisation; Peach-Koehler force; transport equation; dislocation defects; crystal defects.

DOI: 10.1504/IJCSM.2008.019712

International Journal of Computing Science and Mathematics, 2008 Vol.2 No.1/2, pp.28 - 52

Available online: 25 Jul 2008 *

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