A Numerical Study for the Homogenization of One-Dimensional Models describing the Motion of Dislocations

Amin Ghorbel (1) , Philippe Hoch (2) , Régis Monneau (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
2 DAM/DIF - DAM Île-de-France

In this paper we are interested in the collective motion of dislocations defects in crystals. Mathematically we study the homogenization of a non-local Hamilton-Jacobi equation. We prove some qualitative properties on the effective hamiltonian. We also 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 also provide numerical computations of the effective hamiltonian for several models corresponding to the dynamics of dislocations where no theoretical analysis is available.

Mots clés: continuous viscosity solution; Dislocations dynamics; eikonal equation; effective hamiltonian; finite difference scheme; Hamilton-Jacobi equation; non-local equation; numerical homogenization; Peach-Koehler force; transport equation.; transport equation

Domaines: Equations aux dérivées partielles [math.AP]

Amin Ghorbel, Philippe Hoch, Régis Monneau. A Numerical Study for the Homogenization of One-Dimensional Models describing the Motion of Dislocations. 2006. ⟨hal-00110528⟩