Fourier-based numerical approximation of the Weertman equation for moving dislocations

Marc Josien (1, 2) , Yves-Patrick Pellegrini (3) , Frédéric Legoll (2, 4) , Claude Le Bris (1, 2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHERIALS - MATHematics for MatERIALS
3 DAM/DIF - DAM Île-de-France
4 navier umr 8205 - Laboratoire Navier

This work discusses the numerical approximation of a nonlinear reaction-advection-diffusion equation, which is a dimensionless form of the Weertman equation. This equation models steadily-moving dislocations in materials science. It reduces to the celebrated Peierls-Nabarro equation when its advection term is set to zero. The approach rests on considering a time-dependent formulation, which admits the equation under study as its long-time limit. Introducing a Preconditioned Collocation Scheme based on Fourier transforms, the iterative numerical method presented solves the time-dependent problem, delivering at convergence the desired numerical solution to the Weertman equation. Although it rests on an explicit time-evolution scheme, the method allows for large time steps, and captures the solution in a robust manner. Numerical results illustrate the efficiency of the approach for several types of nonlinearities.

Mots clés: Cauchy-type nonlinear integrodifferential equation; Peierls-Nabarro equation; Fractional Laplacian; Dislocations; Preconditioned scheme; Reaction-advection-diffusion equation; Weertman equation

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Marc Josien, Yves-Patrick Pellegrini, Frédéric Legoll, Claude Le Bris. Fourier-based numerical approximation of the Weertman equation for moving dislocations. International Journal for Numerical Methods in Engineering, 2018, Numerical Methods in Engineering, 113 (12), pp.1827-1850. ⟨10.1002/nme.5723⟩. ⟨hal-01510158⟩