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

Abstract : 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.
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Submitted on : Wednesday, April 19, 2017 - 10:39:40 AM
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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, Wiley, 2018, Numerical Methods in Engineering, 113 (12), pp.1827-1850. ⟨10.1002/nme.5723⟩. ⟨hal-01510158⟩



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