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Étude théorique et numérique des systèmes modélisant la dynamique des densités des dislocations

Abstract : In this thesis, we are interested in the theoretical and numerical studies of dislocations densities. Dislocations are linear defects that move in crystals when those are subjected to exterior stress. More generally, the dynamics of dislocations densities are described by a system of transport equations where the velocity field depends non locally on the dislocations densities. First, we are interested in the study of a one dimensional submodel of a (2 × 2) Hamilton-Jacobi system introduced by Groma and Balogh in 1999, proposed in the two dimensional case. For this system, we prove global existence and uniqueness results. Adding to that, considering nondecreasing initial data, we study this problem numerically by proposing a finite difference implicit scheme for which we show the convergence. Then, inspired by the first work, we show a more general theory which allows us to get similar results of existence and uniqueness of solution in the case of one dimensional eikonal systems. By considering nondecreasing initial data, we study this problem numerically. Under certain conditions on the velocity, we propose a finite difference implicit scheme allowing us to calculate the discrete solution and simulate then the dislocations dynamics via this model.
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Submitted on : Saturday, August 6, 2022 - 7:10:11 PM
Last modification on : Tuesday, September 13, 2022 - 3:58:14 PM

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Aya Oussaily. Étude théorique et numérique des systèmes modélisant la dynamique des densités des dislocations. Analyse numérique [math.NA]. Université de Technologie de Compiègne, 2021. Français. ⟨NNT : 2021COMP2634⟩. ⟨tel-03746986⟩

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