Ghorbel, Mohamed-Amin (2007) Numerical analysis of dislocations dynamics and applications to homogenization. PhD thesis, ENPC.
Abstract
The most important part of this work concerns the numerical analysis of the dislocations dynamics. Dislocations are some defects moving in a crystal when exterior stress are applied. Our work focussed essentially on two studies. The first one concerns the numerical and theoretical study of a non-local transport equation; the second one is a numerical study which proposes a computation of the effective hamiltonian for a homogenization problem of the dislocations dynamics. In general, the dynamics of dislocations is described by an eikonal equation with a non-local velocity. Here, we limit our work to a model in one-dimensional space. In the first study we proved some results of existence and uniqueness of solution for long time and an error estimate between the theoretical solution and the discrete solution of a finite difference scheme. In the second study, a monotone scheme is used to compute the homogenized hamiltonian which describe the effective behaviour of densities of dislocations as a limit of model where dislocations are described individually. Numerical results provided here support the theoretical study of the homogenization.
| Item Type: | PhD Thesis (PhD) |
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| Thesis Supervisor: | Monneau, Régis |
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| Date: | January 2007 |
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| Board of examiners: | Ern, Alexandre and Falcone, Maurizio and Mironescu, Elisabeth and Alvarez, Olivier and Imbert, Cyril and Monneau, Régis |
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| Collection (Fonds): | ENPC |
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| Institution: | ENPC |
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| Subjects: | 1. Mathematics and Applications |
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| Uncontrolled Keywords: | Equations aux dérivées partielles (EDP), Analyse numérique, Dislocations, force de Peach-Koehler, Estimationd'erreur, Simulation, Homogénéisation numérique, Hamiltionien effectif, équation Hamilton-Jacobi, équation eikonal, équation de transport, équation non-locale, Solution de viscosité |
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