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We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation whose velocity is a non-local quantity depending on the whole shape of the… (More)

The note studies a nonlocal geometric Hamilton-Jacobi equation that models the motion of a planar dislocation in a crystal. Within the framework of viscosity solutions and of the level-set approach, we show that the equation has a unique solution on a small time interval when the initial curve is the graph of a Lipschitz bounded function. To cite this… (More)

- Philippe Hoch, Michel Rascle
- SIAM J. Scientific Computing
- 2002

In this paper, we consider Hamilton–Jacobi equations on a manifold, typically on the graph of some previously computed function z(x, y), and we show how the corresponding level set method allows us to generate and/or to refine a mesh in regions where this function z has large derivatives. Such as it is, the method needs to be strongly improved and… (More)

- Julien Dambrine, Philippe Hoch, +4 authors Lisl Weynans
- 2012

Abstract. Numerous systems of conservation laws are discretized on Lagrangian meshes where cells nodes move with matter. For complex applications, cells shape or aspect ratio often do not insure sufficient accuracy to provide an acceptable numerical solution and use of ALE technics is necessary. Here we are interested with conduction phenomena depending on… (More)

- Mohamed-Amin Ghorbel, Philippe Hoch, Régis Monneau
- IJCSM
- 2008

Abstract: In this paper we are interested in the collective motion of dislocationsdefects in crystals. Mathematically, we study thehomogenisation of a non-local Hamilton-Jacobi equation. We prove some qualitative properties on the effective Hamiltonian and we provide a numerical scheme which is proved to be monotone under some suitable CFL conditions. Using… (More)

- Elisabetta Carlini, Maurizio Falcone, Philippe Hoch
- SIAM J. Numerical Analysis
- 2013

In this paper we extend the generalized fast marching method (GFMM) presented in [E. Carlini et al., SIAM J. Numer. Anal., 46 (2008), pp. 2920–2952] to unstructured meshes. The GFMM generalizes the classical fast marching method, in the sense that it can be applied to propagate interfaces with time-dependent and changing sign velocity. The main motivation… (More)

- Ronald R. Minor, Philippe Hoch, Thomas R Koszalka, Robert L. Brent, Nicholas A. Kefalides
- Developmental biology
- 1976

Abstract 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.… (More)

In this paper, we consider several high-order schemes in one space dimension. In particular, we compare the second-order relaxation ( << 1) or “relaxed” ( = 0) schemes of Jin and Xin [Comm. Pure Appl. Math., 48 (1995), pp. 235–277] with the second-order Lax–Friedrichs scheme of Nessyahu and Tadmor [J. Comp. Phys., 87 (1990), pp. 408–463] and with… (More)

- Elisabetta Carlini, Philippe Hoch, Y. Le Bouar, Régis Monneau

We study a mathematical model describing dislocation dynamics in crystals. This phase-field model is based on the introduction of a core tensor which mollifies the singular field on the core of the dislocation. We present this model in the case of the motion of a single dislocation, without cross-slip. The dynamics of a single dislocation line, moving in… (More)

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