GO++: A modular Lagrangian/Eulerian software for Hamilton Jacobi equations of geometric optics type

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 dislocation line. We study the special cases where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for HamiltonJacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model.

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“…9 We proceed similarly as in the proof of Proposition 3.8. The only change is in checking thatρ f is a discontinuous solution of (3.32).…”

Section: Proposition 39 (Geometric and Loop Equations)mentioning

confidence: 90%

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

Alvarez

Hoch

Bouar

et al. 2006

Arch Rational Mech Anal

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117

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“…Hydrodynamics codes such GO++ may generate at each time step new polygonal meshes adapted to the distortions of the uid [Hoc02,Hoc11]. Such cells come out from successive renement, derenement and reconnexion steps.…”

Section: Introductionmentioning

confidence: 99%

An asymptotic preserving scheme forP₁model using classical diffusion schemes on unstructured polygonal meshes

Franck

Hoch

Navaro³

et al. 2011

ESAIM: Proc.

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Abstract. A new scheme for discretizing the P1 model on unstructured polygonal meshes is proposed. This scheme is designed such that its limit in the diusion regime is the MPFA-O scheme which is proved to be a consistent variant of the Breil-Maire diusion scheme. Numerical tests compare this scheme with a derived GLACE scheme for the P1 system. Résumé. Un nouveau schéma de discrétisation du modèle P1 sur maillage non structuré composé de polygones est proposé. Ce schéma est construit pour que sa limite en régime diusion soit le schéma MPFA-O qu'on démontre être une variante consistante du schéma de diusion de Breil-Maire. Ce schéma est comparé sur des cas tests avec un schéma dérivé du schéma GLACE pour le modèle P1.

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“…Obviously for a finite volume like discretization of system (1)(2)(3)(4), the domain Ω(t) is replaced by a moving conformal grid M and ω(t) describes the cells of M.…”

Section: Introductionmentioning

confidence: 99%

“…(2) We also want to examine the behavior of ENO-like reconstruction for Glace/Chic schemes. Using these kind of schemes is of interest to us since the node velocity computed by J. Cheng and C.-W. Shu's method may break the compatibility of the gradient in divergence operators of (1)(2)(3)(4), which is a bad property. (3) As said previously, while the Glace scheme can be subject to lack of stability (hourglass modes), the Chic scheme suffers from being too dissipative.…”

Section: Introductionmentioning

confidence: 99%

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Polynomial Least-Squares reconstruction for semi-Lagrangian Cell-Centered Hydrodynamic Schemes

et al. 2009

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Abstract. In Inertial Confinement Fusion (ICF) simulation, use of Lagrangian hydrodynamic numerical schemes is a cornerstone. It avoids mixing of materials and allows for symmetry preservation in dimension two. Recently, [7] and then [9] proposed an interesting alternative to the historical VNR scheme [15]. These two first order schemes are multidimensional generalizations of the Godunov acoustic solver. Alternatively, a WENO Lagrangian scheme was proposed in [6]. This scheme suffers from non-preservation of symmetries and its velocity computation can be discussed.The aim of this work is to evaluate the later scheme on ICF representative test cases and to derive a polynomial reconstruction that preserves symmetries for the three cell-centered scheme. This reconstruction is inspired by [12]. Since this paper focuses on the approximation of Euler equations, considered test cases are purely hydrodynamic and do not illustrate all difficulties encountered in ICF.We first briefly recall different schemes used for this study. We then explain the Least-Squares ENO reconstruction that we chose for symmetry preservation and describe the limiting strategy. We finally illustrates the presented results by some representative numerical experiments.Résumé. La simulation de Fusion par Confinement Inertiel (FCI) utilise souvent des schémas hydrodynamiques Lagrangiens. Cela permet d'éviter le mélange de matériaux et permet de préserver des symétries en dimension deux. Récemment, des alternatives intéressantes au schéma historique VNR [15] ontété proposées dans [7] puis dans [9]. Parallèlement, un schéma WENO aété proposé dans [6]. Ce schéma ne préserve pas les symétries et le calcul des vitesses peutêtre discuté. L'objectif de ce travail est d'évaluer ce dernier schéma pour des cas tests représentatifs de la FCI et d'écrire une reconstruction polynomiale qui préserve les symétries pour les trois schémasétudiés. Cette reconstruction est inspirée de [12]. Puisqu'on se concentre ici sur la résolution deséquations d'Euler, les cas tests présentés sont purement hydrodynamiques et ne prétendent pas couvrir l'ensemble des difficultés rencontrées en FCI.Nous rappelons d'abord les différents schémas utilisés ici. Nous expliquons ensuite la reconstruction au sens des moindres carrés que nous avons choisie ainsi qu'une stratégie de limitation préservant aussi la symétrie. On illustre finalement les résultats au travers de quelques cas tests représentatifs choisis.

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GO++: A modular Lagrangian/Eulerian software for Hamilton Jacobi equations of geometric optics type

Cited by 7 publications

References 16 publications

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

An asymptotic preserving scheme forP₁model using classical diffusion schemes on unstructured polygonal meshes

Polynomial Least-Squares reconstruction for semi-Lagrangian Cell-Centered Hydrodynamic Schemes

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GO++: A modular Lagrangian/Eulerian software for Hamilton Jacobi equations of geometric optics type

Cited by 7 publications

References 16 publications

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

An asymptotic preserving scheme forP1model using classical diffusion schemes on unstructured polygonal meshes

Polynomial Least-Squares reconstruction for semi-Lagrangian Cell-Centered Hydrodynamic Schemes

Contact Info

Product

Resources

About

An asymptotic preserving scheme forP₁model using classical diffusion schemes on unstructured polygonal meshes