Derivation and study of dynamical models  of dislocation
 densities

Hajj, Ahmad El; Ibrahim, Hassan; Monneau, Régis

doi:10.1051/proc/2009029

Cited by 3 publications

(4 citation statements)

References 13 publications

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“…Our approach follows the ideas of Clarke (1990), it is similar to the one developed in de Gournay (2006). Since there is no -strictly speaking-gap of derivative between the operators A and B (and hence no straightforward compactness), we prove the result by hand.…”

Section: Derivability Of the Eigenvaluementioning

confidence: 98%

“…2 The sectional geometry of the generalized Graetz problem This requires a more subtle treatment than the cases where the outer boundary of the domain is varying. Shape sensitivity in the case of an interface between two materials was already studied, see Hettlich and Rundell (1998), Bernardi and Pironneau (2003), Pantz (2005), Conca et al (2009), Allaire et al (2009) and Neittaanmäki and Tiba (2012) for a recent review article. To our knowledge the eigenvalue problem for the Graetz operator, where some coefficient of the operator depends on an auxiliary partial differential equation, was not addressed previously.…”

Section: Definitionmentioning

confidence: 99%

“…The shape optimization problem we address is thus naturally an eigenvalue optimization problem for the Graetz operator. Eigenvalue optimization is a very natural problem in structural design, and it was addressed in many works, see e.g., Osher and Santosa (2001), Conca et al (2009) or de Gournay (2006 for the case of multiple eigenvalues. Topology optimization tools have also been applied to transfer problems over the last few years (Bruns 2007;Iga et al 2009;Canhoto and Reis 2011).…”

mentioning

confidence: 99%

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Shape optimization for the generalized Graetz problem

Gournay

Fehrenbach

Plouraboué

2014

Struct Multidisc Optim

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. Abstract We apply shape optimization tools to the generalized Graetz problem which is a convection-diffusion equation. The problem boils down to the optimization of generalized eigenvalues on a two phases domain. Shape sensitivity analysis is performed with respect to the evolution of the interface between the fluid and solid phase. In particular physical settings, counterexamples where there is no optimal domains are exhibited. Numerical examples of optimal domains with different physical parameters and constraints are presented. Two different numerical methods (level-set and mesh-morphing) are show-cased and compared.

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Section: Derivability Of the Eigenvaluementioning

confidence: 98%

Section: Definitionmentioning

confidence: 99%

mentioning

confidence: 99%

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Shape optimization for the generalized Graetz problem

Gournay

Fehrenbach

Plouraboué

2014

Struct Multidisc Optim

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“…This model is written as a coupled system of nonlinear parabolic equations on a bounded domain in one‐space dimension. Several variants of the GCZ models have been treated in the work of [] where particular assumptions on the exterior stress field have been involved. In fact, in their work, they only considered either constant or bounded space‐time dependent stresses.…”

Section: Introductionmentioning

confidence: 99%

Formal derivation and existence result of an approximate model on dislocation densities

2018

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In this work, we are interested in a mathematical problem arising from the dynamics of dislocation densities in crystals. The model, originally developed by Groma, Czikor, and Zaiser in [], is a coupled singular parabolic system that describes the motion of dislocations in a bounded crystal, taking into account the short‐range interactions and the effect of exterior stresses. We show the derivation and the short time existence and uniqueness of a regular solution in a Hölder space using a fixed point argument and a particular comparison principle.

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Numerical studies of the optimization of the first eigenvalue of the heat diffusion in inhomogeneous media

Matsue

Naito

2015

Japan J. Indust. Appl. Math.

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In this paper, we study optimization of the first eigenvalue of −∇ · (ρ(x)∇u) = λu in a bounded domain Ω ⊂ R n under several constraints for the function ρ. We consider this problem in various boundary conditions and various topologies of domains. As a result, we numerically observe several common criteria for ρ for optimizing eigenvalues in terms of corresponding eigenfunctions, which are independent of topology of domains and boundary conditions. Geometric characterizations of optimizers are also numerically observed.1. The element ρ * ∈ K optimizing λ 1 (ρ) can be characterized by inequalities with respect to ρ * ∇u * , where u * is the eigenfunction associated with λ 1 (ρ * ) of (1.1). As a consequence, the domain S * = {x ∈ Ω | ρ * (x) = c} is given by the super-or the sub-level set of |ρ * ∇u * |. This characterization is independent of topology and geometry of Ω and boundary conditions on ∂Ω.2. If Ω is star-shaped and symmetric in a certain direction, then S * has the same symmetry.3. Optimized region S * depends continuously on a parameter of the boundary condition if the Robin boundary condition is imposed.The precise statements are described in Section 4 (Observation 4.2 -4.4) with various numerical results. This paper is organized as follows. In Section 2, we provide more precise setting of our problems. A numerical method for finding optimizers we apply here, the level set approach, is also derived here. In Section 3, numerical and mathematical known results for a well-considered problem are discussed. Section 4 is where our main discussion is developed. We show several numerical observations about eigenvalue optimization criteria, geometry of the level set S * = {x ∈ Ω | ρ * (x) = c} for the optimizer ρ * and continuous dependence of S * on boundary conditions. 2 Setting 2.

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Derivation and study of dynamical models of dislocation densities

Cited by 3 publications

References 13 publications

Shape optimization for the generalized Graetz problem

Shape optimization for the generalized Graetz problem

Formal derivation and existence result of an approximate model on dislocation densities

Numerical studies of the optimization of the first eigenvalue of the heat diffusion in inhomogeneous media

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