Overall Properties of a Discrete Membrane with Randomly Distributed Defects

Braides, Andrea; Piatnitski, Andrey

doi:10.1007/s00205-008-0114-8

Cited by 20 publications

(24 citation statements)

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“…Our arguments cannot be generalized straightforward to models for linearized elasticity. (viii) A different randomization of the functional (3) has been considered in another context in [25], where a random choice between the potential f (s) = min{s, 1} andf (s) = s is analyzed. However, isotropy of the limit functional remained an open problem.…”

Section: The Main Approximation Resultsmentioning

confidence: 99%

Discrete stochastic approximations of the Mumford–Shah functional

Ruf

2019

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We propose a new Γ-convergent discrete approximation of the Mumford-Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford-Shah functional in any dimension. version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ Remark 2.3. We also make use of the associated Voronoi tessellation V(ω) = {C(x)} x∈L(ω) , where the (random) Voronoi cells with nuclei x ∈ L(ω) are defined asIf L(ω) is admissible, then [6, Lemma 2.3] yields the inclusions B r 2 (x) ⊂ C(x) ⊂ B R (x). Next we introduce some notions from ergodic theory that build the basis for stochastic homogenization.Definition 2.4. We say that a family of measurable functions {τ z } z∈Z d , τ z : Ω → Ω, is an additive group action on Ω if τ 0 = id and τ z1+z2 = τ z2 • τ z1 for all z 1 , z 2 ∈ Z d .An additive group action is called measure preserving ifIn terms of a stochastic lattice the probabilistic properties read as follows:Definition 2.5. A stochastic lattice L is said to be stationary if there exists an additive, measure preserving group action {τ z } z∈Z d on Ω such that for all z ∈ Z d L • τ z = L + z.

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Section: The Main Approximation Resultsmentioning

confidence: 99%

Discrete stochastic approximations of the Mumford–Shah functional

Ruf

2019

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…Note that the energies E ε are still 'of the same order' of the Mumford-Shah energy since their limit is sandwiched between MS and √ 2 MS. In order to overcome the anisotropy due to the lattice symmetries, various corrections have been proposed taking into account long-range interactions (Braides and Gelli [15], Chambolle [27]), adapted grids for finite-elements (Chambolle and Dal Maso [28]), averaged quantities (Bourdin and Chambolle [9]), or random interactions (Braides and Piatnitski [16]). From a standpoint different from that of image processing, the same type of asymptotic analysis is of fundamental interest in physics and continuum mechanics when energies of the same form as E ε are considered, with ψ ε some type of interatomic potential (e.g., Lennard-Jones potentials), and the goal is to derive a corresponding continuous theory.…”

Section: Introductionmentioning

confidence: 99%

A compactness result for a second-order variational discrete model

2011

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Abstract. We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions with bounded hessian, and we give an upper and a lower bound in terms of the Blake and Zisserman energy. We prove a sharp bound by exhibiting the discrete-to-continuous Γ -limit for a special class of functions, showing the appearance new 'shear' terms in the energy, which are a genuinely two-dimensional effect.Mathematics Subject Classification.

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“…random variables. In dimension two an analysis by Braides and Piatnitski [18] shows that the Γ -limit is deterministic and depends almost surely on the probability p of the weak springs. Its form is of 'fracture type' if p is above the percolation threshold, while it coincides with the Dirichlet integral for all values of p below that threshold.…”

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confidence: 99%

“…for any θ ∈ [0, 1], we may obtain both such energies as Γ -limits for suitable choices of f ε i j (see [18] and Sect. 3.4 below).…”

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Models of defects in atomistic systems

Braides

Sigalotti

2010

Calc. Var.

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We analyze the overall behavior of discrete systems in the presence of defects (modeled by truncated quadratic potentials for some of the interactions) by exhibiting approximate free-discontinuity continuous energies. We give bounds on the limit surface energy densities, and we prove that these bounds are sharp in the classes of energy densities depending only on the normal to the discontinuity set, or concave and depending only on the jump across the interface. As a preliminary result we give a continuous descriptions of the ‘discrete Neumann sieve’

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Overall Properties of a Discrete Membrane with Randomly Distributed Defects

Cited by 20 publications

References 24 publications

Discrete stochastic approximations of the Mumford–Shah functional

Discrete stochastic approximations of the Mumford–Shah functional

A compactness result for a second-order variational discrete model

Models of defects in atomistic systems

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