scholarly journals A Γ-Convergence Result for Thin Curved Films Bonded to a Fixed Substrate with a Noninterpenetration Constraint

2006 ◽  
Vol 27 (6) ◽  
pp. 615-636
Author(s):  
Hamdi Zorgati
Keyword(s):  
Convergence Result ◽  
Γ Convergence

HOMOGENIZATION OF ENERGIES DEFINED ON PAIRS SET-FUNCTION

2009 ◽  
Vol 11 (03) ◽  
pp. 459-479 ◽  
Author(s):  
MARGHERITA SOLCI
Keyword(s):  
Asymptotic Behavior ◽  
Energy Density ◽  
Convergence Result ◽  
Energy Scale ◽  
Surface Energy Density ◽  
Surface Energies ◽  
Open Set ◽  
Γ Convergence ◽  
Regular Open ◽  
Set Function

In the present work, we deal with the problem of the asymptotic behavior of a sequence of non-homogeneous energies depending on a pair set-function of the form [Formula: see text] with u ∈ H1(Ω), E regular open set and the energy densities f and φ both 1-periodic in the first variable; this leads, in the Γ-limit, to a problem of homogenization. We prove a Γ-convergence result for the sequence {Fε}, showing that there is no interaction between the homogenized bulk and surface energy density; that is, even though the effect of the bulk and surface energies are at the same energy scale, oscillations in the bulk term can be neglected close to the surfaces ∂*E and S(u), where surface oscillations are dominant.

scholarly journals On the asymptotic behaviour of nonlocal perimeters

2019 ◽  
Vol 25 ◽  
pp. 48
Author(s):  
Judith Berendsen ◽  
Valerio Pagliari
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Behaviour ◽  
Convergence Result ◽  
Integral Functionals ◽  
Multiplicative Constant ◽  
Plateau’S Problem ◽  
Positive Kernel ◽  
Limiting Behaviour ◽  
Plateau's Problem ◽  
Γ Convergence

We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K, which might be singular. In the first part of the paper, we show that these functionals are indeed perimeters in a generalised sense that has been recently introduced by A. Chambolle et al. [Archiv. Rational Mech. Anal. 218 (2015) 1263–1329]. Also, we establish existence of minimisers for the corresponding Plateau’s problem and, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe “flat” boundary conditions. A Γ-convergence result is discussed in the second part of the work. We study the limiting behaviour of the nonlocal perimeters associated with certain rescalings of a given kernel that has faster-than-L1 decay at infinity and we show that the Γ-limit is the classical perimeter, up to a multiplicative constant that we compute explicitly.

A Γ-convergence result for fluid-filled fracture propagation

2020 ◽  
Vol 54 (3) ◽  
pp. 1003-1023
Author(s):  
Annika Bach ◽  
Liesel Sommer
Keyword(s):  
Asymptotic Analysis ◽  
Numerical Simulations ◽  
Discontinuous Galerkin ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Fracture Propagation ◽  
Convergence Result ◽  
Elastic Media ◽  
Γ Convergence

In this paper we provide a rigorous asymptotic analysis of a phase-field model used to simulate pressure-driven fracture propagation in poro-elastic media. More precisely, assuming a given pressure p ∈ W 1,∞ (Ω) we show that functionals of the form $$ E(\vec{u})={\int }_{\mathrm{\Omega }} e(\vec{u}):\mathbb{C}e(\vec{u})+p\nabla \cdot \vec{u}+\left\langle \nabla p,\vec{u}\right\rangle\enspace \mathrm{d}x+{\mathcal{H}}^{n-1}({J}_{\vec{u}}),\enspace \vec{u}\in \mathrm{G}{SBD}(\mathrm{\Omega })\cap {L}^1(\mathrm{\Omega };{\mathbb{R}}^n) $$ can be approximated in terms of Γ-convergence by a sequence of phase-field functionals, which are suitable for numerical simulations. The Γ-convergence result is complemented by a numerical example where the phase-field model is implemented using a Discontinuous Galerkin Discretization.

ASYMPTOTIC ANALYSIS OF PERIODICALLY-PERFORATED NONLINEAR MEDIA AT THE CRITICAL EXPONENT

2009 ◽  
Vol 11 (06) ◽  
pp. 1009-1033 ◽  
Author(s):  
LAURA SIGALOTTI
Keyword(s):  
Asymptotic Analysis ◽  
Critical Exponent ◽  
Space Dimension ◽  
Convergence Result ◽  
Nonlinear Media ◽  
Perforated Domains ◽  
Extra Term ◽  
Γ Convergence ◽  
Vector Valued ◽  
Critical Scale

We give a Γ-convergence result for vector-valued nonlinear energies defined on periodically perforated domains. We consider integrands with n-growth where n is the space dimension, showing that there exists a critical scale for the perforations such that the Γ-limit is non-trivial. We prove that the limit extra-term is given by a formula of homogenization type, which simplifies in the case of n-homogeneous energy densities.

scholarly journals Variational approximation of size-mass energies for k-dimensional currents

2019 ◽  
Vol 25 ◽  
pp. 43 ◽  
Author(s):  
Antonin Chambolle ◽  
Luca A.D. Ferrari ◽  
Benoit Merlet
Keyword(s):  
Unit Length ◽  
Convergence Result ◽  
Variational Approximation ◽  
Plateau Problem ◽  
Numerical Treatment ◽  
Transportation Energy ◽  
Limit Functional ◽  
Γ Convergence

In this paper we produce a Γ-convergence result for a class of energies Fε,ak modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that Fε,a1 Γ-converges to a branched transportation energy whose cost per unit length is a function fan−1 depending on a parameter a > 0 and on the codimension n − 1. The limit cost fa(m) is bounded from below by 1 + m so that the limit functional controls the mass and the length of the limit object. In the limit a ↓ 0 we recover the Steiner energy. We then generalize the approach to any dimension and codimension. The limit objects are now k-currents with prescribed boundary, the limit functional controls both their masses and sizes. In the limit a ↓ 0, we recover the Plateau energy defined on k-currents, k < n. The energies Fε,ak then could be used for the numerical treatment of the k-Plateau problem.

Loss of ellipticity through homogenization in linear elasticity

2015 ◽  
Vol 25 (05) ◽  
pp. 905-928 ◽  
Author(s):  
Marc Briane ◽  
Gilles A. Francfort
Keyword(s):  
Linear Elasticity ◽  
Sufficient Conditions ◽  
Convergence Result ◽  
Strong Ellipticity ◽  
Loss Of Ellipticity ◽  
Linearized Elasticity ◽  
Γ Convergence

In 1993, it was shown by Geymonat, Müller and Triantafyllidis that, in the setting of linearized elasticity, a Γ-convergence result holds for highly oscillating sequences of elastic energies whose functional coercivity constant in ℝNis zero while the corresponding coercivity constant on the torus remains positive. We illustrate the range of applicability of that result by finding sufficient conditions for such a situation to occur. We thereby justify the degenerate laminate construction given by Gutiérrez in 1999. We also demonstrate that the predicted loss of strict strong ellipticity resulting from the construction by Gutiérrez is unique within a "laminate-like" class of microstructures. It will only occur for the specific micro-geometry investigated there. Our results thus confer both a rigorous, and a canonical character to those of Gutiérrez.

scholarly journals The Γ-limit for singularly perturbed functionals of Perona–Malik type in arbitrary dimension

2014 ◽  
Vol 24 (06) ◽  
pp. 1091-1113 ◽  
Author(s):  
Giovanni Bellettini ◽  
Antonin Chambolle ◽  
Michael Goldman
Keyword(s):  
Special Functions ◽  
Arbitrary Dimension ◽  
Convergence Result ◽  
Singularly Perturbed ◽  
Functions Of Bounded Variation ◽  
Second Derivative ◽  
One Dimensional ◽  
Density Result ◽  
Arbitrary Dimensions ◽  
Γ Convergence

In this paper, we generalize to arbitrary dimensions a one-dimensional equicoerciveness and Γ-convergence result for a second derivative perturbation of Perona–Malik type functionals. Our proof relies on a new density result in the space of special functions of bounded variation with vanishing diffuse gradient part. This provides a direction of investigation to derive approximation for functionals with discontinuities penalized with a "cohesive" energy, that is, whose cost depends on the actual opening of the discontinuity.

Variational approximation of functionals defined on 1-dimensional connected sets in ℝn

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mauro Bonafini ◽  
Giandomenico Orlandi ◽  
Édouard Oudet
Keyword(s):  
Variational Problems ◽  
Convergence Result ◽  
Steiner Tree Problem ◽  
Variational Approximation ◽  
Planar Case ◽  
Ginzburg Landau ◽  
Connected Sets ◽  
Steiner Problems ◽  
Γ Convergence ◽  
Euclidean Steiner Tree Problem

AbstractIn this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert–Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in {\mathbb{R}^{n}}. Following the analysis for the planar case presented in [M. Bonafini, G. Orlandi and E. Oudet, Variational approximation of functionals defined on 1-dimensional connected sets: The planar case, SIAM J. Math. Anal. 50 2018, 6, 6307–6332], we provide a variational approximation through Ginzburg–Landau type energies proving a Γ-convergence result for {n\geq 3}.

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