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.

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.

FRACTURE MECHANICS IN PERFORATED DOMAINS: A VARIATIONAL MODEL FOR BRITTLE POROUS MEDIA

2009 ◽  
Vol 19 (11) ◽  
pp. 2065-2100 ◽  
Author(s):  
MATTEO FOCARDI ◽  
M. S. GELLI ◽  
M. PONSIGLIONE
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Fracture Mechanics ◽  
Special Functions ◽  
Variational Model ◽  
Perforated Domain ◽  
Functions Of Bounded Variation ◽  
Perforated Domains ◽  
Γ Convergence ◽  
Internal Scale

This paper deals with fracture mechanics in periodically perforated domains. Our aim is to provide a variational model for brittle porous media in the case of anti-planar elasticity. Given the perforated domain Ωε ⊂ ℝN (ε being an internal scale representing the size of the periodically distributed perforations), we will consider a total energy of the type [Formula: see text] Here u is in SBV(Ωε) (the space of special functions of bounded variation), Su is the set of discontinuities of u, which is identified with a macroscopic crack in the porous medium Ωε, and [Formula: see text] stands for the (N - 1)-Hausdorff measure of the crack Su. We study the asymptotic behavior of the functionals [Formula: see text] in terms of Γ-convergence as ε → 0. As a first (nontrivial) step we show that the domain of any limit functional is SBV(Ω) despite the degeneracies introduced by the perforations. Then we provide explicit formula for the bulk and surface energy densities of the Γ-limit, representing in our model the effective elastic and brittle properties of the porous medium, respectively.

ON A NONLOCAL FUNCTIONAL ARISING IN THE STUDY OF THIN-FILM BLISTERING

2010 ◽  
Vol 08 (02) ◽  
pp. 109-123
Author(s):  
N. ANSINI ◽  
V. VALENTE
Keyword(s):  
Thin Film ◽  
Asymptotic Analysis ◽  
Circular Plate ◽  
One Dimensional ◽  
Von Karman ◽  
Nonlocal Functional ◽  
Γ Convergence ◽  
Von Kármán

The energy of a Von Kármán circular plate is described by a nonlocal nonconvex one-dimensional functional depending on the thickness ε. Here we perform the asymptotic analysis via Γ-convergence as the parameter ε goes to zero.

PARABOLIC LIMIT AND STABILITY OF THE VLASOV–FOKKER–PLANCK SYSTEM

2000 ◽  
Vol 10 (07) ◽  
pp. 1027-1045 ◽  
Author(s):  
F. POUPAUD ◽  
J. SOLER
Keyword(s):  
Free Path ◽  
Space Dimension ◽  
Mass Density ◽  
Mean Free Path ◽  
Convergence Result ◽  
Limit Equation ◽  
Thermal Velocity ◽  
The Stability ◽  
Parabolic Limit ◽  
Fokker Planck

In this paper the stability of the Vlasov–Poisson–Fokker–Planck with respect to the variation of its constant parameters, the scaled thermal velocity and the scaled thermal mean free path, is analyzed. For the case in which the scaled thermal velocity is the inverse of the scaled thermal mean free path and the latter tends to zero, a parabolic limit equation is obtained for the mass density. Depending on the space dimension and on the hypothesis for the initial data, the convergence result in L1 is weak and global in time or strong and local in time.

Slow motion in higher-order systems and Γ-convergence in one space dimension

2001 ◽  
Vol 44 (1) ◽  
pp. 33-57 ◽  
Author(s):  
William D. Kalies ◽  
Robert C.A.M. VanderVorst ◽  
Thomas Wanner
Keyword(s):  
Space Dimension ◽  
Slow Motion ◽  
Higher Order ◽  
Γ Convergence ◽  
One Space Dimension

NEW RESULTS ON THE ASYMPTOTIC BEHAVIOR OF DIRICHLET PROBLEMS IN PERFORATED DOMAINSDIRICHLET PROBLEMS IN PERFORATED DOMAINS

1994 ◽  
Vol 04 (03) ◽  
pp. 373-407 ◽  
Author(s):  
GIANNI DAL MASO ◽  
ADRIANA GARRONI
Keyword(s):  
Adjoint Operator ◽  
Arbitrary Sequence ◽  
Symmetric Part ◽  
Dirichlet Problems ◽  
Limit Measure ◽  
Compactness Result ◽  
Perforated Domains ◽  
Measurable Coefficients ◽  
Open Set ◽  
Γ Convergence

Let A be a linear elliptic operator of the second order with bounded measurable coefficients on a bounded open set Ω of Rn and let (Ωh) be an arbitrary sequence of open subsets of Ω. We prove the following compactness result: there exist a subsequence, still denoted by (Ωh), and a positive Borel measure μ on Ω, not charging polar sets, such that, for every f∈H−1(Ω) the solutions [Formula: see text] of the equations Auh=f in Ωh, extended to 0 on Ω\Ωh, converge weakly in [Formula: see text] to the unique solution [Formula: see text] of the problem [Formula: see text] When A is symmetric, this compactness result is already known and was obtained by Γ-convergence techniques. Our new proof, based on the method of oscillating test functions, extends the result to the non-symmetric case. The new technique, which is completely independent of Γ-convergence, relies on the study of the behavior of the solutions [Formula: see text] of the equations [Formula: see text] where A* is the adjoint operator. We prove also that the limit measure μ does not change if A is replaced by A*. Moreover, we prove that µ depends only on the symmetric part of the operator A, if the coefficients of the skew-symmetric part are continuous, while an explicit example shows that μ may depend also on the skew-symmetric part of A, when the coefficients are discontinuous.

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