scholarly journals On the property of homothetic mean value on periodically perforated domains

2021 ◽  
Vol 15 ◽  
pp. 158
Author(s):  
P.I. Kogut ◽  
T.N. Rudyanova
Keyword(s):  
Weak Limit ◽  
Mean Value ◽  
Perforated Domain ◽  
Periodic Functions ◽  
Perforated Domains ◽  
Boundary Properties

We study boundary properties of one class of periodic functions as $\varepsilon \rightarrow 0$, where $\varepsilon$ is a period of periodically perforated domain. We show that their weak limit is the homothetic mean value of such functions.

scholarly journals Nonlocal problems in perforated domains

2019 ◽  
Vol 150 (1) ◽  
pp. 305-340
Author(s):  
Marcone C. Pereira ◽  
Julio D. Rossi
Keyword(s):  
Critical Radius ◽  
Dirichlet Condition ◽  
Weak Limit ◽  
Limit Problem ◽  
Perforated Domain ◽  
Periodic Case ◽  
Nonlocal Problems ◽  
Perforated Domains ◽  
Fixed Set ◽  
Typical Cell

AbstractIn this paper, we analyse nonlocal equations in perforated domains. We consider nonlocal problems of the form $f(x) = \int \nolimits _{B} J(x-y) (u(y) - u(x)) {\rm d}y$ with x in a perforated domain $\Omega ^\epsilon \subset \Omega $. Here J is a nonsingular kernel. We think about $\Omega ^\epsilon $ as a fixed set Ω from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside Ω. In the latter case we impose that u vanishes in the holes but integrate in the whole ℝN (B = ℝN) and in the former we just consider integrals in ℝN minus the holes ($B={\open R} ^N \setminus (\Omega \setminus \Omega ^\epsilon )$). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of $\Omega ^\epsilon $ has a weak limit, $\chi _{\epsilon } \rightharpoonup {\cal X}$ weakly* in L∞(Ω), we analyse the limit as ε → 0 of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls, we obtain that the critical radius is of the order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behaviour of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems.

scholarly journals Homogenization of the spectral problem on the Riemannian manifold consisting of two domains connected by many tubes

2013 ◽  
Vol 143 (6) ◽  
pp. 1255-1289 ◽  
Author(s):  
Andrii Khrabustovskyi
Keyword(s):  
Riemannian Manifold ◽  
Small Parameter ◽  
Asymptotic Behaviour ◽  
Spectral Problem ◽  
Spectral Parameter ◽  
Beltrami Operator ◽  
Perforated Domain ◽  
Perforated Domains ◽  
Accumulation Points ◽  
Image Position

The paper deals with the asymptotic behaviour as ε → 0 of the spectrum of the Laplace–Beltrami operator Δε on the Riemannian manifold Mε (dim Mε = N ≥ 2) depending on a small parameter ε > 0. Mε consists of two perforated domains, which are connected by an array of tubes of length qε. Each perforated domain is obtained by removing from the fixed domain Ω ⊂ ℝN the system of ε-periodically distributed balls of radius dε = ō(ε). We obtain a variety of homogenized spectral problems in Ω; their type depends on some relations between ε, dε and qε. In particular, if the limitsare positive, then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.

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 the mean value of Haar measurable almost periodic functions

1967 ◽  
Vol 34 (2) ◽  
pp. 201-214
Author(s):  
Henry W. Davis
Keyword(s):  
Mean Value ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
The Mean

Homogenization of singularly perturbed Dirichlet problems in perforated domains

2000 ◽  
Vol 130 (1) ◽  
pp. 35-51
Author(s):  
Viêt Há Hoáng
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behaviour ◽  
Error Estimates ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Perforated Domain ◽  
Dirichlet Problems ◽  
Perforated Domains

We study the singularly perturbed problem —εαΔuε + uε = f (α > 0) with the Dirichlet boundary condition in a perforated domain, in which the holes are distributed periodically with period 2ε throughout a fixed domain Ω. The asymptotic behaviour of uε when ε → 0, together with corrector results and error estimates in L2(Ω), are deduced for all sizes of holes. The behaviour of uε in is obtained for the cases where the size of holes is of order ε or is of a sufficiently smaller order. When the holes' size is of a sufficiently small order, as expected, uε has similar behaviour to that in the case of a non-varying domain.

scholarly journals DG-GMsFEM for Problems in Perforated Domains with Non-Homogeneous Boundary Conditions

Computation ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 75
Author(s):  
Valentin Alekseev ◽  
Maria Vasilyeva ◽  
Uygulaana Kalachikova ◽  
Eric T. Chung
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Homogeneous Boundary Condition ◽  
Multiscale Model ◽  
System Size ◽  
Numerical Results ◽  
Basis Functions ◽  
Perforated Domain ◽  
Perforated Domains ◽  
Grid Construction

Problems in perforated media are complex and require high resolution grid construction to capture complex irregular perforation boundaries leading to the large discrete system of equations. In this paper, we develop a multiscale model reduction technique based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsFEM) for problems in perforated domains with non-homogeneous boundary conditions on perforations. This method implies division of the perforated domain into several non-overlapping subdomains constructing local multiscale basis functions for each. We use two types of multiscale basis functions, which are constructed by imposing suitable non-homogeneous boundary conditions on subdomain boundary and perforation boundary. The construction of these basis functions contains two steps: (1) snapshot space construction and (2) solution of local spectral problems for dimension reduction in the snapshot space. The presented method is used to solve different model problems: elliptic, parabolic, elastic, and thermoelastic equations with non-homogeneous boundary conditions on perforations. The concepts for coarse grid construction and definition of the local domains are presented and investigated numerically. Numerical results for two test cases with homogeneous and non-homogeneous boundary conditions are included, as well. For the case with homogeneous boundary conditions on perforations, results are shown using only local basis functions with non-homogeneous boundary condition on subdomain boundary and homogeneous boundary condition on perforation boundary. Both types of basis functions are needed in order to obtain accurate solutions, and they are shown for problems with non-homogeneous boundary conditions on perforations. The numerical results show that the proposed method provides good results with a significant reduction of the system size.

scholarly journals Homogenized conductivity tensor and absorption function of a locally periodic porous medium

2018 ◽  
Keyword(s):  
Porous Medium ◽  
Boundary Value ◽  
Conductivity Tensor ◽  
Effective Characteristics ◽  
Macroscopic Model ◽  
Perforated Domain ◽  
Absorption Function ◽  
Perforated Domains ◽  
Periodic Porous Medium ◽  
The Media

We study a process of stationary diffusion in locally-periodic porous media with nonlinear absorption at the pore boundary. This process is described by a boundary-value problem for an elliptic equation considered in a complex perforated domain, with a nonlinear third boundary condition on the perforation boundary. In view of the smallness of the local scale of porosity of the media and the complexity of the perforated domain, the direct solution of such boundary-value problems is almost impossible. Therefore, a natural approach in this situation is to study the asymptotic behavior of the solution when the microstructure scale tends to 0, and the transition to the homogenized macroscopic model of the process. Our earlier papers were devoted to homogenization the diffusion equation in a wide class of non-periodically perforated domains: strongly-connected domains, which includes locally-periodically perforated domains. In these works, an homogenized model was obtained, the coefficients of which are expressed in terms of “mesoscopic” (local energy) characteristics of the media, which are determined in small cubes, the size of which, however, are much larger than the microstructure scale. In these papers, convergence theorems were proved under the conditions of the existence of limiting densities of "mesoscopic" characteristics, the fulfillment of which is generally difficult to show, but in a number of specific situations this can be done. In this paper, we show the fulfillment of these conditions and, by studing them, we obtain explicit formulas for the effective characteristics of the locally-periodic porous medium: a conductivity tensor and a function of absorption.

scholarly journals H-convergence of a class of quasilinear equations in perforated domains beyond periodic setting

2021 ◽  
Author(s):  
Hamid Haddadou
Keyword(s):  
Boundary Condition ◽  
Quasilinear Equation ◽  
Neumann Boundary ◽  
Real Variable ◽  
Perforated Domain ◽  
Periodic Case ◽  
Perforated Domains ◽  
Periodicity Condition ◽  
The Family ◽  
Quasilinear Problem

AbstractIn this paper, we aim to study the asymptotic behavior (when $$\varepsilon \;\rightarrow \; 0$$ ε → 0 ) of the solution of a quasilinear problem of the form $$-\mathrm{{div}}\;(A^{\varepsilon }(\cdot ,u^{\varepsilon }) \nabla u^{\varepsilon })=f$$ - div ( A ε ( · , u ε ) ∇ u ε ) = f given in a perforated domain $$\Omega \backslash T_{\varepsilon }$$ Ω \ T ε with a Neumann boundary condition on the holes $$T_{\varepsilon }$$ T ε and a Dirichlet boundary condition on $$\partial \Omega $$ ∂ Ω . We show that, if the holes are admissible in certain sense (without any periodicity condition) and if the family of matrices $$(x,d)\mapsto A^{\varepsilon }(x,d)$$ ( x , d ) ↦ A ε ( x , d ) is uniformly coercive, uniformly bounded and uniformly equicontinuous in the real variable d, the homogenization of the problem considered can be done in two steps. First, we fix the variable d and we homogenize the linear problem associated to $$A^{\varepsilon }(\cdot ,d)$$ A ε ( · , d ) in the perforated domain. Once the $$H^{0}$$ H 0 -limit $$A^{0}(\cdot ,d)$$ A 0 ( · , d ) of the pair $$(A^{\varepsilon },T^{\varepsilon })$$ ( A ε , T ε ) is determined, in the second step, we deduce that the solution $$u^{\varepsilon }$$ u ε converges in some sense to the unique solution $$u^{0}$$ u 0 in $$H^{1}_{0}(\Omega )$$ H 0 1 ( Ω ) of the quasilinear equation $$-\mathrm{{div}}\;(A^{0}(\cdot ,u^{0})\nabla u )=\chi ^{0}f$$ - div ( A 0 ( · , u 0 ) ∇ u ) = χ 0 f (where $$ \chi ^{0}$$ χ 0 is $$L^{\infty }$$ L ∞ weak $$^{\star }$$ ⋆ limit of the characteristic function of the perforated domain). We complete our study by giving two applications, one to the classical periodic case and the second one to a non-periodic one.

Sign in / Sign up

Export Citation Format

Share Document