Homogenization of parabolic equations with contrasting coefficients

Sandrakov, G. V.

doi:10.1070/im1999v063n05abeh000264

Cited by 21 publications

(15 citation statements)

References 7 publications

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“…The first rigorous mathematical result on the subject was obtained in [Arbogast et al 1990], where a linear parabolic equation with asymptotically degenerating coefficients was considered. This result was subsequently generalized in [Panasenko 1991;Bourgeat et al 1996;1998;Sandrakov 1999a;1999b;Pankratov and Piatnitski 2002;Marchenko and Khruslov 2006] also for nonperiodic domains and various rates of contrast. On the physical level of rigor, double-porosity models were studied in [Panfilov 2000].…”

Section: Introductionmentioning

confidence: 77%

Untitled

2016

Math. Mech. Compl. Sys.

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The concept of internal constraints is extended to gradient materials. Here, interesting constraints can be introduced, such as pseudorigid ones. The stresses and the hyperstresses will be given by constitutive equations only up to reactive parts, which do no work during any compatible motion of the body. For the inclusion of thermodynamical effects, the theory is generalized to the case of thermomechanical constraints. Here one obtains reactive parts of the stresses, heat flux, entropy, and energy, which do not contribute to the dissipation. Some critical remarks on the classical concept of internal constraints are finally given. A method to introduce internal constraints in a natural way is described to overcome some conceptual deficiencies of the classical concept.Communicated by Francesco dell'Isola. MSC2010: 74A30. 1 2 ALBRECHT BERTRAM AND RAINER GLÜGE format. It turns out, and will be shown in the sequel, that such an extension is in fact straightforward once a theory of gradient materials has been constructedat least within the mechanical context.The extension to a thermomechanical format is more complicated. This is, however, necessary not only in order to investigate the compatibility of such constraints with the second law of thermodynamics, but also to study the temperature dependence of mechanical constraints.There has been some discussion about a sound format for the inclusion of the thermodynamical variables into such a theory of internal constraints; see [Green et al. ].The starting point for the present approach is a suggestion by [Trapp 1971; Bertram 2005], where a rate form of a thermomechanical constraint is assumed and the possibility for reactive parts of the stresses, heat fluxes, and energies is given that are not dissipative during any process that is compatible with the constraint. Again, the extension of this theory to gradient materials is straightforward.At the end of this contribution, some critical remarks on the standard theory of mechanical constraints are given and a procedure to avoid these shortcomings is suggested.Dedicated to Gérard Maugin, in gratitude for many years of inspiration and friendshipThe differential-geometric underpinnings of a unified theory of material uniformity and evolution are exposed in terms of the language of groupoids subordinate to geometric distributions. Both the standard theory of material uniformity and the extended theory of functionally graded materials are included in the formulation as well as their temporal counterparts in anelastic and aging processes.In this paper we reinvestigate the structure of the solution of a well-known Love's problem, related to the electrostatic field generated by two circular coaxial conducting disks, in terms of orthogonal polynomial expansions, enlightening the role of the recently introduced class of the Lucas-Lehmer polynomials. Moreover we show that the solution can be expanded more conveniently with respect to a Riesz basis obtained starting from Chebyshev polynomials.This paper deals with the modeling of cy...

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Section: Introductionmentioning

confidence: 77%

Untitled

2016

Math. Mech. Compl. Sys.

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“…The usual linear double porosity model assumes that the width of the fracture containing highly permeable porous media is of the same order as the block size. The related homogenization problem was first studied in [11], and was then revisited in the mathematical literature by many other authors (see, e.g., [2,15,[25][26][27][28][29]32,37] and references therein). Let us mention that results on the rate of convergence for the linear double porosity model, for a large range of contrast, were obtained in [32].…”

Section: Introductionmentioning

confidence: 99%

“…The related homogenization problem was first studied in [11], and was then revisited in the mathematical literature by many other authors (see, e.g., [2,15,[25][26][27][28][29]32,37] and references therein). Let us mention that results on the rate of convergence for the linear double porosity model, for a large range of contrast, were obtained in [32]. Linear double porosity models with thin fissures were studied in [4,5,7,16,31].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear flow through double porosity media in variable exponent Sobolev spaces

Amaziane

Pankratov

Piatnitski

2009

Nonlinear Analysis: Real World Applications

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“…The first mathematical results concerning the homogenization of the double-porosity problems were obtained by Arbogast et al [7] and was then revisited in the mathematical literature by many other authors [12,15,16,18,20,25,29,31,34] and the references therein. Double porosity models with thin fissures were considered in [2,3,5,33] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Homogenization of one phase flow in a highly heterogeneous porous medium including a thin layer

Amaziane

Pankratov

Prytula

2010

Asymptotic Analysis

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In this work we consider a model problem describing one phase flow through a thin porous layer made of weakly permeable porous blocks separated by thin fissures. The flow is modeled by a linear parabolic equation considered in a bounded 2D domain with high contrast coefficients. The problem involves three small parameters: the first one characterizes the periodicity of the distribution of the blocks in the layer, the second one stands for the thickness of the layer, the third one characterizes the volume fraction of the fissure part in the layer. Using the notion of two-scale convergence, we derive the homogenized models which govern the global behavior of the flow when the small parameters tend to zero. The global models essentially depend on the relation between the small parameters.

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Homogenization of parabolic equations with contrasting coefficients

Cited by 21 publications

References 7 publications

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Nonlinear flow through double porosity media in variable exponent Sobolev spaces

Homogenization of one phase flow in a highly heterogeneous porous medium including a thin layer

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