Periodic Unfolding Method Research Articles

An optimal control problem for diffusion–precipitation model

We present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Posing controls on the porous part of the medium (distributed control), we set up a convex minimization problem. The main objective of this article is to characterize an arbitrary control to be an optimal control. We establish a relation between the optimal control and the corresponding adjoint state. At first, we analyse the microscopic description of the controlled system, then we homogenised the system by rigorous two-scale convergence method and periodic unfolding method.

Periodic unfolding method for domains with very small inclusions

This work creates a version of the periodic unfolding method suitable for domains with very small inclusions in \(\mathbb{R}^N\) for \(N\geq 3\). In the first part, we explore the properties of the associated operators. The second part involves the application of the method in obtaining the asymptotic behavior of a stationary heat dissipation problem depending on the parameter \( \gamma < 0\). In particular, we consider the cases when \(\gamma \in (-1,0)\), \( \gamma < -1\) and \(\gamma = -1\). We also include here the corresponding corrector results for the solution of the problem, to complete the homogenization process. For more information see https://ejde.math.txstate.edu/Volumes/2023/85/abstr.html

Interface potential in composites with general imperfect transmission conditions

The model analyzed in this paper has its origins in the description of composites made by a hosting medium containing a periodic array of inclusions coated by a thin layer consisting of sublayers of two different materials. This two-phase coating material is such that the external part has a low diffusivity in the orthogonal direction, while the internal one has high diffusivity along the tangential direction. In a previous paper (Amar in IFB 21:41–59, 2019), by means of a concentration procedure, the internal layer was replaced by an imperfect interface. The present paper is concerned with the concentration of the external coating layer and the homogenization, via the periodic unfolding method, of the resulting model, which is far from being a standard one. Despite the fact that the limit problem looks like a classical Dirichlet problem for an elliptic equation, in the construction of the homogenized matrix and of the source term, a very delicate analysis is required.

Homogenization of a semilinear elliptic problem in a thin composite domain with an imperfect interface

In this paper, we consider the asymptotic behavior of a semilinear elliptic problem in a thin two‐composite domain with an imperfect interface, where the flux is discontinuous. For this thin domain, both the height and the period are of order . We first use Minty–Browder theorem to prove the well‐posedness of the problem and then apply the periodic unfolding method to obtain the limit problems and some corrector results for three cases of a real parameter , and , respectively. To deal with the semilinear terms, the extension operator and the averaged function are used.

Asymptotic behavior for textiles with loose contact

The paper is dedicated to the modeling of the elasticity problem for a textile structure. The textile is made of long and thin fibers, crossing each other in a periodic pattern, forming a woven canvas of a square domain. The textile is partially clamped. The fibers cannot penetrate each other but can slide with respect to each other in the in‐plane directions. The sliding is bounded by a contact function, which is chosen loose. The partial clamp and the loose contact lead to a domain partitioning, with different expected behaviors on each of the four subdomains. The homogenization is made via the periodic unfolding method, with an additional dimension reduction. The macroscopic limit problem results in a Leray–Lions problem with only macroconstraints in the plane.

Microscopic tridomain model of electrical activity in the heart with dynamical gap junctions. Part 2 – Derivation of the macroscopic tridomain model by unfolding homogenization method

We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math. 179 (2022) 1–35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov–Riesz compactness’s result.

Homogenization of a class of singular elliptic problems in two-component domains

This paper deals with the homogenization of a quasilinear elliptic problem having a singular lower order term and posed in a two-component domain with an ε-periodic imperfect interface. We prescribe a Dirichlet condition on the exterior boundary, while we assume that the continuous heat flux is proportional to the jump of the solution on the interface via a function of order ε γ . We prove an homogenization result for − 1 < γ < 1 by means of the periodic unfolding method (see SIAM J. Math. Anal. 40 (2008) 1585–1620 and The Periodic Unfolding Method (2018) Springer), adapted to two-component domains in (J. Math. Sci. 176 (2011) 891–927). One of the main tools in the homogenization process is a convergence result for a suitable auxiliary linear problem, associated with the weak limit of the sequence { u ε } of the solutions, as ε → 0. More precisely, our result shows that the gradient of u ε behaves like that of the solution of the auxiliary problem, which allows us to pass to the limit in the quasilinear term, and to study the singular term near its singularity, via an accurate a priori estimate.

Homogenization of a nonlinear stochastic model with nonlinear random forces for chemical reactive flows in porous media

In this study, we establish new homogenization results for chemical reactive fluxes in porous media using a nonlinear stochastic model with nonlinear random forces. This model is mathematically represented by a linear stochastic response equation, a nonlinear boundary condition, a nonlinear stochastic diffusion equation, and external nonlinear random forces that affect the solute concentration and diffusion in the fluid phase. A homogenized model made up of a nonlinear stochastic diffusion equation with extra terms reflecting the impact of the adsorption and chemical reactions occurring on the boundaries of the perforations and a stochastic differential equation with extra nonlinear term representing the limit for the surface problem is derived using the periodic unfolding operator method and probabilistic compactness results. To guarantee that the entire sequence converges, we also show the uniqueness result for the limit problem. Finally, we go over some applications for the interaction of chemical fluxes.

Homogenization of nonlinear hyperbolic problem with a dynamical boundary condition

<abstract><p>In this work, we look at homogenization results for nonlinear hyperbolic problem with a non-local boundary condition. We use the periodic unfolding method to obtain a homogenized nonlinear hyperbolic equation in a fixed domain. Due to the investigation's peculiarity, the unfolding technique must be developed with special attention, creating an unusual two-scale model. We note that the non-local boundary condition caused a damping on the homogenized model.</p></abstract>

Homogenization of Helmholtz equation in a periodic layer to study Faraday cage-like shielding effects

The work is motivated by the Faraday cage effect. We consider the Helmholtz equation over a 3D domain containing a thin heterogeneous interface of thickness . The layer has a δ-periodic structure in the in-plane directions and is cylindrical in the third direction. The periodic layer has one connected component and a collection of isolated regions. The isolated region in the thin layer represents air or liquid, and the connected component represents a solid metal grid with a δ thickness. The main issue is created by the contrast of the coefficients in the air and in the grid and that the zero-order term has a complex-valued coefficient in the connected faze while a real-valued in the complement. An asymptotic analysis with respect to is provided, and the limit Helmholtz problem is obtained with the Dirichlet condition on the interface. The periodic unfolding method is used to find the limit.

Homogenization results for a Landau–Lifshitz–Gilbert equation in composite materials with transmission defects

We study the homogenization of Landau–Lifshitz–Gilbert equation in a ϵ-periodic composite material formed by two constituents, separated by an imperfect interface , on which we prescribe the continuity of the conormal derivatives and a jump of the solution proportional to the conormal derivative, by means of a coefficient of order . We use the periodic unfolding method together with extension operators for handling the nonlinearities to identify the limit problem when tuning up the parameter γ in .

An optimal control problem in a tubular thin domain with rough boundary

In this paper we analyze the asymptotic behavior of a control problem set by a convection-reaction-diffusion equation with mixed boundary conditions and defined in a tubular thin domain with rough boundary. The control term acts on a subset of the rough boundary where a Robin-type boundary condition and a catalyzed reaction mechanism are set. The reaction mechanism depends on a parameter α∈R. Such parameter establishes different regimes which also depend on the profile and geometry of the tube defined by a periodic function g:R2↦R. We see that, if ∂2g is not null (that is, when g really depends on the second variable), then three regimes with respect to α are established: α<2, α=2 (the critical value) and α>2. On the other hand, if ∂2g≡0, similar regimes are obtained but now with a different critical value. Indeed, if ∂2g≡0, then the critical value is achieved at α=1. For each one of these six regimes, we obtain the asymptotic behavior of the control problem when the cylindrical thin domain degenerates to the unit interval. We show that the problem is asymptotically controllable just when α assumes the critical values. Our analysis is mainly performed using the periodic unfolding method adapted to cylindrical coordinates in R3.

Homogenization and Correctors for Stochastic Hyperbolic Equations in Domains with Periodically Distributed Holes

The goal of this paper is to present new results on homogenization and correctors for stochastic linear hyperbolic equations in periodically perforated domains with homogeneous Neumann conditions on the holes. The main tools are the periodic unfolding method, energy estimates, probabilistic and deterministic compactness results. The findings of this paper are stochastic counterparts of the celebrated work [D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Port. Math. (N.S.) 63 (2006) 467–496]. The convergence of the solution of the original problem to a homogenized problem with Dirichlet condition has been shown in suitable topologies. Homogenization and convergence of the associated energies results recover the work in [M. Mohammed and M. Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptot. Anal. 97 (2016) 301–327]. In addition to that, we obtain corrector results.

Modelling wave dispersion in fluid saturating periodic scaffolds

Acoustic waves in a slightly compressible fluid saturating porous periodic structure are studied using two complementary approaches: 1) the periodic homogenization (PH) method provides effective model equations for a general dynamic problem imposed in a bounded medium, 2) harmonic acoustic waves are studied in an infinite medium using the Floquet-Bloch (FB) wave decomposition. In contrast with usual simplifications, the advection phenomenon of the Navier–Stokes equations is accounted for. For this, an acoustic approximation is applied to linearize the advection term. The homogenization results are based the periodic unfolding method combined with the asymptotic expansion technique providing a straight upscaling procedure which leads to the macroscopic model defined in terms of the effective model parameters. These are computed using the characteristic responses of the porous microstructure. Using the FB theory, we derive dispersion equations for the scaffolds saturated by the inviscid, or the viscous, barotropic fluids, whereby the advection due to a permanent flow in the porous structures is respected. A computational study is performed for the numerical models obtained using the finite element discretization. For the FB methods-based dispersion analysis, quadratic eigenvalue problems must be solved. The numerical examples show influences of the microstructure size and of the advection generating an anisotropy of the acoustic waves dispersion.

Homogenization of the fluid–structure interaction in acoustics of porous media perfused by viscous fluid

This paper aims to clarify the homogenization results of the fluid–structure interaction in porous structures under the quasi-static and dynamic loading regimes. In the latter case, the acoustic fluctuations yield naturally a linear model which can be introduced in the configuration deformed as the consequence of the steady permanent flow. We consider a Newtonian slightly compressible fluid under the barotropic acoustic approximation. In contrast with usual simplifications, the advection phenomenon of the Navier–Stokes equations is accounted for. The homogenization results are based on the periodic unfolding method combined with the asymptotic expansion technique which provide a straight procedure leading the local problems for corrector functions yielding the effective model parameters and the macroscopic model. We show that the local problems for the solid and fluid parts are decoupled even in the dynamic interactions including the wall shear stress on the periodic interfaces. The dynamic permeability depends on the fluid flow properties including the advection effects associated with an assumed stationary perfusion of the porous structure.

Homogenization of Perforated Elastic Structures

The paper is dedicated to the asymptotic behavior of varepsilon-periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as varepsilon to 0. In case of plate-like or beam-like structures the asymptotic reduction of dimension from 3D to 2D or 1D respectively takes place. An example of the structure under consideration can be obtained by a periodic repetition of an elementary “flattened” ball or cylinder for plate-like or beam-like structures in such a way that the contact surface between two neighboring balls/cylinders has a non-zero measure. Since the domain occupied by the structure might have a non-Lipschitz boundary, the classical homogenization approach based on the extension cannot be used. Therefore, for obtaining Korn’s inequalities, which are used for the derivation of a priori estimates, we use the approach based on interpolation. In case of plate-like and beam-like structures the proof of Korn’s inequalities is based on the displacement decomposition for a plate or a beam, respectively. In order to pass to the limit as varepsilon to 0 we use the periodic unfolding method.

Two-scale homogenization of abstract linear time-dependent PDEs

Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space approach to evolutionary systems with an operator theoretic reformulation of the well-established periodic unfolding method in homogenization. Regarding the latter, we introduce a well-structured family of unitary operators on a Hilbert space that allows to describe and analyze differential operators with rapidly oscillating (possibly random) coefficients. We illustrate the approach by establishing periodic and stochastic homogenization results for elliptic partial differential equations, Maxwell’s equations, and the wave equation.

Homogenization and correctors for linear stochastic equations via the periodic unfolding methods

In this paper, we use the periodic unfolding method and Prokhorov’s and Skorokhod’s probabilistic compactness results to obtain homogenization and corrector results for stochastic partial differential equations (PDEs) with periodically oscillating coefficients. We show the convergence of the solutions of the original problems to the solutions of the homogenized problems. In contrast to the two-scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems

Homogenization and correctors of Robin problem for linear stochastic equations in periodically perforated domains

In this paper, we present homogenization and corrector results for stochastic linear parabolic equations in periodically perforated domains with non-homogeneous Robin conditions on the holes. We use the periodic unfolding method and probabilistic compactness results. Homogenization results presented in this paper are stochastic counterparts of some fundamental work given in [Cioranescu, Donato and Zaki in Port. Math. (N.S.) 63 (2006), 467–496]. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a parabolic stochastic equation in fixed domain with Dirichlet condition on the boundary. In contrast to the two scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems.

Homogenization and exact controllability for problems with imperfect interface

The first aim of this paper is to study, by means of the periodic unfolding method, the homogenization of elliptic problems with source terms converging in a space of functions less regular than the usual \begin{document}$ L^2 $\end{document} , in an \begin{document}$ \varepsilon $\end{document} -periodic two component composite with an imperfect transmission condition on the interface. Then we exploit this result to describe the asymptotic behaviour of the exact controls and the corresponding states of hyperbolic problems set in composites with the same structure and presenting the same condition on the interface. The exact controllability is developed by applying the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads us to the construction of the exact controls as solutions of suitable transposed problem.