Perforated Media Research Articles

Generalized multiscale discontinuous Galerkin method for convection–diffusion equation in perforated media

In this work, we present a Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for the convection–diffusion equation in perforated media. In order to numerically solve the problem on the fine grid and use the solution as a reference solution, we construct the grid that resolves perforation on the fine grid level and performs finite element approximation using the Interior Penalty Discontinuous Galerkin method (IPDG). To reduce the size of the fine grid system, we present the construction of the coarse grid approximation using a Generalized Multiscale Discontinuous Galerkin Method (GMsDGM). In GMsDGM, we construct the multiscale basis functions in the local domains by solutions of local convection–diffusion problems with different boundary conditions and performing dimension reduction by solutions of local spectral problems. We investigate several types of multiscale space constructions based on the various choices of boundary conditions of the local domains. Numerical results are presented for several test problems with different velocities fields and perforations distributions. We also investigate the influence of various constructions of multiscale basis functions with a different number of basis functions and different model parameters.

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

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.

Homogenization of parabolic problems with dynamical boundary conditions of reactive‐diffusive type in perforated media

AbstractThis paper deals with the homogenization of the reaction‐diffusion equations in a domain containing periodically distributed holes of size ε, with a dynamical boundary condition of reactive‐diffusive type, i.e., we consider the following nonlinear boundary condition on the surface of the holes where denotes the Laplace–Beltrami operator on the surface of the holes, ν is the outward normal to the boundary, plays the role of a surface diffusion coefficient and g is the nonlinear term. We generalize our previous results established in the case of a dynamical boundary condition of pure‐reactive type, i.e., with . We prove the convergence of the homogenization process to a nonlinear reaction‐diffusion equation whose diffusion matrix takes into account the reactive‐diffusive condition on the surface of the holes.

Simulation studies of interface dynamics of secondary electron yield in perforated media

The Secondary Electron Yield (SEY) is one of the key factors of multipactor, which remains a serious potential risk for high power vacuum applications. In this paper, the SEY and electron dynamics at different porous medium interfaces have been analyzed numerically for multipactor simulation and suppression. Due to the existence of an RF field, accumulated charge field, and external bias magnetic field, the previous electron trajectory tracking method in porous metal surfaces is not applicable. The real electromagnetic field distributions in porous medium, including metal and dielectric, are obtained theoretically and numerically for the calculation of electron motion. Based on the Monte Carlo simulation method, a novel numerical algorithm is proposed for the SEY calculation in porous medium. Simulation results show that SEYs of the porous dielectric perforated with cylindrical pores are correctly calculated and match well with experiments. It is demonstrated that the SEY at the porous interface is not only determined by electron motions but is also affected by the geometry structures of micro-pores. Different from the metal surface, the SEY at the porous dielectric interface increases when the width-to-depth ratio of micro-pores is greater than 1. The SEY reduces only since the depth of micro-pores is larger than lateral dimensions. As the depth gets much larger than the width, the “electron trapping” effect becomes remarkable and the SEY reduces considerably, which is promising for multipactor-free design in high power applications.

A Generalized Multiscale Finite Element Method (GMsFEM) for perforated domain flows with Robin boundary conditions

In this work, we develop a multiscale method for elliptic problems in perforated media with Robin boundary conditions. Perforated media are encountered in many applications, e.g., in material science applications, where the material properties are prescribed outside perforation. Such a perforated structure can significantly affect the solution. For the numerical solution of the equation, it is necessary to construct a computational grid with elements resolving the perforations. The computational grid calculated in this way is fine-scale grid and this can lead to a large number of unknowns. For the numerical solution of our problem, we construct an approximation of the equation on a coarse grid using the Generalized Multiscale Finite Element Method (GMsFEM). The main idea of the GMsFEM is to construct multiscale basis functions on a coarse grid, which can reduce the computational cost. The multiscale basis function construction requires snapshots and local spectral problems, which are developed in the paper. In the paper, we study Robin boundary conditions, which arise in many applications. Previous works (Chung et al., 2016) considered Dirichlet and Neumann boundary conditions. The construction of multiscale basis functions differs from those in Chung et al. 2016 [1] , [2] and as we need to impose non-homogeneous Robin boundary conditions. We present several numerical examples. In these examples, perforated domains with many inclusions are considered. Our numerical results show a good agreement between the coarse and the fine-grid simulations.

Asymptotic analysis of a semi-linear elliptic system in perforated domains: Well-posedness and correctors for the homogenization limit

In this study, we prove results on the weak solvability and homogenization of a microscopic semi-linear elliptic system posed in perforated media. The model presented here explores the interplay between stationary diffusion and both surface and volume chemical reactions in porous media. Our interest lies in deriving homogenization limits (upscaling) for alike systems and particularly in justifying rigorously the obtained averaged descriptions. Essentially, we prove the well-posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrations and then use the structure of the two scale expansions to derive corrector estimates delimitating this way the convergence rate of the asymptotic approximates to the macroscopic limit concentrations. Our techniques include Moser-like iteration techniques, a variational formulation, two-scale asymptotic expansions as well as energy-like estimates.

Homogenization of a variational inequality for the p-Laplacian in perforated media with nonlinear restrictions for the flux on the boundary of isoperimetric perforations: p equal to the dimension of the space

We address the homogenization of a variational inequality posed in perforated media issue from a unilateral problem for the p-Laplacian. We consider the n-Laplace operator in a perforated domain of ℝn, n ≥ 3, with restrictions for the solution and its flux (the flux associated with the n-Laplacian) on the boundary of the perforations which are assumed to be isoperimetric. The solution is assumed to be positive on the boundary of the holes and the flux is bounded from above by a negative, nonlinear monotone function multiplied by a large parameter. A certain non periodical distribution of the perforations is allowed while the assumption that their size is much smaller than the periodicity scale is performed. We make it clear that in the average constants of the problem, the perimeter of the perforations appears for any shape.

Homogenization for the p-Laplace operator in perforated media with nonlinear restrictions on the boundary of the perforations: A critical Case

Homogenization for the p-Laplace operator in perforated media with nonlinear restrictions on the boundary of the perforations: A critical Case

Crouzeix-Raviart MsFEM with Bubble Functions for Diffusion and Advection-Diffusion in Perforated Media

AbstractThe adaptation of Crouzeix-Raviart finite element in the context of multi-scale finite element method (MsFEM) is studied and implemented on diffusion and advection-diffusion problems in perforated media. It is known that the approximation of boundary condition on coarse element edges when computing the multiscale basis functions critically influences the eventual accuracy of any MsFEM approaches. The weakly enforced continuity of Crouzeix-Raviart function space across element edges leads to a natural boundary condition for the multiscale basis functions which relaxes the sensitivity of our method to complex patterns of perforations. Another ingredient to our method is the application of bubble functions which is shown to be instrumental in maintaining high accuracy amid dense perforations. Additionally, the application of penalization method makes it possible to avoid complex unstructured domain and allows extensive use of simpler Cartesian meshes.

Homogenization for the p-Laplace operator and nonlinear Robin boundary conditions in perforated media along (n − 1)-dimensional manifolds

Homogenization for the p-Laplace operator and nonlinear Robin boundary conditions in perforated media along (n − 1)-dimensional manifolds

Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics

In the present paper, we study integrodifferential equations with unbounded operator coefficients in Hilbert spaces. The principal part of the equation is an abstract hyperbolic equation perturbed by summands with Volterra integral operators. These equations represent an abstract form of the Gurtin–Pipkin integrodifferential equation describing the process of heat conduction in media with memory and the process of sound conduction in viscoelastic media and arise in averaging problems in perforated media (the Darcy law). The correct solvability of initial-boundary problems for the specified equations is established in weighted Sobolev spaces on a positive semiaxis. Spectral problems for operator-functions are analyzed. Such functions are symbols of these equations. The spectrum of the abstract integrodifferential Gurtin–Pipkin equation is investigated.

Gap in the essential spectrum of an elliptic formally self-adjoint system of differential equations

For a formally self-adjoint elliptic system of partial differential equations with periodic coefficients in the space ℝn, we show that, by conferring contrast properties to the coefficients of differential operators, one can open a gap in the essential spectrum of the system. We suggest a method based on the derivation of an asymptotically sharp generalized Korn inequality and the use of the maximin principle; this method applies to perforated media as well as to periodic layered and quasi-cylindrical waveguides.

Homogenization Results for Enzymatic Dispersion Processes

AbstractThe aim of this paper is to study the asymptotic behavior of the solution of a nonlinear problem arising in the modelling of enzyme catalyzed reactions through perforated media. The effective behavior of the solution of such a problem is governed by a new elliptic boundary‐value problem that captures the effect of the enzymatic reactions. The approach we use is the energy method of L. Tartar [4], coupled with monotonicity methods and results from the theory of semilinear problems [2]. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Homogenization of a convection–diffusion equation in perforated domains with a weak adsorption

The aim of this paper is to study the asymptotic behavior of the solution of a convection–diffusion equation in perforated domains with oscillating velocity and a Robin boundary condition which describes the adsorption on the bord of the obstacles. Without any periodicity assumption, for a large range of perforated media and by mean of variational homogenization, we find the global behavior when the characteristic size e of the perforations tends to zero. The homogenized model, is a convection–diffusion equation but with an extra term coming from the weak adsorption boundary condition. An example is presented to illustrate the methodology.

PARABOLIC PROBLEMS WITH DYNAMICAL BOUNDARY CONDITIONS IN PERFORATED MEDIA

The asymptotic behavior of the solution of a parabolic dynamical boundary‐value problem in a periodically perforated domain is analyzed. The perforations, which are identical and periodically distributed, are of size ϵ. In the perforated domain we consider a heat equation, with a Dirichlet condition on the exterior boundary and a dynamical boundary condition on the surface of the holes. The limit equation, as ϵ ? 0, is a heat equation with extra-terms coming from the influence of the non-homogeneous dynamical boundary condition.

Parabolic problems with dynamical boundary conditions in perforated media

Abstract The asymptotic behavior of the solution of a parabolic dynamical boundary‐value problem in a periodically perforated domain is analyzed. The perforations, which are identical and periodically distributed, are of size ϵ. In the perforated domain we consider a heat equation, with a Dirichlet condition on the exterior boundary and a dynamical boundary condition on the surface of the holes. The limit equation, as ϵ ? 0, is a heat equation with extra‐terms coming from the influence of the non‐homogeneous dynamical boundary condition.

Reaction - diffusion equations in perforated media

A reaction - diffusion equation is studied with linear dynamics in the medium and a nonlinear boundary condition. The diffusion occurs in a domain that is perforated by periodically placed holes. On the boundary of these holes, the nonlinear boundary condition is applied. Wavefront propagation is established by use of a Feynman - Kac representation of the solution and development of an appropriate large deviations principle. The location of the wavefront is described in terms of the Legendre transform of the solution to an eigenvalue problem. The proof uses periodicity in an essential way by application of Doeblin's condition to the associated random process on the torus.

Asymptotic expansions in perforated media with a periodic structure

Asymptotic expansions in perforated media with a periodic structure