Perforated Domains Research Articles

Stochastic homogenisation of free-discontinuity functionals in randomly perforated domains

Abstract In this paper, we study the asymptotic behaviour of a family of random free-discontinuity energies E ε {E_{\varepsilon}} defined in a randomly perforated domain, as ε goes to zero. The functionals E ε {E_{\varepsilon}} model the energy associated to displacements of porous random materials that can develop cracks. To gain compactness for sequences of displacements with bounded energies, we need to overcome the lack of equi-coerciveness of the functionals. We do so by means of an extension result, under the assumption that the random perforations cannot come too close to one another. The limit energy is then obtained in two steps. As a first step, we apply a general result of stochastic convergence of free-discontinuity functionals to a modified, coercive version of E ε {E_{\varepsilon}} . Then the effective volume and surface energy densities are identified by means of a careful limit procedure.

Approximate controllability of a semilinear elliptic problem in periodically perforated domains

In this paper, we study the controllability of a semilinear second order elliptic boundary value problem in a periodically perforated domain, where the control is distributed in an open subset of the domain. Compared to existing literature, the holes in this control problem can intersect the control zone. We study L2 and H1-approximate controllability for the problem and we prove the homogenization results for the control problem in an ε-periodically perforated domain. The holes are of same size as ε with the coefficients of state equation, and the cost functional being rapidly oscillating.

Uniform W1,p estimates and large-scale regularity for Dirichlet problems in perforated domains

In this paper we study the Dirichlet problem for Laplace's equation in a domain ωε,η perforated periodically with small holes in Rd, where ε represents the scale of the minimal distances between holes and η the ratio between the scale of sizes of holes and ε. We establish W1,p estimates for solutions with bounding constants depending explicitly on ε and η. The proof relies on a large-scale Lipschitz estimate for harmonic functions in perforated domains. The results are optimal for d≥2.

Singular behavior for a multi-parameter periodic Dirichlet problem

We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number ϵ > 0, proportional to the radius of the holes, and a map ϕ, which models the shape of the holes. So, if g denotes the Dirichlet boundary datum and f the Poisson datum, we have a solution for each quadruple ( ϵ , ϕ , g , f ). Our aim is to study how the solution depends on ( ϵ , ϕ , g , f ), especially when ϵ is very small and the holes narrow to points. In contrast with previous works, we do not introduce the assumption that f has zero integral on the fundamental periodicity cell. This brings in a certain singular behavior for ϵ close to 0. We show that, when the dimension n of the ambient space is greater than or equal to 3, a suitable restriction of the solution can be represented with an analytic map of the quadruple ( ϵ , ϕ , g , f ) multiplied by the factor 1 / ϵ n − 2 . In case of dimension n = 2, we have to add log ϵ times the integral of f / 2 π.

Asymptotic stability of the spectrum of a parametric family of homogenization problems associated with a perforated waveguide

AbstractIn this paper, we provide uniform bounds for convergence rates of the low frequencies of a parametric family of problems for the Laplace operator posed on a rectangular perforated domain of the plane of height H. The perforations are periodically placed along the ordinate axis at a distance between them, where ε is a parameter that converges toward zero. Another parameter η, the Floquet‐parameter, ranges in the interval . The boundary conditions are quasi‐periodicity conditions on the lateral sides of the rectangle and Neumann over the rest. We obtain precise bounds for convergence rates which are uniform on both parameters ε and η and strongly depend on H. As a model problem associated with a waveguide, one of the main difficulties in our analysis comes near the nodes of the limit dispersion curves.

Darcy’s Law for Porous Media with Multiple Microstructures

In this paper we study the homogenization of the Dirichlet problem for the Stokes equations in a perforated domain with multiple microstructures. First, under the assumption that the interface between subdomains is a union of Lipschitz surfaces, we show that the effective velocity and pressure are governed by a Darcy law, where the permeability matrix is piecewise constant. The key step is to prove that the effective pressure is continuous across the interface, using Tartar’s method of test functions. Secondly, we establish the sharp error estimates for the convergence of the velocity and pressure, assuming the interface satisfies certain smoothness and geometric conditions. This is achieved by constructing two correctors. One of them is used to correct the discontinuity of the two-scale approximation on the interface, while the other is used to correct the discrepancy between boundary values of the solution and its approximation.

Subspace decomposition based DNN algorithm for elliptic type multi-scale PDEs

While deep learning algorithms demonstrate a great potential in scientific computing, its application to multi-scale problems remains to be a big challenge. This is manifested by the “frequency principle” that neural networks tend to learn low frequency components first. Novel architectures such as multi-scale deep neural network (MscaleDNN) were proposed to alleviate this problem to some extent. In this paper, we construct a subspace decomposition based DNN (dubbed SD2NN) architecture for a class of multi-scale problems by combining MscaleDNN algorithms with traditional numerical analysis ideas. The proposed architecture includes one low frequency normal DNN submodule, and one (or a few) high frequency MscaleDNN submodule(s), which are designed to capture the smooth part and the oscillatory part of the multi-scale solutions simultaneously. We demonstrate that the SD2NN outperforms existing models such as MscaleDNN, through several benchmark multi-scale problems in regular or perforated domains.

Operator estimates for non-periodically perforated domains: disappearance of cavities

ABSTRACT We consider a boundary value problem for a general second-order linear equation in a perforated domain. The perforation is made by small cavities, a minimal distance between the cavities is also small. We impose minimal natural geometric conditions on the shapes of the cavities and no conditions on their distribution in the domain. On the boundaries of the cavities, a nonlinear Robin condition is imposed. The sizes of the cavities and the minimal distance between them are supposed to satisfy a certain simple condition ensuring that under the homogenization the cavities disappear and we obtain a similar problem in a non-perforated domain. Our main results state the convergence of the solution of the perturbed problem to that of the homogenized one in W 2 1 - and L 2 -norms uniformly in L 2 -norm of the right-hand side in the equation and provide the estimates for the convergence rates.

Operator estimates for the Neumann sieve problem

Let \(\Omega \) be a domain in \({\mathbb {R}}^n\), \(\Gamma \) be a hyperplane intersecting \(\Omega \), \(\varepsilon >0\) be a small parameter, and \(D_{k,\varepsilon }\), \(k=1,2,3\dots \) be a family of small “holes” in \(\Gamma \cap \Omega \); when \(\varepsilon \rightarrow 0\), the number of holes tends to infinity, while their diameters tends to zero. Let \({\mathscr {A}}_\varepsilon \) be the Neumann Laplacian in the perforated domain \(\Omega _\varepsilon =\Omega \setminus \Gamma _\varepsilon \), where \(\Gamma _\varepsilon =\Gamma \setminus (\cup _k D_{k,\varepsilon })\) (“sieve”). It is well-known that if the sizes of holes are carefully chosen, \({\mathscr {A}}_\varepsilon \) converges in the strong resolvent sense to the Laplacian on \(\Omega \setminus \Gamma \) subject to the so-called \(\delta '\)-conditions on \(\Gamma \cap \Omega \). In the current work we improve this result: under rather general assumptions on the shapes and locations of the holes we derive estimates on the rate of convergence in terms of \(L^2\rightarrow L^2\) and \(L^2\rightarrow H^1\) operator norms; in the latter case a special corrector is required.

Attractors of Ginzburg–Landau equations with oscillating terms in porous media: homogenization procedure

We consider the Ginzburg–Landau equation in the perforated domain, with rapidly oscillating coefficients. We derive the homogenized Ginzburg–Landau equation with a ‘strange term’ (potential) and prove that the trajectory attractors of the given equation tend in a weak sense to the trajectory attractors of the homogenized one. Assuming additional conditions to be satisfied for the coefficients, we provide also a convergence of the global attractor.

Stochastic homogenization on perforated domains III–General estimates for stationary ergodic random connected Lipschitz domains

<abstract><p>This is Part III of a series on the existence of uniformly bounded extension operators on randomly perforated domains in the context of homogenization theory. Recalling that randomly perforated domains are typically not John and hence extension is possible only from $ W^{1, p} $ to $ W^{1, r} $, $ r < p $, we will show that the existence of such extension operators can be guaranteed if the weighted expectations of four geometric characterizing parameters are bounded: The local Lipschitz constant $ M $, the local inverse Lipschitz radius $ \delta^{-1} $ resp. $ \rho^{-1} $, the mesoscopic Voronoi diameter $ {\mathfrak{d}} $ and the local connectivity radius $ {\mathscr{R}} $.</p></abstract>

Corrector results for a class of elliptic problems with nonlinear Robin conditions and $ L^1 $ data

<abstract><p>In this paper, we consider a class of elliptic problems in a periodically perforated domain with $ L^1 $ data and nonlinear Robin conditions on the boundary of the holes. Using the framework of renormalized solutions, which is well adapted to this situation, we show a convergence result for the truncated energy in the quasilinear case. When the operator is linear, we also prove a corrector result. Since we cannot expect to have solutions belonging to $ H^1 $, the main difficulty is to express the corrector result through the truncations of the solutions, together with the fact that the definition of a renormalized solution contains test functions which are nonlinear functions of the solution itself.</p></abstract>

Homogenization of a parabolic equation in a perforated domain with a unilateral dynamic boundary condition: critical case

In this paper, we study the homogenization of a parabolic equation given in a domain perforated by ''tiny'' balls. On the boundary of these perforations, a unilateral dynamic boundary constraints are specified. We address the so-called ''critical'' case that is characterized by a relation between the coefficient in the boundary condition, the period of the structure and the size of the holes. In this case, the homogenized equation contains a nonlocal ''strange'' term. This term is obtained as a solution of the variational problem involving ordinary differential operator.

A Perforated High-Order Element for Fracture Mechanics Problems Using the Hybrid Strain Method

In this paper, a novel hybrid approach with complex and conformal mapping functions is proposed for the first time. This strain-based approach is based on the special trial functions and the assumed natural strain (ANS) theory, which provides a unified framework for solving fracture problems. In the utilized method, the proposed super-element is divided into perforated and solid domains. For the perforated domain, Muskhelishvili–Kolosov potential function is employed to simulate the perforated discontinuity. For the solid domain, the higher-order strain field is assumed as the enriched interpolation functions. Moreover, the rigid body terms are imposed into the kinematic variables to enhance the convergence ability and the element performance. Numerical examples demonstrate that the proposed element (PHSM20) produces high precision results for both displacement and stresses.

Operator estimates for non-periodically perforated domains with Dirichlet and nonlinear Robin conditions: vanishing limit

We consider a general second order linear elliptic equation in a finely perforated domain. The shapes of cavities and their distribution in the domain are arbitrary and non-periodic; they are supposed to satisfy minimal natural geometric conditions. On the boundaries of the cavities we impose either the Dirichlet or a nonlinear Robin condition; the choice of the type of the boundary condition for each cavity is arbitrary. Then we suppose that for some cavities the nonlinear Robin condition is sign-definite in certain sense. Provided such cavities and ones with the Dirichlet condition are distributed rather densely in the domain and the characteristic sizes of the cavities and the minimal distances between the cavities satisfy certain simple condition, we show that a solution to our problem tends to zero as the perforation becomes finer. Our main result are order sharp estimates for the \(L_2\)- and \(W_2^1\)-norms of the solution uniform in the \(L_2\)-norm of the right hand side in the equation.

Homogenization of a mineral dissolution and precipitation model involving free boundaries at the micro scale

In this work we present the homogenization of a reaction-diffusion model that includes an evolving microstructure. Such type of problems model, for example, mineral dissolution and precipitation in a porous medium. In the initial state, the microscopic geometry is a periodically perforated domain, each perforation being a spherical solid grains. A small parameter ϵ is characterizing both the distance between two neighboring grains, and the radii of the grains. For each grain, the radius depends on the unknown (the solute concentration) at its surface. Therefore, the radii of the grains change in time and are model unknowns, so the model involves free boundaries at the micro scale. In a first step, we transform the evolving micro domain to a fixed, periodically domain. Using the Rothe-method, we prove the existence of a weak solution and obtain a priori estimates that are uniform with respect to ϵ. Finally, letting ϵ→0, we derive a macroscopic model, the solution of which approximates the micro-scale solution. For this, we use the method of two-scale convergence, and obtain strong compactness results enabling to pass to the limit in the nonlinear terms.

A phase‐field porous media fracture model based on homogenization theory

AbstractA novel regularized fracture model for crack propagation in porous media is proposed. Our model is obtained through formal asymptotic expansions. We start with a regularized quasi‐static fracture model posed in a periodically perforated domain obtained by periodic extension of a rescaled unit cell with a hole. This setup allows us to write two separated minimality conditions for the primary (displacement) and secondary (or phase field) variables plus a balance of energy relation. Then we apply the usual asymptotic expansion matching to deduce limit relations when the rescaling parameter of the unit cells vanishes. By introducing cell problems solutions and a homogenized tensor, we can recast the obtained relations into a novel model for crack propagation in porous media. The proposed model can be interpreted as a regularized quasi‐static fracture model for porous media. This model yields two separated (homogenized) minimality conditions for the primary and secondary variables and a balance of a homogenized energy relation.

Attractors of 2D Navier–Stokes system of equations in a locally periodic porous medium

This article deals with two-dimensional Navier–Stokes system of equations with rapidly oscillating term in the equations and boundary conditions. Studying the problem in a perforated domain, the authors set homogeneous Dirichlet condition on the outer boundary and the Fourier (Robin) condition on the boundary of the cavities. Under such assumptions it is proved that the trajectory attractors of this system converge in some weak topology to trajectory attractors of the homogenized Navier–Stokes system of equations with an additional potential and nontrivial right hand side in the domain without pores. For this aim, the approaches from the works of A.V. Babin, V.V. Chepyzhov, J.-L. Lions, R. Temam, M.I. Vishik concerning trajectory attractors of evolution equations and homogenization methods appeared at the end of the XX-th century are used. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) system of equations and prove the existence of trajectory attractors for this system. Lastly, we formulate the main theorem and prove it through axillary lemmas.

A homogenized limit for the 2-dimensional Euler equations in a perforated domain

We study the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size $a$ separated by distances $\tilde d$ and the fluid fills the exterior. We analyse the asymptotic behavior of the fluid when $(a,\tilde d) \to (0,0)$. If the inclusions are distributed on the unit square, this issue is studied recently when $\frac{\tilde d}a$ tends to zero or infinity, leaving aside the critical case where the volume fraction of the porous medium is below its possible maximal value but non-zero. In this paper, we provide the first result in this regime. In contrast with former results, we obtain an Euler type equation where a homogenized term appears in the elliptic problem relating the velocity and the vorticity. Our analysis is based on the so-called method of reflections whose convergence provides novel estimates on the solutions to the div-curl problem which is involved in the 2D-Euler equations.

Interaction of scales for a singularly perturbed degenerating nonlinear Robinproblem.

We study the asymptotic behaviour of solutions of a boundary value problem for the Laplace equation in a perforated domain in , , with a (nonlinear) Robin boundary condition on the boundary of the small hole. The problem we wish to consider degenerates in three respects: in the limit case, the Robin boundary condition may degenerate into a Neumann boundary condition, the Robin datum may tend to infinity, and the size ϵ of the small hole where we consider the Robin condition collapses to 0. We study how these three singularities interact and affect the asymptotic behaviour as ϵ tends to 0, and we represent the solution and its energy integral in terms of real analytic maps and known functions of the singular perturbation parameters.This article is part of the theme issue ‘Non-smooth variational problems and applications’.