Perforated Domains Research Articles

Existence results for a nonlinear nonautonomous transmission problem via domain perturbation

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $\phi$. First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $\phi$.

Uniform estimates for Stokes equations in a domain with a small hole and applications in homogenization problems

We consider the Dirichlet problem of the Stokes equations in a domain with a shrinking hole in $${\mathbb {R}}^d, \ d\ge 2$$ . A typical observation is that, the Lipschitz norm of the domain goes to infinity as the size of the hole goes to zero. Thus, if $$p\ne 2$$ , the classical results indicate that the $$W^{1,p}$$ estimate of the solution may go to infinity as the size of the hole tends to zero. With the presence of the shrinking hole in a fixed domain, we give a complete description for the uniform $$W^{1,p}$$ estimates of the solution for all $$1<p<\infty $$ . We show that the uniform $$W^{1,p}$$ estimate holds if and only if $$d'<p<d$$ ( $$p=2$$ when $$d=2$$ ). We then give two applications in the study of homogenization problems in fluid mechanics: a generalization of the restriction operator and a construction of Bogovskii type operator in perforated domains with a quantitative estimate of the operator norm.

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.

Darcy’s law as low Mach and homogenization limit of a compressible fluid in perforated domains

We consider the homogenization limit of the compressible barotropic Navier–Stokes equations in a three-dimensional domain perforated by periodically distributed identical particles. We study the regime of particle sizes and distances such that the volume fraction of particles tends to zero but their resistance density tends to infinity. Assuming that the Mach number is decreasing with a certain rate, the rescaled velocity and pressure of the microscopic system converges to the solution of an effective equation which is given by Darcy’s law. The range of sizes of particles we consider is exactly the same which leads to Darcy’s law in the homogenization limit of incompressible fluids. Unlike previous results for the Darcy regime we estimate the deficit related to the pressure approximation via the Bogovskiĭ operator. This allows for more flexible estimates of the pressure in Lebesgue and Sobolev spaces and allows to proof convergence results for all barotropic exponents [Formula: see text].

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.

Multiscale asymptotic analysis and computations for Steklov eigenvalue problem in periodically perforated domain

A second‐order asymptotic analysis method is developed for the Steklov eigenvalue problem in periodically perforated domain. By the two‐scale expansions of the eigenfunctions and eigenvalues, the first‐ and second‐order cell functions defined on the representative cell are obtained successively, the homogenized elliptic eigenvalue problem is formulated, and effective coefficients are derived. The first‐ and second‐order correctors of the eigenvalues are expressed in terms of the integrations of the cell functions and homogenized eigenfunctions. The error estimations of the expansions of eigenvalues are established, and the corresponding finite element algorithm is proposed. Numerical examples are carried out, and both qualitative and quantitative comparisons with the solutions by classical finite element computations are performed. It is demonstrated that this asymptotic model is effective to capture the local details of the eigenfunctions by considering the second‐order expansion terms, and the algorithm presented in this work is efficient to obtain the accurate spectral properties of the porous material at lower cost.

The effect of algorithm parameters on the number of iterations of the dual fat boundary method

The Dual Fat Boundary Method (DFBM) is an improvement of the Fat Boundary Method (FBM), which is proposed to solve elasticity problems defined on a perforated domain. The DFBM does not need an analytical solution in the holes, so it is suitable for any reasonable hole shapes and Dirichlet boundary conditions on the holes. However, as an iteration method, the relationship between iteration times and algorithmic parameters has not been studied. Here we give a numerical study by taking the perforated infinite plate problem as an example. It reveals that the relaxation parameters for two local domains should be equal to obtain the fastest convergence. Furthermore, the number of iterations for DFBM is smaller than FBM in the larger range of relaxation parameters despite of the decreasing range of convergence.

Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is 8pi k, which is the best upper bound for the k^{text {th}} area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For k=1, the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

English

The goal of this paper is to establish a general homogenization result for linearized elasticity of an eigenvalue problem defined over perforated domains, beyond the periodic setting, within the framework of the H0-convergence theory. Our main homogenization result states that the knowledge of the fourth-order tensor A0, the H0-limit of Ae, is sufficient to determine the homogenized eigenvalue problem and preserve the structure of the spectrum. This theorem is proved essentially by using Tartar’s method of test functions, and some general arguments of spectral analysis used in the literature on the homogenization of eigenvalue problems. Moreover, we give a result on a particular case of a simple eigenvalue of the homogenized problem. We conclude our work by some comments and perspectives.

Appearance of a Nonlocal Monotone Operator in Transmission Conditions when Homogenizing the Poisson Equation in a Domain Perforated Along an (n − 1)-Dimensional Manifold by Sets of Arbitrary Shape and Critical Size with Nonlinear Dynamic Boundary Conditions on the Boundary of Perforations

We construct and justify an efficient model arising when homogenizing the Poisson equation in an e-periodically perforated domain along an (n − 1)-dimensional manifold by sets of an arbitrary shape and critical size on the boundary of which we impose a nonlinear dynamic condition containing an absorption coefficient of the form e−k, where k takes the critical value (n − 1)/(n − 2), n ≥ 3. We show that the transmission conditions on the manifold contain a nonlocal monotone operator.

Convergence rates for linear elasticity systems on perforated domains

In the present work, we established almost-sharp error estimates for linear elasticity systems in periodically perforated domains. The first result was $$L^{\frac{2d}{d-1-\tau }}$$ -error estimates $$O\big (\varepsilon ^{1-\frac{\tau }{2}}\big )$$ for all $$\tau \in (0,1)$$ in a bounded smooth domain, which is new even for homogenization problems on unperforated domains. It followed from weighted Hardy–Sobolev’s inequalities (given by Lehrback and Vahakangas in J Funct Anal 271(2):330–364, 2016) and a suboptimal error estimate for the square function of the first-order approximating corrector (earliest investigated by Kenig et al. in Arch Ration Mech Anal 203(3):1009–1036, 2012) under additional regularity assumption on coefficient). The new approach relied on the weighted quenched Calderon–Zygmund estimate (initially appeared in Gloria et al. work Milan J Math 88(1):99–170, 2020 for a quantitative stochastic homogenization theory). The second effort was $$L^2$$ -error estimates $$O\big (\varepsilon ^{\frac{5}{6}} \ln ^{\frac{2}{3}}(1/\varepsilon )\big )$$ for a Lipschitz domain, followed from a duality scheme coupled with interpolation inequalities. Also, we developed a new weighted extension theorem and local-type Sobolev–Poincare inequalities on perforated domains. Throughout the paper, we do not impose any smoothness assumption on the coefficients.

On effects of perforated domains on parameter-dependent free vibration

Free vibration characteristics of thin perforated shells of revolution vary depending not only on the dimensionless thickness of the shell but also on the perforation structure. All holes are assumed to be free, that is, without any kinematical constraints. For a given configuration there exists a critical value of the dimensionless thickness below which homogenisation fails, since the modes do not have corresponding counterparts in the non-perforated reference shell. For a regular g×g-perforation pattern, the critical thickness is reached when the lowest mode has an angular wave number of g∕2. This observation is supported both by geometric arguments and numerical experiments. The numerical experiments have been computed in 2D with high-order finite element method supporting Pitkäranta’s mathematical shell model.

On the property of homothetic mean value on periodically perforated domains

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.

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

In this paper, we aim to study the asymptotic behavior (when varepsilon ;rightarrow ; 0) of the solution of a quasilinear problem of the form -mathrm{{div}};(A^{varepsilon }(cdot ,u^{varepsilon }) nabla u^{varepsilon })=f given in a perforated domain Omega backslash T_{varepsilon } with a Neumann boundary condition on the holes T_{varepsilon } and a Dirichlet boundary condition on partial Omega . 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) 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) in the perforated domain. Once the H^{0}-limit A^{0}(cdot ,d) of the pair (A^{varepsilon },T^{varepsilon }) is determined, in the second step, we deduce that the solution u^{varepsilon } converges in some sense to the unique solution u^{0} in H^{1}_{0}(Omega ) of the quasilinear equation -mathrm{{div}};(A^{0}(cdot ,u^{0})nabla u )=chi ^{0}f (where chi ^{0} is L^{infty } 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.

A justification of the Darcy law for a suspension of not self-similar solid particles non-periodically distributed

We extend to a non-periodic framework the classical homogenization result permitting to derive the Darcy law from the Stokes or Navier–Stokes system posed in a perforated domain. Mathematically, we study the asymptotic behavior when ɛ tends to zero of the stationary Stokes system posed in a sequence of varying domains Ωɛ=Ω∖∪k∈NTɛk where Ω is a smooth bounded open subset of R3 and Tɛk are closed sets such that each of them is at a distance of order ɛ of the remaining. Moreover ∪k∈NTɛk is at a distance of order at most ɛ of any point of R3. Each set Tɛk is non-empty, smooth and has a size of order δɛ with ɛ3<<δɛ≤rɛ for some r<1. In the classical periodic case, the sets Tɛk are obtained by repeating periodically with period ɛ the set δɛT with T a non-empty smooth closed set in R3. As in this periodic case we show that the limit problem corresponds to a Darcy system. However, even when the sets Tɛk have all the same shape, we show that for δɛ<<ɛ some strong convergence results for the velocity and some capacity formulae for the limit system do not extend from the periodic framework to the non-periodic one.

Homogenization of random convolution energies

We prove a homogenization theorem for a class of quadratic convolution energies with random coefficients. Under suitably stated hypotheses of ergodicity and stationarity we prove that the $\Gamma$-limit of such energy is almost surely a deterministic quadratic Dirichlet-type integral functional, whose integrand can be characterized through an asymptotic formula. The proof of this characterization relies on results on the asymptotic behaviour of subadditive processes. The proof of the limit theorem uses a blow-up technique common for local energies, that can be extended to this `asymptotically-local' case. As a particular application we derive a homogenization theorem on random perforated domains.

Wavelet-based Edge Multiscale Finite Element Method for Helmholtz problems in perforated domains

We introduce a new efficient algorithm for Helmholtz problems in perforated domains with the design of the scheme allowing for possibly large wavenumbers. Our method is based upon the wavelet-based...

A Strange Vertex Condition Coming from Nowhere

We prove norm-resolvent and spectral convergence in $L^2$ of solutions to the Neumann Poisson problem $-\Delta u_\varepsilon = f$ on a domain $\Omega_\varepsilon$ perforated by Dirichlet-holes and shrinking to a 1-dimensional interval. The limit $u$ satisfies an equation of the type $-u''+\mu u = f$ on the interval $(0,1)$, where $\mu$ is a positive constant. As an application we study the convergence of solutions in perforated graph-like domains. We show that if the scaling between the edge neighbourhood and the vertex neighbourhood is chosen correctly, the constant $\mu$ will appear in the vertex condition of the limit problem. In particular, this implies that the spectrum of the resulting quantum graph is altered in a controlled way by the perforation.

Convergence of the CEM-GMsFEM for Stokes flows in heterogeneous perforated domains

In this paper, we consider the incompressible Stokes flow problem in a perforated domain and employ the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to solve this problem. The proposed method provides a flexible and systematical approach to construct crucial divergence-free multiscale basis functions for approximating the velocity field. These basis functions are constructed by solving a class of local energy minimization problems over the eigenspaces that contain local information on the heterogeneities. These multiscale basis functions are shown to have the property of exponential decay outside the corresponding local oversampling regions. By adopting the technique of oversampling, the spectral convergence of the method with error bounds related to the coarse mesh size is proved.

Global representation and multiscale expansion for the Dirichlet problem in a domain with a small hole close to the boundary

For each pair of positive parameters, we define a perforated domain by making a small hole of size in an open regular subset of (). The hole is situated at distance from the outer boundary of the domain. Thus, when both the size of the hole and its distance from tend to zero, but the size shrinks faster than the distance. Next, we consider a Dirichlet problem for the Laplace equation in the perforated domain and we denote its solution by Our aim is to represent the map that takes to in terms of real analytic functions of defined in a neighborhood of (0, 0). In contrast with previous results valid only for restrictions of to suitable subsets of we prove a global representation formula that holds on the whole of Such a formula allows us to rigorously justify multiscale expansions, which we subsequently construct.