Two-scale Asymptotic Method Research Articles

An efficient high-order asymptotic approach for the elastodynamic homogenization of periodic composites

An efficient asymptotic approach, referred to as the Effective Non-Local Asymptotic Model (ENLAM), is developed for the elastodynamic homogenization of periodic composites. In comparison with the relevant two-scale asymptotic method reported in the literature, the effective motion equation obtained by ENLAM has a very compact form which holds for any order of asymptotic approximation. This compact form makes direct use of the effective displacement vector and of its different gradients whose highest order depends on the accuracy degree required. For every wavenumber belonging to the first Brillouin zone, the Floquet-Bloch wave expansion is employed, and compact recursive formulae are derived for determining the effective material parameters of periodic composites. For simplicity, all the results are presented only for a shear wave propagating in a periodic composite. Numerical examples for discussing the effective material parameters, the dispersion relation and the asymptotic approximation error are provided when a periodic composite with inclusions is concerned.

Emission and propagation of sound waves in porous media with active inner heat sources

Wave propagation through porous media with active inner heat sources is investigated in this paper. Through the use of the two-scale asymptotic method of homogenisation, it is found that a macroscopic non-homogeneous wave equation describes the emission and propagation of sound waves in such active porous media. The upscaled model is validated numerically for the cases of single and double porosity media, and shows that the general properties of the effective parameters of the porous media are not altered by the inner heat sources. Instead, the inner heat sources contribute to the non-homogeneous term in the upscaled wave equation. The paper also explores the potential of using active inner heat sources for controlling sound waves incident to the porous media in practical scenarios.

On two elastodynamic homogenization methods for periodic composites

Two-scale asymptotic method and Willis’ method are jointly examined for the elastodynamic homogenization of periodic composites. The effective elastodynamic constitutive law given by Willis’ method, non-local both in time and space, is first reformulated so as to make appear a compact expression for the effective impedance tensor characterizing it. The two-scale asymptotic method is then revisited by exploiting the fact that the results obtained by it constitute an approximation to the general ones delivered by Willis’ method. The solutions for the hierarchical motion equations issuing from asymptotic analysis are shown to admit a compact recursive representation. A generic compact expression is derived for the effective impedance tensor associated with the nth-order approximation of asymptotic analysis. Remarkably, this expression turns out to be formally identical to the one of the effective impedance tensor obtained via Willis’ method. As an example of illustration, the elastodynamic homogenization of a layered composite undergoing anti-plane shear is investigated in detail.

Design and fused filament fabrication of multilayered microchannels for subwavelength and broadband sound absorption

A comprehensive procedure to design and manufacture multilayered microchannels presenting competitive mechanical properties, effective subwavelength and near perfect broadband sound absorption in targeted frequency ranges is presented. The acoustic properties of microchannels are predicted with the Johnson-Champoux-Allard-Lafarge (JCAL) model. The JCAL parameters are calculated with the Two-scale Asymptotic Method (TAM) and the sound absorption coefficient of multilayered microchannels is simulated with the transfer matrix method (TMM). The simplex Nelder-Mead optimization method is used to find the size of the channels and the stacking sequence leading to effective acoustic absorption for three different frequency ranges (500–2400 Hz, 2400–6500 Hz and 500–6500 Hz). 30 mm-thick samples with up to 30-layers of unobstructed and non interconnected channels and microchannels were successfully produced via fused filament fabrication (FFF). The minimum channel size is ~ 100 µm which is very appropriate to produce micro-perforated panels or acoustic liners with optimal absorption and different degrees of freedom. Multilayered microchannels with absorption average up to 0.87 and noise reduction coefficient (NRC) up to 0.49 were produced. The multilayered microchannels offer a good compromise between effective acoustic properties and useful mechanical properties compared to other 3D printed acoustic structures and can be considered as viable candidates for applications where structural resistance is required. • Multilayered microchannels with high broadband sound absorption at three frequency ranges are presented. • Innovative FFF manufacturing technique allows production of 100 μm perforations and structure with up to 30-layer. • 30mm-thick multilayered microchannels present sound absorption average up to 0.87. • PLA 3D printed multilayered microchannels exhibit up to 468MPa compressive modulus and 9MPa compressive yield strength.

Equivalent fluid approach to modeling the acoustical properties of polydisperse heterogeneous porous composites

This paper investigates sound propagation in polydisperse heterogeneous porous composites. The two-scale asymptotic method of homogenization is used to obtain a macroscopic description of the propagation of sound in such composites. The upscaled equations demonstrate that the studied composites can be modeled as equivalent fluids with complex-valued frequency-dependent effective parameters (i.e., dynamic viscous permeability and compressibility) as well as unravel the sound energy dissipation mechanisms involved. The upscaled theory is both exemplified by introducing analytical and hybrid models for the acoustical properties of porous composites with different geometries and constituent materials (e.g., a porous matrix with much less permeable and/or impervious inclusions with simple or complex shapes) and validated through computational experiments successfully. It is concluded that the developed theory rigorously captures the physics of acoustic wave propagation in polydisperse heterogeneous porous composites and shows that the mechanisms that contribute to the dissipation of sound energy in the composite are classical visco-thermal dissipation together with multiple pressure diffusion phenomena in the heterogeneous inclusions. The results show that the combination of two or more permeable materials with highly contrasted permeabilities can improve the acoustic absorption and transmission loss of the composite. This paper provides fundamental insights into the propagation of acoustic waves in complex composites that are expected to guide the rational design of novel acoustic materials.

Bifurcations of self-similar solutions for reversing interfaces in the slow diffusion equation with strong absorption

Bifurcations of self-similar solutions for reversing interfaces are studied in the slow diffusion equation with strong absorption. The self-similar solutions bifurcate from the time-independent solutions for standing interfaces. We show that such bifurcations occur at particular points in parameter space (characterizing the exponents in the diffusion and absorption terms) where the confluent hypergeometric functions satisfying Kummer’s differential equation truncate to finite polynomials. A two-scale asymptotic method is employed to obtain the local dependencies of the self-similar reversing interfaces near the bifurcation points. The asymptotic results are shown to be in excellent agreement with numerical approximations of the self-similar solutions.

Multi-scale asymptotic analysis and computation of the elliptic eigenvalue problems in curvilinear coordinates

A novel second-order two-scale asymptotic method is presented for the eigenvalue problems of the second-order elliptic operator in the general composite domain. The eigenvalue equation is firstly reformulated in curvilinear coordinates with periodic configuration using proper coordinate transformation, and by applying the asymptotic expansion technique, the eigenfunctions of the system are expanded to the second-order terms. Using the argument of the so-called “corrector equations”, the eigenvalues are expressed in terms of the homogenized eigenfunctions and the cell functions are defined in the representative cell domain. The feature of the proposed model is that some homogenized material coefficients and all the microscopic cell functions are dependent on the macroscopic coordinates. Various reduced expressions of the eigenfunctions and eigenvalues are discussed under specific coordinate transformations, and the conditions that the cell functions and homogenized coefficients are decoupled from the macroscopic coordinates are elaborated. The finite element algorithm is developed and three numerical experiments are carried out, which demonstrate the effectiveness of our proposed method in simulating and predicting the vibration behavior of the composite structures. It is also indicated that the second-order correctors are of necessity to capture the locally oscillating behavior within a periodicity of the eigenfunctions. By the coordinate transformation, the asymptotic analysis method can be generalized to more general composite domain with quasi-periodic and non-periodic configurations.

Acoustics of permeable heterogeneous materials with local non-equilibrium pressure states

The key idea developed in this work is the enforcement of local non-equilibrium pressure states in permeable materials by means of introducing geometrical and/or material heterogeneities. The two-scale asymptotic method of homogenisation is used to derive the macroscopic equations that describe sound propagation in the investigated class of materials. This allowed us to conclude that, at the leading order, the macroscopic fluid flow is mostly determined by that occurring in the most permeable fluid network. In contrast, the effective compressibility of the saturating fluid is modified by the non-equilibrium pressure states occurring in the different much less permeable local heterogeneities of the materials. The theory is exemplified by introducing an analytical model for the acoustical properties of a perforated microporous matrix with cylindrical microporous inclusions co-axially inserted in the perforations. The experimental validation of the theory is also provided.

Acoustics of multiscale sorptive porous materials

This paper investigates sound propagation in multiscale rigid-frame porous materials that support mass transfer processes, such as sorption and different types of diffusion, in addition to the usual visco-thermo-inertial interactions. The two-scale asymptotic method of homogenization for periodic media is successively used to derive the macroscopic equations describing sound propagation through the material. This allowed us to conclude that the macroscopic mass balance is significantly modified by sorption, inter-scale (micro- to/from nanopore scales) mass diffusion, and inter-scale (pore to/from micro- and nanopore scales) pressure diffusion. This modification is accounted for by the dynamic compressibility of the effective saturating fluid that presents atypical properties that lead to slower speed of sound and higher sound attenuation, particularly at low frequencies. In contrast, it is shown that the physical processes occurring at the micro-nano-scale do not affect the macroscopic fluid flow through the material. The developed theory is exemplified by introducing an analytical model for multiscale sorptive granular materials, which is experimentally validated by comparing its predictions with acoustic measurements on granular activated carbons. Furthermore, we provide empirical evidence supporting an alternative method for measuring sorption and mass diffusion properties of multiscale sorptive materials using sound waves.

Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern.

Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces thus inspiring applications across a wide range of domains including structural engineering, architectural design and aerospace engineering. The present paper suggests a homogenization-type two-scale asymptotic method which, combined with standard tools from differential geometry of surfaces, yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin-jointed trusses. The outcome of the method is a set of nonlinear differential equations governing the parametrization, metric and curvature of surfaces that the initially discrete structure can fit. The theory is presented through a case study of a fairly generic example: the eggbox pattern. The proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated.

Two-Scale Algorithm of Simulating the Approximation for the Piezoelectric Problem in Small Periodic Composites

The piezoelectric material plays an important role in some special fields. Based on two-scale asymptotic method, the two-scale finite element computational algorithm of simulating the piezoelectric problem in periodic composite is given, the finite element approximate errors are analyzed, and some numerical examples are presented so as to show the effectiveness of the numerical algorithm.

Analytical and numerical analysis of lensing effect for linear surface water waves through a square array of nearly touching rigid square cylinders

This paper describes transport properties of linear water waves propagating within a square array of fixed square cylinders. The main focus is on achieving the conditions for all-angle-negative-refraction (AANR) thanks to anomalous dispersion in fluid-filled periodic structures. Of particular interest are two limit cases when either the edges or the vertices of the cylinders come close to touching. In the former case, the array can be approximated by a lattice of thin water channels (for which dispersion curves are given in closed form and thus frequencies at which AANR occurs) whereas in the latter case, the array behaves as a checkerboard with cells consisting either of water tanks or rigid cylinders (for which standing modes are given in closed form). The tools of choice for the present analysis are, on the one hand, the finite element method which solves numerically spectral problems in periodic media, and on the other hand, a two-scale asymptotic method which provides estimates of dispersion curves and associated eigenfields through a lattice approximation (namely thin water channels between rigid cylinders). Simple duality correspondences are found based on fourfold symmetry of square water checkerboards that allow us to get some insight into their spectra. Last, some numerical evidence is provided for water waves focusing with no astigmatism through such arrays, when they are of finite extent.

A reiterated two-scale probabilistic modeling and numerical simulation for heat equation in a fractal porous media

In this paper, we shall discuss the steady heat equation in fractal structure of a composite porous media. A reiterated two-scale asymptotic method for a deterministic problem with many different length scales is given. For a stationary random structure of a composite porous media, a reiterated two-scale probabilistic modeling is presented. Finally, some numerical results will be reported. © 2005 Wiley Periodicals, Inc. Heat Trans Asian Res, 34(3): 188–196, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20053

Iterated two-scale asymptotic method and numerical algorithm for the elastic structures of composite materials

In this paper, we shall discuss the elasto-static problem of composite materials with two or more different length scales, and shall present iterated two-scale asymptotic expansion of the solution for considering problem. The mathematical proofs of some convergence theorems will be given under some assumptions. Finally, some numerical results are reported.

On nonlinear heat equations and diffusion in porous media

The problem of diffusion of a Brownian particle in a liquid under the influence of an exterior field, taking into account thermal effects, as given by Streater [1–3], is reformulated for the case in which the thermal conductivity coefficient is nonhomogeneous. This modified formulation is applied to the description of diffusion in a liquid filling a porous medium. Macroscopic equations of Brownian diffusion for that case are obtained using the two-scale asymptotic method. As a result, in the macroscopic diffusion equation the friction term is absent, and the external force appears explicitly as a heat production term only; implicitly the drift force is found in the first correction f (1) in the asymptotic expansion of distribution function f.

Homogenized thermoelastic model for a beam of a periodic structure

In this paper, the transition from a 3-D thermoelasticity problem to a 1-D beam model is made on the basis of a two-scale asymptotic method. A non-homogeneous beam of arbitrary periodic structure is considered.

Homogenization and effective properties of plates weakened by partially penetrating fissures: Asymptotic analysis

A method of finding effective properties of elastic plates weakened by periodically distributed part-through-the thickness fissures is presented. The plate behaviour is described by the two-dimensional two-layer plate model. The model involves four kinematical fields standing for: the in-plane displacements of the interface plane, the independent rotations of the cross-sections of the lower and upper layers of the plate and for the common deflection of both layers. Within the framework of this model the bending mode of fissuring is considered. In contrast to the approaches of our previous papers, which were based on the Kirchhoff [ J. Elasticity 19 (1988)] and the Reissner-like [ Arch. Mech. 40 (1988)] plate models, the present approach enables one to describe the opening and closure of the fissures without the undesired effect of possible interpenetration of the adherent faces of the fissures. The homogenized formulae are derived by applying the two-scale asymptotic method. The effective plate is described by the two-layer plate model that turns out to be non-linear and hyperelastic. The hyperelastic potential involves the solution of the basic cell problem. This problem assumes here the form of a variational inequality posed on a rescaled cell of periodicity. Having found the effective two-layer plate model one can construct the effective model of Kirchhoff type. Also this model turns out to be non-linear and hyperelastic.