Two-scale Asymptotic Expansion Method Research Articles

Longitudinal waves in the walls of an annular channel filled with liquid and made of a material with fractional nonlinearity

Purpose of this paper is to study the evolution of longitudinal strain waves in the walls of an annular channel filled with a viscous incompressible fluid. The walls of the channel were represented as coaxial shells with fractional physical nonlinearity. The viscosity of the fluid and its influence on the wave process was taken into account within the study. Metods. The system of two evolutionary equations, which are generalized Schamel equations, was obtained by the two-scale asymptotic expansion method. The fractional nonlinearity of the channel wall material leads to the necessity to use a computational experiment to study the wave dynamics in them. The computational experiment was conducted based on obtaining new difference schemes for the governing equations. These schemes are analogous to the Crank–Nicholson scheme for modeling heat propagation. Results. Numerical simulation showed that over time, the velocity and amplitude of the deformation waves remain unchanged, and the wave propagation direction concurs with the positive direction of the longitudinal axis. The latter specifies that the velocity of the waves is supersonic. For a particular case, the coincidence of the computational experiment with the exact solution is shown. This substantiates the adequacy of the proposed difference scheme for the generalized Schamel equations. In addition, it was shown that solitary deformation waves in the channel walls are solitons.

Two-scale and three-scale asymptotic computations of the Neumann-type eigenvalue problems for hierarchically perforated materials

A top-down strategy is proposed for analyzing the elliptic eigenvalue problems of the hierarchically perforated materials with three-scale periodic configurations. The heterogeneous structure considered is composed of perforated cells in the mesoscopic scale and composite cells in the microscopic scale, and Neumann boundary conditions are imposed on the boundaries of the cavities. By using the classical two-scale asymptotic expansion method, the homogenized eigenfunctions and eigenvalues are obtained and the first- and second-order auxiliary cell functions are defined firstly in the mesoscale. Then, the two-scale asymptotic analysis is furtherly applied to the mesoscopic cell problems and by expanding the meso cell functions to the second-order terms, the homogenized cell functions are derived and the relations between the homogenized coefficients and the coefficients of constituent materials in the three scale levels are established. Finally, the second-order three-scale asymptotic approximations of the eigenfunctions are presented and by the idea of “corrector equation”, the three-scale expressions of the eigenvalues are obtained. The corresponding finite element algorithm is established and the successively up-scaling procedures are established. Typical two-dimensional numerical examples are performed, and both the two-scale and three-scale computed approximations of the eigenvalues are compared with the ones obtained in the classical computation. By the least squares technique, it is demonstrated that the three-scale asymptotic solutions of the eigenfunctions are good approximations of the original eigensolutions corresponding to both the simple and multiple eigenvalues. This study offers an alternative approach to describe the physical and mechanical behaviors of the hierarchically structures with more than two scales and it is indicated that the second-order terms plays an important role not only in the derivation of the expansions but also in the practical computations to capture the local oscillations within the cells.

A two-scale asymptotic expansion method for periodic composite Euler beams

This paper develops a two-scale asymptotic expansion solution method for periodic composite Euler beams for the first time. In this method, a two-scale solution in the asymptotic expansion form is achieved for the fourth-order uniformly elliptic differential equation with periodic oscillating coefficients, and the first-order perturbed displacement is zero. The analytical solutions of unit cell problems are found with the help of four normalization conditions proposed in this work, and it follows that the homogenized elastic modulus of periodic composite beam is exactly the harmonic mean of material moduli. Another contribution of this work is to mathematically verify the convergence of the two-scale asymptotic expansion solution through the two-scale convergence method. In addition, the paper reveals the effects of higher-order expansion terms on the two-scale displacements and concludes that the second order or even higher-order perturbed displacements are necessary for obtaining accurate curvatures relevant to micro stresses. Finally, numerical experiments validate that the present method is rigorous in mathematics and physically acceptable.

Effective piezoelectric coefficients of cement-based 2–2 type piezoelectric composites based on a multiscale homogenization model

The effective piezoelectric properties of the cement-based 2–2 type piezoelectric composites have been investigated both experimentally and theoretically. The two-scale asymptotic expansion method is employed to develop a homogenized model for the effective properties calculation. The validity of the theoretical solution is confirmed through the comparison with the experimental results. The influence of the volume fraction of the piezoelectric phase on the effective piezoelectric coefficients is then examined. It is found that higher volume fraction will induce obvious increment of the magnitude of the effective piezoelectric strain coefficients d31Eff, d32Eff, and d33Eff, however, the hydrostatic piezoelectric strain coefficient dhEff will reach a maximum value at a lower volume fraction.

Acoustics of sorptive porous materials

Sound propagation in conventional rigid-frame porous materials is affected by the viscosity of the saturating gas and the heat transfer between the gas and the solid frame. This paper investigates the acoustical properties of hierarchical porous materials that support mass transfer processes, such as sorption and different types of diffusion, in addition to the previously mentioned effects. The two-scale asymptotic expansion method of homogenisation for periodic media is used to derive the macroscopic acoustic description. This description allowed to conclude that, at the leading order, neither sorption nor diffusion affects the macroscopic fluid flow through the material. However, the effective compressibility is significantly modified by these physical phenomena. Specifically, the real part of the low-frequency bulk modulus can take values much smaller than those found in conventional materials while its imaginary part is increased around the characteristic frequency associated with diffusion. As a consequence, the sound waves are slowed down and more attenuated at low frequencies. The results are exemplified for sorptive porous, fibrous, and granular materials and the influence of their microstructural descriptors on their effective acoustical properties is discussed.

A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures

In this paper we make a comparison between the two-scale asymptotic expansion method for periodic homogenization and the so-called Bloch wave method. It is well-known that the homogenized tensor coincides with the Hessian matrix of the first Bloch eigenvalue when the Bloch parameter vanishes. In the context of the two-scale asymptotic expansion method, there is the notion of high order homogenized equation (Bakhvalov and Panasenko in Homogenization: averaging processes in periodic media. Kluwer, Dordrecht, 1989) where the homogenized equation can be improved by adding small additional higher order differential terms. The next non-zero high order term is a fourth-order term, accounting for dispersion effects (see e.g. Santosa and Symes in SIAM J Appl Math 51:984–1005, 1991; Lamacz in Math Models Methods Appl Sci 21(9):1871–1899, 2011; Dohnal et al. Multiscale Model Simul 12(2):488–513, 2014). Surprisingly, this homogenized fourth-order tensor is not equal to the fourth-order tensor arising in the Taylor expansion of the first Bloch eigenvalue, which is often called Burnett tensor. Here, we establish an exact relation between the homogenized fourth-order tensor and the Burnett fourth-order tensor. It was proved in Conca et al. (J Math Phys 47(3):11, 2006) that the Burnett fourth-order tensor has a sign. For the special case of a simple laminate we prove that the homogenized fourth-order tensor may change sign. In the elliptic case we explain the difference between the homogenized and Burnett fourth-order tensors by a difference in the source term which features an additional corrector term. Finally, for the wave equation, the two fourth-order tensors coincide again, so dispersion is unambiguously defined, and only the source terms differ as in the elliptic case.

Upscaling of an immiscible non-equilibrium two-phase flow in double porosity media

We study an immiscible incompressible two-phase, such as water–oil flow through a -periodic double porosity media. The mesoscopic model consists of equations derived from the mass conservation laws of both fluids, along with a generalized Darcy law in the framework of Kondaurov’s non-equilibrium flow model. The mobility functions as well as the capillary pressure function for each component of the porous medium are the functions of the saturation and an additional non-equilibrium parameter, which satisfies a kinetic equation coming from the definition of the Helmholtz free energy. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the permeability ratio of matrix blocks to fracture planes is of order . Using the method of two-scale asymptotic expansions, we derive the macroscopic model of the flow which is written in terms of the homogenized phase pressures, saturation, and the non-equilibrium parameter. For small relaxation times, we compare our model with the global models obtained earlier by H. Salimi and J. Bruining. We show the novelty of our macroscopic double porosity flow model.

In-plane vibration of rectangular plates with periodic inhomogeneity: Natural frequencies and their adjustment

We consider the in-plane free vibration of a rectangular plate with material properties varying periodically and rapidly in the plate plane. To this end, the two-scale asymptotic expansion method is employed to establish a homogenized model of the original heterogeneous plate, which is then analyzed by the state-space method. For a plate with each repetitive unit composed of two different materials at a fixed volume ratio, we compare the natural frequencies of the homogenized model with those of the original plate calculated by the finite element method, and good agreement is obtained. We also demonstrate that the plate can be designed to have particular frequencies through a proper selection of the material distribution law in each unit.

Two-scale analytical solutions of multilayered composite rectangular plates with in-plane small periodic structure

Abstract We present a two-dimensional analytical solution of an orthotropic multilayered rectangular plate, which, in addition, has a small periodic structure along one in-plane direction. The two-scale asymptotic expansion method is first employed to develop a homogenized model of each individual layer in the plate. The laminated plate with homogenized layers is then analyzed exactly using the state-space approach. Two sets of approximate stresses (i.e. the homogenized model stresses and the zeroth-order two-scale model stresses) are presented analytically. Numerical examples are considered and comparison with the finite element simulation is made. It is shown that the two-scale model can give accurate predictions of the transverse normal stress even it varies sharply in a unit cell. It is also confirmed that the developed analytical solution can be used to investigate the effect of microstructure on the macroscopic behavior of the multilayered composite plate. This work presents an effort to take both advantages of the two-scale asymptotic expansion method and the state space method in the analyses of laminated composite structures with in-plane small periodicity.

Homogenization of a Conductive, Convective, and Radiative Heat Transfer Problem in a Heterogeneous Domain

We are interested in the homogenization of heat transfer in periodic porous media where the fluid part is made of long thin parallel cylinders, the diameter of which is of the same order as the period. The heat is transported by conduction in the solid part of the domain and by conduction, convection, and radiative transfer in the fluid part (the cylinders). A nonlocal boundary condition models the radiative heat transfer on the cylinder walls. To obtain the homogenized problem we first use a formal two-scale asymptotic expansion method. The resulting effective model is a convection-diffusion equation posed in a homogeneous domain with homogenized coefficients evaluated by solving so-called cell problems where radiative transfer is taken into account. In a second step we rigorously justify the homogenization process by using the notion of two-scale convergence. One feature of this work is that it combines homogenization with a three dimensional (3D) to two dimensional (2D) asymptotic analysis since the ra...

Frequency estimate and adjustment of composite beams with small periodicity

It is the first time for us to investigate the estimation and adjustment of the natural frequencies for a composite beam with small periodicity. An approach is presented in which the two-scale asymptotic expansion method is used to establish a homogenized model of the original heterogeneous beam. The accuracy of the homogenized model to predict the natural frequencies is checked by comparing with a fine finite element simulation of the original beam. In the case that each repetitive unit is composed of two different materials and the volume ratio of each material is given, the maximum and minimum frequencies of the composite beam are determined. Then it is shown that the beam can be designed to have particular frequencies within these frequency limits through a proper selection of the material distribution law in the unit. A concrete example is given when the material distribution function is assumed to be piecewise linear.

Analytical solutions of heterogeneous rectangular plates with transverse small periodicity

We consider the cylindrical bending of a simply-supported orthotropic rectangular plate, which is small-periodically heterogeneous in the thickness direction. A homogenized plate model is first established by using the two-scale asymptotic expansion method. The state-space method is then adopted to analyze the homogenized plate exactly. Analytical expressions for two sets of approximate stresses, i.e. the homogenized model stresses and the zeroth-order two-scale model stresses, are presented. To check the accuracy of the approximate stresses, the state-space method is also applied to the original laminate plate or the approximate laminate model of the original heterogeneous plate with continuously varying material properties to obtain an analytical solution. In the latter case, the analytical solution is approximate but approaches the exact solution gradually when the number of layers increases. In order to avoid numerical instability, the joint coupling matrix is utilized. Numerical results illustrate that the two-scale model can predict accurately the realistic stress field in the original plate if it contains enough repeated units along the thickness.

Asymptotic approximations for solutions to quasilinear and linear parabolic problems with different perturbed boundary conditions in perforated domains

We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ω e that is e-periodically perforated by small holes of order . The holes are divided into three e-periodical sets depending on boundary conditions. The homogeneous Dirichlet boundary conditions are imposed for holes of one set, whereas, for holes in the remaining sets, different inhomogeneous Neumann and nonlinear Robin boundary conditions involving additional perturbation parameters are imposed. For a solution to the quasilinear problem we find the leading terms of the asymptotic expansion and prove asymptotic estimates that show the influence of perturbation parameters. In the linear case, we construct and justify a complete asymptotic expansion of the solution by using the two-scale asymptotic expansion method. Bibliography: 25 titles. Illustrations: 1 figure.

Dispersion of elastic waves in microinhomogeneous media and structures

The propagation of elastic waves in periodic stratified media with arbitrary local anisotropy and in anisotropic plates and bars inhomogeneous in thickness is considered under the condition that the ratio of the scale characterizing the inhomogeneity of the medium or the thickness of a plate or bar in thickness to the typical wavelength is small. The propagation of long waves is described using the effective averaged equations with high-order accuracy, which are derived by the method of two-scale asymptotic expansions in ɛ. The results of analytic and numerical studies of their principal terms responsible for the dispersion of waves are presented. The form of the dependence of the wave velocity on the wavelength is studied for structures with different types of symmetry.

Homogenization of a Conductive and Radiative Heat Transfer Problem

This paper is devoted to the homogenization of a heat conduction problem in a periodically perforated domain with a nonlinear and nonlocal boundary condition modeling radiative heat transfer in the perforations. Because of the critical scaling considered it is essential to use a method of two-scale asymptotic expansions inside the variational formulation of the problem. We obtain a nonlinear homogenized problem of heat conduction with effective coefficients which are computed via a cell problem featuring a radiative heat transfer boundary condition. We rigorously justify this homogenization process for the linearized problem by using two-scale convergence. We perform numerical simulations in two dimensions: we reconstruct an approximate temperature field by adding to the homogenized temperature a corrector term. The computed numerical errors agree with the theoretical predicted errors and prove the effectiveness of our method for multiscale simulation of conductive and radiative heat transfer problems in periodically perforated domains.

Multiscale modelling of thermal conductivity in composite materials for cryogenic structures

A multiscale analysis is performed to estimate overall thermal conductivity in a column-type support post used in particle accelerators to sustain cryomagnets. A non-linear homogenisation of thermal conductivity in a unidirectional composite material including phonon mechanism is firstly proposed by using the two-scale asymptotic expansion method. Next, computed conductivity curves are used for the modelling of heat transfer in a global representative volume element of the braided fabric composite. Thermal tests are conducted to measure conductivity of the material. A good agreement is obtained between estimated and measured values.

Microstructure-based evaluation of the influence of woven architecture on permeability by asymptotic homogenization theory

A microstructure-based computational approach is taken to predict the permeability tensor of woven fabric composites, which is a key parameter in resin transfer moulding (RTM) simulation of polymer-matrix composites. An asymptotic homogenization theory is employed to evaluate the permeability from both macro- and microscopic standpoints with the help of the finite-element method (FEM). This theory allows us to study the relation between microscopic woven architecture and macroscopic permeability based on the method of two-scale asymptotic expansions. While the fluid velocity is introduced for Stokes flow microscopically, the macroscopic one is for seepage flow with the Darcy's law. The latter can be characterized for arbitrary configurations of unit microstructures that are analyzed for the former under the assumption of the periodicity. After discussing the validity of this approach, we present a typical numerical example to discuss the permeability characteristics of plain weave fabrics undergoing shear deformation in the preforming in comparison with the undeformed one. Another notable feature of the proposed method is that the correlation between the macroscopic behaviors and the microscopic ones can be analyzed, which is important to analyze and/or design the RTM process. Hence, it is also demonstrated that the microscopic velocity field evaluated with macroscopic pressure gradient provides important information about the flow in RTM processes.

Higher-order effective modeling of periodic heterogeneous beams. I. Asymptotic expansion method

This paper is concerned with the elastostatic behavior of heterogeneous beams with a cross-section and elastic moduli varying periodically along the beam axis. By using the two-scale asymptotic expansion method, the interior solution is formally derived up to an arbitrary desired order. In particular, this method is shown to constitute a systematic way of improving Bernoulli's theory by including higher-order terms, without any assumption, in contrast to Timoshenko's theory or other refined beam models. Moreover, the incompatibility between the interior asymptotic expansions and the real boundary conditions is emphasized, and the necessity of a specific treatment of edge effects is thus underlined.

Flow of electrolyte through porous piezoelectric medium: macroscopic equations

The aim of this contribution is to elaborate a general framework for modelling flows of two-ionic species electrolytes through porous piezoelectric media. By using the method of two-scale asymptotic expansions, the macroscopic phenomenological equations describing electrokinetics of such a structure are derived and the formulae for the effective mechanical and nonmechanical coefficients are given. Natural jump conditions are assumed on the interfaces between the piezoelectric skeleton and conductive fluid.

Macroscopic equations for nonstationary flow of Stokesian fluid through porous elastic medium

The aim of this contribution is to apply homogenization methods in order to describe the nonstationary flow of a viscous fluid through a microperiodic porous elastic medium. By using the method of two-scale asymptotic expansions, the macroscopic phenomenological equations describing such a two-phase structure are derived and the formulae for the effective mechanical coefficients are given. The asymptotic approach is justified by the two-scale convergence