Narrow Results

Let $\mathrm{\Gamma }\subset {ℝ}^d$ be a lattice. For $𝜀>0$, we consider the perforated space ${\mathrm{\Pi }}_{𝜀}\subset {ℝ}^d$ which is an $(𝜀\mathrm{\Gamma })$-periodic open connected set with Lipschitz boundary. In $L_2({\mathrm{\Pi }}_{𝜀};{ℂ}^n)$, we consider a self-adjoint strongly elliptic second-order differential operator ${{𝒜}}_{𝜀}$ with periodic coefficients depending on $x/𝜀$. We study the behavior of the resolvent ${({{𝒜}}_{𝜀}+I)}^{-1}$ for small $𝜀$. Approximations for this resolvent in the $(L_2\to L_2)$ and $(L_2\to H^1)$-operator norms with sharp order error estimates are found. The results are obtained by the operator-theoretic (spectral) approach. General results are applied to particular periodic operators of mathematical physics: the acoustics operator, the elasticity operator, and the Schrödinger operator with a singular potential.

Motivated by the two-scale convergence method introduced by G. Nguetseng and G. Allaire, we study the polarization and relaxation effects induced by two-scale homogenization. By virtue of Boltzmann–Poisson system, we characterize the microscopic properties of electrons in dielectric materials. The homogenized equations are obtained which describe the mean behaviors. By the homogenized equation, we get the polarization and relaxation effects which lead to the modified Gauss law; moreover, the dielectric function and the conductivity are also obtained.

The focus of this paper is on aligned fiber-reinforced composites, where fiber centers were randomly distributed in their cross-sections. The volume fractions of fibers were $V_f=30$% and $V_f=50$%. Samples were built with the help of the simulated annealing technique according to the chosen Radial Distribution Functions (RDFs). For each sample, the fields of local stresses and heat fluxes were simulated by finite element method. Then, homogenization by volume averaging was performed in order to investigate both the effective mechanical and thermal properties. The effect of RDF shape on elastic and thermal properties was quantified along with the influence of the probability of near neighbors of fibers on the physical properties. The more the fiber distributions deviate from Poisson’s Law, the higher the results compared to the lower bound of Hashin–Shtrikman.

In this study, creep behavior of a CFRP laminate subjected to a constant stress is analyzed based on the time-dependent homogenization theory developed by the present authors. The laminate is a unidirectional carbon fiber/epoxy laminate T800H/#3631 manufactured by Toray Industries, Inc. Two kinds of creep analyses are performed. First, 45° off-axis creep deformation of the laminate at high temperature (100°C) is analyzed with three kinds of creep stress levels, respectively. It is shown that the present theory accurately predicts macroscopic creep behavior of the unidirectional CFRP laminate observed in experiments. Then, high temperature creep deformations at a constant creep stress are simulated with seven kinds of off-axis angles, i.e., θ = 0°, 10°, 30°, 45°, 60°, 75°, 90°. It is shown that the laminate has marked in-plane anisotropy with respect to the creep behavior.

In this study, multi-scale creep analysis of plain-woven GFRP laminates is performed using the time-dependent homogenization theory developed by the present authors. First, point-symmetry of internal structures of plain-woven laminates is utilized for a boundary condition of unit cell problems, reducing the domain of analysis to 1/4 and 1/8 for in-phase and out-of-phase laminate configurations, respectively. The time-dependent homogenization theory is then reconstructed for these domains of analysis. Using the present method, in-plane creep behavior of plain-woven glass fiber/epoxy laminates subjected to a constant stress is analyzed. The results are summarized as follows: (1) The in-plane creep behavior of the plain-woven GFRP laminates exhibits marked anisotropy. (2) The laminate configurations considerably affect the creep behavior of the laminates.

The fully implicit incremental homogenization scheme developed by Asada and Ohno (2007) for elastoplastic periodic solids is applied to two-scale analysis of honeycomb blocks subjected to flat punch indentation. To this end, the scheme is rebuilt by introducing half unit cells based on the point-symmetric distributions of stress and strain in unit cells, so that analysis domains in unit cells are reduced by half. Then, by assuming the zigzag and armchair types of cell-arrangements, the two-scale analysis of honeycomb blocks is performed. The corresponding full-scale finite element analysis is also performed to reveal the cell-arrangement dependence of cell deformation in the honeycomb blocks. It is shown that the two-scale analysis is macroscopically successful in spite of microscopic limitations.

As an important class of natural and engineered materials, periodic microstructural composites have drawn substantial attention from the material research community for their excellent flexibility in tailoring various desirable physical behaviors. To develop periodic cellular composites for multifunctional applications, this paper presents a unified design framework for combining stiffness and a range of physical properties governed by quasi-harmonic partial differential equations. A multiphase microstructural configuration is sought within a periodic base-cell design domain using topology optimization. To deal with conflicting properties, e.g. conductivity/permeability versus bulk modulus, the optimum is sought in a Pareto sense. Illustrative examples demonstrate the capability of the presented procedure for the design of multiphysical composites and tissue scaffolds.

In the current stage, left-handed material (LHM) has aroused wide interest based on several basic prototypes such as metamaterials. In this work, we have theoretically investigated the possible existence of granular LHM by using the Bruggeman formalism. The Wiener bounds and the Hashin–Shtrikman bounds were invoked and some examples were represented. We had found that when the dissipations are increased adequately, the estimates of the Bruggeman formalism will lie within the two bounds rigorously, even if the two permittivities have real parts of opposite signs. Thus, one particulate LHM may be achieved by choosing the two component mediums with negative permittivity and negative permeability, respectively.

This paper deals with the homogenization of a linear hyperbolic stochastic partial differential equation (SPDE) with highly oscillating periodic coefficients. We use Tartar’s method of oscillating test functions and deep probabilistic compactness results due to Prokhorov and Skorokhod. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized linear hyperbolic SPDE with constant coefficients. We also prove the convergence of the associated energies.

This paper describes a data-driven approach to predict mechanical properties of auxetic kirigami metamaterials with randomly oriented cuts. The finite element method (FEM) was used to generate datasets, the convolutional neural network (CNN) was introduced to train these data, and an implicit mapping between the input orientations of cuts and the output Young’s modulus and Poisson’s ratio of the kirigami sheets was established. With this input–output relationship in hand, a quick estimation of auxetic behavior of kirigami metamaterials is straightforward. Our examples indicate that if the distributions of training and test datasets are close to each other, a good prediction is achievable. Our efforts provide a fast and reliable way to evaluate the homogenized properties of mechanical metamaterials with various microstructures, and thus accelerate the design of mechanical metamaterials for diverse applications.

We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation [Shroff & Keller, 1993]. The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Padé approximations.

The homogenization of the Dirac-like system is studied. It generates memory effects. The memory (or nonlocal) kernel is described by the Volterra integral equation. When the coefficient is independent of time, the memory kernel can be characterized explicitly in terms of Young's measure. The homogenized equation can be reformulated in the kinetic form by introducing the kinetic variable. We also characterize the memory kernel when the coefficient is of separable variable.

We characterize the functionals which are Mosco-limits, in the L²(Ω) topology, of some sequence of functionals of the kind

In this paper, the existence of solution for a coupled system of variational inequalities with highly oscillating coefficients modelling a micro-elastohydrodynamic lubrication problem is stated. Moreover, by means of two-scale convergence techniques, the limit coupled models when the frequency of oscillation tends to zero are established for different amplitude-frequency rates. The effect of roughness appears to be limited to one of the coupled nonlinear equations, either the elastic one or the hydrodynamic one. Finally, some numerical examples to illustrate both the qualitative behavior of the solution and the theoretical convergence results are presented.

This paper deals with boundary value problems and spectral problems for neutron transport equations involving locally periodic (in space) collision frequencies and collision operators. We show strong convergence results to the solution of homogenized problems when the period goes to zero. The mathematical analysis relies mainly on smoothing effects of velocity averages.

This paper deals with an analysis of a nonlinear monotone conduction problem posed on a medium which we assume to be anisotropic and periodically reinforced by thin fibers also assumed to be anisotropic. We consider the case where the conductivity in the fibers are more important than in the material which surrounds it so that the operator we have to consider loses his uniform ellipticity with respect to the size ε of the period; we then investigate the effect of the anisotropy in the homogenization process.

In this paper, we shall study systems governed by the Neumann problem of second-order elliptic equation with rapidly oscillating coefficients and with control and observations on the boundary. The multiscale asymptotic expansions of the solution for considering problem in the case without any constraints, and homogenized equation in the case with constraints will be given, their rigorous proofs will also be proposed.

We study the homogenization limit of the Mullins–Sekerka problem which serves as a model for late-stage coarsening in a phase transformation. We consider the case that the screening length which describes the effective range of particle interactions is much smaller than the system size, which leads to homogenization problems in unbounded domains.

The present paper deals with deterministic initial particle distributions which are in an average sense homogeneously distributed. Stochastic particle distributions will be considered in a second paper (Part II).

We study a problem set in a finely mixed periodic medium, modelling electrical conduction in biological tissues. The unknown electric potential solves standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by a dielectric surface (the cell membranes), which exhibits both a capacitive and a nonlinear conductive behaviour. Accordingly, dynamical conditions prevail on the membranes, so that the dependence of the solution on the time variable t is not only of parametric character.

As the spatial period of the medium goes to zero, the electric potential approaches in a suitable sense a homogenization limit u₀, which keeps the prescribed boundary data, and solves the equation . This is an elliptic equation containing a term depending on the history of the gradient of u₀; the matrices B⁰, A¹ in it depend on the microstructure of the medium. More exactly, we have that, in the limit, the current is still divergence-free, but it depends on the history of the potential gradient, so that memory effects explicitly appear. The limiting equation also contains a term ℱ keeping trace of the initial data.

The bio-heat transfer equation is a macroscopic model for describing the heat transfer in microvascular tissue. In Ref. 8 the authors applied homogenization techniques to derive the bio-heat transfer equation as asymptotic result of boundary value problems which provide a microscopic description for microvascular tissue. Here those results are generalized to a geometrical setting where the regions of blood are allowed to be connected, which covers more biologically relevant geometries. Moreover, asymptotic corrector results are derived under weaker assumptions.