Two-scale Asymptotic Analysis Research Articles

High-order models for convection–diffusion-reaction transport in multiscale porous media

Developing highly accurate models for predicting the convection–diffusion-reaction (CDR) transport in hierarchical porous media with strong heterogeneities on multiple scales is crucial but not yet available. In this work, an innovative high-order multiscale computational framework is developed to capture the local and global variation characteristics of flow fields and reactant concentration at multiple scales. The homogenized solutions and macro-meso high-order solutions are established by the formal two-scale asymptotic analysis. By directly expanding the mesoscopic cell functions to the microscopic levels, the three-scale high-order models are built by assembling the meso-micro high-order expansions of mesoscopic cell functions and macro-meso low/high-order models. The present approaches follow the reverse thought process of the reiteration homogenization method, and provide a very innovative way to develop highly accurate and efficient solutions for the CDR coupling problems in multiscale porous media. The effectiveness and accuracy of the proposed multiscale models are validated by several representative cases.

Homogenization of Elastomers Filled with Liquid Inclusions: The Small-Deformation Limit

This paper presents the derivation of the homogenized equations that describe the macroscopic mechanical response of elastomers filled with liquid inclusions in the setting of small quasistatic deformations. The derivation is carried out for materials with periodic microstructure by means of a two-scale asymptotic analysis. The focus is on the non-dissipative case when the elastomer is an elastic solid, the liquid making up the inclusions is an elastic fluid, the interfaces separating the solid elastomer from the liquid inclusions are elastic interfaces featuring an initial surface tension, and the inclusions are initially $n$ -spherical ( $n=2,3$ ) in shape. Remarkably, in spite of the presence of local residual stresses within the inclusions due to an initial surface tension at the interfaces, the macroscopic response of such filled elastomers turns out to be that of a linear elastic solid that is free of residual stresses and hence one that is simply characterized by an effective modulus of elasticity $\overline{{\mathbf{L}}}$ . What is more, in spite of the fact that the local moduli of elasticity in the bulk and the interfaces do not possess minor symmetries (due to the presence of residual stresses and the initial surface tension at the interfaces), the resulting effective modulus of elasticity $\overline{{\mathbf{L}}}$ does possess the standard minor symmetries of a conventional linear elastic solid, that is, $\overline{L}_{ijkl}=\overline{L}_{jikl}=\overline{L}_{ijlk}$ . As an illustrative application, numerical results are worked out and analyzed for the effective modulus of elasticity of isotropic suspensions of incompressible liquid 2-spherical inclusions of monodisperse size embedded in an isotropic incompressible elastomer.

High-order asymptotic solutions for gas transport in heterogeneous media with multiple spatial scales

Subsurface structures generally exhibit strong heterogeneities at multiple spatial scales. In this study, two- and three-scale high-order models are developed to effectively predict nonlinear gas transport in heterogeneous porous media with multiscale configurations. The considered heterogeneous porous media are composed of the matrix and inclusions at the mesoscopic and microscopic scales, respectively. First, using the formal two-scale asymptotic analysis, the homogenized solutions, the two-scale first-order and higher-order solutions are derived, with the first-order and second-order cell functions defined at the mesoscopic cell. Second, by further expanding all the mesoscopic cell functions to the microscopic levels, the second-order expansions of the mesoscopic cell functions are established and the upscaled relationships for the permeability tensor from the microscale to the macroscale are developed accordingly. Finally, the three-scale low-order and high-order solutions are constructed by combining the multiscale expansions of the mesoscopic cell functions and the macro–meso two-scale solutions. Several representative cases are simulated to demonstrate the accuracy and reliability of the proposed multiscale solutions. The results show that the high-order solutions can perfectly capture the locally steep pressure fluctuations and non-equilibrium effects caused by the heterogeneities and large permeability contrast in porous media with two- or three-scale configurations. The strategies to obtain the multiscale high-order solutions follow the reverse thought process of the reiteration homogenization method, and can be easily extended to heterogeneous porous media with arbitrary multiple scales.

A strain gradient brittle fracture model based on two-scale asymptotic analysis

Based on a two-scale asymptotic analysis procedure, we rigorously derive a new macroscopic brittle fracture model for materials with micro-cracks inside. Strain gradient effects are naturally involved in the model in the absence of any phenomenological assumptions. Coefficients of the fracture criterion are expressed in form of derivatives of the fourth- and sixth-order effective elastic tensors that are explicitly related to solutions of the first order unit cell problems. The model is numerically implemented into a finite element code. Abilities of the model to predict microstructure size effects and strain gradient effects on macro fracture behaviors are confirmed by numerical simulations. The model is also validated against experimental measurements.

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.

Droplet phase in a nonlocal isoperimetric problem under confinement

<p style='text-indent:20px;'>We address small volume-fraction asymptotic properties of a nonlocal isoperimetric functional with a confinement term, derived as the sharp interface limit of a variational model for self-assembly of diblock copolymers under confinement by nanoparticle inclusion. We introduce a small parameter <inline-formula><tex-math id="M1">\begin{document}$ \eta $\end{document}</tex-math></inline-formula> to represent the size of the domains of the minority phase, and study the resulting droplet regime as <inline-formula><tex-math id="M2">\begin{document}$ \eta\to 0 $\end{document}</tex-math></inline-formula>. By considering confinement densities which are spatially variable and attain a unique nondegenerate maximum, we present a two-scale asymptotic analysis wherein a separation of length scales is captured due to competition between the nonlocal repulsive and confining attractive effects in the energy. A key role is played by a parameter <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> which gives the total volume of the droplets at order <inline-formula><tex-math id="M4">\begin{document}$ \eta^3 $\end{document}</tex-math></inline-formula> and its relation to existence and non-existence of Gamow's Liquid Drop model on <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula>. For large values of <inline-formula><tex-math id="M6">\begin{document}$ M $\end{document}</tex-math></inline-formula>, the minority phase splits into several droplets at an intermediate scale <inline-formula><tex-math id="M7">\begin{document}$ \eta^{1/3} $\end{document}</tex-math></inline-formula>, while for small <inline-formula><tex-math id="M8">\begin{document}$ M $\end{document}</tex-math></inline-formula> minimizers form a single droplet converging to the maximum of the confinement density.

Homogenization of the vibro–acoustic transmission on perforated plates

The paper deals with modelling of acoustic waves which propagate in inviscid fluids interacting with perforated elastic plates. The plate can be replaced by an interface on which transmission conditions are derived by homogenization of a problem describing vibroacoustic fluid-structure interactions in a transmission layer in which the plate is embedded. The Reissner-Mindlin theory of plates is adopted for periodic perforations designed by arbitrary cylindrical holes with axes orthogonal to the plate midplane. The homogenized model of the vibroacoustic transmission is obtained using the two-scale asymptotic analysis with respect to the layer thickness which is proportional to the plate thickness and to the perforation period. The nonlocal, implicit transmission conditions involve a jump in the acoustic potential and its normal one-side derivatives across the interface which represents the plate with a given thickness. The homogenized model was implemented using the finite element method and validated using direct numerical simulations of the non-homogenized problem. Numerical illustrations of the vibroacoustic transmission are presented.

Scattering of elastic bulk waves by periodic arrays of voids or anisotropic inclusions

A two-scale asymptotic analysis coupled with the spatially periodic fundamental solutions is used for analyzing diffraction of elastic bulk waves propagating in anisotropic media containing periodic inclusions or voids. The explicit equations are derived for the scattering cross sections and velocities of bulk waves propagating in spatially periodic media with arbitrary elastic anisotropy.

Modelling resonant arrays of the Helmholtz type in the time domain.

We present a model based on a two-scale asymptotic analysis for resonant arrays of the Helmholtz type, with resonators open at a single extremity (standard resonators) or open at both extremities (double-sided resonators). The effective behaviour of such arrays is that of a homogeneous anisotropic slab replacing the cavity region, associated with transmission, or jump, conditions for the acoustic pressure and for the normal velocity across the region of the necks. The coefficients entering in the effective wave equation are simply related to the fraction of air in the periodic cell of the array. Those entering in the jump conditions are related to near field effects in the vicinity of the necks and they encapsulate the effects of their geometry. The effective problem, which accounts for the coupling of the resonators with the surrounding air, is written in the time domain which allows us to question the equation of energy conservation. This is of practical importance if the numerical implementations of the effective problem in the time domain is sought.

Elastic Waves in Spatially Periodic Anisotropic Media: Diffraction by Periodic Arrays of Voids or Anisotropic Inclusions

Abstract A two-scale asymptotic analysis coupled with the spatially periodic fundamental solutions are used for analyzing diffraction of elastic bulk waves propagating in anisotropic media containing periodic inclusions or voids. Explicit equations are derived for the scattering cross sections and velocities of bulk waves propagating in spatially periodic media with arbitrary elastic anisotropy.

Asymptotic finite fracture mechanics solution for crack onset at elliptical holes in composite plates of finite-width

The onset of cracks at elliptical holes in finite-width laminated plates under uniaxial tension loading is investigated using a finite fracture mechanics approach. A coupled stress and energy criterion is applied in order to determine the effective strength of the notched plate and the size of instantaneously initiating cracks. A semi-analytical approach based on complex potential theory in combination with a two-scale asymptotic analysis is proposed leading to explicit relations for the stress field and energy quantities due to crack formation. This allows for an efficient finite fracture mechanics solution and a significant reduction of the numerical effort compared to a full-numerical treatment. The effect of higher-order terms in the asymptotic procedure and a technique leading to an improved convergence of the expansions are discussed in detail. A thorough comparison with numerical data is performed in order to analyse the validity range of the asymptotic solution. Furthermore, the failure load predictions are compared to experimental data from literature.

Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -- efficient solution strategies based on homogenization theory

Boundary value problems on large periodic networks arise in many applications such as soil mechanics in geophysics or the analysis of photonic crystals in nanotechnology. As a model example, singularly perturbed elliptic differential equations of second order are addressed. Typically, the length of periodicity is very small compared to the size of the covered region. The overall complexity of the networks raises serious problems on the computational side. The high density of the graph, the huge number of edges and vertices and highly oscillating coefficients necessitate solution schemes, where even a numerical approximation is no longer feasible. Realizing that such a system depends on two spatial scales - global scale (full domain) and local scale (microstructure) - a two-scale asymptotic analysis for network differential equations is applied. The limit process leads to a homogenized model on the full domain. The homogenized coefficients cover the micro-oscillations and the topology of the periodic network and characterize the effective behaviour. The approximate model's quality is guaranteed by error estimates. Furthermore, singularly perturbed microscopic models with a decreasing diffusion part and transport-dominant problems are discussed. The effectiveness of the two-scale limit analysis is demonstrated by numerical examples of diffusion-advection-reaction problems on large periodic grids.

Second-order two-scale asymptotic analysis for axisymmetric and spherical symmetric structure with periodic configurations

A new second-order two-scale (SOTS) analysis finite element algorithm is developed for the axisymmetric and spherical symmetric elastic problems with small periodic configurations. The axisymmetric structure considered is periodic in both radial and axial directions and homogeneous in circumferential direction, and the spherical symmetric structure is only periodic in radial direction and homogeneous in other two directions. The SOTS asymptotic expansions for the space problem, plane axisymmetric problem, and spherical symmetric problem are presented, and the main feature is that the anisotropic material is obtained by the homogenization. The analytical expressions of the cell functions and homogenized solutions for plane axisymmetric and spherical symmetric problems are obtained, and the error estimations of the expansions are established. The second-order asymptotic analysis finite-element algorithm is presented and the numerical examples are solved including the hollow cylinder, rotating disk and hollow sphere composed of periodic composite materials. The computational results demonstrate the effectiveness and accuracy of the SOTS asymptotic analysis algorithm, and the converging behavior of the asymptotic analysis algorithm agrees well with the theoretical prediction. It is also indicated that the stress distributions can be correctly computed only by adding the second-order correctors.

Eigenstrain based reduced order homogenization for polycrystalline materials

Abstract In this manuscript, an eigenstrain based reduced order homogenization method is developed for polycrystalline materials. A two-scale asymptotic analysis is used to decompose the original equations of polycrystal plasticity into micro- and macroscale problems. Eigenstrain based representation of the inelastic response field is employed to approximate the microscale boundary value problem using an approximation basis of much smaller order. The reduced order model takes into account the grain-to-grain interactions through influence functions that are numerically computed over the polycrystalline microstructure. The proposed approach is also endowed with a hierarchical model improvement capability that allows accurate representation of stress and deformation state within subgrains. The proposed approach was implemented and its performance was assessed against crystal plasticity finite element simulations. Numerical studies point to the capability to efficiently compute the mechanical response of the polycrystal RVEs with good accuracy and the ability to capture stress risers near grain boundaries.

Field behavior near the edge of a microstrip antenna by the method of matched asymptotic expansions

The cavity model is a wide-spread powerful empirical approach for the numerical simulation of microstrip antennas. It is based on several hypotheses assumed a priori: a dimension reduction in the cavity, that is, the zone limited by a metallic patch and the ground plane in which is fed the antenna, supplied by the additional condition that the open sides of the cavity act as magnetic walls. An additional important assumption of this model consists in an adequate description of the singular field behavior in the proximity of the edge of the patch. A simplified two-dimensional problem incorporating the main features of the field behavior near the edge of the patch and inside the cavity is addressed. The method of matched asymptotic expansions is used to carry out a two-scale asymptotic analysis of the field relatively to the thickness of the cavity. All the empirical hypotheses at the basis of the derivation of the cavity model can thus be recovered. Proved error estimates are given in a simplified framework where the dielectric constants of the substrate are assumed to be 1 in order to avoid some unimportant technical difficulties.

A Two-Scale Asymptotic Analysis of a Time-Harmonic Scattering Problem with a Multi Layered Thin Periodic Domain

A Two-Scale Asymptotic Analysis of a Time-Harmonic Scattering Problem with a Multi Layered Thin Periodic Domain

Two-scale asymptotic analysis for modeling activation of periodic cardiac strand

This work explores the applicability of two-scale asymptotic analysis to model the excitable cardiac strand with periodic discontinuities. The strand is composed of 60 cells connected by junctions. The membrane excitability is modeled using Ebihara-Johnson equations. The problem is described by a non-linear parabolic equation with one parameter, intracellular conductivity, changing periodically in space. Using two-scale asymptotic analysis, the transmembrane potential is given as a two-scale expansion in powers of the period length. The series converges rapidly, and the solution containing only zero- and first-order terms has a negligible error. Each term of the expansion is found by solving the differential equation derived by decomposing the original problem. The periodic, first-order term is given by a linear elliptic equation and has a closed-form solution. The aperiodic, zero-order term is the solution of a non-linear parabolic equation corresponding to the cardiac strand with homogenized intracellular conductivity; this equation is solved numerically using the Crank-Nicolson method modified to account for the presence of the periodic term. The use of the two-scale asymptotic analysis permits a substantial saving in computer resources over the solution from a purely numerical approach: the problem can be solved with a three times coarser discretization step, requires ten times less memory, and uses about half the computation time.

Potential distribution in three-dimensional periodic myocardium. I. Solution with two-scale asymptotic analysis

The use of two-scale asymptotic analysis allows development of a model of the steady-state potential distribution in three-dimensional cardiac muscle preserving the underlying cellular network. The myocardium is modeled as a periodic structure consisting of cylindrical cells embedded in extracellular fluid and connected by longitudinal and side junctions. The method is applicable to cardiac muscle of arbitrary extent since the periodicity of the tissue is dealt with analytically, and thus numerical computations require no more resources than a continuous volume conductor problem. The asymptotic analysis approach reveals that the potential in a periodic myocardium consists of two components. The large-scale component provides the baseline for the total solution and can be determined from the anisotropic monodomain model associated with the original periodic problem. The method provides the formula for calculating the conductivity of the equivalent monodomain model on the basis of cell geometry and conductivity distribution in the cardiac tissue. The small-scale component reflects the periodicity of the underlying structure and oscillates with periods determined by the dimensions of cardiac cells. The magnitude of these oscillations depends upon the gradient of the large-scale component. During stimulation with extracellular electrodes, the small-scale component determines both the shape and the magnitude of the transmembrane potential, while the influence of the large-scale component is negligible. Hence, the small-scale component merits closer attention in pacing and defibrillation studies, especially since the model based on two-scale asymptotic analysis provides an effective means of its computation.

Modelling the periodicity of cardiac muscle

The intractable problem of modelling cardiac muscle of arbitrary extent while preserving cellular structure has been solved using an analytical rather than numerical approach with a method called two-scale asymptotic analysis. In this method, the myocardium was modelled as a collection of bundles arranged periodically in space and connected by junctions, and the distribution of the steady-state potential and current density was determined. The potential both along and across fibers was found to contain a distinct periodic component that determines the transmembrane potential. The magnitude of the transmembrane potential depends on the gradient of applied potential, the dimensions of the bundles, and their internal conductivity. Current flows primarily in the extracellular space, and the extracellular pathway also determines the apparent conductivity of cardiac muscle