Periodic Homogenization Technique Research Articles

A bidomain model for the calcium dynamics in living cells

The aim of this paper is to analyze, via periodic homogenization techniques, the effective behavior of a non-linear system of coupled reaction-diffusion equations appearing in the modeling of calcium dynamics in living cells under the action of buffering proteins. We obtain, at the macroscale, a calcium bidomain model governing the evolution of the concentration of the calcium ions and of the buffers in the cytoplasm.

Investigation of the effect of morphological and crystallographic textures on the ductility limits of thin metal sheets using a CPFEM-based approach

The current contribution investigates the effect of some relevant microstructural parameters (specifically, morphological and crystallographic textures) on the ductility limits of polycrystalline aggregates using the Crystal Plasticity Finite Element Method (CPFEM). The polycrystalline aggregates are assumed to be representative of thin metal sheets and their macroscopic behavior is determined from that of their constituent single crystals on the basis of the periodic homogenization technique. The single crystal behavior is described by a finite strain elastoplastic framework in which the plastic flow rule obeys the classical Schmid law and plastic deformation is solely attributed to the slip on the crystallographic slip systems. The CPFEM is implemented within and in connection with ABAQUS/Standard finite element code. The ductility limits are predicted by the Rice bifurcation theory where strain localization is detected when the macroscopic acoustic tensor becomes singular. Three grain morphologies (namely, cube, random, and elongated morphology) and three initial crystallographic textures (namely, cube, random, and copper orientation) are considered to investigate the effect of morphological and crystallographic textures on the onset of plastic strain localization. The numerical results indicate that the effect of initial crystallographic texture is much more pronounced than that of grain morphology on the predicted ductility limits. In addition, the impact of grain size and sheet thickness are thoroughly analyzed. The research reveals that the trends of the predicted ductility limits are strongly dependent on the size effects.

Homogenized models of conduction-advection-radiation heat transfer in porous media

Upscaled models are derived from the periodic homogenization technique for conduction-convection-radiation problem in porous media for the case of an opaque solid and transparent fluid. Thermal radiation at the pore-scale level is first described through a differential view factor between elementary surfaces. Such heat transfer mode is coupled with the conduction-convection mechanisms by a radiative flux in the fluid phase and a jump of conductive fluxes at the solid/fluid interface. In the framework of the periodic homogenization procedure, a macroscopic model is built with upscaled effective coefficients. Both cases of closed and open pores are studied. In the second case the radiative transfer cannot be restricted to the central unit cell. The radiation at the macroscale only affects the effective conductivity. The validity of the approach is illustrated by some simple examples. The main objective of this work is to establish the macroscopic equation form of the conduction-convection-radiation problem and to understand how the radiation affects the macroscopic behavior.

Semi-numerical micromechanical model for viscoelastic microcracked masonry

This paper proposes a semi-numerical model providing accurate estimates for the mechanical behaviour of viscoelastic microcracked masonry. It relies on two steps. The first one approximates the creep behaviour of the microcracked mortar by coupling - in the Laplace-Carson symbolic space - the Griffith's theory to the Ponte-Castañeda & Willis (PCW) micromechanical scheme accounting for microcracks interactions and spatial distribution. The PCW model is an accurate alternative to the dilute and Mori-Tanaka model frequently used to homogenize microcracked media. Accumulated damage is assumed to follow a power-lawtime evolution according to the Weibull's failure rate function. In the second step, by contrary to the periodic homogenization technique adopted previously in (Rekik, 2020) which approaches the masonry behaviour, finite elements method is used to model the complete geometry of the masonry and compute accurately (up to the numerical error) its local behaviour. The sensitivity of the proposed model to the mesh refinement is validated by comparison to analytical solution. It is worth noting that the proposed model accounts for creep of both brick and mortar constituents following the Generalized Maxwell rheological model. As a first illustration, the proposed model is applied to a compressed masonry wall. For the sake of simplicity, only mortar joints are assumed to be microcracked with frictionless open cracks. Comparisons between panels with elastic and viscoelastic bricks show the additional softening effect induced by the brick's creep increasing thus the risk of failure.

Coupling inverse fin method with infrared thermography to determine the effective thermal conductivity of extruded thermoplastic foams

An inverse method for determining the in-plane effective thermal conductivity of porous thermoplastics was implemented by coupling infrared thermography experiments and numerical simulation of heat transfer in straight fins having temperature-dependent convective heat transfer coefficient. The microstructure heterogeneity of extruded polyethylene foam, in which pores are filled with air with different levels of open and closed porosity, was taken into account. The obtained effective thermal conductivity values were compared with previous results obtained using a numerical solution based on periodic homogenization techniques (NSHT) and the transient plane source technique (TPS) to verify the accuracy of the proposed method. The results show that the suggested method is in good agreement with both NSHT and TPS. Moreover, it is also appropriate for structural materials such as unidirectional fiber-reinforced plastic composites, where heat transfer is very different according to the fiber direction (parallel or transverse to the fibers).

Numerical prediction of elastic properties for alumina green parts printed by stereolithography process

Stereolithography is a process based on the photopolymerization of a UV-reactive system consisting of ceramic particles dispersion in a curable resin. A key issue of this process is the control of the rigidity of green parts, which are strongly related to UV light exposure. This work is focused on the numerical prediction of green part stiffness according to stereolithography manufacturing parameters. A first macroscopic approach, based on the modelling of ceramic suspension polymerization, makes it possible to establish a relationship between the exposure and the Young's modulus. A second microscopic approach, using a periodic homogenization technique based on the strain energy, is applied to a 2D finite element model to evaluate the effective elastic properties. Numerical results show that macroscopic model is able to provide a Young’s modulus with a good level of accuracy. The modelling results from the microscopic model demonstrate an acceptable convergence with the experimental Young’s modulus.

Upscaling of evolving microstructures in a porous medium

Crystal dissolution and precipitation problems often originate in the fields of material sciences, chemical engineering, soil mechanics, hydrology, and other related fields. In this paper, we consider dissolution and precipitation of immobile species (minerals) on the surface of the solid parts and diffusion and reaction of mobile species in the pore space of a porous medium. The mathematical modeling of such processes leads to a system of semilinear parabolic partial differential equations coupled with a system of nonlinear ordinary differential equations, which involves multi-valued right-hand sides. The geometry under consideration here is evolving due to dissolution and precipitation on the interfaces. This moving interface problem is dealt with the help of a level set function. The pore (micro) scale model is upscaled to the macroscale by periodic homogenization and asymptotic expansion techniques.

Effective balance equations for poroelastic composites

We derive the quasi-static governing equations for the macroscale behaviour of a linear elastic porous composite comprising a matrix interacting with inclusions and/or fibres, and an incompressible Newtonian fluid flowing in the pores. We assume that the size of the pores (the microscale) is comparable with the distance between adjacent subphases and is much smaller than the size of the whole domain (the macroscale). We then decouple spatial scales embracing the asymptotic (periodic) homogenization technique to derive the new macroscale model by upscaling the fluid–structure interaction problem between the elastic constituents and the fluid phase. The resulting system of partial differential equations is of poroelastic type and encodes the properties of the microstructure in the coefficients of the model, which are to be computed by solving appropriate cell problems which reflect the complexity of the given microstructure. The model reduces to the limit case of simple composites when there are no pores, and standard Biot’s poroelasticity whenever only the matrix–fluid interaction is considered. We further prove rigorous properties of the coefficients, namely (a) major and minor symmetries of the effective elasticity tensor, (b) positive definiteness of the resulting Biot’s modulus, and (c) analytical identities which allow us to define an effective Biot’s coefficient. This model is applicable when the interactions between multiple solid phases occur at the porescale, as in the case of various systems such as biological aggregates, constructs, bone, tendons, as well as rocks and soil.

Numerical investigation of necking in perforated sheets using the periodic homogenization approach

Due to their attractive properties, perforated sheets are increasingly used in a number of industrial applications, such as automotive, architecture, pollution control, etc. Consequently, the accurate modeling of the mechanical behavior of this kind of sheets still remains a valuable goal to reach. This paper aims to contribute to this effort by developing reliable numerical tools capable of predicting the occurrence of necking in perforated sheets. These tools are based on the coupling between the periodic homogenization technique and three plastic instability criteria. The periodic homogenization technique is used to derive equivalent macroscopic mechanical behavior for a representative volume element of these sheets. On the other hand, the prediction of plastic instability is based on three necking criteria: the maximum force criterion (diffuse necking), the general bifurcation criterion (diffuse necking), and the loss of ellipticity criterion (localized necking). The predictions obtained by applying the three instability criteria are thoroughly analyzed and compared. A sensitivity study is also conducted to numerically investigate the influence on the prediction of necking of the design parameters (dimension, aspect-ratio, orientation, and shape of the holes), the macroscopic boundary conditions and the metal matrix material parameters (plastic anisotropy, hardening).

Multiscale modelling for better hygrothermal prediction of porous building materials

The aim of this work is to understand the influence of the microstructuralgeometric parameters of porous building materials on the mechanisms of coupled heat, air and moisture transfers, in order to predict behavior of the building to control and improve it in its durability. For this a multi-scale approach is implemented. It consists of mastering the dominant physical phenomena and their interactions on the microscopic scale. Followed by a dual-scale modelling, microscopic-macroscopic, of coupled heat, air and moisture transfers that takes into account the intrinsic properties and microstructural topology of the material using X-ray tomography combined with the correlation of 3D images were undertaken. In fact, the hygromorphicbehavior under hydric solicitations was considered. In this context, a model of coupled heat, air and moisture transfer in porous building materials was developed using the periodic homogenization technique. These informations were subsequently implemented in a dynamic computation simulation that model the hygrothermalbehaviourof material at the scale of the envelopes and indoor air quality of building. Results reveals that is essential to consider the local behaviors of materials, but also to be able to measure and quantify the evolution of its properties on a macroscopic scale from the youngest age of the material. In addition, comparisons between experimental and numerical temperature and relative humidity profilesin multilayers wall and in building envelopes were undertaken. Good agreements were observed.

Multiscale modelling for better hygrothermal prediction of porous building materials

The aim of this work is to understand the influence of the microstructuralgeometric parameters of porous building materials on the mechanisms of coupled heat, air and moisture transfers, in order to predict behavior of the building to control and improve it in its durability. For this a multi-scale approach is implemented. It consists of mastering the dominant physical phenomena and their interactions on the microscopic scale. Followed by a dual-scale modelling, microscopic-macroscopic, of coupled heat, air and moisture transfers that takes into account the intrinsic properties and microstructural topology of the material using X-ray tomography combined with the correlation of 3D images were undertaken. In fact, the hygromorphicbehavior under hydric solicitations was considered. In this context, a model of coupled heat, air and moisture transfer in porous building materials was developed using the periodic homogenization technique. These informations were subsequently implemented in a dynamic computation simulation that model the hygrothermalbehaviourof material at the scale of the envelopes and indoor air quality of building. Results reveals that is essential to consider the local behaviors of materials, but also to be able to measure and quantify the evolution of its properties on a macroscopic scale from the youngest age of the material. In addition, comparisons between experimental and numerical temperature and relative humidity profilesin multilayers wall and in building envelopes were undertaken. Good agreements were observed.

A two-scale Stefan problem arising in a model for tree sap exudation

The study of tree sap exudation, in which a (leafless) tree generates elevated stem pressure in response to repeated daily freeze-thaw cycles, gives rise to an interesting multi-scale problem involving heat and multiphase liquid/gas transport. The pressure generation mechanism is a cellular-level process that is governed by differential equations for sap transport through porous cell membranes, phase change, heat transport, and generation of osmotic pressure. By assuming a periodic cellular structure based on an appropriate reference cell, we derive an homogenized heat equation governing the global temperature on the scale of the tree stem, with all the remaining physics relegated to equations defined on the reference cell. We derive a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. Numerical simulations are then performed to validate the results and draw conclusions regarding the phenomenon of sap exudation, which is of great importance in trees such as sugar maple and a few other related species. The particular form of our homogenized temperature equation is obtained using periodic homogenization techniques with two-scale convergence, which we investigate theoretically in the context of a simpler two-phase Stefan-type problem corresponding to a periodic array of melting cylindrical ice bars with a constant thermal diffusion coefficient. For this reduced model, we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, clearly identifying the missing pieces required to extend the proofs to the fully nonlinear sap exudation model. Numerical simulations of the reduced equations are then compared with results from the complete sap exudation model.

Numerical analysis of effective thermal conductivity of plastic foams

This study is dedicated to the development of a numerical finite-element solution to determine the effective thermal conductivity of extruded polymeric foam with different levels of open and closed porosity. The implemented numerical method is derived from periodic homogenization techniques used for the simulation of laser welding of composite materials. For a validation purpose, it was applied to the calculation of the effective thermal conductivity of polyethylene extruded foam for which the input data (density, cell size and distribution, porosity ratio, and extinction coefficient) were characterized experimentally. The computed results were compared with experimental data obtained by the transient plane source technique and analytical results derived from the literature. The deviation from the experimental data is five times lower than that of analytical methods. Moreover, the proposed numerical solution allows reaching higher porosity ratio and makes it possible to reduce the computation time while ensuring the same compactness ratio regardless of the calculation.

Can a continuous mineral foam explain the stiffening of aged bone tissue? A micromechanical approach to mineral fusion in musculoskeletal tissues

Recent experimental data revealed a stiffening of aged cortical bone tissue, which could not be explained by common multiscale elastic material models. We explain this data by incorporating the role of mineral fusion via a new hierarchical modeling approach exploiting the asymptotic (periodic) homogenization (AH) technique for three-dimensional linear elastic composites. We quantify for the first time the stiffening that is obtained by considering a fused mineral structure in a softer matrix in comparison with a composite having non-fused cubic mineral inclusions. We integrate the AH approach in the Eshelby-based hierarchical mineralized turkey leg tendon model (Tiburtius et al 2014 Biomech. Model. Mechanobiol. 13 1003–23), which can be considered as a base for musculoskeletal mineralized tissue modeling. We model the finest scale compartments, i.e. the extrafibrillar space and the mineralized collagen fibril, by replacing the self-consistent scheme with our AH approach. This way, we perform a parametric analysis at increasing mineral volume fraction, by varying the amount of mineral that is fusing in the axial and transverse tissue directions in both compartments. Our effective stiffness results are in good agreement with those reported for aged human radius and support the argument that the axial stiffening in aged bone tissue is caused by the formation of a continuous mineral foam. Moreover, the proposed theoretical and computational approach supports the design of biomimetic materials which require an overall composite stiffening without increasing the amount of the reinforcing material.

Numerical study of a macroscopic finite pulse model of the diffusion MRI signal

Diffusion magnetic resonance imaging (dMRI) is an imaging modality that probes the diffusion characteristics of a sample via the application of magnetic field gradient pulses. The dMRI signal from a heterogeneous sample includes the contribution of the water proton magnetization from all spatial positions in a voxel. If the voxel can be spatially divided into different Gaussian diffusion compartments with inter-compartment exchange governed by linear kinetics, then the dMRI signal can be approximated using the macroscopic Karger model, which is a system of coupled ordinary differential equations (ODEs), under the assumption that the duration of the diffusion-encoding gradient pulses is short compared to the diffusion time (the narrow pulse assumption). Recently, a new macroscopic model of the dMRI signal, without the narrow pulse restriction, was derived from the Bloch–Torrey partial differential equation (PDE) using periodic homogenization techniques. When restricted to narrow pulses, this new homogenized model has the same form as the Karger model.We conduct a numerical study of the new homogenized model for voxels that are made up of periodic copies of a representative volume that contains spherical and cylindrical cells of various sizes and orientations and show that the signal predicted by the new model approaches the reference signal obtained by solving the full Bloch–Torrey PDE in O(ε2), where ε is the ratio between the size of the representative volume and a measure of the diffusion length. When the narrow gradient pulse assumption is not satisfied, the new homogenized model offers a much better approximation of the full PDE signal than the Karger model. Finally, preliminary results of applying the new model to a voxel that is not made up of periodic copies of a representative volume are shown and discussed.

Homogenization of a nonlinear multiscale model of calcium dynamics in biological cells

Calcium is one of the most important intracellular messengers, which occurs in the cytosol and the endoplasmic reticulum of animal cells. While most calcium dynamics models either do not account properly for the fact that the endoplasmic reticulum constitutes a microstructure of the cell or are infeasible by resolving the fine structure very explicitly, Goel et al. [15] derived an effective macroscopic model by formal homogenization. In this paper, this approach is made rigorous using periodic homogenization techniques to upscale the nonlinear coupled system of reaction–diffusion equations and, moreover, the appropriate scaling of the interfacial exchange term is taken into consideration.

Ionic transfer in charged porous media. Periodic homogenization and parametric study on 2D microstructures

In this paper we propose a macroscopic model for ionic diffusion in saturated porous media, taking into account the electrical double layer phenomenon. This model is established by periodic homogenization technique of Nernst–Planck and Poisson–Boltzmann equations at Debye length. The electrical double layer potential appears explicitly in the homogenized ionic diffusion equation and in the homogenized diffusion tensor expression. A numerical parametric study is conducted on more or less complex periodic microstructure to highlight the influence of electrocapillary interactions on the homogenized diffusion tensor and ionic transfer.

Effective elastic properties of auxetic microstructures: anisotropy and structural applications

Materials presenting a negative Poisson’s ratio (auxetics) have drawn attention for the past two decades, especially in the field of lightweight composite structures and cellular media. Studies have shown that auxeticity may result in higher shear modulus, indentation toughness and acoustic damping. In this work, three auxetic periodic microstructures based on 2D geometries are considered for being used as sandwich-core materials. Elastic moduli are computed for each microstructure by using finite elements combined with periodic homogenization technique. Anisotropy of elastic properties is investigated in and out-of-plane. Comparison is made between auxetics and the classical honeycomb cell. A new 3D auxetic lattice is proposed for volumic applications. Cylindrical and spherical elastic indentation tests are simulated in order to conclude on the applicability of such materials to structures. Proof is made that under certain conditions, auxetics can be competitive with honeycomb cells in terms of indentation strength. Their relatively soft response in tension can be compensated, in some situations, by high shear moduli.

Application of the periodic homogenization technique to ionic transfer in cementitious materials. Case of chloride in concrete

Application of the periodic homogenization technique to ionic transfer in cementitious materials. Case of chloride in concrete

Application of simple, periodic homogenization techniques to non-linear heat conduction problems in non-periodic, porous media

Often, detailed simulations of heat conduction in complicated, porous media have large runtimes. Then homogenization is a powerful tool to speed up the calculations by preserving accurate solutions at the same time. Unfortunately real structures are generally non-periodic, which requires unpractical, complicated homogenization techniques. We demonstrate in this paper, that the application of simple, periodic techniques to realistic media, that are just close to periodic, gives accurate, approximative solutions. In order to obtain effective parameters for the homogenized heat equation, we have to solve a so called “cell problem”. In contrast to periodic structures it is not trivial to determine a suitable unit cell, which represents a non-periodic media. To overcome this problem, we give a rule of thumb on how to choose a good cell. Finally we demonstrate the efficiency of our method for virtually generated foams as well as real foams and compare these results to periodic structures.