Classical Homogenization Research Articles

Apprehending the effects of mechanical deformations in cardiac electrophysiology: A homogenization approach

We follow a formal homogenization approach to investigate the effects of mechanical deformations in electrophysiology models relying on a bidomain description of ionic motion at the microscopic level. To that purpose, we extend these microscopic equations to take into account the mechanical deformations, and proceed by recasting the problem in the framework of classical two-scale homogenization in periodic media, and identifying the equations satisfied by the first coefficients in the formal expansions. The homogenized equations reveal some interesting effects related to the microstructure — and associated with a specific cell problem to be solved to obtain the macroscopic conductivity tensors — in which mechanical deformations play a nontrivial role, i.e. they do not simply lead to a standard bidomain problem posed in the deformed configuration. We then present detailed numerical illustrations of the homogenized model with coupled cardiac electrical–mechanical simulations — all the way to ECG simulations — albeit without taking into account the abundantly-investigated effect of mechanical deformations in ionic models, in order to focus here on other effects. And in fact our numerical results indicate that these other effects are numerically of a comparable order, and therefore cannot be disregarded.

Effective calculation of thermophysical properties of composite materials with multiple configurations by asymptotic homogenization technique

In this work a three-scale asymptotic homogenization technique is considered for calculating the thermophysical properties of composite materials with multiple periodic configurations. Heterogenieties of porous structures were considered by the periodic distribution of single cells at the microscale and mesoscale levels. Two effective thermal conductivity coefficients are calculated to obtain an average solution on the global structure. A three-scale algorithm is also compared with the classical two-scale homogenization, where microscale allows all heterogenieties. We provide a numerical experiments which show the efficiency and accuracy for the calculation of heat transfer of composite material with multiple configurations.

Identification of Scale-Independent Material Parameters in the Relaxed Micromorphic Model Through Model-Adapted First Order Homogenization

We rigorously determine the scale-independent short range elastic parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure. This is done using both classical periodic homogenization and a new procedure involving the concept of apparent material stiffness of a unit-cell under affine Dirichlet boundary conditions and Neumann’s principle on the overall representation of anisotropy. We explain our idea of “maximal” stiffness of the unit-cell and use state of the art first order numerical homogenization methods to obtain the needed parameters for a given tetragonal unit-cell. These results are used in the accompanying paper (d’Agostino et al. in J. Elast. 2019. Accepted in this volume) to describe the wave propagation including band-gaps in the same tetragonal metamaterial.

Anisotropic strain gradient thermoelasticity for cellular structures: Plate models, homogenization and isogeometric analysis

For three-dimensional cellular plate-like structures with a triangular (extruded lattice) microarchitecture, the article develops a pair of two-scale plate models relying on the anisotropic form of Mindlin’s strain gradient thermoelasticity theory. Accordingly, a computational homogenization method is proposed for determining the constitutive parameters of the related higher-order constitutive tensors. First, a Reissner–Mindlin plate model is derived by dimension reduction from a general framework of three-dimensional orthotropic strain gradient thermoelasticity and written as a variational formulation. An isogeometric conforming Galerkin method is formulated accordingly. Second, the plate model is modified in order to reduce the number of the constitutive strain gradient parameters. These steps are then repeated by following the kinematical assumptions of the Kirchhoff plate theory. Third, in order to see the cellular microarchitecture as a homogeneous three-dimensional material with classical modulae of transversal isotropy, classical computational homogenization is accomplished for determining the corresponding material parameters. Fourth, in order to see the cellular structures as two-dimensional plates, a non-classical homogenization procedure is proposed for the identification of the strain gradient modulae of the plate models. Finally, a set of numerical examples illustrates the reliability and efficiency of the resulting plate models in homogenizing cellular plate-like structures into strain gradient plate models capturing the bending size effects induced by the microarchitecture.

Wave dispersion curves in discrete lattices derived through asymptotic multi-scale method

This paper falls within the study of dispersion feature of elastic periodic media. In most cases, no analytic description is reachable and the problem is solved via numerical computations of the dispersion curves.We propose in this paper an analytic method dedicated to lattice systems that enables to reconstruct part by part the dispersion curves via an asymptotic multi-scale method. This method is illustrated on periodic reticulated beams. At low frequency, when there is a large scale separation between the length of the cell and the characteristic size of the vibrations, the classical homogenization method allows efficiently to establish the continuous equivalent dynamic description and the associated wave propagation properties. This scale separation is lost for frequencies of the order or higher than the diffraction frequency. However, instead of considering the amplitude of the mean displacement in a unit cell, the concept of scale separation can still be used by considering the amplitude of periodic eigenmodes defined on (multi-)cells. Thus, similar principles of asymptotic multi-scale method enables to describe the large scale modulations around the eigenfrequencies of the mono-and/or multi-cells period. Finally, the properties of the modulation are straightforwardly related to the dispersion curves at the considered frequencies.

A model to predict the elastic properties of gas hydrate-bearing sediments

A simple and accurate model for predicting the elastic properties of gas hydrate-bearing sediment (GHBS) is proposed and validated against experimental data. It is developed on the basis of the homogenization theory for multiphase composite. Unlike the classical homogenization techniques those fix a homogenization scheme (e.g. self-consistent, Mori-Tanaka, DEM, etc.) for a given microstructure, the proposed model considers a flexible homogenization scheme that adapts with the change of the microstructure of the material. The idea is to modify the elastic properties of the reference matrix with respect to the microstructure change of the mixture. Such modification is ensured by adapting the theoretical method to available experimental data. The derived method is proved to be very powerful as it can fit laboratory measured data on gas hydrate-bearing sand formed by different methods including excess gas and excess water methods and log data taken from various sites. Such feature cannot be satisfied by the existent models those consider the hydrate phase as a cement phase or a pore-filling phase. A very good agreement between the proposed method and experimental data offer a simple and accurate ways to predict the saturation of gas hydrate using sonic log data or 3D seismic data.

Bayesian inference as a tool for improving estimates of effective elastic parameters of wood

A simple approach to the identification of geometrical and material uncertainties of wood is presented. This stochastic mechanics problem combines classical micromechanics, computational homogenization and experimental measurements with Bayesian inference to estimate the model parameters including the characteristics of errors in macroscopic elastic properties of wood caused by randomness of microstructural details on the one hand and the experimental errors on the other hand. The former source of uncertainty includes, for example, variability in microfibril angle and growth ring density. Even such limiting consideration of random input illustrates the need for combined computational and experimental approach in a reliable prediction of the desired material properties. Tying the two approaches in the framework of Bayesian statistical method proves useful when addressing their limitations and as such giving better notion on the credibility of the prediction. This is demonstrated here on one particular example of spruce wood.

Asymptotic Behaviour of the Energy Integral of a Two-Parameter Homogenization Problem with Nonlinear Periodic Robin Boundary Conditions

AbstractWe consider a nonlinear Robin problem for the Poisson equation in an unbounded periodically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameterδ. The relative size of each periodic perforation is determined by a positive parameterε. Under suitable assumptions, such a problem admits a family of solutions which depends onεandδ. We analyse the behaviour the energy integral of such a family as (ε,δ) tends to (0, 0) by an approach that represents an alternative to asymptotic expansions and classical homogenization theory.

Symbolic Computation to Derive a Linear-Elastic Buckling Theory for Solids with Periodic Microstructure

abstractA symbolic computation of a double scale asymptotic technique is used to derive a linear-elastic buckling theory for solids with periodic microstructure. From the stationary potential energy, equations are obtained for the classic homogenization problem, and for its extension to the buckling problem at macro, micro and mixed scales. The limitation to #Y-periodic instability modes can be treated afterwards using a Floquet-Bloch theory. The symbolic procedure is worked out step-by-step and results are presented. The obtained equations present a theoretical framework for the topology optimization of structures and microstructures involving linear elastic buckling criteria.

Understanding the role of general interfaces in the overall behavior of composites and size effects

The objective of this contribution is to investigate the role of generalized interfaces in the overall response of particulate composites and the associated size effects. Throughout this work, the effective properties of composites are obtained via three-dimensional computational simulations using the interface-enhanced finite element method for a broad range of parameters. The term interface corresponds to a zero-thickness model representing the interphase region between the constituents and accounting for the interfaces at the micro-scale introduces a physical length-scale to the effective behavior of composites, unlike the classical first-order homogenization that is missing a length-scale. The interface model here is general in the sense that both traction and displacement jumps across the interface are admissible recovering both the cohesive and elastic interface models. Via a comprehensive computational study, we identify extraordinary and uncommon characteristics of particle reinforced composites endowed with interfaces. Notably, we introduce the notion of critical size at which the overall behavior, somewhat surprisingly, shows no sensitivity with respect to the inclusion-to-matrix stiffness ratio. Our study, provides significant insight towards computational design of composites accounting for interfaces and in particular, nano-composites.

Edgeworth expansions for slow-fast systems with finite time-scale separation.

We derive Edgeworth expansions that describe corrections to the Gaussian limiting behaviour of slow-fast systems. The Edgeworth expansion is achieved using a semi-group formalism for the transfer operator, where a Duhamel-Dyson series is used to asymptotically determine the corrections at any desired order of the time-scale parameter ε. The corrections involve integrals over higher-order auto-correlation functions. We develop a diagrammatic representation of the series to control the combinatorial wealth of the asymptotic expansion in ε and provide explicit expressions for the first two orders. At a formal level, the expressions derived are valid in the case when the fast dynamics is stochastic as well as when the fast dynamics is entirely deterministic. We corroborate our analytical results with numerical simulations and show that our method provides an improvement on the classical homogenization limit which is restricted to the limit of infinite time-scale separation.

Combining radial point interpolation meshless method with a new homogenization technique for trabecular bone multiscale structural analyses.

Bone tissue is a dynamic tissue, possessing different functional requirements at different scales. This layered organization indicates the existence of a hierarchical structure, which can be characterized to distinguish macro-scale from micro-scale levels. Structurally, both scales can be linked by the use of classic multiscale homogenization techniques. Since in bone tissue each micro-scale domain is distinct form its neighbour, applying a classic multiscale homogenization technique to a complete bone structure could represent an inadmissible computational cost. Thus, this work proposes a homogenization methodology that is computationally efficient, presenting a reduced computational cost, and is capable to define the homogenized microscale mechanical properties of the trabecular bone highly heterogeneous medium. The methodology uses the fabric tensor concept in order to define the material principal directions. Then, using an anisotropic phenomenological law for bone tissue correlating the local apparent density with directional elasticity moduli, the anisotropic homogenized material properties of the micro-scale patch are fully defined. To validate the developed methodology, several numerical tests were performed, measuring the sensitivity of the technique to changes in the micro-patch size and preferential orientation. The results show that the developed technique is robust and capable to provide a consistent material homogenization. Additionally, the technique was combined with two discrete numerical techniques: the finite element method and radial point interpolation meshless method. Structural analyses were performed using real trabecular patches, showing that the proposed methodology is capable to accurately predict the micro-scale patch mechanical behavior in a fraction of the time required by classic homogenization techniques.

Numerical = periodic homogenization

AbstractIn our work, we try to bridge the existing theory of classical homogenization and the practical methods in numerical homogenization. We suggest an effective coefficient, which stems from the LOD method and which is based on a triangulation of the underlying domain. It turns out that in the case the diffusion coefficient has periodic structure and for a suitable choice of the mesh, the suggested coefficient coincides with the homogenized coefficient from classical homogenization.

Multiscale modeling of the thermomechanical behavior in heterogeneous media embedding Phase Change Materials particles

In this work, a multiscale model for thermomechanical properties of composite structures containing phase change particles is developed. For the mechanical part, a classical linear computational homogenization procedure is employed. For the thermal part, due to the strong nonlinear, history-dependent thermal effects, a concurrent multiscale (FE2) method is extended to take into account the presence of Phase Change Materials particles (PCM) at the microscale. The PCM inclusions change from liquid to solid state in the range of room temperature. This phase change induces a modified macroscopic thermal behavior, which can be used e.g. to design materials with enhanced thermal inertia and reduce energy consumption in civil engineering constructions. The technique allows taking into account accurately the fully nonlinear, history-dependent thermal behavior through numerical calculations at the microscale based on a Representative Volume Element (RVE) and its effect at the macroscale. The method is applied to concrete material including paraffin wax PCM. The results show the benefits of the PCM on the thermal behavior, including shifted and smoothed temperature response as compared to materials without PCM particles.

A double incremental variational procedure for elastoplastic composites with combined isotropic and linear kinematic hardening

We investigate the nonlinear behavior of elasto-plastic composites with isotropic and linear kinematic hardening. We first rely on the incremental variational principles introduced by Lahellec and Suquet (2007a). We also take advantage of an alternative formulation, recently proposed by Agoras et al. (2016) for visco-plastic composites without hardening, which consists in a double application of the variational procedure of Ponte-Castañeda. We extend in this paper this approach to elasto-plastic composites with combined linear kinematic and isotropic work-hardening. The first application of the variational procedure linearizes the local behavior, including hardening, and leads to a thermo-elastic Linear Comparison Composite (LCC) with a heterogeneous polarization field inside the phases. The second one deals with the heterogeneity of the polarization and results in a new thermo-elastic LCC with a per-phase homogeneous polarization field, which effective behavior can then be estimated by classical linear homogenization schemes. We develop and implement this new incremental variational procedure for composites comprised of linear elastic spherical particles isotropically distributed in an elasto-plastic matrix. The predictions of the model are compared with results available in the literature for cyclic proportional and non-proportional loadings. New results for elasto-plastic composites with combined isotropic and kinematic hardening are also provided. They are in good agreement with the numerical computations we carried out, at both local and macroscopic scales.

Homogenization theory for the effective permittivity of a turbulent tokamak plasma in the scrape-off layer

There has been a growing interest, over the past few years, on understanding the effect on radio frequency waves due to turbulence in the scrape-off layer of tokamak plasmas. While the far scrape-off layer density width is of the order of centimetres in contemporary tokamaks, in ITER (International Thermonuclear Experimental Reactor), and in future fusion reactors, the corresponding width will be of the order of tens of centimetres. As such, this could impact the spectral properties of the waves and, consequently, the transport of wave energy and momentum to the core plasma. The turbulence in the scrape-off layer spans a broad range of spatial scales and includes blobs and filaments that are elongated along the magnetic field lines. The propagation of radio frequency waves through this tenuous plasma is given by Maxwell’s equations. The characteristic properties of the plasma appear as a permittivity tensor in the expression for the current in Ampere’s equation. This paper develops a formalism for expressing the permittivity of a turbulent plasma using the homogenization technique. This technique has been extensively used to express the dielectric properties of composite materials that are spatially inhomogeneous, for example, due to the presence of micro-structures. In a similar vein, the turbulent plasma in the scrape-off layer is spatially inhomogeneous and can be considered as a composite material in which the micro-structures are filaments and blobs. The classical homogenization technique is not appropriate for the magnetized plasma in the scrape-off layer, as the radio frequency waves span a broad range of wavelengths and frequencies – from tens of megahertz to hundreds of gigahertz. The formalism in this paper makes use of the Fourier space components of the electric and magnetic fields of the radio frequency waves for the scattered fields and fields inside the filaments and blobs. These are the eigenvectors of the dispersion matrix which, using the Green’s function approach, lead to a homogenized dielectric tensor.

A Homogeneity Test for Comparing Gridded-Spatial-Point Patterns of Human Caused Fires

The statistical evaluation of the spatial similarity of human caused fire patterns is an important issue for wildland fire analysis. This paper proposes a method based on observed data and on a statistical tool (homogeneity test) that is based on non-explicit spatial distribution hypothesis for the human caused fire events. If a tessellation coming from a space filling curve is superimposed on the spatial point patterns, and a linearization mechanism applied, the statistical problem of testing the similarity between the spatial point patterns is equivalent to the one of testing the homogeneity between the two multinomial distributions obtained by modeling the proportions of cases on each cell of the tessellation. This way of comparing spatial point patterns is free of any hypothesis on any spatial point process. Because data are spatially over-dispersed, the existence of many cells of the grid without any count is a problem for classical statistical homogeneity tests. Our work overcomes this problem by applying specific test statistics based on the square Hellinger distance. Simulations and actual data are used in order to tune the process and to demonstrate the capabilities of the proposal. Results indicate that a new and robust method for comparing spatial point patterns of human caused fires is available.

Strain gradient and generalized continua obtained by homogenizing frame lattices

We determine the effective behavior of periodic structures made of welded elastic bars. Taking into account the fact that flexural and torsional stiffnesses are much smaller than the extensional one we overpass classical homogenization formula and obtain totally different types of effective energies. We work in the framework of linear elasticity. We give different examples of two dimensional or three dimensional micro-structures which lead to generalized 1D, 2D or 3D continua like Timoshenko beam, Mindlin-Reissner plate, strain gradient, Cosserat, or micromorphic continua.

Singularly Perturbed Forward-Backward Stochastic Differential Equations: Application to the Optimal Control of Bilinear Systems

We study linear-quadratic stochastic optimal control problems with bilinear state dependence where the underlying stochastic differential equation (SDE) has multiscale features. We show that, in the same way in which the underlying dynamics can be well approximated by a reduced-order dynamics in the scale separation limit (using classical homogenization results), the associated optimal expected cost converges to an effective optimal cost in the scale separation limit. This entails that we can approximate the stochastic optimal control for the whole system by a reduced-order stochastic optimal control, which is easier to compute because of the lower dimensionality of the problem. The approach uses an equivalent formulation of the Hamilton-Jacobi-Bellman (HJB) equation, in terms of forward-backward SDEs (FBSDEs). We exploit the efficient solvability of FBSDEs via a least squares Monte Carlo algorithm and show its applicability by a suitable numerical example.

Size effects in nonlinear periodic materials exhibiting reversible pattern transformations

This paper focuses on size effects in periodic mechanical metamaterials driven by reversible pattern transformations due to local elastic buckling instabilities in their microstructure. Two distinct loading cases are studied: compression and bending, in which the material exhibits pattern transformation in the whole structure or only partially. The ratio between the height of the specimen and the size of a unit cell is defined as the scale ratio. A family of shifted microstructures, corresponding to all possible arrangements of the microstructure relative to the external boundary, is considered in order to determine the ensemble averaged solution computed for each scale ratio. In the compression case, the top and the bottom edges of the specimens are fully constrained, which introduces boundary layers with restricted pattern transformation. In the bending case, the top and bottom edges are free boundaries resulting in compliant boundary layers, whereas additional size effects emerge from imposed strain gradient. For comparison, the classical homogenization solution is computed and shown to match well with the ensemble averaged numerical solution only for very large scale ratios. For smaller scale ratios, where a size effect dominates, the classical homogenization no longer applies.