Equivalent Homogeneous Medium Research Articles

Homogenized Refractive Index and Invariance Property for Photon Propagation in a K-Colored (Isotropic-Poisson) Statistical Mixture

The propagation of photons in complex media, such as K-colored (isotropic-Poisson) statistical mixture, is typically investigated through Monte Carlo methods, which are inherently time-consuming. In this article, we show that under the Lambertian illumination hypothesis and thanks to a generalized invariance property, the K-colored statistical mixture can be fully described by an equivalent homogeneous medium characterized by a homogeneous refractive index. This greatly simplifies the implementation of Monte Carlo codes. To illustrate our findings, we present Monte Carlo simulations carried out on various tutorial examples.

Probability density functions for photon propagation in a binary (isotropic-Poisson) statistical mixture with unmatched positive and negative refractive indexes.

The exact homogenized probability density function, for a photon making a step of length s has been analytically derived for a binary (isotropic-Poisson) statistical mixture with unmatched refractive indexes. The companions, exact, homogenized probability density functions for a photon to change direction ("scatter"), with polar ϑ and azimuthal φ angles, and the homogenized albedo, have also been obtained analytically. These functions also apply to negative refractive indexes and can reduce the number of Monte Carlo simulations needed for photon propagation in complex binary (isotropic-Poisson) statistical mixtures from hundreds to just one, for an equivalent homogeneous medium. Note, that this is not an approximate approach, but a mathematically equivalent and exact result. Additionally, some tutorial examples of homogenized Monte Carlo simulations are also given.

Evaluation of a practical approach for field scale moisture flow modeling in heterogeneous media at a semiarid site

AbstractA practical approach for modeling field‐scale moisture flow in a highly heterogeneous unsaturated medium is described in this study. The validity of this approach is demonstrated through comparison of the numerical simulations with field observations at a semiarid site located in southcentral Washington State. The methodology is based on upscaling the core scale hydraulic properties and combining power‐law and tensorial connectivity‐tortuosity (PA‐TCT) approaches to derive macroscopic anisotropy parameters for each hydrostratigraphic unit (HSU) identified in the field. Each heterogeneous HSU is approximated by an equivalent homogeneous medium (EHM) model for which PA‐TCT parameters are used in the flow simulations. The available field data on moisture content and matric potential are compared with steady‐state flow simulations based on the mean form of Richards' equation. While the homogenization or averaging of heterogeneities, embedded in the EHM modeling approximation, cannot capture all of the field‐scale variability, the simulated steady‐state moisture and matric potential profiles capture well the central tendency of the field data. This approach is deemed practical for assessing the fate and transport of contaminants in highly heterogeneous unsaturated media at the transport scale of hundreds of meters.

Integro-Differential Analysis of Resonant Magnetic Metasurfaces With Equivalent Medium Approximation

This paper proposes a strategy to consider the dielectric effects in the thin-wire integro-differential formulation and applies it to analyzing microstrip metasurfaces that support magnetic resonances. These structures comprise a periodic arrangement of subwavelength unit cells; therefore, the thin-wire approximation can be applied to analyze them. However, the substrate is usually disregarded. This work achieves more accurate results with a homogeneous equivalent medium approximation considering the dielectric properties. Moreover, an experimental methodology is presented to validate this implementation for each device as well as for the complete system. The proposed formulation is numerically solved with the Method of Moments, and the results show close agreement with experimental data. Besides, it leads to a significant reduction in the computation time compared to a full-wave analysis, which makes this approach also compelling for designing inductive wireless power transfer systems and magnetic resonators.

Micromechanical modeling of the elastic-viscoplastic deformation for considering voids and imperfect interfaces in sintered nano-silver under compression

The effects of voids and imperfect interfaces are key issues in the constitutive modeling of porous materials. In the current study, uniaxial compression experiments of short column specimens are performed to investigate the stress–strain relation of porous sintered nano-silver at different temperatures and loading rates. A micromechanical-based constitutive model is proposed to describe the mechanical behaviors of elastic-viscoplastic deformed polycrystalline materials with voids and imperfect interfaces. In the developed model, crystals with different orientations and voids are assumed to be embedded in the homogeneous equivalent medium (HEM). The behavior of single crystal is determined by a crystal plasticity model. A modified self-consistent model is proposed to describe the overall response of HEM by calculating the algorithmic tangent operator and affine strain increment of void and single crystals. The accuracy and rationality of the proposed model is validated by comparing the calculated results with a 3D crystal plasticity finite element (CPFE) model. Influences of the imperfect interface parameter and voids volume fraction on elastic and elastic-viscoplastic deformed media are investigated. The maximum allowable volume fraction of the void is analyzed. The model is applied to simulate the experimental data. The result shows that the proposed model can describe the elastic-viscoplastic deformation of sintered nano-silver with reasonable accuracy.

Identification of MEMS Geometric Uncertainties through Homogenization

Fabrication imperfections strongly influence the functioning of Micro-Electro-Mechanical Systems (MEMS) if not taken into account during the design process. They must be indeed identified or precisely predicted to guarantee a proper compensation during the calibration phase or directly in operation. In this work, we propose an efficient approach for the identification of geometric uncertainties of MEMS, exploiting the asymptotic homogenization technique. In particular, the proposed strategy is experimentally validated on a MEMS filter, a device constituted by a complex periodic geometry, which would require high computational costs if simulated through full-order models. The complex periodic structure is replaced by an equivalent homogeneous medium, allowing a fast optimization procedure to identify imperfections by comparing a simplified analytical model with the experimental data available for the MEMS filter. The actual over-etch, obtained after the release phase, and the electrode offset of a fabricated MEMS filter are effectively identified through the proposed strategy.

Thermo-Mechanical Calculation of Printed Circuit Heat Exchanger Using Homogenization

The Finite-Element Method (FEM) is mainly used to design Compact Heat Exchanger structures. However, the narrow channel structures require fine mesh to accurately compute stress and strain. Combined with the dimensions of the overall component, this leads to an excessively large numerical model and therefore long computing time. This is especially true for transient thermal analyses, which require taking into account the full geometry. It is therefore interesting to reduce the size of the mesh. Periodic homogenization is an efficient method for achieving this goal. It can be applied to the core of the structure when periodic patterns (identified on basis of the arrangement of the channels) exist. A method based on this technique, with some improvements, is proposed in this paper for thermal loading without internal pressure. In addition to the Equivalent Homogenous Medium (EHM) replacing the periodic patterns, some explicit channels are interposed between the EHM and the cover plates. This has two advantages. The first is to smooth the transition of stiffness between the cover plates and the homogenized medium. The second one is to be able to directly compute stress and strain on the most critical channels located in this area. This paper assesses the method’s effectiveness for thermal loadings in order to conduct thermal stress analyses. First, the equivalent elastic constants of the EHM are obtained with a numerical Finite Elements Method. Then, two 2D cases using EHM are compared against a 2D explicit model (i.e. explicit geometry for the core channels). These cases are chosen to validate the EHM itself and its effectiveness for a real section of the heat exchanger. Results show very good agreement with a relative difference lower than 1%. In addition, the sensitivity to the number of layers added between the EHM and the cover plates is analysed. It is recommended interposing at least two layers of patterns to obtain converged results for the considered configuration. Finally, a triangular mesh is considered to reduce the size of the model. No difference with a regular quadrangular mesh can be observed whereas the computing time is reduced. This method can be used to perform the design of any CHE under thermal loading as long as the channel arrangement shows periodicity.

Modeling and simulation of laser shock waves in elasto-plastic 1D layered specimens

The aim of this paper is to study the effect of microstructure heterogeneity upon elasto-plastic wave propagation generated during laser shot peening. We consider a simplified elasto-plastic laminate specimen subjected to uniaxial strain. The microstructure is composed of two phases alternating periodically and perfectly bonded together. The associated PDE system is solved using a high-resolution Godunov scheme, allowing to study the wave propagation in the heterogeneous structure. It is found that, even for a small mechanical contrast between the two phases, the considered laminate microstructure has a significant effect on the distribution of plastic strain. In addition, an elasto-plastic homogenization of the laminate has been carried out, and the resulting Homogeneous Equivalent Medium (HEM) has been used to decrease the computation time of the wave propagation. The HEM-based model is able to reproduce accurately the full-field solution in terms of distribution of mean plastic strain within the specimen and its fluctuation between the two phases.

Temperature dependent radiative properties of semi-transparent fiberglass-epoxy composite materials from 20 °C to 200 °C

The radiative properties of polymer-reinforced composite materials such as laminates may change according to the temperature. The knowledge of the relationship between radiative properties and temperature becomes crucial when designing structural components subjected to thermal stresses, induced for example by a laser irradiation. This study deals with the analysis of the absorption and scattering properties of fiberglass-epoxy composites from room temperature to 200 °C. From the point of view of thermal radiation, the two-phase system (polymer matrix and fibers reinforcements) is treated as an equivalent homogeneous medium characterized by volumetric radiative properties, namely the absorption coefficient, the scattering coefficient, and the scattering phase function. The light interaction with the rough boundaries of the sample is modeled by boundary scattering coefficients. The aim is to determine these volumetric and boundary scattering properties as function of the temperature. We use a robust inverse method based on a Gauss-Newton algorithm to solve the nonlinear least squares problem. It minimizes the sum of the squared difference between the theoretical and experimental data of bidirectional and normal-hemispherical reflectances and transmittances. Scattering measurements are carried out using a visible and near-infrared spectrometer (400 to 2500 nm spectral range) equipped with a goniometric system and a heated sample holder. Theoretical results are retrieved by employing the Monte Carlo method for solving the Radiative Transfer Equation (RTE). We report that reflectance are insensitive to the temperature rise below 150 °C, and increase above this temperature up to 200 °C. Transmittances increase with temperature, reach a maximum around 105–125 °C near the glass transition temperature of the resin (assessed by a differential scanning calorimetry analysis), and decrease until 200 °C. Upon others, this study reveals that radiative properties are strongly temperature-dependent above 100 °C, first because of the dependence of the resin refractive index to the temperature, but also due to the onset of the polymer chemical degradation. The mismatch between the glass fibers and the polymer refractive indexes is at its lowest near the glass transition temperature of the resin, and explains the peak of transmittances and the trough on the extinction coefficient and the scattering albedo. The onset of the resin chemical degradation above 150 °C increases the absorption coefficient. The asymmetry parameter of the Henyey Greenstein phase function decrease notably from 0.94 to 0.3 above 150 °C, due to the enhancement of backscattered light within the sample.

Experimental study of radiative transfer in semi-transparent composite materials at different temperatures

This study deals with the analysis of the propagation of radiation within a diffusing semi-transparent composite medium with rough boundaries. The two-phase medium (resin matrix and glass fibers reinforcement) is treated as an equivalent homogeneous medium characterized by volumetric radiative properties (extinction coefficient, albedo and phase function) and boundary scattering properties. The aim is to identify the radiative properties at different temperatures ranging from room temperature to 200°C. The identification method (Gauss-Newton) uses bidirectional reflectance and transmittance values. The experimental results are obtained using a spectrophotometer equipped with a goniometer and a heated sample holder. The Monte Carlo method is used to solve the Radiative Transfer Equation (RTE) in order to obtain the theoretical values.

Radiative behavior of low-porosity ceramics: II–Experimental and numerical study of porous alumina up to high temperatures

We compare two methods of characterizing the radiative behavior of 22% porous sintered alumina disks containing μm-sized pores on the 1 µm to 7 µm wavelength range up to very high temperatures. The first consists of direct spectroscopic measurements of the normal-hemispherical reflectance and transmittance at room temperature, as well as the normal emittance up to 1300 ∘C. The second is a two-step multi-scale numerical approach: the volume radiative properties and surface reflectivity are first determined from physical optics computations on tomography-reconstructed microstructures, then applied to the resolution of the radiative transfer equation on an equivalent homogeneous medium to obtain the spectral reflectance, transmittance, and emittance. Uncertainties in the numerical results, mainly due to microstructural variability, are quantified through sensitivity studies. The good agreement between the experimental and numerical results confirm the ability of our proposed numerical approach to give accurate predictions of the high-temperature radiative behavior of porous ceramics.

A Multiscale Numerical Modeling Investigation on the Significance of Flow Partitioning for the Development of Quartz c‐Axis Fabrics

AbstractQuartz c‐axis fabrics in natural mylonites can vary to such an extent that they apparently give opposite senses of shear in a single thin section. Many hypotheses have been invoked to explain this. Here, we couple our self‐consistent multiscale approach for flow partitioning with the viscoplastic self‐consistent model for c‐axis fabric simulation to investigate quartz c‐axis fabric development. Quartz aggregates are regarded as microscale Eshelby inhomogeneities embedded in a macroscale medium whose effective rheology is represented by a hypothetical homogeneous equivalent medium which is assumed to be rheologically isotropic or has a planar anisotropy. We reproduced the observed quartz c‐axis fabrics. We found that although the microscale flow fields are distinct from one another and from the macroscale flow, the microscale vorticity in every inhomogeneity has the same sense as the macroscale vorticity. This implies that one can use the average of the microscale vorticity axes determined through the crystallographic vorticity axis analysis to obtain the macroscale vorticity axis. However, quartz c‐axis fabrics cannot be used to determine the vorticity number where flow partitioning is significant.

A Contribution to the Study of the Forming of Dry Unidirectional HiTape® Reinforcements for Primary Aircraft Structures

In the context of developing competitive liquid composites molding processes for primary aircraft structures, modeling the forming stage of automatically-placed initially flat stacks of dry reinforcements is of great interest. In the case of HiTape®, a dry unidirectional carbon fiber reinforcement designed to achieve performances comparable to state-of-the-art pre-impregnated materials, the presence of a thermoplastic veil on each side of the material for both processing and mechanical purposes should also be considered when modeling forming in hot conditions. As a dry unidirectional reinforcement, HiTape® is expected to exhibit a transversely isotropic behavior. Computation cost and strong characterization challenges led us to model its behavior at the forming process temperature (above the thermoplastic veil melting temperature) through a homogeneous equivalent continuous medium exhibiting four ‘classical’ deformation modes and a specific structural mode, namely out-of-plane bending. The response of both single plies and stacks of HiTape® to this latter structural mode was characterized at the forming process temperature using a modifiedPeirce flexometer. Results on single plies showed a non-linear softening moment-curvature behavior and a corresponding flexural stiffness much lower than what can be inferred from continuum mechanics. Moreover, testing stacks revealed that the veil acts as a thin load transfer layer between the plies undergoing relative in-plane displacement,i.e.inter-ply sliding. This inter-ply response was then characterized separately at the forming process temperature thanks to a specific method relying on apull-throughtest. Experiments performed at pressures and speeds representative of the forming stage revealed that a hydrodynamic lubricated friction regime predominates,i.e.a linearly increasing relationship between the friction coefficient and the modified Hersey number. From an industrial point of view, high forming pressures and low speeds are therefore recommended to promote inter-ply slip to limit the occurrence of defects such as wrinkles.

Numerical studies of the statistics of seismic waveform propagation in random heterogeneous media

Incoherent scattered waves are generated when seismic waves propagate through heterogeneous media. These scattered waves contain statistical information, such as random heterogeneity. In order to study the influence of random heterogeneity on seismic waveforms, four series of simulation experiments were performed in the presents study, as follows: (1) Waves were excited by the same source and propagated through four models with different parameters; (2) Waves were excited by different frequencies, but spread in the same model (0.25, 0.50, 1.00, 2 MHz); (3) Waves were excited by the same source and propagated through three different velocity fluctuation models (5%, 10% and 15%); (4) Waves were excited by the same source and propagated through three different propagation distances (30, 60, and 90 mm). In the present study, the correlation between coherent waveforms and observed waveforms, the energy distribution with respect to the elapsed time, and the statistical distribution of the waveform correlation coefficient were calculated. Next, the variation of the waveform parameters with the normalized scale parameter ka was investigated. Stronger scattered waves and more complicated waveforms were observed when the ka value increased, indicating the transition from an equivalent homogeneous medium to a heterogeneous medium.

A self-consistent approach for the acoustical modeling of vegetal wools

Vegetal wools have the capacity to store atmospheric carbon dioxyde, one of the main gases responsible for climate change. So, these insulating materials are used as key elements for green buildings. Moreover, vegetal wools present high sound absorption level performances contributing to the acoustic comfort of indoor living spaces. These properties are directly related to the morphology and the size of their vegetal fibres. Thus, to take their microstructural specificities into account for the modeling of their sound absorption properties, a micro-macro homogenization approach based on a cylindrical geometry is developed. This modeling method, based on a mix between Homogenization of Periodic Media (HPM) and Self-Consistent Method (SCM), is called SCMcyl. The macroscopic behaviour laws of materials are rigorously obtained by using HPM. Then, the SCM leads to the establishment of two possible analytical solutions (a velocity approach v and a pressure approach p) under the fundamental assumption of the energy equivalence between a generic cylindrical inclusion, representative of the vegetal wools physical and geometrical properties at microscopic scale, and the homogeneous equivalent medium at the macroscopic scale. The two modeling approaches developed in this paper, SCMcyl−v and SCMcyl−p, can be used to determine the sound absorption of fibrous materials using only two parameters, an equivalent fibre radius value and the material porosity. Finally, these solutions are validated for the vegetal wools case by comparison with experimental measurements.

The Quasi-optical Equation in a Medium with Weak Dissipation

The form of a quasi-optical equation for a pulse of quasi-monochromatic radiation propagating in a homogeneous or one-dimensionally inhomogeneous linear medium with dissipation is analyzed. For a layer of an inhomogeneous medium, the effective parameters of an equivalent homogeneous medium are obtained.

An Effective Mixed Extracting Method for Electromagnetic Parameters of Periodically Loaded Substrate Integrated Waveguide Units and Its Applications

In this article, we propose an effective mixed extracting method for anisotropic constitutive parameters of periodically loaded substrate integrated waveguide (SIW). This method takes the advantage of both scattering parameters retrieval theory and the equivalent homogeneous medium dispersion analysis, which could accurately and effectively extract the equivalent constitutive parameters of the periodically loaded SIW. With this method, the corresponding constitutive parameters of the six typical patterns of periodically loaded SIW schemes are accurately extracted, verified, explained, and compared for their performance. It can be used to quantitatively predict the miniaturization potential of each unit cell and sequentially evaluate the performance as an integrated component. In addition, in terms of application, with the aid of equivalent parameters extracted by the proposed method, a slow-wave SIW evanescent mode filter centered at 12.64 GHz with a frequency bandwidth of 300 MHz is synthesized, fabricated, and measured. In the meanwhile, the theory of slow-wave evanescent mode filter is further consummated, by introducing the dual factors of enhanced permittivity and permeability, as well as analyzing respective advantages in filter synthesis. Measurements and simulation results show good agreement, which embodies the role of the method in guiding the design of actual microwave devices.

Simplified Micro-Mechanical Approach for Coated Viscoelastic Inclusion

The present work deals with the analytical resolution of the problem of viscoelastic coated inclusion embedded in a viscoelastic matrix.In a first step, we will study the problem of a linear viscoelastic inclusion, without coating, embedded in a linear viscoelastic matrix.Then, the problem of coated viscoelastic inclusion considering the coating as a thin layer whose viscoelastic properties are different from those of the inclusion and the matrix is performed.The resolution of this problem will be based simultaneously on the Green function technique as well as the interface operator. The analytical expression of the solution is obtained by assuming the isotropy of the matrix as well as the spherical shape of the coatedinclusion.These results are used to determine the effective properties of a heterogeneous medium from a self-consistent approach taking into accountthe interactions between coated inclusions and the equivalent homogeneous medium.

A 3D reflection ray-tracing method based on linear traveltime perturbation interpolation

The linear traveltime interpolation (LTI) method, one of the common methods to calculate the wavefront traveltimes and the raypaths in complex media, is based on the assumption that the traveltime varies linearly along each cell boundary of a discrete model. The linear assumption may result in large calculation errors when the cell size is large. To solve this problem, we have adopted a linear traveltime perturbation interpolation (LTPI) method. In LTPI, the original traveltime is decomposed into two parts: a reference traveltime and a traveltime perturbation. The reference traveltime, being one propagating in the equivalent homogeneous medium, varies nonlinearly. The traveltime perturbation, defined as the difference between the original and reference traveltimes, is assumed to be linear along each cell boundary. The traveltime perturbation is much less than the reference traveltime; therefore, the original traveltime keeps its nonlinearity. We modify LTPI to suit to irregular hexahedral cells for simulating the undulating interfaces more precisely. By combining the modified LTPI with the wavefront group-marching method, we have developed a reflection ray-tracing method in 3D complex media. Numerical experiments indicate that the modified LTPI is more accurate than LTI in computing wavefront traveltimes and raypaths. Besides, the modified LTPI performs more consistently than LTI in different grid spacing. Therefore, under a certain accuracy requirement, the modified LTPI is more efficient than LTI.

Multi-scale shape optimisation of lattice structures: an evolutionary-based approach

This study deals with the problem of the least-weight design of a lattice structure subject to constraints of different nature. To face this problem, a general multi-scale optimisation procedure is proposed. This approach aims at optimising both global and local geometric parameters defining the shape of the representative volume element of the lattice at the mesoscopic scale. The optimisation procedure involves design requirements defined at different scales: geometric and manufacturing constraints are involved at the mesoscopic scale, whilst thermodynamic constraints on the positive definiteness of the stiffness tensor of the lattice (modelled as an equivalent homogeneous anisotropic medium) intervene at the macroscopic scale. Finally, since lattice structures usually undergo compressive loads, a requirement on the first local buckling load is considered too. The proposed approach is based on (a) the non-uniform rational basis splines curves theory to describe the shape of the struts composing the lattice, (b) the strain energy homogenisation technique of periodic media to perform the scale transition and (c) a special genetic algorithm to perform optimisation calculations. The optimised solutions provided by the presented method are characterised by a weight saving of about $$39\%$$ with slightly enhanced mechanical properties when compared to conventional octahedral lattice structures.