Classical Homogenization Research Articles

Stress-driven nonlocal homogenization method for cellular structures

Strong size-dependent mechanical behaviors can be observed in cellular structures violating the principle of scale separation and cannot be captured by classical homogenization methods. This paper proposes a stress-driven nonlocal homogenization method to capture the size-dependent mechanical behavior due to the nonlocal force of cellular structures. A nonlocal discrete element model is proposed first to describe the nonlocal deformation mechanism of cellular structures. Then, a continuum stress-driven nonlocal homogenization method is calibrated by deriving the continuum limit from the nonlocal discrete element model. The continuum homogenization method releases the assumption of scale separation by introducing an intrinsic length, which can be calibrated by high throughput numerical computation. Also, for efficient prediction of size-dependent mechanical behaviors, an offline dataset of the intrinsic length is constructed for different unit cells. With the help of the offline dataset, the proposed homogenization method improves the accuracy of the classic homogenization method and reduces the computational cost of the high-fidelity finite element method. Finally, a cellular rod under tension is used as an application to illustrate the efficiency and accuracy of the proposed homogenization method. Results indicate that compared with the classic homogenization method, the relative error of the proposed homogenization method is less than 1% which has a good consistency with the high-fidelity method. Moreover, the computational efficiency of the proposed homogenization method is more than five times that of the high-fidelity finite element method.

Material Homogenization of Human Femur Bone Material Using a New Weighted Approach

Abstract The design of efficient and safe biocompatible artificial implants for bone replacement and reconstructions requires a thorough understanding of natural bone material. A new homogenization approach is therefore used to predict effective material properties of the human femur bone. The bone structure is idealized as a functionally graded composite made of different quantities of a collagen protein matrix reinforced by hydroxyapatite mineral acting as the filler. Two distinct composite models were investigated, one where the mineral reinforcements were assumed to be randomly dispersed inclusions in a matrix of collagen, and a second model wherein the minerals were assumed to be unidirectional fibers. Predicted results for longitudinal and transverse elastic modulus from the models compare very well with range of values from published experimental data. It is also compared to those predicted using classical homogenization models. These material properties are then used to obtain the structural response of a bone plate section under transverse load.

Effective Electromagnetic Properties of Composite Material Computed From Neural Network Approach

ABSTRACTThanks to their lightweight, composite materials have become widely used in the automotive and aerospace industries. The design of components made from these materials is carried out by numerical modeling which can sometimes be tedious because of the need to take into account the internal structure of these materials. Obtaining the effective properties of an equivalent homogeneous material to replace the composite in our numerical models makes modeling easier. Classical homogenization approaches are not always suitable to obtain these effective properties. This article deals with an inverse problem that consists in computing the electromagnetic properties from the knowledge of the magnetic shielding effectiveness values. For different composite samples, an artificial neural network method is used to predict the effective conductivities from the magnetic shielding effectiveness measurements. The magnetic shielding effectiveness values computed from the predicted conductivities are close to those obtained from the measurements.

The time horizon for stochastic homogenization of the one-dimensional wave equation

The wave equation with stochastic rapidly oscillating coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity ε, classical homogenization fails for times of the order ε − 2 . We consider the one-dimensional wave equation with random rapidly oscillation coefficients on scale ε and are interested in the critical time scale ε − β from where on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ε − 1 .

Wave power calculation of a large periodic array of bottom-hinged paddle wave energy converters using Floquet–Bloch theory

In this study we consider a wave energy farm consisting of a large number of bottom-hinged paddles. The paddles are arranged in a doubly-periodic manner, with a finite number of identical rows and each row containing an infinite number of paddles. Each paddle is attached to its own damper and spring, allowing power to be generated through its pitching motion. Unlike previous studies into paddle arrays, we envisage compact arrays consisting of small devices arranged across a large number of rows. The mathematical analysis of this proposed configuration is best suited to methods that exploit the periodicity of the array. It is shown that the velocity potential everywhere within the array can be described by an expansion in terms of suitably-defined Floquet–Bloch eigenmodes and this allows the solution to the scattering problem involving waves incident upon the array to be determined by a simple low-order system of equations. The method used in this study is exact within the setting of linearised water waves and has all of the advantages of classical low-frequency multi-scale homogenisation without the low-frequency restriction. It may also be regarded as a natural extension of the well-known wide-spacing approximation without the large separation restriction. Although the focus of this paper is on the mathematical approach to the solution of the problem, we provide a range of results to illustrate the performance of this proposed wave energy converter concept.

Asymptotic, second-order homogenization of linear elastic beam networks

We propose a general approach to the higher-order homogenization of discrete elastic networks made up of linear elastic beams or springs in dimension 2 or 3. The network may be nearly (rather than exactly) periodic: its elastic and geometric properties are allowed to vary slowly in space, in addition to being periodic at the scale of the unit cell. The reference configuration may be prestressed. A homogenized strain energy depending on both the macroscopic strain ɛ and its gradient ∇ɛ is obtained by means of a two-scale expansion. The homogenized energy is asymptotically exact two orders beyond that obtained by classical homogenization. The homogenization method is implemented in a symbolic calculation language and applied to various types of networks, such as a 2D honeycomb, a 2D Kagome lattice, a 3D truss and a 1D pantograph. It is validated by comparing the predictions of the microscopic displacement to that obtained by full, discrete simulations. This second-order method remains highly accurate even when the strain gradient effects are significant, such as near the lips of a crack tip or in regions where a gradient of pre-strain is imposed.

A computational approach to identify the material parameters of the relaxed micromorphic model

We determine the material parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure in this work. This is achieved through a least squares fitting of the total energy of the relaxed micromorphic homogeneous continuum to the total energy of the fully-resolved heterogeneous microstructure, governed by classical linear elasticity. We avoid establishing exact micro–macro transition relations, as in classical homogenization theory, because defining a representative volume element is not feasible in the absence of scale separation, as such an element does not exist. The relaxed micromorphic model is a generalized continuum that utilizes the Curl of a micro-distortion field instead of its full gradient as in the classical micromorphic theory, leading to several advantages and differences. The most crucial advantage is that it operates between two well-defined scales. These scales are determined by linear elasticity with microscopic and macroscopic elasticity tensors, which respectively bound the stiffness of the relaxed micromorphic continuum from above and below. While the macroscopic elasticity tensor is established a priori through standard periodic first-order homogenization, the microscopic elasticity tensor remains to be determined. Additionally, the characteristic length parameter, associated with curvature measurement, controls the transition between the micro- and macro-scales. Both the microscopic elasticity tensor and the characteristic length parameter are here determined using a computational approach based on the least squares fitting of energies. This process involves the consideration of an adequate number of quadratic deformation modes and different specimen sizes. We conduct a comparative analysis between the least square fitting results of the relaxed micromorphic model, the fitting of a skew-symmetric micro-distortion field (Cosserat-micropolar model), and the fitting of the classical micromorphic model with two different formulations for the curvature; one simplified formulation involving only one single characteristic length and a simplified isotropic curvature with three parameters. The relaxed micromorphic model demonstrates good agreement with the fully-resolved heterogeneous solution after optimizing only four parameters. The “simplified” full micromorphic model, which includes isotropic curvature and involves the optimization of seven parameters, does not achieve superior results, while the Cosserat model exhibits the poorest fitting.

Solver-free reduced order homogenization for nonlinear periodic heterogeneous media

Reduced-order homogenization (ROH) and related methods are important computational tools for simulating the material behavior of composites. These methods generally sacrifice accuracy in exchange for superior computational efficiency, relative to methods such as classical computational homogenization (CCH). In this study, building on the recently developed solver-free CCH, we propose a fine-scale solver-free reduced order homogenization approach that avoids solving the fine-scale equilibrium equations and approximates the phase-average eigenstrains by sampling the fine-scale eigenstrain at a small number of points. The proposed method, which we call solver-free ROH, works by pre-computing history-dependent eigenstrain influence function tensors and sampling point contribution factors based on training data from a small set of CCH simulations. Then, during the online stage of the computation, phase-average eigenstrains are computed from sampling point eigenstrains and used to compute homogenized stresses and strains. Focusing on small-deformation problems, this paper formulates and verifies the solver-free ROH approach for nonlinear periodic heterogeneous media. First, in the formulation, we delineate a sampling point approximation of the phase eigenstrains and describe the use of this approximation within the coarse-scale stress update. Next, we verify the solver-free ROH using loading cases outside the training data set. Finally, we use the proposed method to simulate a multilayer composite plate in three point bending (3pt-bend) and open hole tension (OHT), demonstrating the method’s efficiency and accuracy relative to the CCH.

Microstructure-based modeling to characterize low pore density open-cell foams and its experimental validation.

Microlattices with large pore sizes are involved in many multifunctional applications, so it is essential to understand their acoustic properties. However, for these low pore density microlattice foams, the classical homogenization or "equivalent fluid" methods fail abruptly. This paper proposes and discusses a microstructure-based direct fluid model (DFM) that would help to predict the acoustic performance of low pore density periodic open-cell foams with spherical pores. The DFM is simulated directly, including the microscale geometric features inherent in the unit cell. A comparative study is performed for designed three-dimensional (3D) body-centered-cubic (BCC) porous foams having pores per inch (PPI) ranging from 1 to 12 over the frequency range of 500-4100 Hz with equivalent fluid models and experiments. The study shows the extent of deviation in homogenization-based methods from the experiment for PPI < 5. On the other hand, the acoustic performance parameters predicted with the DFM agree well with experiments on 3D-printed samples fabricated by additive manufacturing of varying PPI starting from 1. This study shows that the DFM is a valid method to predict the acoustics of low PPI microlattices. Furthermore, the gradual transition from dissipative to the reactive regime with a decrease in PPI is also brought out.

FAST FOURIER TRANSFORM METHOD FOR PERIDYNAMIC BAR OF PERIODIC STRUCTURE

The basic feature of the peridynamics [introduced by Silling (2000)] considered is a continuum description of material behavior as the integrated nonlocal force interactions between infinitesimal material points. A heterogeneous bar of the periodic structure of constituents with peridynamic mechanical properties is analyzed. One introduces the volumetric periodic boundary conditions (PBCs) at the interaction boundary of a representative unit cell (UC), whose local limit implies the known locally elastic PBCs. This permits us to generalize the classical computational homogenization approach to its counterpart in peridynamic micromechanics (PM). Alternative to the finite element methods (FEM) for solving computational homogenization problems are the fast Fourier transforms (FFTs) methods developed in local micromechanics (LM). The Lippmann-Schwinger (L-S) equation-based approach of the FFT method in the LM is generalized to the PM counterpart. Instead of one convolution kernel in the L-S equation, we use three convolution kernels corresponding to the properties of the matrix, inclusions, and interaction interface. The Eshelby tensor in LM depending on the inclusion shape is replaced by PM counterparts depending on the inclusion size and interaction interface (although the Eshelby concept of homogeneous eigenfields does no work in PM). The mentioned tensors are estimated one time (as in LM) in a frequency domain (also by the FFT method). Numerical examples for 1-D peridynamic inhomogeneous bar are considered. Computational complexities O (N &lt;i&gt;log&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt; N) of the FFT methods are the same in both LM and PM.

Solver‐free classical computational homogenization for nonlinear periodic heterogeneous media

AbstractModeling the behavior of composite materials is an important application of computational homogenization methods. Classical computational homogenization (CCH), based on asymptotic analysis, is such a method. In CCH, equilibrium equations are separated into two length scales and solved numerically. Solving the fine‐scale equilibrium equations at every coarse‐scale Gauss point, in every iteration of a Newton–Raphson loop, is often too computationally expensive for real engineering applications. In this study, we propose a modified CCH approach that avoids solving the fine‐scale equilibrium equations. The proposed method, which we call solver‐free CCH, works by pre‐computing a set of eigenstrain influence function tensors based on data from a small number of numerical experiments. Then, during the online stage of the computation, these eigenstrain influence tensors are used in the fine‐scale problem to evaluate and homogenize strains and stresses. This article begins by formulating the solver‐free CCH approach for small‐deformation problems involving composites with nonlinear constituent phase material models, including computation of the eigenstrain influence tensors and their application within the online stage. To verify the proposed approach, we consider loading cases outside the training set used to derive the eigenstrain influence tensors. The combinational efficiency and accuracy of the solver‐free CCH in comparison to the conventional CCH is studied on a multilayer composite plate in three point bending (3pt‐bend) and open hole tension.

Homogenization of Porous Thin Layers With Internal Stratification for the Estimation of Seismic Reflection Coefficients

AbstractStratified thin layers often present a prominent mechanical contrast with regard to the embedding background and, hence, are important targets for seismic reflection studies. An efficient way to study the reflectivity response of these thin layers is to employ their homogenized viscoelastic equivalents. We aim to homogenize a simple, yet realistic, thin‐layer model, which is composed of a finite non‐periodic sequence of homogeneous porous strata embedded in a background deemed impermeable at the seismic frequencies. The overarching objective is to reproduce the reflectivity response of such stratified thin layers. However, the estimation of the equivalent moduli is inherently affected by the boundary conditions (BC) associated with the embedding background. Therefore, classical homogenization procedures, which assume the existence of a periodic structure, are not readily applicable. We, therefore, propose a novel homogenization procedure that incorporates naturally the appropriate BC. To this end, we consider a sample that includes both a part of the background and a section of the thin layer, to which we apply classical oscillatory relaxation tests. However, we estimate the average of stress and strain components only over the thin layer section of interest. To test the accuracy of the method, we consider a sandstone composed of two strata saturated with different fluids embedded in impermeable half‐spaces. After estimating the corresponding equivalent moduli, we compare the resulting P‐wave reflectivities with those obtained using the original model. Our results show that the inferred viscoelastic equivalent closely reproduces the reflectivities of the stratified thin layer in the seismic frequency range.

Empirical Interscale Finite Element Method (EIFEM) for modeling heterogeneous structures via localized hyperreduction

This work proposes a special type of Finite Element (FE) technology – the Empirical Interscale FE method – for modeling heterogeneous structures in the small strain regime, for both dynamic and static analyses. The method combines a domain decomposition framework, where interface conditions are established through “fictitious” frames, with dimensional hyperreduction at subdomain level. Similar to other multiscale FE methods, the structure is assumed to be partitioned into coarse-scale elements, each of these elements is equipped with a fine-scale subgrid, and the displacements of the boundaries of the coarse-scale elements are described by standard polynomial FE shape functions. The distinguishing feature of the proposed method is the employed “interscale” variational formulation, which directly relates coarse-scale nodal internal forces with fine-scale stresses, thereby avoiding the typical nested local/global problems that appear, in the nonlinear regime, in other multiscale methods. This distinctive feature, along with hyperreduction schemes for nodal internal and external body forces , greatly facilitate the implementation of the proposed formulation in existing FE codes for solid elements. Indeed, one only has to change the location and weights of the integration points, and to replace a few polynomial-based FE matrices with “empirical” operators, i.e., derived from the information obtained in appropriately chosen computational experiments. We demonstrate that the elements resulting from this formulation are not afflicted by volumetric locking when dealing with nearly-incompressible materials, and that they can handle non-matching fine-scale grids as well as curved structures. Last but not least, we show that, for periodic structures, this method converges upon mesh refinement to the solution delivered by classical first-order computational homogenization. Thus, although the method does not presuppose scale separation, it can represent solutions in this limiting case by taking sufficiently small coarse-scale elements.

A continuum mixture model for transient heat conduction in multi-phase composites

The difference between transient thermal responses of each constituent in composites is one of the important factors to cause significant nonuniform deformation, high stress gradient and failure of composites. While direct numerical simulations for transient heat conduction in multi-phase composites are highly costly and even formidable, classical homogenization methods are focused on predicting the effective conductivities and the macroscopic thermal responses of a whole representative volume element, leaving the prediction of this difference at the macroscopic level to be a challenging issue during structural analysis of composite materials. In order to solve this issue, we propose a continuum mixture model for the transient heat conduction in multi-phase composites through dynamic homogenization methods. The derived governing equations retain the identity of each constituent behaviour and relate them to the overall response of each constituent. All the coefficients in the equations can be directly determined by the constituent phases and geometric parameters through explicit relations. The precision of the model is verified by comparison with direct numerical simulations. The results indicate that the model not only captures the disparate thermal responses of each constituent at the macroscopic level, but also accurately predicates the macro-average responses. The methodology can be generalized to other non-equilibrium thermodynamic phenomena such as diffusion and other kinds of conduction in multi-phase systems.

A mixed micromechanical homogenisation scheme for the prediction of the 3D orthotropic elastic properties of different masonry bond typologies

Masonry structures composed of two distinct material phases (units and mortar) with potentially vastly different elastic properties are characterised by strong orthotropy. The variety of commonly employed bond typologies for masonry walls, featuring unit groups and mortar joints with different orientation with respect to the plane of the wall, result in complex stress and strain interactions between the material phases. This interaction is further complicated by the high volume ratio of the units within the masonry composite. Therefore, classical homogenisation schemes for matrix-inclusion composite materials, mortar being the matrix and units the inclusions, are suitable for only a narrow range of relative unit and mortar stiffness, expressed as the ratio of the Young’s moduli of the two phases. This feature makes their application in nonlinear analysis of masonry structures, which have a strong tendency to crack under mechanical loading, problematic.The paper presents a novel homogenisation scheme based on different mean-field homogenisation schemes combined through a process derived from the discretisation of the masonry cells using a method-of-cells approach. This approach allows the intuitive interpretation of the stress and strain interaction of the material phases and yields good predictions of the elastic properties of a variety of masonry bonds over an extensive range of relative unit and mortar stiffness.

Automated Symbolic Upscaling: 1. Model Generation for Extended Applicability Regimes

AbstractIn porous media theory, upscaling techniques are fundamental to deriving rigorous Darcy‐scale models for flow and reactive transport in subsurface systems. Due to limitations in classical techniques, a number of ad hoc approaches have been proposed to address physical regimes in which reactive time scales are similar to, or faster than, diffusive time scales. In Part 1 of this two part series, we present a strategy for expanding the applicability of classical homogenization theory by generalizing the assumed closure form. We detail the implementation of this strategy on two reactive mass transport problems with moderately reactive physics. The strategy produces nontrivial homogenized models with emergent terms and effective parameters that couple reactive, diffusive, and advective transport. The differences in equation forms between the macroscopic and pore‐scale descriptions advise caution to further studies where the forms of macroscopic equations are assumed, as opposed to rigorously derived. Numerical validation is provided for each problem to show that the error estimates of homogenization theory are satisfied, and to justify the implemented strategy. In Part 2, the presented strategy is automated using symbolic computing to expedite its implementation.

Automated Symbolic Upscaling: 2. Model Generation for Extended Applicability Regimes

AbstractIn this second part of the two paper series, we detail an algorithmic procedure for systematically implementing the generalized closure form strategy presented in Part 1. This strategy extends the applicability of homogenized models with respect to classical homogenization theory, as demonstrated in Part 1 where upscaled models are rigorously derived in moderately reactive physical regimes. After encoding the algorithm into Symbolica, an automated upscaling framework, we upscale two reactive mass transport problems and numerically validate the resulting nonlinear homogenized models by showing the absolute error estimates predicted by homogenization theory are satisfied. In both problems, nontrivial closure forms and closure problems are automatically formulated using the encoded strategy with no human interaction, nor prior knowledge regarding the closure required for the systems. We hope these demonstrations spark further interest in automated analytical frameworks for multiscale modeling, as such capabilities are invaluable for generating rigorous multiscale models of complex phenomena in porous media.

Numerical Modeling of the Dynamic Elastic Modulus of Concrete.

This article introduces simulations of theoretical material with controlled properties for the evaluation of the effect of key parameters, as volumetric fractions, elastic properties of each phase and transition zone on the effective dynamic elastic modulus. The accuracy level of classical homogenization models was checked regarding the prediction of dynamic elastic modulus. Numerical simulations were performed with the finite element method for evaluations of the natural frequencies and their correlation with Ed through frequency equations. An acoustic test validated the numerical results and obtained the elastic modulus of concretes and mortars at 0.3, 0.5 and 0.7 water-cement ratios. Hirsch calibrated according to the numerical simulation (x = 0.27) exhibited a realistic behavior for concretes of w/c = 0.3 and 0.5, with a 5% error. However, when the water-to-cement ratio (w/c) was set to 0.7, Young's modulus displayed a resemblance to the Reuss model, akin to the simulated theoretical triphasic materials, considering matrix, coarse aggregate and a transition zone. Hashin-Shtrikman bounds is not perfectly applied to theoretical biphasic materials under dynamic situations.

A simple analytical estimate for the elastic-plastic behavior of two-phase bi-continuous isotropic composites

A simple modeling is proposed to estimate the elastic-plastic behavior of two-phase isotropic composites made of interpenetrated co-continuous phases. This nonlinear modeling, without equivalent so far, is based on the extension of an explicit Laminate System (LS) scheme proposed previously for linear elastic behavior of such co-continuous composites. Considering monotonic radial loadings, the effective nonlinear stiffness of bi-continuous elastic-plastic composites is estimated from stepwise linearizing the composite overall and phase behaviors using a classical secant linearization procedure. The efficiency of this proposed modeling is checked on some comparisons with rare experimental literature data and with other classical analytical homogenization estimate schemes.

Assessment of temporal homogeneity of long-term rainfall time-series datasets by applying classical homogeneity tests

Assessment of temporal homogeneity of long-term rainfall time-series datasets by applying classical homogeneity tests