Second-order Homogenization Research Articles

Propagation and non-reciprocity in time-modulated diffusion through the lens of high-order homogenization

The homogenization procedure developed here is conducted on a laminate with periodic space–time modulation on the fine scale: at the leading order, this modulation creates convection in the low-wavelength regime if both parameters are modulated. However, if only one parameter is modulated, which is more realistic, this convective term disappears and one recovers a standard diffusion equation with effective homogeneous parameters; this does not describe the non-reciprocity and the propagation of the field observed from exact dispersion diagrams. This inconsistency is corrected here by considering second-order homogenization that results in a non-reciprocal propagation term that is proved to be non-zero for any laminate and verified via numerical simulation. The same methodology is also applied to the case when the density is modulated in the heat equation, leading, therefore, to a corrective advective term that cancels out non-reciprocity at the leading order but not at the second order.

Effective boundary conditions for second-order homogenization

Using matched asymptotic expansions, we derive an equivalent bar model for a periodic, one-dimensional lattice made up of linear elastic springs connecting both nearest and next-nearest neighbors. We obtain a strain-gradient model with effective boundary conditions accounting for the boundary layers forming at the endpoints. It is accurate to second order in the scale separation parameter ɛ≪1, as shown by a comparison with the solution to the discrete lattice problem. The homogenized modulus associated with the gradient effect (gradient stiffness) is found negative, as is often the case in second-order homogenization. Negative gradient stiffnesses are widely viewed as paradoxical as they can induce short-wavelength oscillations in the homogenized solution. In the one-dimensional lattice, the asymptotically correct boundary conditions are shown to suppress the oscillations, thereby restoring consistency. By contrast, most of the existing work on second-order homogenization makes use of postulated boundary conditions which, we argue, not only ruin the order of the approximation but are also the root cause of the undesirable oscillations.

Stress-constrained optimization of multiscale structures with parameterized microarchitectures using machine learning

A multiscale topology optimization framework for stress-constrained design is presented. Spatially varying microstructures are distributed in the macroscale where their material properties are estimated using a neural network surrogate model for homogenized constitutive relations. Meanwhile, the local stress state of each microstructure is evaluated with another neural network trained to emulate second-order homogenization. This combination of two surrogate models — one for effective properties, one for local stress evaluation — is shown to accurately and efficiently predict relevant stress values in structures with spatially varying microstructures. An augmented lagrangian approach to stress-constrained optimization is then implemented to minimize the volume of multiscale structures subjected to stress constraints in each microstructure. Several examples show that the approach can produce designs with varied microarchitectures that respect local stress constraints. As expected, the distributed microstructures cannot surpass density-based topology optimization designs in canonical volume minimization problems. Despite this, the stress-constrained design of hierarchical structures remains an important component in the development of multiphysics and multifunctional design. This work presents an effective approach to multiscale optimization where a machine learning approach to local analysis has increased the information exchange between micro- and macroscales.

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.

Fully optimized second-order estimates for the macroscopic behavior and field statistics of particle-reinforced viscoplastic composites

This paper is concerned with the characterization of the macroscopic behavior and statistics for the distribution of the stress and strain-rate fields in composites consisting of random and isotropic suspensions of rigid spherical particles in power-law viscoplastic materials. For this purpose, use is made of the Fully Optimized Second-Order (FOSO) homogenization method (Ponte Castañeda, 2016) in combination with recently developed estimates (Kammer and Ponte Castañeda, 2022) for the macroscopic properties of the associated ‘linear comparison composite’ (LCC). Special attention is devoted to the method’s ability to account for the dependence of the homogenized properties of the nonlinear composite on the Lode angle (third invariant) of the applied loading. It is found that, while, for large particle volume fractions c, the effective flow stress is only weakly dependent on the Lode angle, for dilute volume fractions, the dependence on the Lode angle becomes more pronounced. In the ideally plastic limit, as c tends to zero, the effective yield stress is shown to depend linearly on c for axisymmetric shear, while this dependence becomes weaker with a non-analytic leading-order correction of O(c/lnc) for pure shear loading. This strong dependence on the Lode angle at dilute concentrations is shown to be due to significant differences in the local deformation patterns, which become strongly anisotropic and localize for pure shear conditions, but do not for axisymmetric shear. In turn, the FOSO homogenization method is able to capture the statistical features of these different deformation patterns by providing consistent estimates for the covariance tensor of the strain-rate field fluctuations in the matrix phase, which tend to become more strongly anisotropic for the pure shear case. As c increases, the shear bands are deflected by the randomly dispersed spheres leading to a more isotropic distribution of the stress and strain-rate fields, which is consistent with a weaker Lode angle effect. The estimates can also capture the effect of strong particle interactions, including the existence of a rigidity threshold where the macroscopic flow stress and field fluctuations blow up.

Hysteretic magnetic field analysis with Second-Order homogenization

A dynamic hysteresis model represented by the Cauer circuit was combined with the finite element analysis to efficiently compute magnetic field in micro-structured iron-cores. A finite element eddy-current analysis with higher order homogenization method is reformulated to implement the Cauer circuit representation. Three types of hysteretic Cauer circuits for the 2nd-order homogenization were examined. The analysis of wounded core under sinusoidal excitation showed that the 2nd order homogenization scheme with finite difference approximation improved the accuracy of representation of iron loss and current-flux loops. The 2nd-order scheme also improved the accuracy of homogenized representation under pulse-width-modulation excitation. The computation time was greatly reduced by the homogenization schemes.

Automated Extraction of Textural Features From Segmented Sentinel-1ASynthetic Aperture Radar Satellite Image Using Grey Level Co-Occurrence Matrix

This paper focuses on the automated extraction of textural features from segmented Sentinel-1A Synthetic Aperture Radar (SAR) imagery, which was captured on 25th August 2017. Textural properties play a pivotal role in discerning regions of interest in satellite imagery, essential for precise classification in diverse applications, including remote sensing, medical image analysis, biometric analysis, and document image analysis. To achieve this, the Local Adaptive Threshold (LAT) technique was employed for segmenting the foreground and background areas of the pre-processed vertical transmit-vertical receive (VV) polarized Sentinel-1A SAR data. The approach relies on second-order statistics, specifically co-occurrence measures, which evaluate the relationships between pairs of pixels within their local neighbourhood’s in the input image. Four fundamental second-order statistical measures, namely correlation, energy, homogeneity, and contrast, were computed using the Grey Level Co-occurrence Matrix (GLCM). The GLCM was generated by quantizing the foreground region of the segmented grey-scale image into different levels, including 8, 16, and 64 grey levels, considering two inter-pixel distances, d=1 and d=2, and adopting an omni-directional orientation (θ = 00, 450, 900, 1350).The results underscore the effectiveness of the GLCM-based approach in computing second-order statistical texture measures from the segmented SAR image. Notably, the findings highlight that quantizing the image to Ng=8 with d=1 and considering omni-directional statistical measures (θ = 00, 450, 900, 1350) significantly enhances performance.

Frame Duplication Forgery Detection in Surveillance Video Sequences Using Textural Features

Frame duplication forgery is the most common inter-frame video forgery type to alter the contents of digital video sequences. It can be used for removing or duplicating some events within the same video sequences. Most of the existing frame duplication forgery detection methods fail to detect highly similar frames in the surveillance videos. In this paper, we propose a frame duplication forgery detection method based on textural feature analysis of video frames for digital video sequences. Firstly, we compute the single-level 2-D wavelet decomposition for each frame in the forged video sequences. Secondly, textural features of each frame are extracted using the Gray Level of the Co-Occurrence Matrix (GLCM). Four second-order statistical descriptors, Contrast, Correlation, Energy, and Homogeneity, are computed for the extracted textural features of GLCM. Furthermore, we calculate four statistical features from each frame (standard deviation, entropy, Root-Mean-Square RMS, and variance). Finally, the combination of GLCM’s parameters and the other statistical features are then used to detect and localize the duplicated frames in the video sequences using the correlation between features. Experimental results demonstrate that the proposed approach outperforms other state-of-the-art (SOTA) methods in terms of Precision, Recall, and F1Score rates. Furthermore, the use of statistical features combined with GLCM features improves the performance of frame duplication forgery detection.

Willis couplings in continuously varying cross-sectional area duct.

Acoustic wave propagation in a one-dimensional periodic and asymmetric duct is studied theoretically and numerically to derive the effective properties. Closed form expressions for these effective properties, including the asymmetric Willis coupling, are derived through truncation of the Peano-Baker series expansion of the matricant (which links the state vectors at the two sides of the unit-cell) and Padé's approximation of the matrix exponential. The results of the first-order and second-order homogenization (with Willis coupling) procedures are compared with the numerical results. The second-order homogenization procedure provides scattering coefficients that are valid over a much larger frequency range than the usual first-order procedure. The frequency well below which the effective description is valid is compared with the lower bound of the first Bragg bandgap when the profile is approximated by a two-step function of identical indicator function, i.e., two different cross-sectional areas over the same length. This validity limit is then questioned, particularly with a focus on impedance modeling. This article attempts to facilitate the engineering use of Willis materials.

Second-order homogenization of 3-D lattice materials towards strain gradient media: numerical modelling and experimental verification

The literature in the field of higher-order homogenization is mainly focused on 2-D models aimed at composite materials, while it lacks a comprehensive model targeting 3-D lattice materials (with void being the inclusion) with complex cell topologies. For that, a computational homogenization scheme based on Mindlin (type II) strain gradient elasticity theory is developed here. The model is based on variational formulation with periodic boundary conditions, implemented in the open-source software FreeFEM to fully characterize the effective classical elastic, coupling, and gradient elastic matrices in lattice materials. Rigorous mathematical derivations based on equilibrium equations and Hill–Mandel lemma are provided, resulting in the introduction of macroscopic body forces and modifications in gradient elasticity tensors which eliminate the spurious gradient effects in the homogeneous material. The obtained homogenized classical and strain gradient elasticity matrices are positive definite, leading to a positive macroscopic strain energy density value—an important criterion that sometimes is overlooked. The model is employed to study the size effects in 2-D square and 3-D cubic lattice materials. For the case of 3-D cubic material, the model is verified using full-field simulations, isogeometric analysis, and experimental three-point bending tests. The results of computational homogenization scheme implemented through isogeometric simulations show a good agreement with full-field simulations and mechanical tests. The developed model is generic and can be used to derive the effective second-grade continuum for any 3-D architectured material with arbitrary geometry. However, the identification of the proper type of generalized continua for the mechanical analysis of different cell architectures is yet an open question.

Second-Order Homogenization of Periodic Schrödinger Operators with Highly Oscillating Potentials

Second-Order Homogenization of Periodic Schrödinger Operators with Highly Oscillating Potentials

A homogenized model accounting for dispersion, interfaces and source points for transient waves in 1D periodic media

A homogenized model is proposed for linear waves in 1D microstructured media. It combines second-order asymptotic homogenization (to account for dispersion) and interface correctors (for transmission from or towards homogeneous media). A new bound on a second-order effective coefficient is proven, ensuring well-posedness of the homogenized model whatever the microstructure. Based on an analogy with existing enriched continua, the evolution equations are reformulated as a dispersive hyperbolic system. The efficiency of the model is illustrated via time-domain numerical simulations. An extension to Dirac source terms is also proposed.

Towards the design of nature-inspired materials: Impact of complex pore morphologies via higher-order homogenization

Even though the development of novel materials that mimic nature is widely used in a variety of engineering and scientific fields, the relationship between effective material properties and underlying, often complex pore morphology is still not fully understood. To address this knowledge gap and accelerate the development of novel nature-inspired materials, this paper adopts a higher-order asymptotic homogenization method to numerically investigate the effect of complex micropore morphology on the effective mechanical properties of a porous system. Specifically, we create unique pore morphologies with varying levels of complexity that serve as a more realistic representation of natural materials. We then use the second-order homogenization method to capture the role of pore size, shape, orientation, and distribution on effective properties. By creating different pore morphologies, we systematically studied the relationship between morphology and effective mechanical properties. The results highlight the necessity of higher-order parameters to fully capture the role of realistic pore morphologies on effective mechanical properties and provide a path forward in the design of nature-inspired materials.

The effects of habitat heterogeneity, as measured by satellite image texture, on tropical forest bird distributions

Global biodiversity loss is most pronounced in the tropics. Monitoring of broad-scale patterns of habitat is essential for biodiversity conservation. Image texture measures derived from satellite data are proxies for habitat heterogeneity, but have not been tested in tropical forests. Our goal was to evaluate image texture to predict tropical forest bird distributions across Thailand for different guilds. We calculated a suite of texture measures from cumulative productivity (1-km fPAR-MODIS data) for Thailand's forests, and assessed how well texture measures predicted distributions of 86 tropical forest bird species in relation to body size, and nesting guild. Finally, we compared the predictive performance of combining (a) satellite image texture measures, (b) habitat composition, and (c) habitat fragmentation. We found that texture measures predicted occurrences of tropical forest birds well (AUC = 0.801 ± 0.063). Second-order homogeneity was the most predictive texture measure. Our models based on texture were significantly better for birds with larger body size (p < 0.05), but did not differ among nesting guilds (p > 0.05). Models that combined texture with habitat composition measures (AUC = 0.928 ± 0.038) outperformed models that combined fragmentation with habitat composition measures (AUC = 0.905 ± 0.047) (p < 0.05). The incorporation of texture, composition, and fragmentation variables significantly improved model accuracy over texture-only models (AUC = 0.801 ± 0.063 to AUC = 0.938 ± 0.034; p < 0.05). We suggest that texture measures are a valuable tool to predict bird distributions at broad scales in tropical forests.

Computational simulation of rolling process of polycrystalline plane strain model using second-order homogenization finite element method

Computational simulation of rolling process of polycrystalline plane strain model using second-order homogenization finite element method

Multi-scale study on interfacial bond failure between normal concrete (NC) and ultra-high performance concrete (UHPC)

It is a promising way to strengthen and repair the damaged normal concrete (NC) structure with ultra-high performance concrete (UHPC), and the interfacial performance between NC and UHPC is a critical issue. This paper aims to reveal the debonding failure performance between UHPC (repair material) and NC (substrate material) under tensile loading and shear loading. A multi-scale analysis method is established to simulate the mechanical behavior of the NC-UHPC interface, considering the complex interface geometry on the meso-scale. The failure of NC matrix is described by the elastic-plastic damage model in form of coupled gradient-enhanced damage evolution, the interface is represented by the cohesive zone model (CZM), and the UHPC is characterized by the second-order mean-field homogenization (MFH) method. After extracting the roughness characteristic parameters of the tested interfaces, the representative volume element (RVE) corresponding to the roughness is established, the load transfer and the debonding failure behavior are investigated, and the effective traction-separation law (TSL) is obtained. Finally, the tensile and the shear separation mechanisms of the NC-UHPC interface are revealed, and the effects of the roughness and strength of the interface on the macro shear and tensile strength are evaluated. Both the interface roughness and the substrate concrete strength have positive effects on the mechanical properties of the interface.

Thermoelastic wave propagation damping in a hollow FG-GPLRC cylinder with the spinning motion

This analysis deals with generalized thermoelasticity via Lord–Shulman’s theory for hollow cylinders reinforced with graphene platelets (GPL). The viscosity effect is also considered for applied materials. The second-order correlation homogenization techniques are utilized to obtain the equivalent thermo-mechanical properties. The hollow cylinder with spinning motion is modeled based on the linear thermoelastic constitutive law. The GPLs are functionally distributed along the cylinder’s radius based upon a power-law function. The Kelvin–Voigt type of viscosity behavior which is a differential state of viscous materials, is employed. The energy equation is attained based on the coupled and generalized framework. After obtaining the motion and energy equations in dimensionless form, the generalized differential quadrature (GDQ) and Newmark time marching methods are implemented to extract the temporal evolution and wave propagation of temperature, displacement, and stresses that occurred in the structure. After validating the responses with the available theoretical article, some numerical results are demonstrated to examine the effect of thermo-mechanical properties on the thermoelastic behavior of chosen nanocomposite cylinder.

Self-consistent assessments for the effective properties of two-phase composites within strain gradient elasticity

Analytical method for the second-order homogenization of two-phase composites within Mindlin–Toupin strain gradient elasticity theory is proposed. Direct approach and self-consistent approximation are used to reduce the homogenization problem to the problem of determination of averaged Cauchy stresses, double stresses and static moments of Cauchy stresses inside the inclusions under prescribed quadratic boundary conditions. The ellipsoidal shape of inclusions and orthotropic properties of phases are assumed. Extended equivalent inclusion method with linear eigenstrain is proposed to derive the explicit relations between the Eshelby-like tensors and corresponding concentration tensors that are used to define the averaged field variables inside the inclusions. Obtained analytical solutions allow to evaluate the effective classical and gradient elastic moduli of composite materials accounting for the phases properties, volume fraction, shape and size of inclusions. Presented solution for the effective gradient moduli covers the full range of volume fraction and correctly predicts the absence of gradient effects for the homogeneous classical Cauchy medium. Examples of calculations for the composites with spherical inclusions are given. Micro-scale definition of the well-known simplified strain gradient elasticity theory is provided. Namely, it is shown that this theory is the phenomenological continuum model for the composites with isotropic matrix and with the small volume fraction of stiff isotropic spherical inclusions, which have the same Poisson’s ratio to those one of matrix phase.

A hierarchical multi-scale analytical approach for predicting the elastic behavior of short fiber reinforced polymers under triaxial and flexural loading conditions

This paper presents a computationally efficient multi-scale analytical framework for predicting the effective elastic response of short fiber reinforced polymers (SFRPs) under triaxial and flexural loading conditions where the details of microstructure such as core/shell thickness, volume fraction distribution, fiber misalignment and fiber length variation are objectively taken into account. To this end, the mean-field homogenization and finite element approaches are compared to calculate the elastic response of SFRPs at the microscopic level while the orientation averaging approach is used to address the effects of fiber misalignment. The obtained mechanical behavior is then linked to an enhanced laminate theory to predict the effective triaxial and bending macrostructural behavior considering the core/shell effects and variation of volume fraction through the thickness. Using the second-order homogenization technique, the numerical validation of the proposed analytical approach is investigated based on the micro- and meso-scale analyses. Furthermore, the potential of the proposed strategy is demonstrated for hybrid composites. Finally, the accuracy of the suggested model is thoroughly studied using the available experimental tests in literature where the statistical information about the details of SFRP microstructures is presented.

On the free vibrations of FG-GPLRC folded plates using GDQE procedure

The present study theoretically investigates the performance of the generalized differential quadrature element (GDQE) method to obtain the free vibration response of folded plates. The considered folded plate is assumed as a laminated functionally graded composite augmented with graphene platelets (GPLs). This kind of composite is identified as the FG-GPLRC structure. Each lamina is strengthened with randomly oriented and uniformly distributed GPLs. The GPL weight fraction of each layer changes based on several functionally graded (FG) models. The effective Young modulus of each ply is calculated utilizing a second-order correlation homogenization technique called Halpin–Tsai. Based on the GDQE method, the folded plate is split into two plate elements. The motion equations of each element are derived based on the first-order shear deformation theory and utilizing Hamilton’s principle. Equations of motion for each element are solved by employing the conventional GDQ method. After that, the suitable continuity conditions are applied to the common boundaries of the two elements. In this research, the influences of plate dimensions, crank angle, FG patterns, boundary conditions, and GPL weight fraction on the natural frequencies and corresponding mode shapes are examined.