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In this article we study the homogenization and uniform decay rates estimates of the energy associated to the damped nonlinear wave equation ∂ t t u ε − Δ u ε + f (u ε) + a (x) g (∂ t u ε) = 0 in Ω ε × (0 , ∞) where Ω ε is a domain containing holes with small capacity (i.e. the holes are smaller than a critical size). The homogenization's proofs are based on the abstract framework introduced by Cioranescu and Murat [14] for the study of homogenization of elliptic problems. The main goal of this article is to prove, in one shot, uniform decay rate estimates of the energy associated to solutions of the problem posed in the perforated domain Ω ε as well as for the limit case Ω when ε → 0 by using refined arguments of microlocal analysis. [ABSTRACT FROM AUTHOR]

This work investigates asymptotic homogenization method (AHM) for axially graded beams that are mapped from periodic ones. The unit cell problems, the homogenized constitutive and governing equations are first established theoretically. Then a novel FE formulation of unit cell problems and effective stiffness, distinct from that of the periodic beams, is derived and resolved for solid elements. Besides, to improve analysis efficiency, an updated concise formulation is acquired for shell elements with proper handle of in-plane rotational DOFs, and a MATLAB code is presented to show implementation details. At last, four numerical examples show the correctness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we present the homogenization for nonlinear viscoelastic second-grade non-simple perforated materials at large strain in the quasistatic setting. The reference domain Ω ε is periodically perforated and is depending on the scaling parameter ε which describes the ratio between the size of the whole domain and the small periodic perforations. The mechanical energy depends on the gradient and also the second gradient of the deformation, and also respects positivity of the determinant of the deformation gradient. For the viscous stress we assume dynamic frame indifference and it is therefore depending on the rate of the Cauchy-stress tensor. For the derivation of the homogenized model for ε → 0 we use the method of two-scale convergence. For this uniform a priori estimates with respect to ε are necessary. The most crucial part is to estimate the rate of the deformation gradient. Due to the time-dependent frame indifference of the viscous term, we only get coercivity with respect to the rate of the Cauchy-stress tensor. To overcome this problem we derive a Korn inequality for non-constant coefficients on the perforated domain. The crucial point is to verify that the constant in this inequality, which is usually depending on the domain, can be chosen independently of the parameter ε . Further, we construct an extension operator for second order Sobolev spaces on perforated domains with operator norm independent of ε . [ABSTRACT FROM AUTHOR]

This article discusses the use of non-smooth variational methods in a wide range of applications, particularly in fields that involve partial differential equations. The authors focus on nonlinear problems in solid and fluid mechanics, as well as inverse and optimal control problems. They aim to develop non-standard analysis and effective numerical methods for solving these problems. The article also highlights the contributions of various authors from different countries and discusses the physical applications of non-smooth variational methods in areas such as elastic and inelastic solids, heat transfer, and fracture mechanics. [Extracted from the article]

Two hypotheses divide experts on determining the effective properties of composite materials using multi–scale homogenization methods. The first hypothesis states that multi-scale homogenization methods can ensure the direct determination of effective properties, at the macro level, of composite materials from a single representation of the medium at the lowest possible scale that allows for a good representation of all heterogeneities. The second hypothesis states that the determination cannot be ensured directly from a single scale but rather through multistep homogenization where each step represents the medium at a different scale from the lowest to the macroscale. To answer this question, a rigorous study is carried out; it includes calculating the two effective elastic properties, bulk, and shear moduli of three phases of a multi–layered sphere composite model by studying three phases. A multistep homogenization method is used to determine the effective properties of the composite and the obtained results are compared with those of the direct homogenization. Two different studies are considered: the first is based on an analytical model and the second on the numerical homogenization based on finite element calculation. To consider the effect of some influential parameters, several situations are treated by the combination of the variation of the volume fractions of the three phases and their property contrasts. The analytical calculations are performed using the Python 3.10 commercial software. It could be concluded that the effective elastic properties obtained either by the multistep or by the direct homogenization show no significant difference. [ABSTRACT FROM AUTHOR]

We derive effective models for a heterogeneous second-gradient elastic material taking into account chiral scale-size effects. Our classification of the effective equations depends on the hierarchy of four characteristic lengths: The size of the heterogeneities ℓ, the intrinsic lengths of the constituents ℓ SG and ℓ chiral , and the overall characteristic length of the domain L. Depending on the different scale interactions between ℓ SG , ℓ chiral , ℓ, and L we obtain either an effective Cauchy continuum or an effective second-gradient continuum. The working technique combines scaling arguments with the periodic homogenization asymptotic procedure. Both the passage to the homogenization limit and the unveiling of the correctors' structure rely on a suitable use of the periodic unfolding operator. [ABSTRACT FROM AUTHOR]

The aim of the present study is to design a solid material with specific and tailored mechanical properties through a suitably defined design framework and to evaluate the effectiveness of different microstructure geometries in an engineering perspective. To these ends, topology optimization algorithms are applied on a 2D homogenized equivalent model of a periodic structure. The design framework, developed in a previous work for 2D lattices made of regular hexagons, is here expanded and validated also in the cases of circular and square unit cells. The proposed approach involves optimizing porosity distribution of a homogenized equivalent solid, obtained through a Bloch–Floquet-based analysis, within a 2D lattice of regular unit cells forming the core element of a sandwich panel. Finite-element analyses on homogenized and fine structural models are carried out in order to validate the procedure, beyond the particular choice of the unit cell geometry and to detect its effectiveness and limits. [ABSTRACT FROM AUTHOR]

Purpose: Transient response of continuous composite material (CCM) fin made of high thermally conductive composite material is presented. The continuously varying effective properties of composite material such as thermal conductivity, heat capacity and density have been modelled using the Mori-Tanaka homogenization theory and rule of mixture. Additionally, temperature dependency of thermal conductivity, heat generation (composite materials) and convection coefficient (fluid properties) have also been incorporated. Different base boundary conditions are addressed such as oscillating heat flow, oscillating temperature, step-changing heat flow and step-changing temperature. At the other boundary, the fin is assumed to have a convective tip. Design/methodology/approach: Lattice Boltzmann method is implemented using an in-house source code for obtaining the numerical solution of typical non-linear heat balance equation of the aforementioned problem under various transient base boundary conditions. Findings: The effects of various thermal parameters such as material diffusivity ratio and conductivity ratio, area ratio and Biot number on transient response of fin and temperature distribution of fins are studied and interpreted. The heat transfer rate and time for attainment of steady state temperature of metal matrix composite (MMC) fin are found to be proportionally dependent on their diffusivity ratio. Additionally for higher values of area ratio and biot number, MMC fins are reported to dissipate the heat more efficiently in comparision to homogeneous fins in terms of time required to attain the steady state and surface temperature. Practical implications: Response of transient fin associated with advanced class of material can facilitates the practicing engineers for designing high-performance and/or miniaturized thermal management devices as used in electronic packaging industries. Originality/value: Studies of composite fin consisting of laminating second layer of material over the first layer have been reported previously, however transient response of CCM fin fabricated by continuously varying the volume fraction of two materials along the fin length has not been reported till date. Such material finds its application in thermal management and electronic packaging industries. Results are plotted in form of a graph for different application-wise material combinations that have not been reported earlier, and it can be treated as design data. [ABSTRACT FROM AUTHOR]

Exact correspondence relations between the bending solutions of a simply supported rectangular functionally graded material plate based on the four variable refined higher-order shear deformation theory and those of the corresponding reference homogenous Kirchhoff plate based on the classical plate theory are derived analytically for the material properties varying continuously in the thickness direction. The deflection, stress components, the resultant forces and bending moments of a thick functionally graded material plate are expressed analytically in terms of the deflection of the reference homogenous Kirchhoff plate with the same geometry, loadings and boundary constraints. Consequently, the bending solution of a functionally graded material plate based on the higher-order shear deformation theory is simplified as calculations of three scaling factors which can be easily determined analytically for the specified material gradient profile, the shear stress shape function and the aspect ratio of the functionally graded material plate, because the solution of the reference homogenous Kirchhoff plate can be easily found even in the text book. As examples, particular solutions for a functionally graded material plate subjected to both uniformly and sinusoidally distributed loads are presented, which illustrate the validity of this new approach. Accuracy of the present solutions are demonstrated by comparing them with those obtained by different palate theories with different shear stress shape functions available in the literature. The analytical solutions can be used as benchmarks to check numerical solutions of static bending of functionally graded material plates based on different higher-order shear deformation theories. [ABSTRACT FROM AUTHOR]

A spectro-hierarchical algorithm is proposed to determine an approximate solution to the elasto-dynamic problem for periodic heterogeneous materials. Such a homogenization scheme is derived by employing tools from perturbation theory in finite dimension used in the context of the celebrated Theorem of Nekhoroshev. In particular, it is shown how a classical algorithm based on a suitable hierarchy of harmonics can be implemented for the problem at hand, leading to the explicit construction of functions approximating the solution of the original problem with an error that is superexponentially small in the cell dimension. According to this approach, the fully homogenized model turns out to be naturally related to the "integrable case" of perturbation theory. Furthermore, all the featured constants are estimated explicitly. More importantly, a fully detailed bound of the threshold for the cell dimension is presented to ensure the validity of the theory. An example of application of the described theory is given in the final section for the case of a layered material. • The paper deals with the system of PDEs arising from elasto-dynamic problems in Cauchy materials. • Focuses on a novel approach based on perturbation theory to deal with homogenisation-type problems. • Provides an explicit scheme to approximate the solutions of the problem at hand. • Gives explicit estimates of validity of the theory. [ABSTRACT FROM AUTHOR]

We extend the applicability of the embedded unit cell (EUC) method to three-dimensional (3D) fracture problems, which are modeled by the extended finite element method (XFEM). The EUC method is a concurrent multiscale method based on the computational homogenization theory for nonperiodic domains. Herein, we show that this method can accurately estimate fracture parameters and, in particular, stress intensity factors using the J-integral method. Additionally, the method is shown to capture crack propagation within the microscale domain, as well as cracks initiating at the microscale and propagating outwards onto the macroscale through the internal subdomain boundaries. To demonstrate the accuracy, robustness, and computational efficiency of the proposed method, several 3D numerical benchmark examples, including planar cracks with single and mixed-mode fractures, are considered. In particular, we analyze horizontal, inclined, square, and penny-shaped cracks embedded in a homogeneous material. The results are verified against full FEM models and known analytical solutions if available. Practical Applications: The insights of this research offer practical application for engineers and scientists in designing more resilient and durable structures. By extending the EUC method to 3D fracture problems, the study addresses the ability to forecast and access fracture phenomena in materials. This method is shown to be an effective approach for exploring the interaction between local microscopic discontinuities and cross-scale crack propagation, crucial for evaluating the durability of engineering structures. The EUC approach, integrated with XFEM, provides a comprehensive methodology for analyzing different fracture scenarios, including stationary mixed-mode cracks and crack-propagation examples. Its ability to accurately transition from microscale to macroscale analysis without remeshing introduces a valuable computational advantage, making it a more cost-efficient solution in fracture analysis. This research's application offers tangible benefits in industrial context, especially in aerospace, automotive, and construction industries, where precise evaluation of structure and material failure can lead to more cost-efficient and safer designs by reducing the maintenance costs and failure risks. [ABSTRACT FROM AUTHOR]

Sandwich structures with lattice cores are novel, lightweight composite structures and are widely used in the aerospace industry. Besides, the aeroelastic behavior of sandwich panels in a supersonic flow regime still needs to be thoroughly studied. This work investigates the supersonic flutter of a sandwich panel whose core is topology-optimized. A finite element model of a sandwich panel based on the layerwise theory, coupled with the first-order piston theory, is presented. The sandwich panel core is assessed using a topology optimization approach with flutter loading constraints. The subsequent analytical homogenization scheme is developed to provide the equivalent mechanical properties of the topology-optimized panel. The modeling approach is fully validated, and the results demonstrate that the sandwich panel is capable of enlarging the flutter-free operational flight range when compared with other conventional panel designs. A parametric analysis of the topology-optimized sandwich panel regarding the critical flutter conditions is performed. [ABSTRACT FROM AUTHOR]

This paper describes several different variational formulations of the so-called 'cellular problem'' which is a system of partial differential equations arising in the theory of homogenization, subject to periodicity boundary conditions. These variational formulations of the cellular problem, all of them equivalent, have as main unknown the displacement, the stress or the strain, respectively. For each of these three cases, an equivalent minimization problem is introduced. The variational formulation in stress proves to have a distinguished role and it gives rise to two dual formulations, one in displacement-stress and another one in strain-stress. The corresponding Lagrangians may be used in numerical optimization for developing algorithms based on alternated directions, of Uzawa type. [ABSTRACT FROM AUTHOR]

In this paper we consider a nonlinear reaction‐diffusion problem with low diffusion scales of order (εδ(ε))2$(\epsilon \delta (\epsilon))^{2}$ in a porous domain which periodically perforated by two different size obstacles. A nonlinear Neumann condition is prescribed on the boundaries of smaller obstacles, while a homogeneous Neumann condition is imposed on the other boundaries. Our aim is to derive the homogenized model as the small parameters tend to zero by using the method of three‐scale convergence and periodic reiterated unfolding operator. The final homogenized macroscopic model depends on three scales. [ABSTRACT FROM AUTHOR]

The present paper studies the homogenization of the parabolic equation stated in a domain perforated by "tiny" balls. On the boundary of these perforations, a unilateral dynamic boundary constraints are specified. We address the so-called "critical" case that is characterized by a relation between the coefficient in the boundary condition, the period of the structure and the size of the holes. In this case, the homogenized equation contains a nonlocal "strange" term. This term is obtained as a solution to the variational problem involving an ordinary differential operator. [ABSTRACT FROM AUTHOR]

In the quest to elevate the mechanical properties of cellulose nanofiber-reinforced poly (lactic acid) composite materials for robust industrial applications, we conducted multiscale nonlinear finite element analysis rooted in homogenization theory. Our study delved into the intricate web of mechanical mechanisms and their pivotal influencers. This paper zeroed in on three paramount factors: the distribution of fibers, the fiber–matrix interface, and the aspect ratio of the fibers. For fiber distribution, we assumed fiber orientation control in fused filament fabrication and clarified the limit of material improvement by setting one-dimensional orientation as the upper limit. Our analysis of the fiber–matrix interface proposed a comprehensive three-scale structure, encompassing microstructures featuring intricate interfacial phases around the fibers, sub-microstructures with myriad dispersed fibers, and macrostructures subjected to external loads. The profound impact of the fiber–matrix interface on material properties was unveiled through a meticulous two-step homogenization process. Furthermore, we examined the substantial influence of the fiber aspect ratio on material properties. By unraveling the complex interplay among these critical factors, this research provided new design guidelines for fiber morphology to the performance of cellulose composites. [ABSTRACT FROM AUTHOR]

A multi-scale modeling framework is proposed for the prediction of the chemo-mechanical degradation of paper, with the particular aim of uncovering the key factors affecting the degradation process. Paper is represented as a two-dimensional, periodic repetition of a fibrous network unit cell, where the fibers are characterized by a moisture-dependent chemo-hygro-mechanical constitutive behavior. The degradation of paper occurs primarily as a result of the hydrolysis of cellulose, which causes a reduction of the degree of polymerization and a consequent decrease of the effective mechanical properties, ultimately leading to fiber embrittlement and a loss of material integrity. The interplay between the acidity of the paper, the ambient environmental conditions, and its chemo-mechanical degradation behaviour is a complex process. In the model, these interactions are accounted for by determining the coupled temporal evolution of the degree of polymerization, the acidity of the paper, and the moisture content, from which the time-dependent tensile strength of the paper is calculated. The internal stresses developing in the fibrous network under a change in moisture content lead to brittle fiber fracture once they reach the fiber tensile strength. The successive breakage of individual fibers results in damage development in the fibrous network, altering its effective constitutive properties. The temporal evolution of the effective hygro-mechanical properties of the fibrous network is calculated by employing asymptotic homogenization. For obtaining accurate model input, the strength and stiffness properties of individual fibers and the degree of polymerization of paper samples are measured at different ageing times by carrying out dedicated experiments. Subsequently, a series of numerical simulations is performed to analyze the chemo-mechanical degradation process of paper, highlighting the influence of the time-evolving acidity and moisture content. The numerical study further considers the effects of micro-structural features (i.e., the anisotropy of the fibrous network orientation and the fiber longitudinal elastic modulus) on the macroscopic degradation response of paper. The results of this work may help conservators of cultural heritage institutions determining optimal environmental conditions to limit or delay the time-dependent degradation of valuable historical paper artefacts. [ABSTRACT FROM AUTHOR]

The intractable multiscale constitutives and the high computational cost in direct numerical simulations are the bottlenecks in fracture analysis of heterogeneous materials. In an attempt to achieve a balance between accuracy and efficiency, we propose a mathematically rigorous phase‐field model for multiscale fracture. Leveraging the phase‐field theory, the difficulty of discrete‐continuous coupling in conventional cross‐scale crack propagation analysis is resolved by constructing a continuum description of the crack. Based on the asymptotic expansion, an equivalent two‐field coupled boundary‐value problem is well‐defined, from which we rigorously derive the macroscopic equivalent parameters, including the equivalent elasticity tensor and the equivalent fracture toughness tensor. In our approach, both the displacement field and the phase‐field are simultaneously expanded, allowing us to obtain a fracture toughness tensor with diagonal elements of the corresponding matrix controlling anisotropic fracture behavior and non‐diagonal elements governing crack deflection. This enables multiscale finite element homogenization procedure to accurately reproduce microstructural information, and capture the crack deflection angle in anisotropic materials without any a priori knowledge. From the numerical results, the proposed multiscale phase‐field method demonstrates a significant reduction in computation time with respect to full‐field simulations. Moreover, the method accurately reproduces physical consistent anisotropic fracture of non‐centrosymmetric porous media, and the experimentally consistent damage response of fiber‐reinforced composites. This work fuses well‐established mathematical homogenization theory with the cutting‐edge fracture phase‐field method, sparking a fresh perspective for the fracture of heterogeneous media. [ABSTRACT FROM AUTHOR]

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 . [ABSTRACT FROM AUTHOR]

We consider the homogenisation of the quasi-stationary Stokes equations in a porous medium that evolves over time. The evolution is a priori given. At the interface of the pore space and the solid part, we prescribe an inhomogeneous Dirichlet boundary condition, which enables a no-slip boundary condition at the evolving boundary. We pass rigorously to the homogenisation limit employing the two-scale transformation method. In order to derive uniform a priori estimates, we show a Korn-type inequality for the two-scale transformation method. The homogenisation result is a new version of Darcy's law. It features a time- and space-dependent permeability tensor, which accounts for the local pore structure, and a macroscopic inhomogeneous divergence condition, which induces a new source term for the pressure. In the case of a no-slip boundary condition at the interface, this source term relates to the change of the local pore volume. [ABSTRACT FROM AUTHOR]

Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting in simplified equations that are computationally tractable while accurately predicting the macroscopic response. In the field of continuum mechanics, homogenization is crucial for deriving constitutive laws that incorporate microscale physics in order to formulate balance laws for the macroscopic quantities of interest. However, obtaining homogenized constitutive laws is often challenging as they do not in general have an analytic form and can exhibit phenomena not present on the microscale. In response, data-driven learning of the constitutive law has been proposed as appropriate for this task. However, a major challenge in data-driven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material. These discontinuities in the coefficients affect the smoothness of the solutions of the underlying equations. Given the prevalence of discontinuous materials in continuum mechanics applications, it is important to address the challenge of learning in this context, in particular, to develop underpinning theory that establishes the reliability of data-driven methods in this scientific domain. The paper addresses this unexplored challenge by investigating the learnability of homogenized constitutive laws for elliptic operators in the presence of such complexities. Approximation theory is presented, and numerical experiments are performed which validate the theory in the context of learning the solution operator defined by the cell problem arising in homogenization for elliptic PDEs. [ABSTRACT FROM AUTHOR]

This paper illustrates how particle size affects the initial yield stress of particle-reinforced composites. A formulation in a closed form is presented to demonstrate the size effect of yielding of the composites. This paper also demonstrates that there is an upper bound and a lower bound for the size-dependent yield stress with the change of particle size. This means that decreasing particle size increases its yield stress up to an upper bound. Similarly, increasing particle size decrease its yield stress up to a lower bound. In this paper the asymptotic homogenization method is used in framework of the Cosserat elasticity. A virtual "unreinforced matrix" is introduced as a reference configuration. As a numerical example, the size effect of yielding of Sicp/Al is predicted. [ABSTRACT FROM AUTHOR]

Elastomeric mechanical metamaterials exhibit unconventional behaviour, emerging from their microstructures often deforming in a highly nonlinear and unstable manner. Such microstructural pattern transformations lead to non-local behaviour and induce abrupt changes in the effective properties, beneficial for engineering applications. To avoid expensive simulations fully resolving the underlying microstructure, homogenization methods are employed. In this contribution, a systematic comparative study is performed, assessing the predictive capability of several computational homogenization schemes in the realm of two-dimensional elastomeric metamaterials with a square stacking of circular holes. In particular, classical first-order and two enriched schemes of second-order and micromorphic computational homogenization type are compared with ensemble-averaged full direct numerical simulations on three examples: uniform compression and bending of an infinite specimen, and compression of a finite specimen. It is shown that although the second-order scheme provides good qualitative predictions, it fails in accurately capturing bifurcation strains and slightly over-predicts the homogenized response. The micromorphic method provides the most accurate prediction for tested examples, although soft boundary layers induce large errors at small scale ratios. The first-order scheme yields good predictions for high separations of scales, but suffers from convergence issues, especially when localization occurs. [ABSTRACT FROM AUTHOR]

• Thermomechanical problem of FGMs sphere with material uncertainties is analyzed. • The interval random uncertainty model is recommended for material parameters. • A data-driven ANN fast calculation for FGMs sphere uncertainties analysis is established. • The volume fraction distribution is optimized for the safety design of the FGMs sphere. Mechanical analysis and optimal design for functionally graded materials (FGMs) structures are significant. The present paper analyzes the parameter uncertainties problem for the thermomechanical response of the FGMs spherical shells with volume fraction power distribution. Here, the interval random uncertainty model is recommended to describe the uncertainties of each component of material parameters in FGMs, and a data-driven artificial neural network (ANN) fast calculation for FGMs spherical shells thermomechanical problems with uncertainties analysis is established and trained. The thermomechanical coupling response analysis and volume fraction optimization analysis of the FGMs spherical shell with material uncertainties are realized for zirconia-titanium FGMs, with the maximum normalized Von Mises stress as the structural safety index and the ANN as the fast calculation method. In addition, the Von Mises stress of the FGMs spherical shell and a ZrO 2 -Ti alloy double-layer spherical shell under different loads is compared. The results show that material parameter uncertainties, especially those in Young's modulus, thermal expansion coefficient, and uniaxial yield limit significantly affect structural safety. The structural safety index λ calculated based on the volume fraction optimization design of FGMs spherical shell is significantly lower than that of the ZrO 2 -Ti alloy double-layer. The ANN method greatly improves the computational efficiency for uncertain problems and maintains high accuracy, compared with the statistical analysis after many calculations by randomly selecting material parameters. This study provides a path for the safety design of FGMs structures with uncertain parameters. [ABSTRACT FROM AUTHOR]

Surface-based inflatables are composed of two thin layers of nearly inextensible sheet material joined together along carefully selected fusing curves. During inflation, pressure forces separate the two sheets to maximize the enclosed volume. The fusing curves restrict this expansion, leading to a spatially varying in-plane contraction and hence metric frustration. The inflated structure settles into a 3D equilibrium that balances pressure forces with the internal elastic forces of the sheets. We present a computational framework for analyzing and designing surface-based inflatable structures with arbitrary fusing patterns. Our approach employs numerical homogenization to characterize the behavior of parametric families of periodic inflatable patch geometries, which can then be combined to tessellate the sheet with smoothly varying patterns. We propose a novel parametrization of the underlying deformation space that allows accurate, efficient, and systematical analysis of the stretching and bending behavior of inflated patches with potentially open boundaries. We apply our homogenization algorithm to create a database of geometrically diverse fusing patterns spanning a wide range of material properties and deformation characteristics. This database is employed in an inverse design algorithm that solves for fusing curves to best approximate a given input target surface. Local patches are selected and blended to form a global network of curves based on a geometric flattening algorithm. These fusing curves are then further optimized to minimize the distance of the deployed structure to target surface. We show that this approach offers greater flexibility to approximate given target geometries compared to previous work while significantly improving structural performance. [ABSTRACT FROM AUTHOR]

In this paper we study critical sets of solutions uε$u_\varepsilon$ of second‐order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. Under some condition on the first‐order correctors, we show that the (d−2)$(d-2)$‐dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period ε$\varepsilon$, provided that doubling indices for solutions are bounded. The key step is an estimate of "turning" of an approximate tangent map, the projection of a non‐constant solution uε$u_\varepsilon$ onto the subspace of spherical harmonics of order ℓ$\ell$, when the doubling index for uε$u_\varepsilon$ on a sphere ∂B(0,r)$\partial B(0, r)$ is trapped between ℓ−δ$\ell -\delta$ and ℓ+δ$\ell +\delta$, for r$r$ between 1 and a minimal radius r∗≥C0ε$r^*\ge C_0\varepsilon$. This estimate is proved by using harmonic approximation successively. With a suitable L2$L^2$ renormalization as well as rescaling we are able to control the accumulated errors introduced by homogenization and projection. Our proof also gives uniform bounds for Minkowski contents of the critical sets. [ABSTRACT FROM AUTHOR]

In this article, we propose a novel homogenization method for piezoelectric composites with periodic microstructures. In the proposed homogenization method, one can find the analytical expressions for the material properties of the composites in the representative volume element (RVE) via diffused material interface method. The availability of analytical expressions for the material properties in the entire domain of the RVE enables one to determine the effective homogenized material properties with the standard finite element method. As the proposed homogenization method regularizes the discontinuity in material properties across the interfaces, there is no need to explicitly track the material interfaces while implementing the finite element method for determining the effective material properties. Hence, one can implement the proposed homogenized method for any piezoelectric composites with complicated material interfaces and determine the effective material properties. In this study, we have considered the piezoelectric composites to be comprised of periodic microstructures where the RVE constitutes a matrix with an inclusion of a specific shape. We have carried out a study on the effect of the shape of inclusion viz. square-shaped, I-shaped, T-shaped, and plus-shaped, and the size of inclusion on the effective piezoelectric material properties. We have also studied the influence of shape and size of inclusion on the electro-mechanical response of a homogenized piezoelectric continuum. To implement the proposed homogenization method, we have used Gridap, an open-source finite element toolbox in Julia that provides very compact codes freely available to all the researchers and makes a third-party verification of the proposed method straightforward. [ABSTRACT FROM AUTHOR]

• Reliability-based non-probabilistic multiscale topology optimization (NRBMTO) with stress constrained is proposed. • The p-norm function aggregates global unit stresses. • Mechanical and thermo-mechanical loads are considered as non-probabilistic uncertain parameters. • Ellipsoid modeling describes uncertainty in non-probabilistic random variables. • NRBMTO results in more security than the traditional deterministic multiscale topology optimization (DMTO) approach. A reliability-based non-probabilistic multiscale topology optimization (NRBMTO) method with stress constraints is proposed for thermo-mechanical continuous structures with uncontrollable stresses. The physical parameters, external loads and temperature values at the macro scale, are regarded as non-probabilistic uncertain parameters in the optimization of structural topologies with complex physical fields at multi-scale. The homogenization-based finite element method is employed to quantify thermo-mechanical structures with multi-scale uncertain parameters in the established multi-scale topology model. The ellipsoid model is applied to describe the uncertainty of non-probabilistic random variables, and the non-probabilistic reliability index is obtained by estimating the failure probability based on the first-order reliability method (FORM). The unit stresses are aggregated to the global maximum stresses with the normalized p-norm function, taking into account the mechanical and thermal stresses. The sensitivity information of the compliance and stress constraint to the macro- and micro-design variables and uncertain variables are derived simultaneously. The macro- and micro- design variables are solved by the method of moving asymptotes (MMA), respectively. Several numerical examples are given to verify the effectiveness and feasibility of the proposed NRBMTO method. The results demonstrate that the optimized structure based on the NRBMTO method provides better security with reliability index β =3 and minimum compliance (244.39) while stress is controlled below 235 MPa compared to the classical deterministic multiscale topology optimization (DMTO) method. [ABSTRACT FROM AUTHOR]

This article extends recently developed finite variation sensitivity analysis (FVSA) approaches to an inverse homogenization problem. The design of metamaterials with prescribed mechanical properties is stated as a discrete density‐based topology optimization problem, in which the design variables define the microstructure of the periodic base cell. The FVSA consists in estimating the finite variations of the objective and constraint functions after independently switching the state of each variable. It is used to properly linearize the functions of binary variables so the optimization problem can be solved through sequential integer linear programming. Novel sensitivity expressions were developed and it was shown that they are more accurate than the ones conventionally used in literature. Referred to as the conjugate gradient sensitivity (CGS) approach, the proposed strategy was quantitatively evaluated through numerical examples. In these examples, metamaterials with prescribed homogenized Poisson's ratios and minimal homogenized Young's moduli were obtained. A hexagonal base cell with dihedral D3$$ {D}_3 $$ symmetry was used to produce only metamaterials with isotropic properties. It was shown that, by using the CGS approach instead of the conventional sensitivity analysis, the sensitivity error was substantially reduced for the considered problem. The proposed developments effectively improved the stability and robustness of the discrete optimization procedures. In all the considered examples, when more accurate sensitivity analyses were performed, the parameters of the topology optimization method could be tuned more easily, yielding effective solutions even if the settings were not ideal. [ABSTRACT FROM AUTHOR]

We propose a homogenized supremal functional rigorously derived via L p -approximation by functionals of the type ess-sup x ∈ Ω f (x , D u) , when Ω is a bounded open set of ℝ n and u ∈ W 1 , ∞ (Ω ; ℝ d). The homogenized functional is also deduced directly in the case where the sublevel sets of f (x , ⋅) satisfy suitable convexity properties, as a corollary of homogenization results dealing with pointwise gradient constrained integral functionals. [ABSTRACT FROM AUTHOR]

We derive, through the deterministic homogenization theory in thin domains, a new model consisting of Hele–Shaw equation with memory coupled with the convective Cahn–Hilliard equation. The obtained system, which models in particular tumor growth, is then analyzed and we prove its well-posedness in dimension 2. To achieve our goal, we develop and use the new concept of sigma-convergence in thin heterogeneous media, and we prove some regularity results for the upscaled model. [ABSTRACT FROM AUTHOR]

Weak form equation learning and surrogate modeling has proven to be computationally efficient and robust to measurement noise in a wide range of applications including ODE, PDE, and SDE discovery, as well as in coarse-graining applications, such as homogenization and mean-field descriptions of interacting particle systems. In this work we extend this coarse-graining capability to the setting of Hamiltonian dynamics which possess approximate symmetries associated with timescale separation. A smooth ε -dependent Hamiltonian vector field X ε possesses an approximate symmetry if the limiting vector field X 0 = lim ε → 0 X ε possesses an exact symmetry. Such approximate symmetries often lead to the existence of a Hamiltonian system of reduced dimension that may be used to efficiently capture the dynamics of the symmetry-invariant dependent variables. Deriving such reduced systems, or approximating them numerically, is an ongoing challenge. We demonstrate that WSINDy can successfully identify this reduced Hamiltonian system in the presence of large perturbations imparted in the ε > 0 regime, while remaining robust to extrinsic noise. This is significant in part due to the nontrivial means by which such systems are derived analytically. WSINDy naturally preserves the Hamiltonian structure by restricting to a trial basis of Hamiltonian vector fields. The methodology is computationally efficient, often requiring only a single trajectory to learn the global reduced Hamiltonian, and avoiding forward solves in the learning process. In this way, we argue that weak-form equation learning is particularly well-suited for Hamiltonian coarse-graining. Using nearly-periodic Hamiltonian systems as a prototypical class of systems with approximate symmetries, we show that WSINDy robustly identifies the correct leading-order system, with dimension reduced by at least two, upon observation of the relevant degrees of freedom. While our main contribution is computational, we also provide a contribution to the literature on averaging theory by proving that first-order averaging at the level of vector fields preserves Hamiltonian structure in nearly-periodic Hamiltonian systems. This provides theoretical justification for our approach as WSINDy's computations occur at the level of Hamiltonian vector fields. We illustrate the efficacy of our proposed method using physically relevant examples, including coupled oscillator dynamics, the Hénon–Heiles system for stellar motion within a galaxy, and the dynamics of charged particles. [ABSTRACT FROM AUTHOR]

This paper deals with the long-wavelength behaviour of a Euler beam periodically supported by co-located rotation and compression springs. An asymptotic homogenization method is applied to derive the several macroscopic models according to the stiffness contrasts between the elastic supports and the beam. Effective models of differential order two or four are obtained, which can be merged into a single unified model whose dispersion relations at long and medium wavelengths fit those derived by Floquet-Bloch. Moreover, the essential role of rotation supports is clearly evidenced. A mixed "discrete/continuous" approach to the boundary conditions is proposed, which allows the boundary conditions actually applied at the local scale to be expressed in terms of Robin-type boundary conditions on macroscopic variables. This approach can be applied to both dominant-order and higher-order models. The modal analysis performed with these boundary conditions and the homogenised models gives results in good agreement with a full finite element calculation, with great economy of numerical resources. [ABSTRACT FROM AUTHOR]

In this article a FE2 shell model for thin and thick shells within a first order homogenization scheme is presented. A variational formulation for the two‐scale boundary value problem and the associated finite element formulation is developed. Constraints with 5 or 9 Lagrange parameters are derived which eliminate both rigid body movements and dependencies of the shear stiffness on the size of the representative volume elements (RVEs). At the bottom and top surface of the RVEs which extend through the total thickness of the shell stress boundary conditions are present. The periodic boundary conditions at the lateral surfaces are applied in such a way that particular membrane, bending and shear modes are not restrained. This is shown by means of a homogeneous RVE. The first of all linear formulation is extended to finite strain problems introducing transformation relations for the stress resultants and the material matrix. The transformations are performed at the Gauss points on macro level. Several boundary value problems including large deformations, stability and inelasticity are computed and compared with 3D reference solutions. [ABSTRACT FROM AUTHOR]

Studying convection–diffusion problems of delayed type in physics helps us to understand transport phenomena and has practical applications in various fields. The mathematical analysis of this model has practical applications in various fields, such as flow dynamics, material science, and environmental modeling. In this paper, the theory of reproducing kernel spaces (RKS) is utilized to solve singularly perturbed 2D parabolic convection–diffusion problems of delayed type. To this end, a series form for the solution is first constructed in reproducing kernel Hilbert space, and then, the approximate solution is given as an N -term summation. The main contribution of the present research is that, for the first time, a novel formula is found for the homogenization of 2D initial-boundary-value problems. Furthermore, a semi-analytical RKS method is employed without employing the Gram–Schmidt orthogonalization algorithm. We derive theorems to reveal stability and convergence properties which are examined by numerical experiments. The technique is especially suited for problems having boundary-layer behavior. Numerical results are provided to demonstrate the efficiency, stability, and superiority of the proposed technique. [ABSTRACT FROM AUTHOR]

We study the asymptotic behaviour of a linear optimal control problem posed on a locally periodic rapidly oscillating domain. We consider an $ L^2 $ L 2 -cost functional constrained by a Poisson problem having a mixed boundary condition: we assume a homogeneous Neumann condition on the oscillating part of the boundary and a homogeneous Dirichlet condition on the remaining part. [ABSTRACT FROM AUTHOR]

The present investigation has analyzed the buckling of bi-directional functionally graded conical micro-shells bearing axial loading. The dominant equations serve on the basis of the theory of first-order shear deformation in the modified couple stress theory (MCST). The Ritz technique is used so as to solve the governing equations. Bi-directional functionally graded material is the material used to construct the shell, and in a predefined composition profile, the volume fractions pertaining to the constituent materials constantly vary through the conical edge directions and thickness. The model predictions are compared and successfully validated using the ones available in the literature. According to the results, in comparison with the classical theory, the micro-shell buckling load is higher in the MCST. In addition, the contribution of a number of geometrical and mechanical parameters, for example, radius-to-thickness ratio, thickness-to-length scale ratio, length-to-radius ratio, semi-vertex angle, homogenization schemes, and material gradient indexes, to the buckling performance of conical micro-shells has been investigated. It is noteworthy that this investigation is the first effort to present a buckling analysis for the purpose of bi-directional functionally graded truncated conical micro-shells in accordance with the MCST. [ABSTRACT FROM AUTHOR]

The nonlinear frequency responses of functionally graded (ceramic/metal/ceramic) straight/curved blade-type structures are examined under rotation and thermal environment. Here, the geometrical nonlinearity is introduced using Green-Lagrange's strain via the higher-order shear deformation theory. The constituent materials, i.e. ceramic and metal-alloy, are considered to be temperature-dependent, whereas, the overall blade material properties are evaluated using Voigt's homogenization scheme via a modified power-law function. The equations of motion are obtained using Hamilton's principle by including the geometrical nonlinearity and stress stiffening effects. The linear and the nonlinear frequency responses are computed through the developed nonlinear finite element method and Picard's successive iteration scheme. An exhaustive computation is carried out to report the linear and the nonlinear vibrational behavior of the proposed composite straight/curved blade model at small-to-large amplitudes for various sets of combinations of geometric and material parameters, such as blade span, blade thickness, volume fractions, rotational speed, and temperature. [ABSTRACT FROM AUTHOR]

The present study deals with shear wave propagation in a fully coupled Magneto-Electro-Elastic (MEE) multi-laminated periodic structure having non-uniform and imperfect interfaces. As a solution methodology, we applied a more general low-frequency dynamic asymptotic homogenization technique where the solution will be single-frequency dependent and the obtained results generalize those published in Chaki and Bravo-Castillero (Compos Struct 322:117410, 2023b) where the perfect contact case was studied. Effective homogenized dispersive equations of motion in second- and fourth-order approximations, also known as "Good" Boussinesq equations in elastic case, are derived. Local problems, closed-form expression of dispersion equations in second and fourth-order approximations and closed-form solutions of first and second local problems in second-order approximation for tri-laminated MEE periodic structure have been obtained and also validated for elastic laminates with imperfect contact case and MEE laminates with perfect contact case. The effect of non-uniform and imperfect contact, angle of incidence, unit cell size, volume fraction and ME-coupling on the wave propagation is illustrated through dispersion graphs. The effect of non-uniform and imperfect contact on dispersion curve serves as the highlight of the present work. [ABSTRACT FROM AUTHOR]

The present work aims to propose an interface-enriched isogeometric analysis strategy for predicting the size-dependent effective moduli and local stress field of periodic arrays of nanosize inhomogeneity. The proposed framework allows for an exact representation of the curved boundary of inhomogeneity inside the matrix due to the representation of the geometry of repeating unit cells for microstructured materials with nonuniform rational B-splines. The energetic surface was characterized by the Gurtin–Murdoch model, and it was incorporated into the proposed framework by introducing additional surface energies linked to the bulk elements neighbouring the interface. The surface-enhanced isogeometric homogenization method was verified through comparisons with existing solutions found in the literature. It is demonstrated that the proposed framework enables the satisfaction of higher-order continuity of the displacement fields, leading to smooth and accurate predictions of the stress fields and homogenized moduli of nanocomposites, without encountering the convergence problems associated with conventional finite-element methods in the literature. [ABSTRACT FROM AUTHOR]

This article deals with an inverse problem of determining two space-time-dependent sources in an integerfractional mobile-immobile two-region solute transport system by additional Dirichlet-Neumann data. The unique existence of a solution to the forward problem is obtained by the method of Laplace transform, and a dynamical system connecting the known data with the unknown sources is established by variational method and boundary homogenization. The dynamical system is discretized to a linear system at a given time in a homogenous polynomial space, and the sources are reconstructed by alternative iterations and Tikhonov regularization. Numerical examples are presented to illustrate the validity of the inversion algorithm. [ABSTRACT FROM AUTHOR]

We study in this paper a variation of the acclaimed carrier sense multiple access (CSMA) protocol. A random access algorithm is where back-off rates depend on the state of the network through queue lengths. We provide the first case where full proof of convergence to fluid limits can be obtained. Although this has been often justified informally, this is the first proof dedicated to obtaining this result formally. We then outline the difficulties arising when queues reach zero and solve them in the case of a complete interference graph. The main contribution of this paper is to provide a formal justification of the fluid limits for a complete interference graph in the case where some initial queue lengths are zero. This is done with a coupling argument. This paper also proves convergence to the fluid limits in the general case up to the time a queue reaches 0 on the fluid scale. [ABSTRACT FROM AUTHOR]

In this paper we consider the homogenization of the evolution problem associated with a jump process that involves three different smooth kernels that govern the jumps to/from different parts of the domain. We assume that the spacial domain is divided into a sequence of two subdomains A n ∪ B n and we have three different smooth kernels, one that controls the jumps from A n to A n , a second one that controls the jumps from B n to B n and the third one that governs the interactions between A n and B n . Assuming that χ A n (x) → X (x) weakly in L ∞ (and then χ B n (x) → 1 - X (x) weakly in L ∞ ) as n → ∞ and that the initial condition is given by a density u 0 in L 2 we show that there is an homogenized limit system in which the three kernels and the limit function X appear. When the initial condition is a delta at one point, δ x ¯ (this corresponds to the process that starts at x ¯ ) we show that there is convergence along subsequences such that x ¯ ∈ A n j or x ¯ ∈ B n j for every n j large enough. We also provide a probabilistic interpretation of this evolution equation in terms of a stochastic process that describes the movement of a particle that jumps in Ω according to the three different kernels and show that the underlying process converges in distribution to a limit process associated with the limit equation. We focus our analysis in Neumann type boundary conditions and briefly describe at the end how to deal with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

Spatial network models are used as a simplified discrete representation in a wide range of applications, e.g., flow in blood vessels, elasticity of fiber based materials, and pore network models of porous materials. Nevertheless, the resulting linear systems are typically large and poorly conditioned and their numerical solution is challenging. This paper proposes a numerical homogenization technique for spatial network models which is based on the super-localized orthogonal decomposition (SLOD), recently introduced for elliptic multiscale partial differential equations. It provides accurate coarse solution spaces with approximation properties independent of the smoothness of the material data. A unique selling point of the SLOD is that it constructs an almost local basis of these coarse spaces, requiring less computations on the fine scale and achieving improved sparsity on the coarse scale compared to other state-of-the-art methods. We provide an a posteriori analysis of the proposed method and numerically confirm the method's unique localization properties. In addition, we show its applicability also for high-contrast channeled material data. [ABSTRACT FROM AUTHOR]

Crystal plasticity finite element method (CPFEM) has been an integrated computational materials engineering (ICME) workhorse to study materials behaviors and structure-property relationships for the last few decades. These relations are mappings from the microstructure space to the materials properties space. Due to the stochastic and random nature of microstructures, there is always some uncertainty associated with materials properties, for example, in homogenized stress-strain curves. For critical applications with strong reliability needs, it is often desirable to quantify the microstructure-induced uncertainty in the context of structure-property relationships. However, this uncertainty quantification (UQ) problem often incurs a large computational cost because many statistically equivalent representative volume elements (SERVEs) are needed. In this article, we apply a multi-level Monte Carlo (MLMC) method to CPFEM to study the uncertainty in stress-strain curves, given an ensemble of SERVEs at multiple mesh resolutions. By using the information at coarse meshes, we show that it is possible to approximate the response at fine meshes with a much reduced computational cost. We focus on problems where the model output is multi-dimensional, which requires us to track multiple quantities of interest (QoIs) at the same time. Our numerical results show that MLMC can accelerate UQ tasks around 2.23 × , compared to the classical Monte Carlo (MC) method, which is widely known as ensemble average in the CPFEM literature. [ABSTRACT FROM AUTHOR]

In this paper we propose a consensus group decision making scheme under model uncertainty consisting of an iterative two-stage procedure based on the concept of Fréchet barycenter. Each stage consists of two steps: the agents first update their position in the opinion metric space adopting a local barycenter characterized by the agents' immediate interactions and then a moderator makes a proposal in terms of a global barycenter, checking for consensus at each stage. In cases of large heterogeneous groups, the procedure can be complemented by an auxiliary initial homogenization stage, consisting of a clustering procedure in opinion space, leading to large homogeneous groups for which the aforementioned procedure will be applied. The scheme is illustrated in examples motivated from environmental economics. [ABSTRACT FROM AUTHOR]