Macroscopic Problem Research Articles

Asymptotic analysis of sign-changing transmission problems with rapidly oscillating interface

We study the asymptotic behavior of a sign-changing transmission problem, stated in a symmetric oscillating domain obtained by gluing together a positive and a negative material, separated by an imperfect and rapidly oscillating interface. The interface separating the two heterogeneous materials has a periodic microstructure and is a small perturbation of a flat interface. The solution of the transmission problem is continuous and its flux has a jump on the oscillating interface. Under certain conditions on the properties of the two materials, we derive the limit problem and we prove the convergence result. The T-coercivity method is used to handle the lack of coercivity for both the microscopic and the macroscopic limit problems. For more information see https://ejde.math.txstate.edu/Volumes/2024/60/abstr.html

Development and Calibration of a Phenomenological Material Model for Steel-Fiber-Reinforced High-Performance Concrete Based on Unit Cell Calculations.

A phenomenological material model has been developed to facilitate the efficient numerical analysis of fiber-reinforced high-performance concrete (HPC). The formulation integrates an elasto-plastic phase-field model for simulating fractures within the HPC matrix, along with a superimposed one-dimensional elasto-plasticity model that represents the behavior of the embedded fibers. The Drucker-Prager plasticity and one-dimensional von-Mises plasticity formulations are incorporated to describe the nonlinear material behavior of both the HPC matrix and the fibers, respectively. Specific steps are undertaken during the development and calibration of the phenomenological material model. In the initial step, an experimental and numerical analysis of the pullout test of steel fibers embedded in an HPC matrix is conducted. This process is used to calibrate the micro-mechanical model based on the elasto-plastic phase-field formulation for fracture. In the subsequent step, virtual experiments based on an ellipsoidal unit cell, also with the resolution of fibers (used as a representative volume element, RVE), are simulated to analyze the impact of fiber-matrix interactions and their physical properties on the effective material behavior of fiber-reinforced HPC. In the final step, macroscopic boundary value problems (BVPs) based on a cuboid are simulated on a single scale using the developed phenomenological material model. The resulting macroscopic stress-strain characteristics obtained from both types of simulations, namely simulations of virtual experiments and macroscopic BVPs, are compared. This comparison is utilized for the calibration of material parameters to obtain a regularized solution and to assess the effectiveness of the presented phenomenological material model.

Multi-scale FE[formula omitted] investigation of non linear rate dependent 3D composite structures accounting for fiber–matrix damage

In the present paper, a two-scale FE2 technique based on periodic homogenization theory is investigated to predict the macroscopic non-linear behavior of polymer matrix composite structures.The computational technique accounts for the fiber/matrix interfacial damage, the matrix ductile damage and the effect of 3D periodic microstructure. The developed approach integrates the geometric description and the non-linear time-dependent local behavior of the different constituents (fibers, matrix and interface). For numerical calculations, advanced User Defined Material and User Element subroutines are developed at the two scales, simultaneously activated to solve macroscopic and microscopic problems through an incremental scheme in the finite element commercial code Abaqus/Implicit The computational efficiency of the developed multi-scale approach is demonstrated by predicting the overall response of 3D composite structures under complex loading paths. The composite structures consist of thermoplastic polymer matrix with elasto-viscoplastic behavior and ductile damage, reinforced by elastic aligned short fibers that are coated by a cohesive zone, which obeys the general unified potential. The numerical results obtained by FE2 simulation with and without accounting for interface effect are analyzed and compared for two examples: Meuwissen-like and 3D corner shape structures. The main benefits of the developed approach lie in accessing the microscopic strain fields, the distributions of the internal variables and the damage evolution in both polymer matrix and interface, as well as identifying their repercussions on the macroscopic response.

The Sharp Interface Limit of an Ising Game

The Ising model of statistical physics has served as a keystone example of phase transitions, thermodynamic limits, scaling laws, and many other phenomena and mathematical methods. We introduce and explore an Ising game, a variant of the Ising model that features competing agents influencing the behavior of the spins. With long-range interactions, we consider a mean-field limit resulting in a nonlocal potential game at the mesoscopic scale. This game exhibits a phase transition and multiple constant Nash-equilibria in the supercritical regime. Our analysis focuses on a sharp interface limit for which potential minimizing solutions to the Ising game concentrate on two of the constant Nash-equilibria. We show that the mesoscopic problem can be recast as a mixed local/nonlocal space-time Allen-Cahn type minimization problem. We prove, using a Γ-convergence argument, that the limiting interface minimizes a space-time anisotropic perimeter type energy functional. This macroscopic scale problem could also be viewed as a problem of optimal control of interface motion. Sharp interface limits of Allen-Cahn type functionals have been well studied. We build on that literature with new techniques to handle a mixture of local derivative terms and nonlocal interactions. The boundary conditions imposed by the game theoretic considerations also appear as novel terms and require special treatment.

Uniformity in Tophet Ceramics?

Abstract Burial urns and lids found in the Carthage tophet precinct play a pivotal role in our understanding of material trajectories. Significant differences in macroscopic features present problems as to where and how these vessels were made. In general, issues in sourcing north African ceramic materials stem from their compositional homogeneity, having mostly sand and calcareous fractions varying to a greater or lesser degree. In this paper we present the results of a petrographic study of ceramic materials in order to understand the different fabrics present within the tophet assemblage, with the ultimate aim of determining whether or not their production took place in Carthage itself and to trace transformations in local and regional ceramic fabric types.

Multifield asymptotic homogenization for periodic materials in non-standard thermoelasticity

This article presents a multifield asymptotic homogenization scheme for the analysis of Bloch wave propagation in non-standard thermoelastic periodic materials, leveraging on the Green-Lindsay theory that accounts for two relaxation times. The procedure involves several steps. Firstly, an asymptotic expansion of the micro-fields is performed, considering the characteristic size of the microstructure. By utilizing the derived microscale field equations and asymptotic expansions, a series of recursive differential problems are solved within the repetitive unit cell Q. These problems are then expressed in terms of perturbation functions, which incorporate the material’s geometric, physical, and mechanical properties, as well as the microstructural heterogeneities. The down-scaling relation, which connects the microscopic and macroscopic fields along with their gradients through the perturbation functions, is then established in a consistent manner. Subsequently, the average field equations of infinite order are obtained by substituting the down-scaling relation into the microscale field equations. To solve these average field equations, an asymptotic expansion of the macroscopic fields is performed based on the microstructural size, resulting in a sequence of macroscopic recursive problems. To illustrate the methodology, a bi-phase layered material is introduced as an example. The dispersion curves obtained from the non-local homogenization scheme are compared with those generated from the Floquet-Bloch theory. This analysis helps validate the effectiveness and accuracy of the proposed approach in predicting the wave propagation behavior in the considered non-standard thermoelastic periodic materials.

Review of deep learning algorithms in molecular simulations and perspective applications on petroleum engineering

In the last few decades, deep learning (DL) has afforded solutions to macroscopic problems in petroleum engineering, but mechanistic problems at the microscale have not benefited from it. Mechanism studies have been the strong demands for the emerging projects, such as the gas storage and hydrate production, and for some problems encountered in the storage process, which are common found as the chemical interaction between injected gas and mineral, and the formation of hydrate. Emerging advances in DL technology enable solving molecular dynamics (MD) with quantum accuracy. The conventional quantum chemical method is computational expensive, whereas the classical MD method cannot guarantee high accuracy because of its empirical force field parameters. With the help of the DL force field, precision at the quantum chemistry level can be achieved in MD. Moreover, the DL force field promotes the computational speed compared with first-principles calculations. In this review, the basic knowledge of the molecular force field and deep neural network (DNN) is first introduced. Then, three representative open-source packages relevant to the DL force field are introduced. As the most common components in the development of oil and gas reservoirs, water and methane are studied from the aspects of computational efficiency and chemical reaction respectively, providing the foundation of oil and gas researches. However, in the oil and gas problems, the complex molecular topo structures and various element types set a high challenge for the DL techniques in MD. Regarding the computational efficiency, it needs improvement via GPU and parallel accelerations to compete with classical MD. Even with such difficulties, the booming of this technique in the area of petroleum engineering can be predictable.

Modeling the material behavior of heterogeneous materials considering a two‐scale FE–FFT‐based simulation framework

AbstractIn various engineering applications, components subjected to high mechanical or multi‐physical loads such as thermal, thermo‐mechanical, or electro‐thermal loads are predominantly made of metals or composites. These materials are characterized by polycrystalline or multi‐phase microstructures which determine the overall material response. However, since the microscopic material behavior is directly related to the distribution, size, and morphology of the individual grains or phase constituents, precise predictions of the macroscopic material response require simulations of the microstructural behavior. Hence, this work considers a two‐scale finite element (FE) and fast Fourier transform (FFT)‐based simulation framework, which is an efficient alternative to the classical method for the simulation of periodic unit cells. Assuming scale separation, the macroscopic and microscopic boundary value problems are first solved individually and subsequently linked by performing a scale transition in terms of averaging the macroscopic quantities over the corresponding local fields. While the macroscopic solution is computed by means of the finite element method, the microscopic boundary value problem, which is embedded as a periodic unit cell in each macroscopic integration point, is solved using an FFT‐based simulation method. In general, these microscopic simulations allow to model grain‐scale phenomena such as martensitic phase transformation or plastic deformation caused by dislocation glide which can be observed in polycrystalline materials. In this work, we present a two‐scale FE–FFT‐based model to capture the martensitic phase transformations in shape memory alloys. In order to demonstrate the applicability of the model, a numerical example is given.

Asymptotic behavior for textiles with loose contact

The paper is dedicated to the modeling of the elasticity problem for a textile structure. The textile is made of long and thin fibers, crossing each other in a periodic pattern, forming a woven canvas of a square domain. The textile is partially clamped. The fibers cannot penetrate each other but can slide with respect to each other in the in‐plane directions. The sliding is bounded by a contact function, which is chosen loose. The partial clamp and the loose contact lead to a domain partitioning, with different expected behaviors on each of the four subdomains. The homogenization is made via the periodic unfolding method, with an additional dimension reduction. The macroscopic limit problem results in a Leray–Lions problem with only macroconstraints in the plane.

Deep learning framework for multiscale finite element analysis based on data-driven mechanics and data augmentation

In this study, a deep learning framework for multiscale finite element analysis (FE2) is proposed. To overcome the inefficiency of the concurrent classical FE2 method induced by the repetitive analysis at each macroscopic integration points, the distance-minimizing data-driven computational mechanics is adopted for the FE2 analysis. Macroscopic strain and stress data are directly assigned to the material points of the macroscopic finite element models without constitutive model. Here, the macroscopic problems are solved with the offline macroscopic material genome database, without solving the microscopic problems simultaneously. The novelty of the proposed approach lies in using a deep neural network to enable adaptive sampling points without prior knowledge of the specific mechanical problem. The proposed data augmentation framework updates the sampling points gradually using the distance minimization algorithm with the mechanistic constraints, including the equilibrium and compatibility equations. Particularly, the deep neural network plays a crucial role as a guide in the phase space sampling process, facilitating the efficient use of sparse data. Thus, this method shows high feasibility and significantly improves the computational efficiency of offline computing.

Learning macroscopic internal variables and history dependence from microscopic models

This paper concerns the study of history dependent phenomena in heterogeneous materials in a two-scale setting where the material is specified at a fine microscopic scale of heterogeneities that is much smaller than the coarse macroscopic scale of application. We specifically study a polycrystalline medium where each grain is governed by crystal plasticity while the solid is subjected to macroscopic dynamic loads. The theory of homogenization allows us to solve the macroscale problem directly with a constitutive relation that is defined implicitly by the solution of the microscale problem. However, the homogenization leads to a highly complex history dependence at the macroscale, one that can be quite different from that at the microscale. In this paper, we examine the use of machine-learning, and especially deep neural networks, to harness data generated by repeatedly solving the finer scale model to: (i) gain insights into the history dependence and the macroscopic internal variables that govern the overall response; and (ii) to create a computationally efficient surrogate of its solution operator, that can directly be used at the coarser scale with no further modeling. We do so by introducing a recurrent neural operator (RNO), and show that: (i) the architecture and the learned internal variables can provide insight into the physics of the macroscopic problem; and (ii) that the RNO can provide multiscale, specifically FE2, accuracy at a cost comparable to a conventional empirical constitutive relation.

A numerical two-scale approach for nonlinear hyperelastic beams and beam networks

We introduce a numerical framework for modeling hyperelastic slender trusses, oftentimes used as elementary building blocks in architected materials, which accounts for both the geometric nonlinearity inherent in thin structures and the nonlinear constitutive behavior of the base material. Akin to the FE2 method in homogenization, our approach is based on a formal, two-scale expansion. We decompose the three-dimensional (3D) description of a slender structure into a macroscale problem solving for the deformation of the beam center-line (based on an effective strain energy density, which depends on the stretching, bending, and torsional strains of the beam’s center-line) and a series of two-dimensional (2D) microscale boundary value problems (defined over the beam cross-section). We solve a series of 2D problems (covering a range of macroscopic strain combinations) for a given cross-sectional geometry and material distribution in an offline, pre-processing step. Using this pre-computed energy landscape, we solve the macroscopic boundary value problem within a geometrically exact, nonlinear discrete beam framework. We demonstrate the accuracy of this approach through a set of benchmark problems highlighting the nonlinear effects of the cross-sectional geometry and constitutive material, including material heterogeneity and pre-strains. We further illustrate how this technique is applied to truss-based architected materials consisting of networks of slender struts, which are typically made of polymeric base materials and thus allow for large nonlinear elastic deformation.

Modified sodium acetate trihydrate/expanded perlite composite phase change material encapsulated by epoxy resin for radiant floor heating

In this study, sodium acetate trihydrate was modified with urea, ethylene glycol, and disodium hydrogen phosphate dodecahydrate. The modified phase change material was impregnated into expanded perlite to obtain a composite phase change material, which was encapsulated with epoxy resin as the coating to obtain a coated composite phase change material with a phase change temperature of 46.4 °C, a latent heat of phase change of 132.7 J/g, and a low supercooling temperature of 0.6 °C. Leak tests and thermal cycle stability showed that the coating could effectively slow down the escape rate of the liquid phase and solve the macroscopic leakage problem. Moreover, the decrease in latent heat was only 1.8 % after 100 cycles, demonstrating excellent thermal cycle stability. The morphology and elemental distribution of the modified phase change materials and coatings were studied. In addition, laboratory insulation simulation tests were conducted, and the coated composite phase change material panels showed a significant delay in temperature reduction, while the insulation box showed longer lasting thermal performance. The excellent properties of the material find their application in radiant floor heating systems and play a key role in the expansion of coated phase change materials.

Two‐scale FE‐FFT‐based thermo‐mechanically coupled modeling of elasto‐viscoplastic polycrystalline materials at finite strains

AbstractDue to the general pursuit of technological advancement, structural components need to meet increasingly higher standards. In order to optimize the performance behavior of the used materials, detailed knowledge of the overall as well as microscopic material behavior under certain mechanical and thermal loading conditions is required. Hence, we present a two‐scale finite element (FE) and fast Fourier transformation (FFT)‐based method incorporating finite strains and a thermo‐mechanically coupled constitutive model for elasto‐viscoplastic polycrystalline materials. Assuming that the length scale of the microscale is sufficiently smaller compared to the length scale of the macroscale, we consider the macroscopic and microscopic boundary value problem as two coupled subproblems. The macroscopic boundary value problem is solved utilizing the finite element method. In each macroscopic integration point, the microscopic boundary value problem is embedded as a periodic unit cell whose solution fields are computed utilizing fast Fourier transforms and a Newton‐Krylov solver. The scale transition is performed by defining the macroscopic quantities via the volume averages of their microscopic counterparts. In order to demonstrate the use of the proposed framework, we predict the macroscopic and microscopic fields of a polycrystalline material within a numerical example using an efficient and accurate FE‐FFT‐based two‐scale method.

Two‐scale computational homogenization of calcified hydrogels

The development of new type of hydrogels featuring enhanced properties is a recent topic and of particular interest for a wide range of industrial applications, especially in biomedical sector. The present contribution focuses on the mutliscale modeling of hydrogels that are treated by the enzymatic mineralization and thus enriched with calcium phosphate. More specifically, the calcium phosphate forms spherical or honeycomb structures in the hydrogel matrix, which significantly improves effective material properties such as stiffness and strength. The chosen multiscale finite element method (FEM) homogenization strategy uses the Hill–Mandel macrohomogeneity condition for bridging two scales: the macroscopic boundary value problem (BVP) simulates the specimen behavior, whereas the microscopic BVP investigates the representative volume element (RVE) depicting the heterogeneous multiphase microstructure. The approach proposed uses the Ogden model to simulate the hydrogel and the neo‐Hooke model for the calcium phosphate phase. It varies the RVE type and the macroscopic tests in order to study the influence of the microstructure on the effective behavior and uses experimental data to determine missing microscopic material parameters. Chosen numerical examples demonstrate the applicability of the numerical tool for the estimation of the optimal microscopic arrangement of phases.

Asymptotic homogenization of flexoelectric composite plates with periodically varying thickness

A micromechanical model for the analysis of structurally periodic flexoelectric plates with periodically varying thickness is developed on the basis of asymptotic homogenization. The period of thickness variation is small and comparable to the plate thickness; accordingly, the plate is usually referred to as having a “rapidly varying thickness.” The stipulation for rapidly varying thickness is important because it is envisioned that the developed model can be applied to a broad range of thin plates, both stratified as well as plates endowed with an arbitrary distribution of reinforcements attached to the surface or embedded within the plate. The microscopic problem is implemented in two steps pertaining, respectively, to first- and second-gradient asymptotic homogenization. Each level of homogenization culminates in its own set of unit cell problems from which the effective coefficients of the homogenized structure can eventually be obtained. These effective coefficients couple the force and moment resultants as well as the averaged electric displacement with the first and second gradients of the macroscopic displacement and electric potential. Once the effective coefficients are obtained, the macroscopic problem is invoked, which provides a set of four differential equations from which the macroscopic variables of mechanical displacement and electric potential can be obtained. The model is illustrated by means of laminated flexoelectric composites as well as simple rib-reinforced plates. It is shown that in the limiting case of a thin, purely elastic plate, the derived model converges to the familiar classical plate model.

Traffic assignment problem for footpath networks with bidirectional links

The estimation of pedestrian traffic in urban areas is often performed with computationally intensive microscopic models that usually suffer from scalability issues in large-scale footpath networks. In this study, we present a new macroscopic user equilibrium traffic assignment problem (UE-pTAP) framework for pedestrian networks while taking into account fundamental microscopic properties such as self-organization in bidirectional streams and stochastic walking travel times. We propose four different types of pedestrian volume-delay functions (pVDFs), calibrate them with empirical data, and discuss their implications on the existence and uniqueness of the traffic assignment solution. We demonstrate the applicability of the developed UE-pTAP framework in a small network as well as a large scale network of Sydney footpaths.

Homogenization of discrete mesoscale model of concrete for coupled mass transport and mechanics by asymptotic expansion

Mass transport phenomenon in concrete structures is strongly coupled with their mechanical behavior. The first coupling fabric is the Biot's theory according to which fluid pressure interacts with solid stress state and volumetric deformation rate of the solid induces changes in fluid pressure. Another coupling mechanism emerges with cracks which serve as channels for the fluid to flow through them and provide volume for fluid storage. Especially the second coupling mechanism presents a challenge for numerical modeling as it requires detailed knowledge about cracking process. Discrete mesoscale mechanical models coupled with mass transport offer simple and robust way to solve the problem. On the other hand, however, they are computationally demanding. In order to reduce this computational burden, the present paper applies the asymptotic expansion homogenization technique to the coupled problem to deliver (i) continuous and homogeneous description of the macroscopic problem which can be easily solved by the finite element method, (ii) discrete and heterogeneous mesoscale problem in the periodic setup attached to each integration point of the macroscale along with (iii) equations providing communication between these two scales. The transient terms appear at the macroscale only, as well as the Biot's coupling terms. The coupling through cracking is treated at the mesoscale by changing conductivity of the conduit elements according to the mechanical solution, otherwise the two mesoscale steady state problems are decoupled and can be therefore solved in a sequence. This paper presents verification studies showing performance of the homogenized solution.

A Novel Efficient Prediction Method for Microscopic Stresses of Periodic Beam-like Structures

This paper presents a novel superposition method for effectively predicting the microscopic stresses of heterogeneous periodic beam-like structures. The efficiency is attributed to using the microscopic stresses of the unit cell problem under six generalized strain states to construct the structural microscopic stresses. The six generalized strain states include one unit tension strain, two unit bending strains, one unit torsion strain, and two linear curvature strains of a Timoshenko beam. The six microscopic stress solutions of the unit cell problem under these six strain states have previously been used for the homogenization of composite beams to equivalent Timoshenko beams (Acta. Mech. Sin. 2022, 38, 421520), and they are employed in this work. In the first step of achieving structural stresses, two stress solutions concerning linear curvatures are transformed into two stress solutions concerning unit shear strains by linearly combining the stresses under two unit bending strains. Then, the six stress solutions corresponding to six generalized unit beam strains are combined together to predict the structural microscopic stresses, in which the six stress solutions serve as basic stresses. The last step is to determine the coefficients of these six basic stress solutions by the principle of the internal work equivalence. It is found that the six coefficients, in terms of the product of the inverse of the effective stiffness matrix and the macroscopic internal force column vector, are the actual generalized strains of the equivalent beam under real loads. The obtained coefficients are physically reasonable because the basic stress solutions are produced by the generalized unit strains. Several numerical examples show that the present method, combining the solutions of the microscopic unit cell problem with the solutions of the macroscopic equivalent beam problem, can accurately and effectively predict the microscopic stresses of whole composite beams. The present method is applicable to composite beams with arbitrary periodic microstructures and load conditions.

Concurrent multiscale topology optimization: A hybrid approach

This paper presents a hybrid approach for multiscale topology optimization of structures. The topological shape of both macro-structure and micro-structure are concurrently optimized, based on the solid isotropic material with penalization (SIMP) technique in combination with finite element method (FEM). The material is assumed to have periodically patterned micro-structures, such that the effective properties can be evaluated via energy-based homogenization method (EBHM). In every iteration, the effective properties of material are passed to the macroscopic problem, and the macroscopic behavior (e.g. strain energy) is transferred back to the micro-scale problem, where the unit cell representing the micro-structure of material is determined for the next iteration. It is found that the update process can be done separately, i.e., the sensitivity of macro-scale design variables is not required during the update of micro-scale design variables, and vice versa. Hence, the proposal is that the macro-structure is updated by the gradient-free Proportional Topology Optimization (PTO) algorithm to utilize the computational efficiency of PTO. The micro-structure is still updated by the common gradient-based algorithm, namely Optimality Criteria (OC). Three benchmark numerical examples are investigated, demonstrating the feasibility and efficiency of the proposed hybrid approach.