Two-scale Model Research Articles

Numerical simulations for a two-scale model in a porous medium

This paper deals with the numerical simulations of a system of diffusion-reaction equations in the context of a porous medium. We start by giving a microscopic model and then the upscaled version (i.e., homogenized or continuum model) of it from the previous works of the author. Since with the help of homogenization we obtain the macroscopic description of amodel that is microscopically heterogeneous, via these numerical simulations, we show that this macroscopic description approximates the microscopicmodel, which contains the heterogeneities and oscillating terms at the pore scale such as diffusion coefficients.

A multi-scale method for modeling degradation of bioresorbable polyesters

The original work in this paper is a multi-scale method (including micro scale, mesoscopic scale, macro scale and their coupling) for modeling degradation of bioresorbable polyesters and provides understanding to the process of degradation of biodegradable polymers. The result denotes the solution is reliable. As we know, there have no papers recently to implement three scales modeling and its coupling. There is a two-scale model of amorphous polyester degradation described by Han and Pan (Acta Biomaterialia 2011), our model accounts for effects of re-crystallization to explain the degradation process from three scales and takes into account of copolymers. From our model, the molecular weight distribution with time, chain number with time, degree of crystallinity with time, the evolution of polymer inner shape, weight loss with time (which is found from calculation that both oligomer diffusion and small molecules solution work to the weight loss) can be obtained from the calculation of the three scale model.

From Surface Protrusion to Tether Extraction: A Mechanistic Model.

Human leukocyte rolling on the endothelium is essential for leukocyte emigration and it is a process regulated by many factors including shear stress, receptor-ligand kinetics, and mechanical properties of cells and molecules. During this process, both leukocytes and endothelial cells (ECs) are pulled by forces due to blood flow and both may experience surface protrusion and tether extraction. In this study, we established a two-scale (cellular and molecular) model of cellular deformation because of a point pulling force and illustrated how surface protrusion makes the transition to tether extraction, either gradually or abruptly. Our simulation results matched well with what was observed in the experiments conducted with the optical trap and the atomic force microscope. We found that, although the traditional method of determining the force loading rate and the protrusional stiffness were still reasonable, the crossover force should not be simply interpreted as the rupture force of the receptor-cytoskeleton linkage. With little modification, this model can be incorporated into any leukocyte rolling model as a module for more accurate and realistic simulation.

Tensile behavior of 2D-C/SiC composites at elevated temperatures: Experiment and modeling

In this paper, on-axis tensile behavior of a coated 2D-C/SiC composite at elevated temperatures was studied experimentally and theoretically. The measured data reveals that the tensile modulus and strength increase continuously with increasing temperature till 1273K. Contrarily, the failure strains decrease sharply at high temperatures than the counterpart at room temperature, manifesting the significant influence of thermal residual stresses (TRS) on mechanical behavior of C/SiC composites. Simulation of stress-strain response is based on a two-scale analytical model, in which the plain-weave element is idealized as a cross-ply laminate and its macroscopic mechanical parameters are evaluated by shear-lag approach. The primary calculation was concentrated on TRS of the composite. And, a new crack evolution model was introduced to describe the stochastic cracking process. The total strain response including residual strain and elastic strain from the loading-unloading-reloading conception was finally formulated through micromechanical analysis involving the influence of TRS on matrix cracking and interface debonding. Additionally, a strength model was developed for plain-weave structures by using shear-lag theory, statistical theory and rule of mixture. Both of the proposed constitutive and strength models can give accurate predictions for 2D-C/SiC composites at elevated temperatures.

A 3D two-scale multiplane cohesive-zone model for mixed-mode fracture with finite dilation

A 3D multi-scale cohesive-zone model (CZM) combining friction and finite dilation by a multi-plane approach (M-CZM), based on the concept of Representative Multiplane Element (RME), is developed within the mechanics of generalized continua for the analysis of mixed-mode fracture.The proposed M-CZM formulation captures the increase of measured fracture energy in mode II as a natural effect of multi-scale coupling between cohesion, friction and interlocking, employing a reduced set of micromechanical parameters characterized by a well-defined micromechanical interpretation. This permits to devise clear calibration and identification procedures for 3D fracture problems.Upon assessing the retrieval, by a regular 5-plane RME, of a quasi-isotropic response for fracture resistance and for dilation, the M-CZM is employed in FEM simulations of Double-Cantilever Beam (DCB) tests to obtain predictions of mixed mode I–II and mixed mode I–III fracture resistance.The DCB analyses show the key role of the characteristic height of asperities in determining the macroscopic fracture resistance in both mixed mode I–II and I–III interactions. Numerical results also show the independence of the mode-I fracture resistance on the geometry of the beam section and a marked dependence of the measured mixed-mode fracture resistance on the section aspect ratio.

The extended distributed microstructure model for gradient-driven transport: A two-scale model for bypassing effective parameters

Numerous problems involving gradient-driven transport processes—e.g., Fourier's and Darcy's law—in heterogeneous materials concern a physical domain that is much larger than the scale at which the coefficients vary spatially. To overcome the prohibitive computational cost associated with such problems, the well-established Distributed Microstructure Model (DMM) provides a two-scale description of the transport process that produces a computationally cheap approximation to the fine-scale solution. This is achieved via the introduction of sparsely distributed micro-cells that together resolve small patches of the fine-scale structure: a macroscopic equation with an effective coefficient describes the global transport and a microscopic equation governs the local transport within each micro-cell. In this paper, we propose a new formulation, the Extended Distributed Microstructure Model (EDMM), where the macroscopic flux is instead defined as the average of the microscopic fluxes within the micro-cells. This avoids the need for any effective parameters and more accurately accounts for a non-equilibrium field in the micro-cells. Another important contribution of the work is the presentation of a new and improved numerical scheme for performing the two-scale computations using control volume, Krylov subspace and parallel computing techniques. Numerical tests are carried out on two challenging test problems: heat conduction in a composite medium and unsaturated water flow in heterogeneous soils. The results indicate that while DMM is more efficient, EDMM is more accurate and is able to capture additional fine-scale features in the solution.

Upscaling of a tri-phase phase-field model for precipitation in porous media

The last part of the paper presents an efficient numerical scheme to approximate the solution of the twoscale model. This scheme has as starting point the algorithm in [M. Redeker and C. Eck. A fast and accurate adaptive solution strategy for two-scale models with continuous inter-scale dependencies. J. Comput. Phys., 240:268‐283, 2013]. After some test cases validating the method, we finally present computations for several realistic scenarios. The results demonstrate the interdependence of the change of the pore structure due to precepitaion/dissolution and the evolution of the Darcy scale concentration of the dissolved particles in the one fluid.

Complete global analysis of a two-scale network SIRS epidemic dynamic model with distributed delay and random perturbations

A stochastic SIRS epidemic dynamic model with distributed time delay for vector-borne diseases in two-scale network structured populations is presented. The distributed delay accounts for the varying incubation period of the disease. Furthermore, the disease dynamics is influenced by random environmental perturbations in the disease transmission process as well as the two-scale human mobility process. The basic reproduction numbers at three human–vector contact levels in the two-scale population are computed, and the results for the stochastic asymptotic stability of the equilibria are presented. Moreover, the asymptotic stability results are exhibited in several real life scenarios and the significance of the results are presented. Numerical simulation results are presented.

$$\varvec{N}$$-Layer concentric cylinder model (NCYL): an extended micromechanics-based multiscale model for nonlinear composites

A validated multi-scale computational mechanics framework for progressive failure modeling of composite structures is developed. Zhang and Waas (Acta Mech 225(4–5):1391–1417, 2014) proposed a micromechanics-based two-scale multiscale model which is able to predict the nonlinear response of a composite. Their proposed micromechanics analysis uses the analytical solutions derived from the concentric cylinder model along with the generalized self-consistent model to find the effective properties of the unidirectional fiber reinforced composites, which are dependent on constituent level fiber–matrix mechanical properties and fiber volume fraction. In this model, the composite material is represented by an inner fiber core and an outer matrix annulus. It should be noted that the 2-CYL fiber–matrix concentric cylinder micromechanics model can be extended to a concentric fiber and any number of \((N-1)\) matrix layers in general, keeping the volume fraction constant, hence called the N-cylinder model (NCYL), which can analytically calculate the strain and stress field within the fiber and matrix cylinders for a given applied remote composite strain field, acting on the outer boundary of the matrix cylinder. The spatial variation of strain and stress fields within the cylinder are the key determinants for progressive damage and failure analysis of composite structures and hence, they need to be predicted accurately. The advantage of the NCYL concentric cylinder model is that the matrix strain and stress fields can be found in a discrete manner for each layer and the evolution of damage can be localized at a particular layer of matrix. These results can be used at the sub-scale in a multi-scale analysis to calculate the effective nonlinear composite response at a global scale. This work is based on a full analytical solution, and hence, it delivers a distinct computational advantage in composite failure analysis. That is, high fidelity and computational efficiency are gained at the same time.

A theory of electrical conductivity, dielectric constant, and electromagnetic interference shielding for lightweight graphene composite foams

This work was driven by the need to understand the electromagnetic interference (EMI) shielding effectiveness (SE) of light weight, flexible, and high performance graphene composite foams, but as EMI SE of a material depends on its electrical conductivity, dielectric permittivity, and magnetic permeability, the investigation of these three properties also became a priority. In this paper, we first present a continuum theory to determine these three electromagnetic properties, and then use the obtained properties to evaluate the EMI SE of the foam. A two-scale composite model is conceived to evaluate these three properties, with the large one being the skeleton-void composite and the small one being the graphene-polymer composite that serves as the skeleton of the foam. To evaluate the properties of the skeleton, the effective-medium approach is taken as the starting point. Subsequently, the effect of an imperfect interface and the contributions of electron tunneling to the interfacial conductivity and Maxwell-Wagner-Sillars polarization mechanism to the dielectric constant are also implemented. The derived skeleton properties are then utilized on the large scale to determine the three properties of the composite foam at a given porosity. Then a uniform plane electromagnetic wave is considered to evaluate the EMI SE of the foam. It is demonstrated that the electrical conductivity, dielectric constant, and EMI SE of the foam calculated from the developed theory are in general agreement with the reported experimental data of graphene/PDMS composite foams. The theory is further proven to be valid for the EMI SE of solid graphene/epoxy and solid carbon nanotube/epoxy nanocomposites. It is also shown that, among the three electromagnetic properties, electrical conductivity has the strongest influence on the EMI shielding effectiveness.

Simplified energy-based analysis of collapse risk of reinforced concrete buildings

This paper presents a simplified method for evaluating the risk of progressive collapse of reinforced concrete (RC) buildings. The method is formulated based on a recently developed two-scale numerical model for RC structures. In this model, all structural members are modeled by a set of coarse-scale cohesive elements representing the potential damage zones (PDZs). The cohesive constitutive behavior is determined by fine-scale finite element (FE) simulations of the corresponding PDZs. In the present study, a new energy-equivalent linear elastic cohesive model is developed for RC buildings. The damage status of the PDZ is determined by comparing the elastic energy stored in the cohesive element with the actual energy dissipation capacity of the PDZ. This model is applied to analyze the behavior of a two-dimensional frame subassemblage under a column removal scenario, and it is shown the model is capable of capturing the total energy dissipation of the entire failure process. This cohesive model is combined with a sequential analysis method to identify different possible failure paths leading to collapse initiation. The present model is applied to analyze the collapse initiation risk of a prototype RC building, where the randomness in both gravity loads and material properties is taken into account. The results are compared with the recent simulations using a nonlinear dynamic model. It is shown that the present analysis is much more efficient than the conventional nonlinear dynamic analysis, and it yields a reasonable upper bound of the collapse probability.

Hydrodynamic modeling of laser interaction with micro-structured targets

A model is developed for numerical simulations of laser absorption in plasmas made of porous materials, with particular interest in low-density foams. Laser absorption is treated on two spatial scales simultaneously. At the microscale, the expansion of a thin solid pore wall is modeled in one dimension and the information obtained is used in the macroscale fluid simulations for the description of the plasma homogenization behind the ionization front. This two-scale laser absorption model is implemented in the arbitrary Lagrangian–Eulerian hydrocode PALE. The numerical simulations of laser penetration into low-density foams compare favorably with published experimental data.

Statistical second-order two-scale analysis and computation for heat conduction problem with radiation boundary condition in porous materials**Project supported by the China Postdoctoral Science Foundation (Grant Nos. 2015M580256 and 2016T90276).

This paper discusses a statistical second-order two-scale (SSOTS) analysis and computation for a heat conduction problem with a radiation boundary condition in random porous materials. Firstly, the microscopic configuration for the structure with random distribution is briefly characterized. Secondly, the SSOTS formulae for computing the heat transfer problem are derived successively by means of the construction way for each cell. Then, the statistical prediction algorithm based on the proposed two-scale model is described in detail. Finally, some numerical experiments are proposed, which show that the SSOTS method developed in this paper is effective for predicting the heat transfer performance of porous materials and demonstrating its significant applications in actual engineering computation.

A POD–EIM reduced two-scale model for precipitation in porous media

ABSTRACTA time-dependent two-scale multiphase model for precipitation in porous media is considered, which has recently been proposed and investigated numerically. For numerical treatment, the microscale model needs to be finely resolved due to moving discontinuities modelled by several phase-field functions. This results in high computational demands due to the need of resolving many such highly resolved cell problems in course of the two-scale simulation. In this article, we present a model order reduction technique for this model, which combines different ingredients such as proper orthogonal decomposition for construction of the approximating spaces, the empirical interpolation method for parameter dependency and multiple basis sets for treating the high solution variability. The resulting reduced model experimentally demonstrates considerable acceleration and good accuracy both in reproduction as well as generalization experiments.

A stochastic two-scale model for pressure-driven flow between rough surfaces

Seal surface topography typically consists of global-scale geometric features as well as local-scale roughness details and homogenization-based approaches are, therefore, readily applied. These provide for resolving the global scale (large domain) with a relatively coarse mesh, while resolving the local scale (small domain) in high detail. As the total flow decreases, however, the flow pattern becomes tortuous and this requires a larger local-scale domain to obtain a converged solution. Therefore, a classical homogenization-based approach might not be feasible for simulation of very small flows. In order to study small flows, a model allowing feasibly-sized local domains, for really small flow rates, is developed. Realization was made possible by coupling the two scales with a stochastic element. Results from numerical experiments, show that the present model is in better agreement with the direct deterministic one than the conventional homogenization type of model, both quantitatively in terms of flow rate and qualitatively in reflecting the flow pattern.

Dynamic localization of damage and microstructural length influence

In the present contribution a localization analysis is performed for a two-scale dynamic damage model. The damage evolution law considered here results by homogenization from micro-crack propagation criteria and involve a characteristic length of the microstructure. For the one-dimensional dynamic system, we study the localization of strain and damage using analytical and numerical methods. An instability analysis based on harmonic perturbations of a homogeneous state for the linearized equations reveals wave dispersion properties and an intrinsic length of the model is deduced in the high frequency regime. The explicit dependence of this intrinsic length on the microstructural length and the damage level is obtained. Numerical simulations of dynamic tests illustrate the localization of damage, strain-rate sensitivity of the tensile strength and the influence of the size of the microstructure. The model predictions are compared with experimental results for spalling tests on concrete specimens. It is shown that the damage model is able to reproduce the observed influences of the strain rate and the size of the microstructure on the dynamic tensile strength.

A stochastic approximation approach to improve the convergence behavior of hierarchical atomistic-to-continuum multiscale models

The aim of this work is to provide an improved information exchange in hierarchical atomistic-to-continuum settings by applying stochastic approximation methods. For this purpose a typical model belonging to this class is chosen and enhanced. On the macroscale of this particular two-scale model, the balance equations of continuum mechanics are solved using a nonlinear finite element formulation. The microscale, on which a canonical ensemble of statistical mechanics is simulated using molecular dynamics, replaces a classic material formulation. The constitutive behavior is computed on the microscale by computing time averages. However, these time averages are thermal noise-corrupted as the microscale may practically not be tracked for a sufficiently long period of time due to limited computational resources. This noise prevents the model from a classical convergence behavior and creates a setting that shows remarkable resemblance to iteration schemes known from stochastic approximation. This resemblance justifies the use of two averaging strategies known to improve the convergence behavior in stochastic approximation schemes under certain, fairly general, conditions. To demonstrate the effectiveness of the proposed strategies, three numerical examples are studied.

A new methodology for computing ionic profiles and disjoining pressure in swelling porous media

A new two-scale computational model is proposed to construct the constitutive law of the swelling pressure which appears in the modified form of the macroscopic effective stress principle for expansive clays saturated by an aqueous electrolyte solution containing multivalent ionic species. The microscopic non-local nanoscale model is constructed based on a coupled Poisson-Fredholm integral equation arising from the thermodynamics of inhomogeneous fluids in nanopores (Density Functional Theory), which governs the local electric double layer potential profile coupled with the ion-particle correlation function in an electrolytic solution of finite size ions. The local problem is discretized by invoking the eigenvalue expansion of the convolution kernel in conjunction with the Galerkin method for the Gauss-Poisson equation. The discretization of the Fredholm equation is accomplished by a collocation scheme employing eigenfunction basis. Numerical simulations of the local ionic profiles in rectangular cell geometries are obtained showing considerable discrepancies with those computed with Poisson-Boltzmann based models for point charges, particularly for divalent ions in calcium montmorillonite. The constitutive law for the disjoining pressure is reconstructed numerically by invoking the contact theorem within a post-processing approach. The resultant computational model is capable of capturing ranges of particle attraction characterized by negative values of the disjoining pressure overlooked by the classical electric double layer theory. Such results provide further insight in the role the swelling pressure plays in the modified macroscopic effective stress principle for expansive porous media.

Transient heat conduction problem with radiation boundary condition of statistically inhomogeneous materials by second-order two-scale method

This paper develops a statistical second-order two-scale (SSOTS) method for predicting transient heat transfer performance of statistically inhomogeneous porous materials. In these materials, the complicated microscopic information of inclusions, including their shape, size, orientation, spatial distribution, volume fraction and so on, leads to changes of the macroscopic thermal properties. Also, heat radiation effect at microscale has an important impact on the macroscopic temperature field. A novel unified homogenization procedure, based on two-scale asymptotic expressions depending on the position variables, has been established. The second-order two-scale formulations for computing the transient heat transfer problem are given successively. Then, the statistical prediction algorithm based on the proposed two-scale model is described in details. Finally, numerical examples for porous materials with varying probability distribution models are calculated by the algorithm given, and compared with the data by the theoretical results. The comparison shows that the statistical two-scale computational method developed is useful for determination of the heat transfer properties of porous materials and demonstrates its potential applications in actual engineering computation.

A two-scale method for predicting the mechanical properties of 3D braided composites with internal defects

Pore defects have significant effect on the mechanical properties of 3D braided composites. Two-scale finite element models of the fiber tows and the braided composites are developed to predict the elastic properties and the micro stress of 3D braided composites. Two basic types of pore defects, the voids in resin matrix and the dry patches in fiber tows, have been taken into account. Periodical boundary conditions are applied on the two-scale FE models by the coupling and constraint equation defining commands in ANSYS. The predicted effective elastic constants agree well with the available experimental data, demonstrating the validity of the two-scale FE models. The effects of braiding angle and fiber volume fraction on the engineering elastic constants are discussed. Furthermore, the effects of two types of defects on the effective elastic properties of 3D braided composites are also discussed in detail. Some useful conclusions are drawn herein. From simulation, dry patches in fiber tows have more significant effect on the elastic properties than voids in matrix.