Numerical Approaches for microscopic modelling of solute redistribution during solidification of binary alloys

In the present study, two numerical approaches for single-domain modelling of microsegregation during solidification of binary alloys are presented. In the first approach, the concentration jump at the moving solid/liquid interface is formulated using a volumetric term and a Boolean function. The governing solute redistribution equation, valid for the whole domain comprising the solid and liquid regions, is derived in terms of the liquid phase composition. The effects of microstructure coarsening on microsegregation has been described and included in the model. In the second approach, the continuum mixture theory is utilized to derive a single domain solute redistribution equation in terms of the mixture composition. The solidification front motion and dendrite arm coarsening effects are accommodated by considering the representative elementary volume to consist of solid, interdendritic, and extradendritic liquid phases. Numerical solutions have been obtained using a control-volume based finite-difference method with a fixed grid. Good agreement has been observed between the predictions of the present fixed-domain models and the exact analytical and experimental results.     

A fixed domain model for microsegregation during solidification of binary alloys

A fixed-domain numerical model for microsegregation during alloy solidification is developed. The phenomena of solute partitioning at the moving solid/liquid interface and subsequent redistribution by diffusion in the solid and liquid phases have been formulated using volumetric terms. A solute balance equation valid for the whole domain comprising the solid and liquid phases has been obtained in terms of the liquid concentration. The effects of microstructure coarsening on microsegregation has been described and included in the present model. Numerical experiments and comparisons have been carried out between the present fixed-domain model, previous deforming-domain models, and the exact analytical solutions available in the literature. Good agreement has been observed between the predictions of the present fixed-domain model and the exact analytical solutions. Further extensions of the present model for the analysis of two-dimensional microsegregation have been also reported.     

The effect of solutal undercooling on double‐diffusive convection and macrosegregation during binary alloy solidification: a numerical investigation

In this paper, we present a macroscopic numerical model that is capable of capturing the interaction between the double‐diffusive convective field and a localized fluid flow on account of solutal undercooling during non‐equilibrium solidification of binary alloys. The model is essentially based on a fixed‐grid enthalpy based control volume approach. In the present model, microscopic features pertaining to non‐equilibrium effects on account of solutal undercooling are incorporated through the formulation of a modified partition‐coefficient. The effective partition‐coefficient is numerically modelled by means of a number of macroscopically observable parameters related to the solidifying domain. This feature has made the present treatment different from micro‐macro modelling of alloy solidification, which involves certain parameters that may not be macroscopically resolvable. Numerical simulations are performed for the case of two‐dimensional transient solidification of Pb–Sn alloys (both hypoeutectic and hypereutectic) in a rectangular cavity, employing the present model. The simulation results are also compared with the corresponding experimental results quoted in the literature, and the agreement is excellent. From the results, it can be concluded that non‐equilibrium effects on account of solutal undercooling result in a more enhanced macrosegregation. Copyright © 2002 John Wiley & Sons, Ltd.     

Derivation of Micro/Macro Lithium Battery Models from Homogenization

In this article, we develop a micro–macroscopic coupled model aimed at studying the interplay between electrokinetics and transport in lithium ion batteries. The system studied consists of a solid (electrode material) and a liquid phase (electrolyte) with periodic microscopic features. In this work, homogenization of generalized Poisson–Nernst–Planck (PNP) equation set leads to a micro/macro formulation similar in nature to the one developed in Newman’s model for lithium batteries. Underlying conservation equations are derived for each phase using asymptotic expansions and mathematical tools from homogenization theory, starting from a PNP micromodel, and in particular Newman’s model is obtained as a corollary of the micro/macro approach developed here. The advantage of homogenization lies in the fact that effective parameters can be derived directly from the analysis of the periodic microstructure and from the application of the theory developed in this article. In addition, the advantages of using homogenization in Lithium ion battery modeling are outlined. Lastly, this work is a necessary step toward more general homogenized models and toward mathematical proofs, and it is also needed preliminary analysis for multiscale computational schemes.     

3D macrosegregation simulation with anisotropic remeshing

The article presents a three-dimensional coupled numerical solution of momentum, mass, energy and solute conservation equations, for binary alloy solidification. The interdendritic flow in the mushy zone is assumed to obey the Darcy's law. Microsegregation is governed by the lever rule, assuming local equilibrium at phase interfaces. The resulting energy and solute advection–diffusion equations are solved using the Streamline-Upwind/Petrov–Galerkin (SUPG) finite element method. A SUPG-PSPG velocity-pressure formulation is applied for the momentum equation. The full algorithm was implemented in the 3D code THERCAST, together with an anisotropic remeshing method. Two applications have been considered: a small ingot of Pb-48wt%Sn alloy and a large steel ingot. The numerical results of these two cases are presented with the evolution of temperature, liquid velocity, and solute concentration fields during solidification. To cite this article: S. Gouttebroze et al., C. R. Mecanique 335 (2007).     

Viscoelastic flow past confined objects using a micro–macro approach

This paper is concerned with the numerical prediction of viscoelastic flow past a cylinder in a channel and a sphere in a cylinder using molecular-based models. The basis of the numerical method employed is a micro–macro model in which the polymer dynamics is described by the evolution of an ensemble of Brownian configuration fields. The spectral element method is used to discretize the equations in space. Comparisons are made between the macroscopic simulations based on the Oldroyd B constitutive model and microscopic simulations based on Hookean dumbbells, and excellent agreement is found. The micro–macro approach can be used to simulate models, such as the finitely extensible nonlinear elastic (FENE) dumbbell model, which do not possess a closed-form constitutive equation. Numerical simulations are performed for the FENE model. The influence of the model parameters on the flow is described and, in particular, the dependence of the drag as a function of the Weissenberg number.     

基于多尺度模型解释固体中的挠曲电效应

挠曲电效应是一种跨尺度的多场耦合现象。当前的宏观挠曲电理论均是基于应变梯度局部破坏晶体反演对称这一微观机理对该现象进行唯象描述。该宏观理论与基于晶格动力学及密度泛函理论的微观挠曲电理论模型之间存在较大差异。难以将两者结合用以跨尺度地研究材料中的挠曲电效应。针对该现状,本文基于前人提出的原子场理论,建立了一种新的多尺度挠曲电模型。并在该多尺度模型框架下解释了应变梯度诱发极化的微观机理。一方面,与基于连续介质力学的唯象理论不同,本文从材料微结构演化的角度推导了原子位移与极化的关系。另一方面,与通过晶格波假设原子位移的微观理论不同,本文得到的极化表达式更加真实和广义地解释了挠曲电效应。其能够适用于材料边界存在机械力作用,材料内部存在缺陷等复杂的情况。本文所建立的多尺度挠曲电模型能够为后续多尺度挠曲电效应的研究提供一些思路。     

A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media

A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of three-dimensional elasticity, but, since the assumption of strict separation of scales is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which higher-order terms, (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. An energy based micro–macro transition is then proposed for upscaling and constitutes, in fact, a generalization of the Hill–Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. As an illustration purpose, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed in the last part of the paper. Its efficiency is shown in the particular case of composites reinforced by long fibers.     

Coupled Upscaling Approaches For Conduction, Convection, and Radiation in Porous Media: Theoretical Developments

This study deals with macroscopic modeling of heat transfer in porous media subjected to high temperature. The derivation of the macroscopic model, based on thermal non-equilibrium, includes coupling of radiation with the other heat transfer modes. In order to account for non-Beerian homogenized phases, the radiation model is based on the generalized radiation transfer equation and, under some conditions, on the radiative Fourier law. The originality of the present upscaling procedure lies in the application of the volume averaging method to local energy conservation equations in which radiation transfer is included. This coupled homogenization mainly raises three challenges. First, the physical natures of the coupled heat transfer modes are different. We have to deal with the coexistence of both the material system (where heat conduction and/or convection take place) and the non-material radiation field composed of photons. This radiation field is homogenized using a statistical approach leading to the definition of radiation properties characterized by statistical functions continuously defined in the whole volume of the porous medium. The second difficulty concerns the different scales involved in the upscaling procedure. Scale separation, required by the volume averaging method, must be compatible with the characteristic length scale of the statistical approach. The third challenge lies in radiation emission modeling, which depends on the temperature of the material system. For a semi-transparent phase, this temperature is obtained by averaging the local-scale temperature using a radiation intrinsic average while a radiation interface average is used for an opaque phase. This coupled upscaling procedure is applied to different combinations of opaque, transparent, or semi-transparent phases. The resulting macroscopic models involve several effective transport properties which are obtained by solving closure problems derived from the local-scale physics.     

Mathematical Models for Metal Matrix Composites Manufacturing

Infiltration by a pure liquid metal into a rigid preform in the presence of partial solidification is analyzed. The mathematical model which is developed takes into account the coupling between the flow through the porous medium and the thermal effects. The model has a two-scale formulation: phase change occurs at the microscopic scale, while macroscopic flow and heat transfer involve large scale quantities.     

Évolution des propriétés élastiques en poroplasticité finie

The formulation of poroplasticity at large strains requires to clarify the influence of plasticity on the poroelastic properties. This effect is modelled within the framework of a micro–macro approach which takes into account the geometry changes of the microstructure. It is shown that the coupling between elasticity and plasticity introduces an additional term in the rate-type form of the macroscopic state equation which yields a modified form of the Jaumann rate.     

橡胶材料弹性的一种新的螺旋管模型

建立统计力学模型正确描述材料微观结构与宏观力学特性之间的关系是软物质类材料的最大挑战之一,已有的橡胶材料统计模型尚存在一些不足.文章根据橡胶类材料宏观各向同性、连续均匀和不可压缩特性,结合分子链的非高斯统计模型,提出一种橡胶材料网络结构的力学特性模型.该模型将代表体元上对应点之间的传力路径用一个类螺旋管区域约束的分子链子网络来描述,螺旋管的表面随材料的宏观变形做仿射变形,分子链子网络由方向和长度随机的分子链或链段首尾链接而成,在此基础上由分子链的熵推导出描述材料宏观力学特性的本构关系.通过大量的材料测试数据对本构模型进行拟合验证,拟合结果表明该模型具有非常好的精度,并且在采用两个参数时模型具有非常高的可靠性,仅用单轴拉伸实验数据拟合模型就能准确预测全部3类实验数据.该模型使用了仿射的弯曲管假设,能从微观结构尺度上说明材料的不可压缩特性,避免了直管模型的近似性,为微观尺度的随机性和宏观的均匀性的联系提出一个新的模型.     

Bridging techniques in a multi-scale modeling of pattern formation

Bridging techniques between microscopic and macroscopic models are discussed in the case of wrinkling analysis. The considered macroscopic models are related to envelope equations of Ginzburg–Landau type, but generally, they are not valid up to the boundary. To this end, a multi-scale approach is considered: the reduced model is implemented in the bulk while the full model is applied near the boundary and these two models are coupled with the Arlequin method (Ben Dhia, 1998). This paper focuses on the definition of the coupling model and the transition between two scales. Especially, a new nonlocal bridging technique is presented and compared with another recent one (Hu et al., 2011). The present method can also be seen as a guide for coupling techniques involving other reduced order models.     

Fluid flow in fractured and fracturing porous media: A unified view

Fluid flow in fractures that pre-exist or propagate in a porous medium can have a major influence on the deformation and flow characteristics. With the aim of carrying out large-scale calculations at reasonable computing costs, a sub-grid scale model has been developed. While this model was originally embedded in extended finite element methods, thereby exploiting some special properties of the enrichment functions, we will herein show that, using proper micro–macro relations, in particular for the mass balance, sub-grid scale models can be coupled to a range of discretisation methods at the macroscopic scale, from standard interface elements to isogeometric finite element analysis.     

一种化学-力学耦合连续介质理论与数值格式

对化学驱动的连续介质化学-力学耦合系统进行研究,从热力学定律和化学势角度出发,推导了等温过程的化学-力学耦合本构关系和控制方程,利用变分方法建立了化学-力学耦合系统的能量泛函,得到化学-力学耦合控制方程的等效积分形式和相应的有限元列式. 结合算例,对连续介质的化学-力学耦合行为进行了数值计算,数值结果反映了化学与力学系统的相互耦合作用,即浓度变化能引起介质的变形,同样力学作用也能引起浓度重分布. 从全新的角度建立了描述连续介质的化学-力学耦合行为的基本理论和数值方法,能够较好地反映一类连续介质的化学-力学耦合行为.      

A Three-Scale Model of pH-Dependent Flows and Ion Transport with Equilibrium Adsorption in Kaolinite Clays: I. Homogenization Analysis

A new three-scale model to describe the coupling between pH-dependent flows and transient ion transport including sorption phenomena in kaolinite clays is proposed. The kaolinite is characterized by three separate nano-micro and macroscopic length scales. The (micro)-scale consists of micro-pores saturated by an aqueous solution containing four monovalent ionic species (Na⁺, H⁺, Cl^?, OH^?) and charged solid particles surrounded by thin electrical double layers. The movement of the ions is governed by the Nernst-Planck equations and the influence of the double layers upon the flow is dictated by the Helmholtz–Smoluchowski slip boundary condition in the tangential velocity. In addition, sorption interface conditions for ion transport are postulated in the sense of Auriault and Lewandowska (Eur. J. Mech. A 15:681–704, 1996) to capture the immobilization of the ions in the electrical double layer and on particle surface due to protonation/deprotonation reactions. The intensity of sorption relative to diffusion effects is quantified by the Damköhler number, whose order of magnitude is estimated by invoking the nanoscopic modeling of the thin EDL based on Poisson–Boltzmann problem for the local electric potential coupled with a non-linear surface charge density with constitutive law dictated by the protonation/deprotonation reactions. The two-scale nano/micro model including sorption and slip boundary condition is homogenized to the core scale leading to a derivation of macroscopic governing equations.     

Comparison between Lever and Scheil Rules for Modeling of Microporosity Formation during Solidification

Segregation and microporosity formation are two important physicalphenomena that occur during solidification of binary alloys. The aim ofthis study is to investigate the effect of the model of solute diffusionat the local scale (which means at the local scale of the microscopicrepresentative elemental average volume (REV)) on solute transport andthe microporosity formation during this process. The Scheil rule and thelever rule are used to describe the solute diffusion at the local scale.Results indicate that solute diffusion at the local scale is animportant factor in microporosity formation. Also, microporosityformation slightly reduces inverse segregation because it partiallycompensates for shrinkage. The increase of the external pressure at thefree surface or the decrease of the initial hydrogen concentration inthe molten alloy can be effectively utilized to control microporosityformation.     

Electroelastic waves in dielectrics modeled as polarizable continua

On the basis of a dielectric microcontinuum model, we investigate the problem of bulk wave propagation in a dielectric crystal with hexagonal material symmetry. The present linear micropolar model allows to express electric polarization via mechanical macro and micro-strain measures so that the coupling between acoustic and polarization modes can be described in terms of intrinsic dipole and quadrupole densities. The governing differential systems for different coupled modes are equivalent to some previous results of the classical phenomenological approach to ferroelectrics but also hold for piezoelectric solids with null intrinsic polarization. Resonance couplings between polaritons and acoustic waves arise from the dispersion equations depending on suitable relations among the micropolar constitutive parameters. Exploiting the dynamical representation of polarization for the admitted modes, we obtain piezoelectric coefficients and electromechanical coupling factors as functions of the wavelength (or frequency). As an application, a numerical example is given for the hexagonal phase of zinc sulfide.     

A two‐grid fictitious domain method for direct simulation of flows involving non‐interacting particles of a very small size

The full resolution of flows involving particles whose scale is hundreds or thousands of times smaller than the size of the flow domain is a challenging problem. A naive approach would require a tremendous number of degrees of freedom in order to bridge the gap between the two spatial scales involved. The approach used in the present study employs two grids whose grid size fits the two different scales involved, one of them (the micro‐scale grid) being embedded into the other (the macro‐scale grid). Then resolving first the larger scale on the macro‐scale grid, we transfer the so obtained data to the boundary of the micro‐scale grid and solve the smaller size problem. Since the particle is moving throughout the macro‐scale domain, the micro‐scale grid is fixed at the centroid of the moving particle and therefore moves with it. In this study we combine such an approach with a fictitious domain formulation of the problem resulting in a very efficient algorithm that is also easy to implement in an existing CFD code. We validate the method against existing experimental data for a sedimenting sphere, as well as analytical results for motion of an inertia‐less ellipsoid in a shear flow. Finally, we apply the method to the flow of a high aspect ratio ellipsoid in a model of a human lung airway bifurcation. Copyright © 2009 John Wiley & Sons, Ltd.