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.     

General mean-field homogenization schemes for viscoelastic composites containing multiple phases of coated inclusions

We consider matrix materials reinforced with multiple phases of coated inclusions. All materials are linear viscoelastic. We present general schemes for the prediction of the effective properties based on mean-field homogenization. There are four contributions in this work. First, we present a two-step homogenization procedure in a general setting which besides the usual assumptions of Eshelby-based models, does not suffer any restriction in terms of material properties, aspect ratio or orientation. Second, for a matrix reinforced with coated inclusions, we propose two general homogenization schemes, a two-step method and a two-level recursive scheme. We develop and compare the mathematical expressions obtained by the two schemes and a generalized Mori–Tanaka (M–T) model. Third, for a two-phase composite, either standalone or stemming from two-step or two-level schemes, we use a double-inclusion model based on a closed-form but non-trivial interpolation between M–T and inverse M–T estimates. Fourth, we conduct an extensive validation of the proposed schemes as well as others against experimental data and unit cell finite element simulations for a variety of viscoelastic composite materials. Under severe conditions, the proposed schemes perform much better than other existing homogenization methods.     

Novel numerical implementation of asymptotic homogenization method for periodic plate structures

The present paper develops and implements finite element formulation for the asymptotic homogenization theory for periodic composite plate and shell structures, earlier developed in  and , and thus adopts this analytical method for the analysis of periodic inhomogeneous plates and shells with more complicated periodic microstructures. It provides a benchmark test platform for evaluating various methods such as representative volume approaches to calculate effective properties. Furthermore, the new numerical implementation (Cheng et al., 2013) of asymptotic homogenization method of 2D and 3D materials with periodic microstructure is shown to be directly applicable to predict effective properties of periodic plates without any complicated mathematical derivation. The new numerical implementation is based on the rigorous mathematical foundation of the asymptotic homogenization method, and also simplicity similar to the representative volume method. It can be applied easily using commercial software as a black box. Different kinds of elements and modeling techniques available in commercial software can be used to discretize the unit cell. Several numerical examples are given to demonstrate the validity of the proposed methods.     

Generalized Macroscopic Models for Fluid Flow in Deformable Porous Media I: Theories

This study developed generalized mathematical models to describe the motion of fluids in porous media, and applied these models to harmonic excitation applications. The problem of fluid flow in small channels of a periodic elastic solid matrix was studied at the pore scale, and the homogenization technique was applied to predict the macroscopic behavior of reservoirs. Based on the homogenization study, five separate characteristic macroscopic models were identified according to the relation between a length scale parameter and a property contrast number. These five models can be used to interpret the corresponding responses of a saturated porous medium. The relation to existing theories, such as Darcy's law, the Telegrapher's equation and Biot's theory, was investigated. The numerical results and applications are presented in Part II of the study.     

On effective moduli of inhomogeneous media characterized by several small parameters

Composites and porous media of elongated structure, as well as materials with pores or inclusions having the shape of parallelepipeds or channels of rectangular cross-section, are considered under certain conditions on the inclusion-to-matrix modulus ratio and the volume fraction of inclusions (pores). The effective moduli are calculated by the method of mathematical homogenization theory. Numerical results on the dependence of the effective moduli on the prolateness of the structure, the shape of the inclusions (pores), the inclusion-to-matrix modulus ratio, and the volume fraction of inclusions (pores) are given. The effective moduli computed according to the algorithm of mathematical homogenization theory are compared with those given by the explicit approximate formulas earlier developed by the authors.     

STRENGTH PROPERTIES OF JOINTED ROCK MASSES BASED ON THE HOMOGENIZATION METHOD

This paper aims to determine the strength properties of jointed rock masses by means of the homogenization method.To reflect the microstructure of jointed rock masses,a representative element volume (R...     

基于均匀化理论韧性复合材料塑性极限分析

运用细观力学中的均匀化方法分析了韧性复合材料的塑性极限承载能力.从反映复合材料细观结构的代表性胞元入手,将均匀化理论运用到塑性极限分析中,计算由理想刚塑性、Mises组分材料构成的复合材料的极限承载能力.运用机动极限方法和有限元技术,最终将上述问题归结为求解一组带等式约束的非线性数学规划问题,并采用一种无搜索直接迭代算法求解.为复合材料的强度分析提供了一个有效手段.     

Thermoelastic characteristics of a composite solid with initial stresses

Thermoelastic problem for a composite solid with initial stresses is considered on the basis of the asymptotic homogenization method. The homogenized model is constructed by means of the two-scale asymptotic homogenization techniques. The major result of a present paper is that the effective (homogenized) thermoelastic characteristics of the composite material depend not only on local distributions of all types of material characteristics: local elastic properties, local thermoelastic properties, but also on local initial stresses. Therefore it is shown that for the inhomogeneous (composite) material local initial stresses contribute towards values of the effective characteristics of the material. This kind of interaction is not possible for the homogeneous materials. From the mathematical viewpoint, the asymptotic homogenization procedure is equivalent to the computation of G-limit of the corresponding operator. And the above noted phenomenon is based on the fact that in the considering case the G-limit of a sum is not equal to the sum of G-limits. The developed general homogenized model is illustrated in the particular case of the small initial stresses, which is common for the practical mechanical problems. The explicit formulas for the effective thermoelastic characteristics and numerical results are obtained for a laminated composite solid with the initial stresses.     

Functionally graded rod with small concentration of inclusions: Homogenization and optimization

This work is devoted to strain analysis and optimal design of a Functionally Graded (FG) rods and beams with small inclusions. The homogenization procedure plays a key role in our investigations. The method is illustrated using an example of the rod longitudinal deformation and bending of a beam. We consider the cases of FG inclusion sizes and FG steps between inclusions separately. Particular problems of optimal design are discussed in some details. The mathematical model of the bending beam, which adapts to the external load action, is proposed and an illustrative example of the adaptation process is given.     

Limit analysis of composite materials based on an ellipsoid yield criterion

Employing repeating unit cell (RUC) to represent the microstructure of periodic composite materials, this paper develops a numerical technique to calculate the plastic limit loads and failure modes of composites by means of homogenization technique and limit analysis in conjunction with the displacement-based finite element method. With the aid of homogenization theory, the classical kinematic limit theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. Based on nonlinear mathematical programming techniques, the finite element modelling of kinematic limit analysis is then developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the limit load of a composite is then obtained. The nonlinear formulation has a very small number of constraints and requires much less computational effort than a linear formulation. An effective, direct iterative algorithm is proposed to solve the resulting nonlinear programming problem. The effectiveness and efficiency of the proposed method have been validated by several numerical examples. The proposed method can provide theoretical foundation and serve as a powerful numerical tool for the engineering design of composite materials.     

Micro/macromechanical plastic limit analyses of composite materials and structures

The load-bearing capacities of ductile composite materials and structures are studied by means of a combined micro/macromechanics approach. Firstly, on the microscopic scale, the aim is to get the macroscopic strength domains by means of the homogenization theory of micromechanics. A representative volume element (RVE) is selected to reflect the microstructures of the composite materials. By introducing the homogenization theory into the kinematic limit theorem of plastic limit analysis, an optimization format to directly calculate the limit loads of the RVE is obtained. And the macroscopic yield criterion can be determined according to the relation between macroscopic and microscopic fields. Secondly, on the macroscopic scale, by introducing the Hill's yield criterion into the kinematic limit theorem, the limit loads of orthotropic structures such as unidirectional fiber-reinforced composite structures are worked out. The finite element modeling of the kinematic limit analysis is deduced into a nonlinear mathematical programming with equality-constraint conditions that can be solved by means of a direct iterative algorithm. Finally, some examples are illustrated to show the application of the present approach. Project supported by the National Natural Science Foundation of China (No. 19902007), the National Foundation for Excellent Doctoral Dissertation of China (No. 200025), the Fund of the Ministry of Education of China for Returned Oversea Scholars and the Basic Research Foundation of Tsinghua University.     

复合材料周期结构数学均匀化方法的一种新型单胞边界条件

数学均匀化方法是计算周期复合材料结构的有效方法之一,单胞边界条件施加的合理性直接决定了影响函数控制方程的计算效率和精度,进而影响均匀化弹性参数和摄动位移的计算精度.本文首先将单胞影响函数作为虚拟位移处理,给出了单胞在结构中真实的边界条件,结果表明,四边固支适合作为二维结构单胞边界条件;其次,针对二维结构提出了超单胞周期边界条件,有效提高了影响函数的计算精度,并使用与虚拟位移相对应的虚拟势能泛函验证超单胞周期边界条件的有效性;最后,利用数值分析验证多尺度渐进展开方法的计算精度,强调了二阶摄动的必要性.     

An enhanced affine formulation and the corresponding numerical algorithms for the mean-field homogenization of elasto-viscoplastic composites

This paper deals with the prediction of the overall behavior of a class of two-phase elasto-viscoplastic composites, based on mean-field homogenization. For this, important improvements are made to the recently-proposed affine formulation. The latter theory linearizes the rate-dependent inelastic constitutive equations of each phase’s material and transforms them into fictitious linear thermo-elastic relations in the Laplace–Carson domain. The main contributions of the present work are threefold. Firstly, complete mathematical developments including a full treatment of internal variables are carried out, enabling the modeling of the response under unloading and cyclic histories. Secondly, robust and accurate computational algorithms are proposed. Thirdly, an extensive validation of the predictions against reference unit cell finite element results is conducted for a variety of materials and loadings. A good agreement between predictions and reference results is observed.     

Global–local analysis of granular media in quasi-static equilibrium

A method of global–local analysis is developed for quasi-static equilibrium problems for granular media. The two-scale modeling based on mathematical homogenization theory enables us to formulate two separate boundary value problems in terms of macro- and microscales. The macroscale problem governs the equilibrium of a global structure composed of granular assemblies, while the microscale one is posed for the particulate nature of a local structure with the friction-contact mechanism between particles. The local structure is identified with a periodic representative volume element, or equivalently, a unit cell, over which averaging is performed. The mechanical behavior of unit cells is analyzed by a discrete numerical model, in which spring and friction devices connect rigid particles, whereas the continuum-based finite element method is used for the macroscopic one. Representative numerical examples are presented to demonstrate the capability of the proposed two-scale analysis method for granular materials.     

Variational asymptotic method for unit cell homogenization of periodically heterogeneous materials

A new micromechanics model, namely, the variational asymptotic method for unit cell homogenization (VAMUCH), is developed to predict the effective properties of periodically heterogeneous materials and recover the local fields. Considering the periodicity as a small parameter, we can formulate a variational statement of the unit cell through an asymptotic expansion of the energy functional. It is shown that the governing differential equations and periodic boundary conditions of mathematical homogenization theories (MHT) can be reproduced from this variational statement. In comparison to other approaches, VAMUCH does not rely on ad hoc assumptions, has the same rigor as MHT, has a straightforward numerical implementation, and can calculate the complete set of properties simultaneously without using multiple loadings. This theory is implemented using the finite element method and an engineering program, VAMUCH, is developed for micromechanical analysis of unit cells. Many examples of binary composites, fiber reinforced composites, and particle reinforced composites are used to demonstrate the application, power, and accuracy of the theory and the code of VAMUCH.     

Modeling Effective Interface Laws for Transport Phenomena Between an Unconfined Fluid and a Porous Medium Using Homogenization

In this chapter of the special issue of the journal “Transport in Porous Media,” on the topic “Flow and transport above permeable domains,” we present modeling of flow and transport above permeable domains using the homogenization method. Our goal is to develop a heuristic approach which can be used by the engineering community for treating this type of problems and which has a solid mathematical background. The rigorous mathematical justification of the presented results is given in the corresponding articles of the authors. The plan is as follows: We start with the section “Introduction” where we give an overview and comparison with interface conditions obtained using other approaches. In Sect. 2, we give a very short derivation of the Darcy law by homogenization, using the two-scale expansion in the typical pore size parameter ε. It gives us the definition of various auxiliary functions and typical effective properties as permeability. In Sect. 3, we introduce our approach to the effective interface laws on a simple 1D example. The approximation is obtained heuristically using the two steps strategy. For the 1D problem we calculate the approximation and the effective interface law explicitly and show that it is valid at order O(ε ²). Next, in Sect. 4 we give a derivation of the Beavers–Joseph–Saffman interface condition and of the pressure jump condition, using homogenization. We construct the corresponding boundary layer and present a heuristic calculation, leading to the interface law and being based on the rigorous mathematical result. In addition, we show the invariance of the law with respect to the small variations in the choice of the interface position. Finally, there is a short concluding section. The research of A.M. was partially supported by the GDR MOMAS (Modélisation Mathématique et Simulations numériques liées aux problèmes de gestion des déchets nucléaires) (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN).     

Topology optimization design of continuum structures under stress and displacement constraints

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Analytic solutions of a class of nonlinear partial differential equations

An approach is presented for computing the adjoint operator vector of a class of nonlinear （that is, partial-nonlinear） operator matrices by using the properties of conjugate operators to generalize a previous method proposed by the author. A unified theory is then given to solve a class of nonlinear （partial-nonlinear and including all linear） and non-homogeneous differential equations with a mathematical mechanization method. In other words, a transformation is constructed by homogenization and triangulation, which reduces the original system to a simpler diagonal system. Applications are given to solve some elasticity equations.     

Objective evaluation of linearization procedures in nonlinear homogenization: A methodology and some implications on the accuracy of micromechanical schemes

A systematic methodology for an accurate evaluation of various existing linearization procedures sustaining mean fields theories for nonlinear composites is proposed and applied to recent homogenization methods. It relies on the analysis of a periodic composite for which an exact resolution of both the original nonlinear homogenization problem and the linear homogenization problems associated with the chosen linear comparison composite (LCC) with an identical microstructure is possible. The effects of the sole linearization scheme can then be evaluated without ambiguity. This methodology is applied to three different two-phase materials in which the constitutive behavior of at least one constituent is nonlinear elastic (or viscoplastic): a reinforced composite, a material in which both phases are nonlinear and a porous material. Comparisons performed on these three materials between the considered homogenization schemes and the reference solution bear out the relevance and the performances of the modified second-order procedure introduced by Ponte Castañeda in terms of prediction of the effective responses. However, under the assumption that the field statistics (first and second moments) are given by the local fields in the LCC, all the recent nonlinear homogenization procedures still fail to provide an accurate enough estimate of the strain statistics, especially for composites with high contrast.     

Flux Norm Approach to Finite Dimensional Homogenization Approximations with Non-Separated Scales and High Contrast

We consider linear divergence-form scalar elliptic equations and vectorial equations for elasticity with rough (L ^∞(Ω), W ì \mathbb R^d{\Omega \subset \mathbb R^d}) coefficients a(x) that, in particular, model media with non-separated scales and high contrast in material properties. While the homogenization of PDEs with periodic or ergodic coefficients and well separated scales is now well understood, we consider here the most general case of arbitrary bounded coefficients. For such problems, we introduce explicit and optimal finite dimensional approximations of solutions that can be viewed as a theoretical Galerkin method with controlled error estimates, analogous to classical homogenization approximations. In particular, this approach allows one to analyze a given medium directly without introducing the mathematical concept of an e{\epsilon} family of media as in classical homogenization. We define the flux norm as the L ² norm of the potential part of the fluxes of solutions, which is equivalent to the usual H ¹-norm. We show that in the flux norm, the error associated with approximating, in a properly defined finite-dimensional space, the set of solutions of the aforementioned PDEs with rough coefficients is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard space (for example, piecewise polynomial). We refer to this property as the transfer property. A simple application of this property is the construction of finite dimensional approximation spaces with errors independent of the regularity and contrast of the coefficients and with optimal and explicit convergence rates. This transfer property also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that can be extended to elasticity equations. The proofs of these homogenization results are based on a new class of elliptic inequalities. These inequalities play the same role in our approach as the div-curl lemma in classical homogenization.