On the homogenization of a diffusion–deformation problem in strongly heterogeneous media

We consider a porous fluid-saturated medium with periodic distribution of heterogeneities where the value of permeability decreases with the scale parameters. Homogenization of such double-porous material is performed using the method of periodic unfolding. The resulting homogenized macroscopic model is featured by the fading memory effect in the viscoelastic behaviour. This paper is based upon the work sponsored by the Ministry of Education of the Czech Republic under the research project MSM 49777513 03.     

Dislocation dynamics: from microscopic models to macroscopic crystal plasticity

In this paper we study the connection between four models describing dislocation dynamics: a generalized 2D Frenkel-Kontorova model at the atomic level, the Peierls-Nabarro model, the discrete dislocation dynamics and a macroscopic model with dislocation densities. We show how each model can be deduced from the previous one at a smaller scale.      

Numerical Simulation for Convection-Diffusion Problem with Periodic Micro-Structure

In this article, the convection dominated convection-diffusion problems with the periodic micro-structure are discussed. A two-scale finite element scheme based on the homogenization technique for this kind of problems is provided. The error estimates between the exact solution and the approximation solution, of the homogenized equation or the two-scale finite element scheme are analyzed. It is shown that the scheme provided in this article is convergent for any fixed diffusion coefficient 5, and it may be convergent independent of δ under some conditions. The numerical results demonstrating the theoretical results are presented in this article.     

Multiscale stochastic homogenization of convection-diffusion equations

Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form ∂u _ɛ ^ω / ∂t+1 / ɛ ₃ C(T ₃(x/ɛ ₃)ω ₃) · ∇u _ɛ ^ω − div(α(T ₂(x/ɛ ₂)ω ₂, t) ∇u _ɛ ^ω ) = f. It is shown, under certain structure assumptions on the random vector field C(ω ₃) and the random map α(ω ₁, ω ₂, t), that the sequence {u _ɛ ^ω } of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem ∂u/∂t − div (B(t)∇u= f).     

Multiresolution Analysis of Elastic Degradation in Heterogeneous Materials

In this study we examine the stiffness properties of heterogeneous elastic materials and their degradation at different levels of observations. To this end we explore the opportunities and limitations of multiresolution wavelet analysis, where successive Haar transformations lead to a recursive separation of the stiffness properties and the response into coarse- and fine-scale features. In the limit, this recursive process results in a homogenization parameter which is an average measure of stiffness and strain energy capacity at the coarse scale. The basic concept of multiresolution analysis is illustrated with one- and two-dimensional model problems of a two-phase particulate composite representative of the morphology of concrete materials. The computational studies include the microstructural features of concrete in the form of a bi-material system of aggregate particles which are immersed in a hardened cement paste taking due account of the mismatch of the two elastic constituents. Sommario. In questo studio si esaminano le proprietà di rigidezza di materiali elastici eterogenei ed il loro degrado a diverse scale di osservazione. A questo scopo si esplorano le opportunità e le limitazioni di analisi con wavelets a risoluzione multipla, dove successive trasformazioni di Haar conducono ad una separazione ricorsiva della proprietà della rigidezza e della risposta nelle loro caratteristiche di scala fine e grossolana. Al limite, questo processo ricorsivo dà luogo ad un parametro di omogeneizzazione che rappresenta una misura media della rigidezza e della capacità di immagazzinare energia di deformazione alla grande scala. Il concetto di base dell'analisi a risoluzione multipla è illustrato per mezzo di problemi modello mono- e bi-dimensionali che si riferiscono ad un composito particolato a due fasi rappresentativo della morfologia del calcestruzzo. Le caratteristiche microstrutturali del calcestruzzo sono modellate nello studio computazionale sotto forma di un sistema a due materiali di particelle aggregate, immerse in una pasta di cemento indurita e tenendo conto della mencata congruenza tra i due constituenti elastici.     

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.     

Homogenization and Upscaling for Diffusion,Heat Conduction,and Wave Propagation in Heterogeneous Materials

We present a general homogenization method for diffusion,heat conduction,and wave propagation in a periodic heterogeneous material with piecewise constants.The method is relevant to the frequently encountered upscaling issues for heterogeneous materials.The dispersion relation for each problem is first expressed in the general form where the frequency ω (or wavenumber k) is expanded in terms of the wavenumber k (or frequency ω).A general homogenization model can be directly obtained with any given dispersion relation.Next step we study the unit cell of the heterogeneous material and derive the exact dispersion relation.The final homogenized equations include both leading order terms (effective properties) and high order contributions that represent the effect of the microscopic heterogeneity on the macroscopic behavior.That effect can be lumped into a single dimensionless heterogeneity parameter β,which is bounded between 1/12 ≤β≤ 0 and has a universal expression for all three problems.Numerical examples validate the proposed method and demonstrate a significant computational saving.     

连续能量蒙特卡罗方法组件均匀化中的临界效应

研究蒙特卡罗均匀化中的临界效应,应用B_N理论修正临界效应.简化压水堆模型和临界试验装置的验证表明:在连续能量蒙特卡罗方法均匀化中,B_N方法很好的考虑了群常数的临界效应,曲率基模成功度量了堆芯的有限性.     

Homogenization for Periodic Heterogeneous Materials with Arbitrary Position-Dependent Material Properties

We present a rigorous homogenization approach for efficient computation of a class of physical problems in a one-dimensional periodic heterogeneous material. This material is represented by a spatially periodic array of unit cells with a length of ε. More specifically, the method is applied to the diffusion, heat conduction, and wave propagation problems. Heterogeneous materials can have arbitrary position-dependent continuous or discontinuous materials properties (for example heat conductivity) within the unit cell. The final effective model includes both effective properties at the leading order and high-order contributions due to the microscopic heterogeneity. A dimensionless heterogeneity parameter β is defined to represent high-order contributions, shown to be in the range of [-1/12, 0], and has a universal expression for all three problems. Both effective properties and heterogeneity parameter β are independent of ε, the microscopic scale of heterogeneity. The homogenized solution describing macroscopic variations can be obtained from the effective model. Solution with sub-unit-cell accuracy can be constructed based on the homogenized solution and its spatial derivatives. The paper represents a general approach to obtain the effective model for arbitrary periodic heterogeneous materials with position-dependent properties.     

周期结构区域中压电问题的双尺度有限元方法

近年来纤维压电复合材料的力电性能预测已发展为一个重要的研究领域．对力电耦合周期结构的复合材料问题,通过引入匹配的边界层得到了电势与位移解的新型双尺度有限元计算方法,建立了电势与位移的双尺度耦合关系,分析了双尺度有限元解的误差.数值算例验证了方法的有效性．