Analytical formulas for complex permittivity of periodic composites. Estimation of gain and loss enhancement in active and passive composites

ABSTRACT

The asymptotic homogenization method is applied to complex dielectric periodic composites. An equivalence to coupled dielectric problems with real coefficients is shown. This is similar to a piezoelectric problem: an out-plane mechanical displacement and an in-plane electric potential establishing a correspondence principle. Closed-form formulas for the complex dielectric effective tensor in the case of a square array of circular inclusions embedded in a matrix are given. These formulas are written in terms of a real and symmetric matrix which facilitates the implementation of the computational scheme. We also get similar formulas for multilayered complex dielectric composites. The real closed-form formulas are advantageous for estimating gain and loss enhancement properties of active and passive composites in certain volume fraction intervals. Numerical computations are performed and the results are compared with other approaches showing the usefulness of the obtained formulas. This may be of interest in the context of metamaterials.     

一种基于自适应单元删除率的BESO方法

在ESO中采用动态删除率能有效地提高优化效率和稳定性，但现有的动态删除率策略都含有经验参数，确定删除率的过程较为复杂。本文提出了一种用于BESO的无经验参数自适应单元删除率确定方法。通过分析单元删除率对优化稳定性的影响，得到了结构优化过程中单元删除率的理想变化规律和单元灵敏度均匀化信息对删除率的影响情况，并据此分析了经验参数引入的原因，从而构造了评价单迭代步的单元灵敏度均匀化程度指标。然后，基于单迭代步的单元灵敏度均匀化程度指标，构造了全部迭代步信息下的单元灵敏度均匀化程度相对指标，结合单元删除率的推荐范围值，给出了一种自适应于结构优化进程的单元删除率自适应函数。最后，给出了基于自适应单元删除率的BESO方法实现流程。经典算例的结果对比说明，本文方法在保证优化质量相近的情况下，具有更好的优化效率和稳定性。     

Linearized viscoelastic Oldroyd fluid motion in an almost periodic environment

In most of the linear homogenization problems involving convolution terms so far studied, the main tool used to derive the homogenized problem is the Laplace transform. Here we propose a direct approach enabling one to tackle both linear and nonlinear homogenization problems that involve convolution sequences without using Laplace transform. To illustrate this, we investigate in this paper the asymptotic behavior of the solutions of a Stokes–Volterra problem with rapidly oscillating coefficients describing the viscoelastic fluid flow in a fixed domain. Under the almost periodicity assumption on the coefficients of the problem, we prove that the sequence of solutions of our ?‐problem converges in L² to a solution of a rather classical Stokes system. One important fact is that the memory disappears in the limit. To achieve our goal, we use some very recent results about the sigma‐convergence of convolution sequences. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal control problem in a domain with branched structure and homogenization

We consider a linear parabolic problem in a thick junction domain which is the union of a fixed domain and a collection of periodic branched trees of height of order 1 and small width connected on a part of the boundary. We consider a three‐branched structure, but the analysis can be extended to n‐branched structures. We use unfolding operator to study the asymptotic behavior of the solution of the problem. In the limit problem, we get a multi‐sheeted function in which each sheet is the limit of restriction of the solution to various branches of the domain. Homogenization of an optimal control problem posed on the above setting is also investigated. One of the novelty of the paper is the characterization of the optimal control via the appropriately defined unfolding operators. Finally, we obtain the limit of the optimal control problem. Copyright © 2016 John Wiley & Sons, Ltd.     

A simple and rapid route for synthesizing the nanosized g-C3N4 materials with narrow bandgap and their photocatalytic activity

The graphitic carbon nitride (g-C₃N₄) materials with many intriguing properties have attracted much attention in photocatalysis. The photocatalytic activity of g-C₃N₄ is hindered by serious aggregation and limited exposed active sites. Herein is shown that nanosized g-C₃N₄ can be simply obtained by a superfast high-pressure homogenization approach. The high-pressure homogenization treatment can provide strong force to cut and/or to exfoliate the bulk g-C₃N₄ into nanosized g-C₃N₄ with good dispersion. Moreover, choosing different solvents during treatment can cause a different surface structure of as-prepared nanosized g-C₃N₄. In addition, the narrow bandgap properties, the high photogenerated charge carrier separation, and the transport abilities are achieved in as-prepared nanosized g-C₃N₄ because of the retaining conjugated C₃N₄ system. Specifically, the photocatalytic activities of as-prepared nanosized g-C₃N₄ have been significantly enhanced in terms of degradation of organic dye Rhodamine B (RhB) under visible light irradiation (10 times higher than that of bulk g-C₃N₄). These findings can provide a promising and simple approach to the exfoliation, nanonization, and surface functionalization of 2D layered materials.     

HOMOGENIZATION—BASED TOPOLOGY DESIGN FOR PURE TORSION OF COMPOSITE SHAFTS

In conjunction with the homogenization theory and the finite element method, the mathematical models for designing the corss-section of composite shafts by maximizing the torsion rigidity are developed in this paper. To obtain the extremal torsion rigidity, both the cross-section of the macro scale shaft and the representative microstructure of the composite material are optimized using the new models. The micro scale computational model addresses the problem of finding the periodic microstructures with extreme shear moduli. The optimal microstructure obtained with the new model and the homogenization method can be used to improve and optimize natural or artificial materials. In order to be more practical for engineering applications, cellular materials rather than ranked materials are used in the optimal process in the existence of optimal bounds for the elastic properties. Moreover, the macro scale model is proposed to optimize the cross-section of the torsional shaft based on the tailared composites. The validating optimal results show that the models are very effective in obtaining composites with extreme elastic properties, and the cross-section of the composite shaft with the extremal torsion rigidity. The project supported by the National Natural Science Foundation of China (10172078 and 10102018)     

Exact Results for the Homogenization of Elastic Fiber-Reinforced Solids at Finite Strain

This work is concerned with the homogenization of solids reinforced by aligned parallel continuous fibers or weakened by aligned parallel cylindrical pores and undergoing large deformations. By alternatively exploiting the nominal and material formulations of the corresponding homogenization problem and by applying the implicit function theorem, it is shown that locally homogeneous deformations can be produced in such inhomogeneous materials and form a differentiable manifold. For every macroscopic strain associated to a locally homogeneous deformation field, the effective nominal or material stress–strain relation is exactly determined and connections are also exactly established between the effective nominal and material elastic tangent moduli. These results are microstructure-independent in the sense that they hold irrespectively of the transverse geometry and distribution of the fibers or pores. A porous medium consisting of a compressible Mooney–Rivlin material with cylindrical pores is studied in detail to illustrate the general results. This work was the first time presented at the Euromech Colloqium 464 on “Fiber-reinforced Solids: Constitutive Laws and Instabilities”, September 28–October 1, 2004, Cantabria, Spain.     

结构拓扑优化研究方法综述

结构拓扑优化研究方法目前有解析方法和数值方法两大类.首先介绍了解析方法中的 Michell理论,它在结构拓扑优化领域研究较早,影响最为深远.随后着重讨论了杆系和连 续体结构拓扑优化的数值方法.杆系结构常采用基结构方法,通过删除部分杆件达到结构 拓扑优化的目的.连续体结构一般要划分为有限单元,通过删除单元形成带孔的连续体, 以实现拓扑优化.介绍了连续体结构拓扑优化常采用的材料模型:各向同性、各向异性和 带微结构材料.并对连续体结构(0-1)拓扑优化中的数值计算不稳定问题的机理进行了分 析,给出了解决方法.此外,对应力约束问题存在解的奇异性现象也作了简要介绍.最后, 对数值方法中的主要数学求解方法进行了简单介绍.     

HT—7U超导托克马克纵场线圈等效模量的数值分析

HT－7U托克马克装置的纵场线圈是由多种材料组成的具有周期性分布的大型复杂结构，线圈可视为由超导线、支撑结构和绝缘材料组成的复合材料，结构极其复杂。整个线圈工作在液氦温区。在设计阶段对其宏观等效力学性能进行数值分析计算是十分必要的，但要对整个线圈直接进行有限元分析或实验是极其困难的，主要利用均匀化方法对其进行等效处理，从而分析计算其宏观等效模量，为线圈的设计和评估提供参考依据。另外本文对均匀化方法的边界条件进行了一些改进，提出了更加合理的三维边界条件。