循环对称结构的多尺度拓扑优化方法

研究了循环对称结构多尺度拓扑优化问题,阐明了均匀化等效性能的基本性质,提出了循环对称单胞的特征参数概念,建立了均匀化映射计算方法,构造了等效性能的三元参数插值模型,有效简化了不同特征参数、微结构构型与体分比下的单胞均匀化等效过程.通过微结构的特征驱动建模与B样条参数化,克服了微结构拓扑优化变量多和变量离散难以保证其光滑连接的问题.给出了典型数值算例,比较了等比例排列单胞与等间距排列单胞对结构优化结果的影响,验证了多尺度优化方法的有效性.     

基于参数化水平集方法的微结构拓扑优化设计

相比于单一材料,复合材料具有轻质高强等优点,拓扑优化方法是设计复合材料的方法之一.本文采用改进的参数化水平集方法,更新了水平集迭代格式,并应用水平集带方法在优化过程中引入中间密度,使水平集方法与变密度法无缝结合以改善水平集方法的拓扑寻优能力,降低其初始设计依赖性.本文以最大化体积模量、剪切模量和负泊松比作为材料设计目标,结合均匀化方法预测材料的宏观等效性能,研究了不同体积分数、多种初始设计及水平集带方法的引入对优化结果的影响,并得到了多种满足不同目标函数的微结构拓扑形式.数值算例验证了本文方法在复合材料微结构设计问题中的有效性.     

On nonstandard Padé approximants suitable for effective properties of two-phase composite materials

This article investigates the existence of the nonstandard Padé approximants introduced by Cherkaev and Zhang [D.-L. Zhang and E. Cherkaev, Reconstruction of spectral function from effective permittivity of a composite material using rational function approximations, J. Comput. Phys. 228 (2009), pp. 5390–5409] for approximating the spectral function of composites from effective properties at different frequencies. The spectral functions contain information on microstructure of composites. Since this reconstruction problem is ill-posed Cherkaev [Inverse homogenization for evaluation of effective properties of a mixture, Inverse Probl. 17 (2001), pp. 1203–1218], the well-performed Padé approach is noteworthy and deserves further investigations. In this article, we validate the assumption that the effective dielectric component of interest of all two-phase composites can be approximated by Padé approximants whose denominator has nonzero power one term. We refer to this as the nonstandard Padé approximant, in contrast to the standard approximants whose denominators have nonzero constant terms. For composites whose spectral function assumes infinitely many different values such as the checkerboard microstructure, the proof is carried by using classical results for Markov–Stieltjes functions (also referred to as Stieltjes functions) Golden and Papanicolaou [Bounds on effective parameters of heterogeneous media by analytic continuation, Commun. Math. Phys. 90 (1983), pp. 473–491] and Cherkaev and Ou [De-homogenization: Reconstruction of moments of the spectral measure of the composite, Inverse Probl. 24 (2008), p. 065008]. However, it is well-known that spectral functions for microstructure such as rank-n laminates assume only finitely many different values, i.e. the measure in the Markov–Stieltjes function is supported at only finitely many points. For this case, we cannot find any existence results for nonstandard Padé approximants in the literature. The proof for this case is the focus of this article. It is done by utilizing a special product decomposition of the coefficient matrix of the Padé system. The results in this article can be considered as an extension of the Padé theory for Markov–Stieltjes functions whose spectral function take infinitely many different values to those taking only finitely many values. In the literature, the latter is usually excluded from the definition of Markov–Stieltjes functions because they correspond to rational functions, hence convergence of their Padé approximants is trivial. However, from an inverse problem point of view, we need to assure both the existence and convergence of the nonstandard Padé approximants, for all microstructures. The results in this article provide a mathematical foundation for applying the Padé approach for reconstructing the spectral functions of composites whose microstructure is not a priori known.     

Two-scale homogenization in transmission problems of elasticity with interface jumps

The elasticity problem in a periodic structure with prescribed interface jumps in displacements and tractions and oscillating Neumann condition on a part of the external boundary is considered. This work is just a generalization of inhomogeneous Dirichlet and Neumann conditions on the oscillating interface. Such interface jumps arise, e.g. in contact problems with known periodic contact interface. Two-scale approach was applied to the problem and the two-scale convergence was proven. This article also provides a detailed auxiliary analysis for Sobolev functions with interface jumps.     

Asymptotic heat equation for crossing superconductive thin walls

This work deals with the homogenization of the nonstationary heat conduction which takes place in a binary three-dimensional medium consisting of an ambiental phase having conductivity of unity order and a set of highly conductive thin walls crossing orthogonally and periodically. This situation covers in fact three types of microstructures, usually called: box-type (or honeycomb), gridwork and layered. The study is based on the energetic procedure of homogenization associated to a control-zone method, specific to the geometry of the microstructure and to the singularity of the conductivity coefficients. In the present case the main result is the system that governs the asymptotic behaviour of the temperature distribution in this binary medium. It displays a significant increase of the conductivity due to the superconductive thin walls, revealing their seemingly paradoxal behaviour of having an everlasting action on the environment, in spite of an obvious vanishing volume. Moreover, the dependence of this behaviour with respect to the relative thicknesses of the walls can be detailed..     

Homogenization of fluid–porous interface coupling in a biconnected fractured media

The modelization of mass transfer through biconnected fractured porous media is studied by homogenizing the coupling between the Darcyan percolation and the viscous Stokes flow on their interface governed by the Beavers–Joseph law. The case of high transmission coefficients is considered. The asymptotic behaviour is completely described with the help of the solutions of some specific local problems and of a nonhomogeneous Neumann problem defined by the effective permeability tensor.     

Homogenization of a nonlinear degenerate parabolic differential equation

This paper is devoted to the homogenization of a nonlinear degenerate parabolic problem ɑtu∈-div(D(x/∈, u∈,▽u∈)+ K(x/∈, u∈))= f(x) with Dirichlet boundary condition. Here the operator D(y, s,s) is periodic in y and degenerated in ▽s. In the paper, under the two-scale convergence theory, we obtain the limit equation as ∈→ 0 and also prove the corrector results of ▽u∈ to strong convergence.