小周期复合材料热传导问题的双尺度渐近展开及收敛性分析

利用双尺度渐近展开和均匀化思想讨论了小周期复合材料的热传导问题,得到了具有高阶震荡系数的抛物型方程的渐近展开式,并证明了当Ω为R~2中的光滑的区域时渐近展开式在空间L~2(0,T;H~1(Ω))中具有较好的收敛性.     

非齐次小周期复合材料稳态热问题的多尺度渐近展开和收敛性分析

利用多尺度渐近展开和均匀化思想讨论了小周期复合材料的稳态热问题,得到了非齐次边界条件下二阶椭圆型方程的渐近解,并给出了原始解与渐近解之间的误差估计,数值结果表明了结论的正确性.     

复合材料稳态热传导问题多尺度计算的一个数学模型

本文给出小周期复合材料稳态热传导问题的一种多尺度渐近展开方法，区别于传统方法中一次项和二次项系数都用解Hper^1（Q）周期边值问题得到，新展式构造时一次项系数仍通过解关于单胞Hper^1（Q）周期边值问题求得，而二次项系数用齐次边值问题求得，所构造渐近解属于H^1（n）．对光滑凸区域Ω，渐近解在H^1（Ω）空间仍具有较好的收敛性．优点为数值方法求解时，解一个齐次边界问题要比解一个Hper^1（Q）周期边值问题简单．     

一类小周期结构带有阻尼项热力耦合问题的双尺度渐近误差分析

通过构造适当的单胞函数对一类小周期结构带有阻尼项热力耦合的偏微分方程组进行双尺度渐近展开,得到了对应问题的均匀化方程和均匀化常数,并分析了双尺度渐近解的误差估计.     

整周期复合材料弹性结构的双尺度渐近分析

本文讨论了仅包含完整基本构造的小周期弹性结构的双尺度渐近分析方法,并给出了采用有限项表示式的截断误差。     

具有小周期孔洞弹性结构的双尺度渐近分析

本文对具有小周期孔洞的复合材料弹性结构进行研究,得到了位移函数一类可计算的双尺度渐近展开式,并给予严格的理论证明．     

改进的椭圆型问题一阶渐近展开误差估计

1引言考虑下述多尺度椭圆问题:■(1)其中椭圆算子A_ε定义为A_ε=-■/(■x_i)(a_(ij)~ε■/(■x_j).(2)本文使用爱因斯坦求和约定,重复指标表示求和.系数a_(ij)~ε(x)=a_(ij)(x/ε)满足下列条件:     

一类小周期椭圆方程的三重尺度渐近分析

利用三重尺度方法对一类小周期椭圆方程进行了三重尺度渐近展开分析,构造了对应的三重尺度形式渐近展开式,得到了均匀化常数和均匀化方程.在形式渐近展开的基础上,构造了对应边值问题解的三重尺度渐近近似解,并分析了对应三重尺度形式渐近误差估计.     

具有小周期孔隙复合材料弹性结构的双尺度有限元分析

对于具有小周期孔隙复合材料弹性结构,在双尺度渐近分析理论结果的基础上提出了双尺度有限元计算格式,并给出了严格的误差估计.     

交错排列结构的多尺度渐近分析

针对一类交错排列结构上的具有快速振荡系数的椭圆问题进行了多尺度渐近分析.证明了多尺度渐近展开方法的相关基础定理和多尺度解的误差估计.数值算例验证了所提出的多尺度有限元算法的有效性.进一步地,讨论了不同交错排列方式对材料等效性能的影响.     

Analysis of Multiscale Methods

The heterogeneous multiscale method gives a general framework for the analysis of multiscale methods. In this paper, we demonstrate this by applying this framework to two canonical problems: The elliptic problem with multiscale coefficients and the quasicontinuum method.     

Multiscale algorithm with high accuracy for the elastic equations in three-dimensional honeycomb structures

In this paper, we consider the elastomechanical problems of a honeycomb structure of composite materials. A multiscale finite element method and the postprocessing technique with high accuracy are presented. We will derive the proofs of all theoretical results. Finally, some numerical tests validate the theoretical results of this paper.     

Multiscale computation method for parabolic problems of composite materials

In this paper, we consider the initial-boundary value problem of parabolic type equation with rapidly oscillating coefficients in both time and space. A multiscale asymptotic expansion of solution for this kind of problem is presented. The full discrete finite element method for computing above problem is introduced. This method can apply to heat conduction analysis of composite materials. The main advantages of this method are that it can greatly save computer memory and CPU time, and it has good precision at the same time. Finally numerical results show that the method presented in this paper is effective and reliable.     

颗粒随机分布复合材料热传导问题均匀化方法的理论分析

针对区域内颗粒随机分布复合材料的热传导问题给出了一种均匀化理论计算温度场.首先根据复合材料的特性以及通过用多尺度方法预测复合材料热传导参数的要求定义了一些基本的概率空间,然后结合材料的物理特性做合理的假设得到了在整个随机复合材料区域上的期望温度场与均匀化温度场之间的一种理论估计,从而说明了此均匀化温度场可以作为预测此类随机颗粒分布复合材料期望温度场的理论基础.     

Convergence of a Ritz Approximation for the Steady State Heat Flow Problem

Let u be the true solution of the steady state heat flow problemwith Dirichlet boundary conditions. Let U be the Ritz approximationto u, from the smooth Hermite space of order two. We establishthat the L2-norm of the discretization error, U—u, tendsto zero like h⁴ as h 0. This order of convergence is independentof the sequence of partitions chosen, as long as the mesh sizeh 0.     

Multiscale analysis and numerical algorithm for the Schrödinger equations in heterogeneous media

In solid state physics, the most widely used techniques to calculate the electronic levels in nanostructures are the effective masses approximation (EMA) and its extension the multiband k · p method (see [9]). They have been particularly successful in the case of heterostructures (see, e.g. [4], [9] and [11]). This paper discusses the multiscale analysis of the Schrödinger equation with rapidly oscillating coefficients. The new contributions obtained in this paper are the determination of the convergence rate for the approximate solutions, the definition of boundary layer solutions, and higher-order correctors. Consequently, a multiscale finite element method and some numerical results are presented. As one of the main results of this paper, we give a reasonable interpretation why the effective mass approximation is very accurate for calculating the band structures in semiconductor in the vicinity of Γ point, from the viewpoint of mathematics.     

The Method of Fundamental Solution for a Radially Symmetric Heat Conduction Problem with Variable Coefficient

We consider an inverse heat conduction problem with variable coefficient on an annulus domain. In many practice applications, we cannot know the initial temperature during heat process, therefore we consider a non-characteristic Cauchy problem for the heat equation. The method of fundamental solutions is applied to solve this problem. Due to ill-posedness of this problem, we first discretize the problem and then regularize it in the form of discrete equation. Numerical tests are conducted for showing the effectiveness of the proposed method.     

混合互补问题的光滑算法及收敛性

利用Fischer-Burmeister函数将混合互补问题转化为非线性方程组,由光滑函数逼近FB函数来求解非线性方程组．文中将信赖域方法和梯度法相结合,提出了Jacobian光滑化方法．算法在一定条件下的全局收敛性得到了证明,数值试验表明算法切实有效,有一定的优越性．