多孔复合材料周期结构的多尺度模型与高精度算法
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摘要
本文主要研究复合材料周期结构的多尺度渐近展开与高精度算法,特别对多孔复合材料的多尺度渐进展开与高精度算法作了比较详细的研究。
第一章介绍了本文的目的与意义以及复合材料和多尺度有限元方法的重要性,给出了本文的一个概要。
第二章的内容是:在均匀化方法的基础上,采用双元渐近展开与投影型插值相结合的方法,得到具有小周期系数两点边值问题的高精度算法。
第三章考虑了一类具有周期振荡系数二阶椭圆型方程边值问题,提出了基于双尺度渐近展开式的高精度有限元算法,并给出严格的证明。
第四章讨论多孔区域上具有振荡周期系数二阶椭圆型方程的Neumman边值问题,采用多尺度展开与投影型插值方法,得到高精度计算格式。
第五章讨论二维蜂窝结构热方程边值问题,给出了一个多尺度渐近展开式和有限元计算格式。
第六章研究二维蜂窝结构线弹性方程组的多尺度高精度算法,利用后处理技巧,提出了一个高精度的多尺度有限元计算格式。
第七章研究多孔复合介质的热传导方程(与时间有关),这是一个非常重要和非常难于研究的问题。由于材料的高度不均匀性,直接应用有限元方法或其他数值方法难以得到其数值解。当然难以得到其解析解。在这一章,我们将对几何与物理参数具有某些拓扑周期结构的问题进行讨论,给出多尺度有限元计算格式。
第八章是数值算例
In this thesis, we mainly study the multi-scale asymptotic expansion and high accuracy algorithm for the periodic structure of composite materials. Especially, we study the multi-scale asymptotic expansion and high accuracy algorithm for perforated composite materials in details.In Chapter 1, we introduce the aim of this thesis and the importance of composite materials and the multi-scale FE methods. A summary of this thesis is given.In Chapter 2, we consider the two point boundary value problem with rough periodic coefficients and obtain a high accuracy algorithm by using homogeniza-tion method and two-scale asymptotic expansion and projective interpolationIn Chapter 3, the boundary value problem of second order elliptic type equation with rough periodic coefficients is considered, which it comes from mechanical problem of composite materials and the heat equation of porous media and so on. A two-scale finite element method with high accuracy and its rigorous theoretical verification are reported.In Chapter 4, we discuss the Neumman boundary value problem of second order elliptic equation with rough periodic coefficients in perforated domains. Using homogenization method and two-scale asymptotic expansion and projective interpolation, a high accuracy algorithm is obtained.In Chapter 5, we study the multi-scale finite element method for the heat equation of composite materials with honeycomb structure in two dimension domain.In Chapter 6, we study elastic problem of composite materials with honeycomb structure in two dimension domain. A multiscale FE computing scheme and the post-processing technique with high accuracy are proposed.In Chapter 7, we study the computation of the heat transfer equations of composite media with cavities, witch is a very important and very difficult to study. For the high degree of heterogeneity, it is very difficult to obtain the numerical solution by using directly finite element method or other numerical
method. Of cause, it is difficult to obtain its analysis solution. In this paper, we shall discuss these problems of which geometric and physical parameters have some topological periodic properties, and shall give a multiscale finite element computational schemes.In Chapter 8, some numerical examples are given.
第一章介绍了本文的目的与意义以及复合材料和多尺度有限元方法的重要性,给出了本文的一个概要。
第二章的内容是:在均匀化方法的基础上,采用双元渐近展开与投影型插值相结合的方法,得到具有小周期系数两点边值问题的高精度算法。
第三章考虑了一类具有周期振荡系数二阶椭圆型方程边值问题,提出了基于双尺度渐近展开式的高精度有限元算法,并给出严格的证明。
第四章讨论多孔区域上具有振荡周期系数二阶椭圆型方程的Neumman边值问题,采用多尺度展开与投影型插值方法,得到高精度计算格式。
第五章讨论二维蜂窝结构热方程边值问题,给出了一个多尺度渐近展开式和有限元计算格式。
第六章研究二维蜂窝结构线弹性方程组的多尺度高精度算法,利用后处理技巧,提出了一个高精度的多尺度有限元计算格式。
第七章研究多孔复合介质的热传导方程(与时间有关),这是一个非常重要和非常难于研究的问题。由于材料的高度不均匀性,直接应用有限元方法或其他数值方法难以得到其数值解。当然难以得到其解析解。在这一章,我们将对几何与物理参数具有某些拓扑周期结构的问题进行讨论,给出多尺度有限元计算格式。
第八章是数值算例
In this thesis, we mainly study the multi-scale asymptotic expansion and high accuracy algorithm for the periodic structure of composite materials. Especially, we study the multi-scale asymptotic expansion and high accuracy algorithm for perforated composite materials in details.In Chapter 1, we introduce the aim of this thesis and the importance of composite materials and the multi-scale FE methods. A summary of this thesis is given.In Chapter 2, we consider the two point boundary value problem with rough periodic coefficients and obtain a high accuracy algorithm by using homogeniza-tion method and two-scale asymptotic expansion and projective interpolationIn Chapter 3, the boundary value problem of second order elliptic type equation with rough periodic coefficients is considered, which it comes from mechanical problem of composite materials and the heat equation of porous media and so on. A two-scale finite element method with high accuracy and its rigorous theoretical verification are reported.In Chapter 4, we discuss the Neumman boundary value problem of second order elliptic equation with rough periodic coefficients in perforated domains. Using homogenization method and two-scale asymptotic expansion and projective interpolation, a high accuracy algorithm is obtained.In Chapter 5, we study the multi-scale finite element method for the heat equation of composite materials with honeycomb structure in two dimension domain.In Chapter 6, we study elastic problem of composite materials with honeycomb structure in two dimension domain. A multiscale FE computing scheme and the post-processing technique with high accuracy are proposed.In Chapter 7, we study the computation of the heat transfer equations of composite media with cavities, witch is a very important and very difficult to study. For the high degree of heterogeneity, it is very difficult to obtain the numerical solution by using directly finite element method or other numerical
method. Of cause, it is difficult to obtain its analysis solution. In this paper, we shall discuss these problems of which geometric and physical parameters have some topological periodic properties, and shall give a multiscale finite element computational schemes.In Chapter 8, some numerical examples are given.
引文
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17. L.Q.Cao, Multiscale asymptotic expansion and finite element methods for the mixed boundary value problems of second order elliptic equation in performted domains, to be published in Numer. Math., 2006.
18. L. Q. Cao, J. Z. Cui, Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains, Numer.Math., (2004)96: 525-581
19. L. Q. Cao, J. Z. Cui, L. J. Luo, Multiscale asymptotic expansion and a post-processing algorithm for second order elliptic problems with highly osillatory coefficients over general convex domains, J. Comp. and Appl. Math., (2003), 157: 1-29
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