Multi-scale Model And High Accuracy Algorithm For Period Structure Of Perforated Composite Materials

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 numericalmethod. 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.