Soluções assintóticas formais de segunda ordem na homogeneização assintótica de meios microperiódicos

Abstract

In this dissertation, the asymptotic homogenization method is used as a tool to obtain solutions for some physical, linear, one-dimensional and multidimensional problems. The study is based on mathematical models that describe the behavior of heterogeneous solids in the microstructure. The method looks for a new problem, called homogenized problem (homogeneous medium), which is equivalent to the original problem (heterogeneous medium). Thus, from a small geometric parameter, the macroscopic and microscopic scales are separated and the asymptotic homogenization method is applied from the basic cell. This cell presents a regularity and periodicity in the microstructure of the material, and it contains all the physical characteristics of the medium. Therefore, the results obtained can be extended over the domain of the structure. Thus, a solution to the original problem is proposed as a formal asymptotic solution. In the development of the method the local problems, the effective coefficients and the homogenized problem are found. The local problems are related to the behavior of the material in the basic cell and must be solved to obtain the solutions of the terms of the formal asymptotic solution and the effective coefficients. These coefficients, together with the solution of the homogenized problem, provide the effective (global) behavior, equivalent to what occurs physically in the heterogeneous medium. The original problem presents partial differential equations with rapidly oscillating coefficients. Therefore, effective coefficients of the homogenized problem, corresponding to the coefficients of the original problem are sought. Finally, the solution of the original problem must converge to the solution of the homogenized problem, and this is shown mathematically through the proximity relation between the solutions. In particular, second-order formal asymptotic solutions are obtained to improve the approximation of the local details of the exact solutions, as the traditional approach of the asymptotic homogenization method seeks only first-order formal asymptotic solutions which do not provide such details. In order to illustrate the results obtained, examples are proposed to verify the proximity between the solutions found.