Some recent developments of the homogenization theory with applications

Abstract: This thesis in applied mathematics is devoted to theoretical and some of the numerical aspects of the homogenization theory. The thesis consists of five papers. In the first paper we give a brief presentation of some basic ideas, which can serve as an introduction and frame to the author´s contributions to the papers in this thesis. The second paper presents some theory and numerical results concerning the effective elastic moduli of heterogeneous structures. In the third paper we consider in-plane stiffness properties of square symmetric unidirectional two-phase composites with given volume fractions. The fourth and fifth papers are concerned with the heat conduction in checkerboard structures. Very few microstructures yield explicit formulae for their effective conductivity. One type of such structures is checkerboards. Due to the behavior of the solutions near the corner points it is difficult to solve the corresponding variational problems by usual numerical methods, even for the standard checkerboard. In these papers we consider a generalized version of the standard checkerboard and focus on these difficulties, both theoretically and experimentally (numerical experiments). Moreover, we present a new numerical method for determining the corresponding field which converges in the energy norm independent of the local conductivities.