| Description |
In this dissertation, several problems are solved using different approaches. The first problem is the two-dimensional three-material G-closure problem: finding all possible effective tensors from given conductivities of three materials and volume fractions. We solve this problem by establishing lower bounds on effective tensors, and finding the (sequences of) microstructures that attain the lower bounds. The lower bound is a piece wise analytic function that depends on the conductivity and volume fraction of each component. They are derived using a combination of the translation method, and additional constraints on the field in the materials. The found bound extend their results to the anisotropic case. Furthermore, the lower bound obtained in this dissertation is also the improvement of the Hashin-Shtrikman and translation bounds, in the sense that it is optimal in a range of parameters where previously known bounds are not; and in the region where both the new bound and previously known bounds are not optimal, the bound derived here is tighter. In the case when the established bound is optimal, structures that attain the bound are presented. All structures are laminates of finite rank. While the bound cannot be obtained by laminate structures, we estimate the bound by comparing it with some particular structure. The numerical experiment shows that the gap between the two is rather small, hence the bound is very close to the optimal bound. The next two problems are typical problems in optimal design, and are solved using the variational method in the frame of Young measures developed by Pedregal. The key idea of this approach is to find the quasiconvex envelope of sets and functions. Those ideas have been used before for optimal design problems with two materials at disposal. Our goal here is to explore how those ideas can be extended to three or more materials situations. In particular, we focus on two paradigmatic cases, where we consider a linear-in-the-gradient cost functional and a typical quadratic situation. In both cases, we are able to formulate, quite explicitly, a full relaxation of the problem problem through which optimal microstructures for the original nonconvex problem can be understood. In principle, this approach can be also used to find the G-closure problem as long as one can find the quasiconvex hull of the set, composed of the gradient fields and their associated divergence free fields determined by the governing equations in the G-closure problem. |
