Optimal composites show with extremal effective properties.
Our main interest is to find structures of these composites and to find bounds of the range of their properties. The extremal composites are used in structural optimization , the bounds provide a priori information of the possible range of achievable properties. The set of effective properties of all possible composites made from given materials is called G-closure of the set of original properties. We have now a pretty detailed picture of structures of two-phase conducting and elastic materials, but the best structures of multiphase composites are mainly unknown. I am working on this extremely interesting and knotty problem.