Andrej Cherkaev, [email protected] Optimal multimaterial composites: Bounds and structures The paper suggests a method for funding exact bounds for the effective conductivity moduli of multimaterial composites. These bounds expand and refine Hashin–Shtrikman and Nesi bounds. We prove that the fields in the materials within optimal structures vary in restricted domains and take this into account, obtaining more restricted bounds. The new bounds are solutions of a formulated relaxed finite-dimensional constrained optimization problem. For two-dimensional conducting three-material composites, bounds for effective conductivity are explicitly computed. These bounds are exact: Three-material isotropic microstructures of extremal conductivity are found that realize the bounds for all values of parameters. The optimal structures are laminates of a finite rank, their parameters vary with the volume fractions and they experience two topological transitions: For large values of material of minimal conductivity, its subdomain percolates (is connected), for intermediate values of that fraction, no material forms a connected domain, and for small values of that fraction, the domain of intermediate material percolates. Another type of isotropic optimal three–material structures is the “wheel assemblages” that replaces the Hashin–Shtrikman coated circles.