On Hashin–Shtrikman bounds for mixtures of two isotropic materials

In the context of stationary diffusion equation we calculate explicitly the optimal microstructure for the Hashin–Shtrikman energy bound in the case of two isotropic phases with prescribed ratio, in three dimensions. A similar, but more general problem arises in the study of optimal design in conductivity with multiple state equations. Here, the necessary condition of optimality leads to a finite-dimensional optimisation problem which extends the problem of Hashin–Shtrikman bounds, which can be solved explicitly, as well.These calculations have important applications to the optimality criteria method for numerical solution of optimal design problems with multiple state equations. In this iterative algorithm, the presented results enable one to calculate explicitly the update of design variables, similar to the problems with one state equation. Therefore, its implementation is simple, showing nice convergence results on a number of examples, two of them being demonstrated here.