Variational Methods for Optimal Multimaterial Composites and Optimal   Design
  The paper outlines novel variational technique for finding microstructures of optimal multimaterial composites, bounds of composites properties, and multimaterial optimal designs. The translation method that is used for the exact two-material bounds is complemented by additional pointwise inequalities on stresses in materials within an optimal composites. The method leads to exact multimaterial bounds and provides a hint for optimal structures that may be multi-rank laminates or, for isotropic composites, ``wheel assemblages". The Lagrangian of the formulated nonconvex multiwell variational problem is equal to the energy of the best adapted to the loading microstructures plus the cost of the used materials; the technique improves both the lower and upper bounds for the quasiconvex envelope of that Lagrangian. The problem of 2d elastic composites is described in some details; on particular, the isotropic component of the quasiconvex envelope of three-well Lagrangian for elastic energy is computed. The obtained results are applied for computing of optimal multimaterial elastic designs; an example of such a design is demonstrated. Finally, the optimal ``wheel assemblages" are generalized and novel types of exotic microstructures with unusual properties are described. 
Andrej Cherkaev
math.OC