Material interpolation schemes in topology optimization

In topology optimization of structures, materials and mechanisms, parametri zation of geometry is often performed by a grey-scale density-like interpol ation function. In this paper we analyze and compare the various approaches to this concept in the light of variational bounds on effective properties of composite materials. This allows us to derive simple necessary conditio ns for the possible realization of grey-scale via composites, leading to a physical interpretation of all feasible designs as well as the optimal desi gn. Thus it is shown that the so-called artificial interpolation model in m any circumstances actually falls within the framework of microstructurally based models. Single material and multi-material structural design in elast icity as well as in multi-physics problems is discussed.