This chapter addresses the rigorous treatment of the tunability effect (DC electric field driven variation of the permittivity) in a high-contrast composite of periodic structure (a nonlinear ferroelectric matrix with large dielectric constant filled with linear dielectric inclusions with normal dielectric constant). A combination of the homogenization theory and classical small parameter method makes possible carrying out the detailed analysis of the nonlinear problem for weak fields (the weak field theory attracts the attention of many researchers due to its potential applications in engineering, see, e.g., [395, 396]). We present recent results of numerical analysis of the nonlinear problem for strong fields [190]. The results presented in this chapter also give some information about the material design problem as applied to an engineering problem (about various aspects of material design problems see, e.g., [27, 29, 161]).