Stress formulation and duality approach in periodic homogenization

This paper describes several different formulations of the so-called “cellular problem” which is a system of partial differential equations arising in the theory of homogenization, subject to periodicity boundary conditions. Variational formulations of the cellular problem are presented where the main unknown is the displacement, the stress or the strain, as well as several different formulations as minimization problems. Two dual formulations are also presented, one in displacement-stress and another one in strain-stress. The corresponding Lagrangians may be used in numerical optimization algorithms based on alternated directions.