FFT-based homogenization technique is a class of numerical methods devoted to the simulation of the behavior of heterogeneous materials on a regular grid. Most frequently used variants of the methods are originated from the works of Moulinec-Suquet based on integral equations where unknowns are distributions defined in the periodic simulation domain. In this paper, we generalize and extend our recent work by deriving boundary domain integral equations where the unknowns, strain and displacement, inside the domains and on the boundary are connected. Resolution methods based on regular grid and discrete Green tensors are also presented. The displacement formulation can be used to solve problems in arbitrary domains under periodic and non periodic boundary conditions. In the case of strain formulation, the simulation domain can be decomposed into subdomains with different reference parameters and the boundary terms appear due to the reference material mismatch. Applications to tomography images of a cement material also show that by decomposing suitably the domain and using adaptive reference parameters, the fixed point iteration scheme converges much faster than the classical FFT scheme (special case without domain decomposition)