Applying periodic boundary conditions in computational homogenization method

I want to apply this type of boundary on a cube : T= G*x +t
where the fluctuation t is periodic. It takes the same values at two homologous points on opposite faces of the boundaries of the elementary volume V, and G is a constant vector independent of the vector position x.
can you help me please ?

Hi, You have to rewrite your variational problem so as to shift the unknown function from T to S =T - Gx (using your notation) so that in the new formulation you have only to apply periodic boundary conditions, which can be done easily. Even in this case the solution is not unique, so you have to fix one of its values in one point (or impose zero average).