  We study AKSZ-type BV constructions for the topological A- and B-models
within a double field theory formulation that incorporates backgrounds with
geometric and non-geometric fluxes. We relate them to a Courant sigma-model, on
an open membrane, corresponding to a generalized complex structure, which
reduces to the A- or B-models on the boundary. We introduce S-duality at the
level of the membrane sigma-model based on the generalized complex structure,
which exchanges the related AKSZ field theories, and interpret it as
topological S-duality of the A- and B-models. Our approach leads to new classes
of Courant algebroids associated to (generalized) complex geometry.
