A novel post-Friedmann formalism (cosmological perturbation theory) covering all spatial scales and permitting of nonlinear contrasts of the matter density has been recently formulated within the concordance cosmological model. This formalism is suitable for relativistic simulations of the cosmic structure growth. It passes various corroborative tests and possesses advantages over prior approaches. In addition, it predicts Yukawa screening of interparticle gravitational attraction in the globally expanding universe. The corresponding range (screening length) is related to the homogeneity scale and dimensions of the observed largest cosmic structures (such as Hercules-Corona Borealis Great Wall). The objective of this project consists in development of a similar all-scale perturbation theory for such popular extensions of the conventional paradigm as models with anisotropic topologies or nonzero spatial curvature (aiming at revealing the universe shape and volume). The proposed theory will provide a well-grounded description of gravitational interaction at arbitrary (sub-horizon and super-horizon) distances in the inhomogeneous universe within the discussed models. The role of Yukawa screening in weakening the impact of periodic boundary conditions in cosmological N-body simulations and resolving the ultraviolet divergence problem in lattice universes will be clarified along with a possibility to estimate the cosmological backreaction effects.