  The mechanical and electronic properties of a graphene membrane placed on top
of a superlattice of nanopillars are investigated. We use molecular dynamics
(MD) simulations to access the deformation fields and the tight-binding
approaches to calculate the electronic properties. The system of interest
consists of a triangular lattice of nanopillars with a period of a=750 nm over
which the graphene layer is deposited. Ripples form in the graphene layer that
span across the unit cell, connecting neighbouring pillars, in agreement with
recent experiments. We find that the resulting pseudo-magnetic field (PMF)
varies strongly across the unit cell. We investigate the dependence of PMF on
unit cell boundary conditions, height of the pillars, and the strength of the
van der Waals interaction between graphene and the substrate. We find direct
correspondence with typical experiments on pillars, showing intrinsic "slack"
in the graphene membrane. PMF values are confirmed by the local density of
states calculations performed at different positions of the unit cell showing
pseudo-Landau levels at varying spacings. Our findings regarding the relaxed
membrane configuration and the induced strains are transferable to other
flexible 2D membranes.
