  Much work has been done in topology optimization of multiscale structures for
maximum stiffness or minimum compliance design. Such approaches date back to
the original homogenization-based work by Bends{\o}e and Kikuchi from 1988,
which lately has been revived due to advances in manufacturing methods like
additive manufacturing. Orthotropic microstructures locally oriented in
principal stress directions provide for highly efficient stiffness optimal
designs, whereas for the pure stiffness objective, porous isotropic
microstructures are sub-optimal and hence not useful. It has, however, been
postulated and exemplified that isotropic microstructures (infill) may enhance
structural buckling stability but this has yet to be directly proven and
optimized. In this work, we optimize buckling stability of multiscale
structures with isotropic porous infill. To do this, we establish local density
dependent Willam-Warnke yield surfaces based on local buckling estimates from
Bloch-Floquet-based cell analysis to predict local instability of the
homogenized materials. These local buckling-based stress constraints are
combined with a global buckling criterion to obtain topology optimized designs
that take both local and global buckling stability into account. De-homogenized
structures with small and large cell sizes confirm validity of the approach and
demonstrate huge structural gains as well as time savings compared to standard
singlescale approaches.
