  In this paper, a two-scale approach for elastic shape optimization of
fine-scale structures in additive manufacturing is investigated. To this end, a
free material optimization is performed on the macro-scale using elasticity
tensors in a set of microscopically realizable tensors. A database of these
realizable tensors and their cost values is obtained with a shape and topology
optimization on microscopic cells, working within a fixed set of elasticity
tensors samples. This microscopic optimization takes into account
manufacturability constraints via predefined material bridges to neighbouring
cells at the faces of the microscopic fundamental cell. For the actual additive
manufacturing on a chosen fine-scale, a piece-wise constant elasticity tensor
ansatz on grid cells of a macroscopic mesh is applied. The macroscopic
optimization is performed in an efficient online phase, whereas the associated
cell-wise optimal material patterns are retrieved from the database that was
computed offline. For that, the set of admissible realizable elasticity tensors
is parametrized using tensor product cubic B-splines over the unit square
matching the precomputed samples. This representation is then efficiently used
in an interior point method for the free material optimization on the
macro-scale.
