Two-Scale Elastic Shape Optimization for Additive Manufacturing

In this paper, a two-scale approach for elastic shape optimization of fine-scale struc-tures 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 optimiza-tion takes into account manufacturability constraints via predefined material bridges to neighboring cells at the faces of the microscopic fundamental cell. For the actual additive manufacturing on a chosen fine-scale, a piecewise 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 asso-ciated cellwise 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 cu-bic 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.