  We present a novel continuous optimization method to the discrete problem of
quadtree optimization. The optimization aims at achieving a quadtree structure
with the highest mechanical stiffness, where the edges in the quadtree are
interpreted as structural elements carrying mechanical loads. We formulate
quadtree optimization as a continuous material distribution problem. The
discrete design variables (i.e., to refine or not to refine) are replaced by
continuous variables on multiple levels in the quadtree hierarchy. In discrete
quadtree optimization, a cell is only eligible for refinement if its parent
cell has been refined. We propose a continuous analogue to this dependency for
continuous multi-level design variables, and integrate it in the iterative
optimization process. Our results show that the continuously optimized quadtree
structures perform much stiffer than uniform patterns and the heuristically
optimized counterparts. We demonstrate the use of adaptive structures as
lightweight infill for 3D printed parts, where uniform geometric patterns have
been typically used in practice.
