  The fabrication flexibility of 3D printing has sparked a lot of interest in
designing structures with spatially graded material properties. In this paper,
we propose a new type of density graded structure that is particularly designed
for 3D printing systems based on filament extrusion. In order to ensure
high-quality fabrication results, extrusion-based 3D printing requires not only
that the structures are self-supporting, but also that extrusion toolpaths are
continuous and free of self-overlap. The structure proposed in this paper,
called CrossFill, complies with these requirements. In particular, CrossFill is
a self-supporting foam structure, for which each layer is fabricated by a
single, continuous and overlap-free path of material extrusion. Our method for
generating CrossFill is based on a space-filling surface that employs spatially
varying subdivision levels. Dithering of the subdivision levels is performed to
accurately reproduce a prescribed density distribution. We demonstrate the
effectiveness of CrossFill on a number of experimental tests and applications.
