A Low-Parametric Rhombic Microstructure Family for Irregular Lattices
ACM Transactions on Graphics (Proceedings of SIGGRAPH), 2020
Abstract
New fabrication technologies have significantly decreased the cost of
fabrication of shapes with highly complex geometric structure. One important
application of complex fine-scale geometric structures is to create variable
effective elastic material properties in shapes manufactured from a single
material. Modification of material properties has a variety of uses, from
aerospace applications to soft robotics and prosthetic devices. Due to its
scalability and effectiveness, an increasingly common approach to creating
spatially varying materials is to partition a shape into cells and use a
parametric family of small-scale geometric structures with known effective
properties to fill the cells.
We propose a new approach to solving this problem for extruded, planar
microstructures. Differently from existing methods for two-scale optimization
based on regular grids with square periodic cells, which cannot conform to an
arbitrary boundary, we introduce cell decompositions consisting of (nearly)
rhombic cells. These meshes have far greater flexibility than those with square
cells in terms of approximating arbitrary shapes, and, at the same time, have a
number of properties simplifying small-scale structure construction. Our main
contributions include a new family of 2D cell geometry structures, explicitly
parameterized by their effective Young’s moduli 𝐸, Poisson’s ratios 𝜈, and
rhombic angle 𝛼 with the geometry parameters expressed directly as smooth
spline functions of 𝐸, 𝜈, and 𝛼. This family leads to smooth transitions
between the tiles and can handle a broad range of rhombic cell shapes. We
introduce a complete material design pipeline based on this microstructure
family, composed of an algorithm to generate rhombic tessellation from
quadrilateral meshes and an algorithm to synthesize the microstructure
geometry. We fabricated a number of models and experimentally demonstrated how
our method, in combination with material optimization, can be used to achieve
the desired deformation behavior.