Topology Optimization
Cellular materials such as truss materials achieve excellent mechanical properties at a considerably lower weight than conventional solids. They span a wide material property space through a careful choice of the base material(s) and, moreover, the topology and geometry of the cellular architecture. The rapid development of modern additive manufacturing techniques has enabled the integration of cellular design principles across length scales down to the nanoscale, which has opened an unprecedented design space. Based on an efficient and accurate continuum description of discrete lattice structures, we systematically utilize this design space and optimize the topology of spatially variant cellular structures such as truss lattices.
Using our effective large-scale continuum description of trusses, we replace the discrete truss by an effective continuum to be treated by finite elements in a macroscale boundary value problem. Inspired by the atomic arrangement in crystals, we parametrize the truss unit cell via a set of Bravais lattice vectors along with Voronoi tessellation. Hence, we formulate the optimization problem with regards to the spatially varying basis vectors. Unlike most competing approaches, both the displacement field and the topology are continuously varying unknown fields on the macroscale. The specific homogenization scheme chosen here introduces the regularization required for unique solutions. The outlined approach results in heterogeneous truss architectures with a smoothly varying unit cell, enabling easy fabrication with a tunable length scale (the latter avoiding the ill-posedness stemming from classical nonconvex methods without an intrinsic length scale).