Simple optimal lattice structures for arbitrary loadings

This paper identifies four categories of optimal truss lattice structures (TLSs) that together provide ultimate stiffness for arbitrary multi-loading scenarios in the low volume fraction limit. Each category consists of 7 periodic sets of straight bars, forming periodic parallelepiped unit cells. Compared to other optimal TLSs, the identified TLSs most probably have the simplest possible geometries with the least number of bar sets. Macroscopic properties of a TLS are estimated using a superposition model, and an optimization problem is solved to determine the exact geometries of the optimal TLSs. Systematic optimization results, run for thousands of random multi-loading conditions, are compared to (postulated) theoretical bounds for both truss and plate lattice structures. The results clearly demonstrate near-optimality of the identified TLSs (relative difference mostly within machine precision except in few cases up to 0.1%) for any loading scenarios in linear elasticity. At the same time, the optimal anisotropic TLSs always have inferior stiffness to the corresponding optimal plate lattice structures and this inferiority is bounded between 1 (single uniaxial load) and a factor of 3 (optimal isotropy).