Convergence of topological patterns of optimal periodic structures under multiple scales

The current techniques for topology optimization of material microstructure are typically based on infinitely small and periodically repeating base cells. These base cells have no actual size. It is uncertain whether the topology of the microstructure obtained from such a material design approach could be translated into real structures of macroscale. In this work we have carried out a first systematic study on the convergence of topological patterns of optimal periodic structures, the extreme case of which is a material microstructure with infinitesimal base cells. In a series of numerical experiments, periodic structures under various loading and boundary conditions are optimized for stiffness and frequency. By increasing the number of unit cells, we have found that the topologies of the unit cells converge rapidly to certain patterns. It is envisaged that if we continue to increase the number of unit cells and thus reduce the size of each unit cell until it becomes the infinitesimal material base cell, the optimal topology of the unit cell would remain the same. The finding from this work is of significant practical importance and theoretical implication because the same topological pattern designed for given loading and boundary conditions could be used as the optimal solution for the periodic structure of vastly different scales, from a structure with a few (e.g. 20) repetitive modules to a material microstructure with an infinite number of base cells.