Homogenization-based space-time topology optimization of tunable microstructures

Abstract

A topology optimization framework is developed for smart materials with tunable microstructures. The framework addresses spatial and temporal design variables in a unified setting so as to deliver the optimal periodic microstructure with stimulus-sensitive constituents. The optimal topology allows the macroscopic response of the microstructure to track a time-dependent cyclic path in the stress-strain space with minimal error. The relevant homogenization-based variational analysis for the sensitivity-based optimization framework incorporates not only material variables but also the geometry information regarding the unit cell. Extensive numerical investigations demonstrate the ability of the developed approach to deliver optimal topologies for realizable target macroscopic paths. The error in optimization increases monotonically with the degree of unrealizability, yet the critical role of the microstructure in minimizing the error in comparison to a pure time optimization approach is demonstrated in all cases.