Binary Optimization Techniques for Linear PDE-governed Material Design

The design of materials is currently a fertile research domain. However, most of the material designs described in the literature arise from physical intuition, and often assume infinite periodicity. There is a need for a design methodology capable of computing patterns and designs involving two different materials where the underlying design variables correspond to a finite set of pixels in a 2-dimensional mesh, and where the goal is a design with prescribed material properties. This naturally leads to the consideration of binary optimization models in contrast to classical (continuous) gradient-based methods, which generically provide continuous solutions that then need to be "rounded" to binary values. While the potential drawback of binary optimization is that its computational complexity is usually NP-hard, and hence theoretically unattractive, we show herein that binary optimization combined with a reduced basis approach can relatively efficiently produce good solutions to material design problems of interest.