Spectral integration of microstructure and design

Mechanical design can be conducted in a framework where consideration of microstructure as a continuous design variable is facilitated by the use of a Fourier space. Selection of the mechanical framework for the problem (e.g., mechanical constitutive model and homogenization relations) dictates the dimensionality of the pertinent microstructure representation. Microstructure is comprised of basic elements that belong to the local state space. Local state includes crystallographic phase and orientation, and other parameters such as composition. The local state space is transformed into an isomorphic set in Fourier space. The universe of pertinent microstructures is found to be the convex hull of the local state space in the Fourier space, and is named the microstructure hull. Bounds on material properties are represented by one or more families of bounding hyper-surfaces (often hyper-planes) of finite dimension that intersect the microstructure hull. Consideration of the full range of these hypersurfaces gives rise to properties closures, representing the full range of combined properties that are predicted to be possible by considering the entire microstructure hull. We describe how properties closures can be introduced into design optimization systems, thereby introducing microstructure as a continuous variable in the mechanical design methodology for highly-constrained systems. Also, we describe the challenges and possibilities for extending the methodology to processing to achieve the prescribed optimal microstructures.