Reconstruction, optimization, and design of heterogeneous materials and media: Basic principles, computational algorithms, and applications

Modeling of heterogeneous materials and media is a problem of fundamental importance to a wide variety of phenomena with applications to many disciplines, ranging from condensed and soft materials, fuel cells, alloys, and composite media, to biological materials such as proteins, and even such large-scale structure as field-scale porous media and clusters of galaxies. While several approaches have been developed over the past several decades to address this class of problems, it has become increasingly clear that a most fruitful approach to the problem is by reconstruction : given a certain amount of data for a given medium, how should one develop a model for the medium that not only honors (reproduces) the data, but also provides accurate predictions for those properties of the medium for which no data are available, or are hard to measure. Although deterministic approaches were developed to address the problem, given that the amount of available data is typically limited, the uncertainty in the models is large and, at the same time the deterministic approaches cannot provide any quantitative measure of the uncertainty in the models. Thus, it became clear that the best that one can hope for is developing stochastic methods of reconstruction that not only can generate plausible realizations of the media of interest, but also provide quantitative measures of the uncertainty in the realizations. The literature on the subject is rich. Some have formulated the problem as one of optimization: one begins with a “guess” for the structure of a medium, and defines an “energy” or “cost” function that represents a quantitative measure of the difference between the data and the corresponding “predictions” of the model. There are many optimization techniques, ranging from simulated annealing and its variants, to Nature-inspired methods, such as the genetic algorithm. Through optimization one varies systematically the initial guess for the morphology of the medium in order to finally identify a structure for the medium that minimizes the energy in a global sense. the cost or energy function may be based on direct measurements of some properties, or based on various two- or three-point statistical correlation functions. A second approach attempts to mimic the physical process that gave rise to the medium. The third approach uses a global cross-correlation function, together with some concepts from graph theory in order to achieve the goal of reconstruction. This Review describes and discusses the current important methods of reconstruction. After describing the relevant theoretical and computational aspects of reconstruction and the various approaches to the problem, a wide variety of applications are described, and comparison is made between the reconstructed systems and their properties and the existing data. Finally, the problem of developing new classes of materials with specific properties, based on reconstruction techniques is discussed.