Efficient generation of anisotropic N-field microstructures from 2-point statistics using multi-output Gaussian random fields

The ability to efficiently generate microstructure instances corresponding to specified two-point statistics is a crucial capability in rigorously studying random heterogeneous materials within the Integrated Computational Materials Engineering and Materials Informatics frameworks. However, the lack of computationally efficient, statistically expressive models for achieving this transformation is a recurring roadblock in many foundational Materials Informatics challenges. In this article, we present a theoretical and computational framework for generating stationary, periodic microstructural instances corresponding to specified stationary, periodic two-point statistics by stochastically modeling the microstructure as an N-output Gaussian Random Field. First, we illustrate how two-point statistics can be used to parameterize anisotropic Gaussian Random Fields. Second, we derive analytic relationships between the two-point statistics and the spatially resolved sampled microstructures, within the approximation of a N-output Gaussian Random Field. Finally, we propose the algorithms necessary to efficiently sample these fields in O (S ln S) computational complexity and while incurring O (S) memory cost. We also discuss the current limitations of the proposed framework, and its usefulness to future Materials Informatics workflows.