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# Autonomous and Scalable Computation of Spatial Correlation Function from Spatiotemporal Materials Data

Material science is experiencing pressure to deliver new and improvement materials to the marketplace in half the time and at half the cost.[ref2] To reach this goal, a hurdle that will need to be overcome is the increasing dissonance between the strength of material science in generating massive stores of empirical and simulated physical information and its weakness to utilize the data at scale. Materials science is challenged by the 3 V’s of Big Data as the Volumes of the unstructured datasets are reaching terabyte sizes, as high performance computing is increasing the Velocity which information is generated, and with the Variety of information being generated by conduits in the field from both experiment and simulation. The current subjective, undocumented, and ad-hoc data analysis tools cannot be scaled to suit these needs. New data science techniques will need to be explored to offer objective statistical representations of materials science information.[ref]

Microstructure Informatics is an emerging framework that provides a growing suite of data science tools to extract bi-directional structure-property/processing relationships for most classes of materials science information. Microstructure Informatics uses digital signal processing and advanced statistics to properly encode material information into useful forms for machine learning, data structures, and algorithms that extract added value from the information. In microstructure informatics, the material structure, or microstructure, is considered to be the independent of the dependent property or processing response. Spatial statistics (i.e. Pair Correlations, N-Point statistics) are commonly used statistical utilities as it provides an objective statistical description of the material structure. The following successful case studies have been illustrated to shown the effective of spatial statistics in (1) determining objective microstructure comparisons in heat treated α-β Titanium [ref], (2) building regression models for the homogenized structure-property connections between the internal structure of fuel cell materials and their diffusivity [ref], and (3) determining the variance of properties associated with individual microstructures [ref].

This paper discusses fast algorithms to encode material structure information using parameterized basis function and spatial correlation functions. The concept of the microstructure function is employed to parameterize the material information and produce a digital microstructure signal; encoding can be performed on most classes of experimental and simulated material structure information.[ref] The digital signals are convolved using embarrassingly parallel Fast Fourier methods to compute the spatial statistics of the digitized microstructure. The following properties of the spatial correlation functions make them a worthy candidate for an objective material structure descriptor: several widely used statistical metrics are embedded in the correlations such as volume fraction [ref] and specific surface area [ref]; for binary images, they contain information about the original material structure information within a translation [ref]; and they can describe most types of materials science information in raw signal is processed appropriately. N case studies will be presented to illustrate the generality of the technique when applied to things.

This paper will begin with a discussion on classifications of materials science information and their proposed conversions to a digital signal using the microstructure function. Once the raw material structure information is digitized, they will form the foundations that allow spatial statistics to be computed using fast, scalable FFT algorithms. N case studies will be shown to express the diversity and general applicability of the spatial correlation functions in structure-structure comparison.

# A Generic Framework to Quantify Materials Science Information

## The Microstructure Function

The microstructure function expresses spatially resolved material structure information from physics-based models as digital signals. A physics-based model extracts material structure-response behaviors with either simulated simulation or empirical experimental techniques; materials information generated by either technique, or source, are uniquely identified by a set of boundary conditions and control parametersset of modeling conditions (i.e. control parameters). In a model, A model includes spatially resolved raw model information is output about the material structure, $$\beta(x)$$, as either an input or output with local material states, $$\beta(x)$$, at disparate positions, $$x$$, in the sample volume, $$L$$. The sample volume has dimensions $$L_1×⋯×L_d$$ where $$d$$ is the dimensionality of the dataset; the dimensionality corresponds to the number of independent axes in the spatial domain and will have the property, $$d$$ ≤ 3. The local material state is an ordered set of salient material features such as phase, classification, grain orientation, volume fraction, spin, curvature, etc. The generality of the local material state definition enables a framework to describe most material information from most sources. The microstructure function digitizes the raw model information by the following equation $\beta(x) = \sum_{h=1}^{H}{m_{s}^{h} \, \chi_{s}^{h}(x) \, \chi^{h}(\beta(x))}$

where $$m_{s}^{h}$$ is a digitized coefficients of the raw material information corresponding to any normalized basis functions $$\chi_{s}(x)$$ and $$\chi^{h}(\beta(x))$$ applied to represent the spatial domain and local state of the material, respectively. The spatial domain and local state space are orthogonal to one another and will require different basis functions. Each basis function is normalized requiring that $$m_{s}^{h}$$ is bounded between zero and one. The selection of the basis functions is an extremely important matter and will be discussed extensively throughout this paper. From this point on, $$m_{s}^{h}$$ will be referred to as the microstructure function where $$h$$ is a local state index corresponding to the $$h^{th}$$ function in the local material state basis; similarly, $$s$$ is an index pertaining to a volume contained within the sample volume prescribed by the spatial basis function .

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