In 1969, M.I. Mendelson published a paper in the Journal of the American Ceramic Society that introduced a proportionality constant of 1.558 for estimating the average 3D grain size from the mean 2D lineal intercept under the assumption of lognormally distributed grains. Recent simulations by the authors revealed that the lognormal parameterization in the original work actually calculates the median grain size instead of the mean. The relationship between the mean caliper diameter and mean lineal intercept was found to be 1.60 when using common parameterizations. In addition, it is demonstrated through simulations that the correct proportionality constant can range from 1.776 to below unity depending on a material's grain size dispersion, such that 1.60 should only be used as a crude approximation. K E Y W O R D Scharacterization, grain size distribution microstructure, particle size distribution, simulations, stereology

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Accurately estimating the mean size of features within an opaque material using only 2D observations is a common requirement in the materials and medical communities. Attempting to employ numerical methods to obtain an accurate estimate of the full 3D size distribution, which is often important for calculating structure-dependent material properties, is substantially more challenging. This paper circumvents the error propagation issues observed with classical numerical approaches, such as those formulated by Saltikov, Johnson, and Schwartz, through comparison of two calculation methods: (i) the direct simulation of synthetic particles and (ii) inversion of the conventional Saltikov analysis. Each method was used to generate dispersions whose 3D particle diameters followed lognormal distributions of varying shape parameters; the mean and variance were then predicted for 2D cross-sectional measurements. Parameters from both the 2D and 3D distributions were analytically coupled to produce a set of equations for accurately transforming cross-sectional observations into estimates of the full 3D distributions. Accuracy of the 2D-to-3D transformation is, for the first time, characterized as a function of the population size and measurement resolution. Specifically, included figures allow for rapid conversion from 2D to 3D metrics with the corresponding margin of error. The Inverse Saltikov method is shown to provide superior transformation accuracy for the idealized cases of lognormally distributed and randomly dispersed spherical particles when compared to either the Direct Simulation method or classical numerical approaches.

In computational materials research, uncertainty analysis (more specifically, uncertainty propagation, UP) in the outcomes of model predictions is essential in order to establish confidence in the models as well as to validate them against the ground truth (experiments or higher fidelity simulations). Unfortunately, conventional UP models relying on exhaustive sampling from the distributions of input parameters may be impractical, particularly when the models are computationally expensive. In these cases, investigators must sacrifice accuracy in the propagated uncertainty by down-sampling the input distribution. Recently, a method was developed to correct for these inaccuracies by re-weighing the input distributions to create more statistically representative samples. In this work, the method is applied to computational models for the response of materials under high strain rates. The method is shown to effectively approximate converged output distributions at a lower cost than using conventional sampling approaches.