Simulation-Free Determination of Microstructure Representative Volume Element Size via Fisher Scores

TL;DR

The paper tackles the cost of determining RVE size by proposing a simulation-free method that uses micrographs and Fisher-score based nonstationarity monitoring. It fits a supervised ML model to the micrograph, computes pixel-wise score vectors, and assesses the moving-window nonstationarity through $\overline{D}_k$ as a function of window size $w_k$, identifying the elbow to select the RVE size. Across several 2D two-phase microstructures, the RVE sizes chosen by this elbow criterion closely agree with FE-convergence thresholds, demonstrating robustness to ML model choice and image resolution. This approach yields a generally applicable, FE-independent RVE sizing framework that can apply to any property governed by microstructure morphology, with potential extension to 3D and nonlinear materials.

Abstract

A representative volume element (RVE) is a reasonably small unit of microstructure that can be simulated to obtain the same effective properties as the entire microstructure sample. Finite element (FE) simulation of RVEs, as opposed to much larger samples, saves computational expense, especially in multiscale modeling. Therefore, it is desirable to have a framework that determines RVE size prior to FE simulations. Existing methods select the RVE size based on when the FE-simulated properties of samples of increasing size converge with insignificant statistical variations, with the drawback that many samples must be simulated. We propose a simulation-free alternative that determines RVE size based only on a micrograph. The approach utilizes a machine learning model trained to implicitly characterize the stochastic nature of the input micrograph. The underlying rationale is to view RVE size as the smallest moving window size for which the stochastic nature of the microstructure within the window is stationary as the window moves across a large micrograph. For this purpose, we adapt a recently developed Fisher score-based framework for microstructure nonstationarity monitoring. Because the resulting RVE size is based solely on the micrograph and does not involve any FE simulation of specific properties, it constitutes an RVE for any property of interest that solely depends on the microstructure characteristics. Through numerical experiments of simple and complex microstructures, we validate our approach and show that our selected RVE sizes are consistent with when the chosen FE-simulated properties converge.

Figures (8)

• Figure 1: Illustration of how the training dataset is constructed from the micrograph. Each pixel $i$ in the red dashed box is associated with one data point $(\bm{x}_i, y_i )$. The parameter $l_s$ represents the neighborhood size and determines the dimension ($l_s^2-1$) of $\bm{x}_i$.
• Figure 2: Comparison of two candidate RVE sizes, $w_1=3$ pixels and $w_2=5$ pixels, for moving windows (depicted as blue shaded squares at two locations in each image). The interiors of the red dashed boxes represent the pixels around which the moving window can be centered without extending outside the micrograph, the total number of which is $N_k$. Each $\bar{\bm{s}}_{i,k}$ denotes the average score vector across all pixels in the shaded moving window of size $w_k$, centered at pixel $i$.
• Figure 3: (a) A $2000 \times 2000$ pixel micrograph that represents a $2\mu m\times 2\mu m$ PMMA/SiO$_2$ microstructure with $3\%$ vf. (b - k) Magnified subregions of (a) for various window sizes.
• Figure 4: (a) FE-simulated Young's modulus and yield stress vs. sample size, with five samples simulated for each sample size. Green dots are the simulated properties of individual samples, and the blue line is the mean of the five individual sample results. (b - c) Plots of $\overline{D}_k^2$ vs. $w_k$ for a neural network ML model (b) and logistic regression ML model (c). Left column used $\bm{A} = \widehat{\bm{\Sigma}}_s$ and right column used $\bm{A}= diag(\widehat{\bm{\Sigma}}_s)$. The red vertical lines indicate the ground truth RVE size based on the FE results in (a).
• Figure 5: Micrographs with irregularly-shaped particles with the top and bottom rows representing higher- and lower-resolution images, respectively, of the same microstructure. The $2000\times2000$ pixel images in (a) and (e) represent $15 \mu m\times 15 \mu m$ and $30 \mu m\times 30\mu m$ physical regions. (b - d) and (f - h) are magnified versions of smaller windows from (a) and (e), respectively.