Reduced-order structure-property linkages for stochastic metamaterials

TL;DR

This work tackles the problem of efficiently mapping stochastic metamaterial unit-cell geometries to their homogenized stiffness $C_{11}$ without requiring massive data. It combines the MKS framework with FFT-based homogenization, 2-point spatial statistics, PCA, and Gaussian process regression to build reduced-order structure–property maps, augmented by uncertainty-based active learning to minimize labeled data. The study demonstrates that combining solid and interface statistics (via a three-state representation) with six principal components yields the most accurate surrogate (parity $R^2\approx0.92$, MAE ~ $2.3\times10^{-2}$), and that active learning can reach near-full accuracy with only $0.61\%$ of the data. Compared to neural-network baselines, the proposed approach achieves competitive performance with far fewer training samples and provides intrinsic uncertainty quantification, enabling robust, data-efficient design guidance for stochastic metamaterials.

Abstract

The capabilities of additive manufacturing have facilitated the design and production of mechanical metamaterials with diverse unit cell geometries. Establishing linkages between the vast design space of unit cells and their effective mechanical properties is critical for the efficient design and performance evaluation of such metamaterials. However, physics-based simulations of metamaterial unit cells across the entire design space are computationally expensive, necessitating a materials informatics framework to efficiently capture complex structure-property relationships. In this work, principal component analysis of 2-point correlation functions is performed to extract the salient features from a large dataset of randomly generated 2D metamaterials. Physics-based simulations are performed using a fast Fourier transform (FFT)-based homogenization approach to efficiently compute the homogenized effective elastic stiffness across the extensive unit cell designs. Subsequently, Gaussian process regression is used to generate reduced-order surrogates, mapping unit cell designs to their homogenized effective elastic constant. It is demonstrated that the adopted workflow enables a high-value low-dimensional representation of the voluminous stochastic metamaterial dataset, facilitating the construction of robust structure-property maps. Finally, an uncertainty-based active learning framework is utilized to train a surrogate model with a significantly smaller number of data points compared to the original full dataset. It is shown that a dataset as small as $0.61\%$ of the entire dataset is sufficient to generate accurate and robust structure-property maps.

Figures (8)

• Figure 1: Example unit cells from the dataset of periodic 2D stochastic metamaterials.
• Figure 2: Interface extraction workflow: the solid microstructure array $m_{\mathbf{s}}^{1}$ (left) is convolved with a 5-point kernel to produce the filtered field $m_{\mathbf{s}}^{1,\text{conv}}$ (middle), where each pixel value indicates the number of neighboring pixels in the solid phase. The interface microstructure array $m_{\mathbf{s}}^{2}$ (right) is obtained by labeling void pixels that are adjacent to at least one solid pixel.
• Figure 3: 2-point spatial correlations for an example unit cell, including the solid–solid auto-correlation $f^{11}_{\mathbf{r}}$ (left), the interface–interface auto-correlation $f^{22}_{\mathbf{r}}$ (middle), and the solid–interface cross-correlation $f^{12}_{\mathbf{r}}$ (right).
• Figure 4: PC representation of all metamaterial unit cells for the combination $\{ \tilde{f}^{11}_{\mathbf{r}}, \tilde{f}^{22}_{\mathbf{r}} \}$ in the space of the first two PCs, colored by different quantities: solid volume fraction $f^{11}_{\mathbf{0}}$ (left), interface volume fraction $\tilde{f}^{22}_{\mathbf{0}}$ (middle), and effective elastic constant $C_{11}$ (right).
• Figure 5: MAE of the test set for GPR models trained with varying numbers of PC scores (from 1 to 8) obtained from three different combinations of 2-point statistics, including $\tilde{f}^{11}_{\mathbf{r}}$, $\{ \tilde{f}^{11}_{\mathbf{r}}, \tilde{f}^{22}_{\mathbf{r}} \}$, and $\{ \tilde{f}^{11}_{\mathbf{r}}, \tilde{f}^{22}_{\mathbf{r}}, \tilde{f}^{12}_{\mathbf{r}} \}$.