Bayesian-guided inverse design of hyperelastic microstructures: Application to stochastic metamaterials

From a given pool of all feasible design variants, our aim is to identify a structure that achieves a target macroscopic stress response. For each candidate design, the response is obtained from a high-fidelity oracle, in particular, time- and resource-intensive computational homogenization or experiments. We consider the case where (i) the geometry cannot be conveniently parameterized, rendering gradient-based optimization inapplicable, and (ii) brute-force evaluation of all candidates is infeasible due to the cost of oracle queries. To tackle this challenge, we propose a Bayesian-guided inverse design framework that proceeds as follows. First, the dimensionality of the design variants is reduced through statistical feature engineering, and the resulting low-dimensional descriptors are mapped to effective constitutive parameters describing the macroscopic hyperelastic response. This mapping is modeled using a multi-output Gaussian process surrogate that accounts for correlations between the parameters. The surrogate is trained using uncertainty-driven active learning under severe budget constraints, allowing only a very limited number of high-fidelity oracle evaluations. Based on surrogate predictions, a finite number of promising candidates are shortlisted. Since the surrogate accuracy is inherently limited, the final selection of the optimal design is performed through high-fidelity oracle evaluations within the shortlist. In numerical test cases, we consider a dataset of 50,000 candidate structures. Active learning requires labeling less than half a percent of the full dataset. Bayesian-guided inverse design under unseen loading conditions reaches a prescribed error threshold with only a handful of oracle evaluations in the majority of cases.

Reduced-order structure-property linkages for stochastic metamaterials

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.

FFT-based surrogate modeling of auxetic metamaterials with real-time prediction of effective elastic properties and swift inverse design

Auxetic structures, known for their negative Poisson's ratio, exhibit effective elastic properties heavily influenced by their underlying structural geometry and base material properties. While periodic homogenization of auxetic unit cells can be used to investigate these properties, it is computationally expensive and limits design space exploration and inverse analysis. In this paper, surrogate models are developed for the real-time prediction of the effective elastic properties of auxetic unit cells with orthogonal voids of different shapes. The unit cells feature orthogonal voids in four distinct shapes, including rectangular, diamond, oval, and peanut-shaped voids, each characterized by specific void diameters. The generated surrogate models accept geometric parameters and the elastic properties of the base material as inputs to predict the effective elastic constants in real-time. This rapid evaluation enables a practical inverse analysis framework for obtaining the optimal design parameters that yield the desired effective response. The fast Fourier transform (FFT)-based homogenization approach is adopted to efficiently generate data for developing the surrogate models, bypassing concerns about periodic mesh generation and boundary conditions typically associated with the finite element method (FEM). The performance of the generated surrogate models is rigorously examined through a train/test split methodology, a parametric study, and an inverse problem. Finally, a graphical user interface (GUI) is developed, offering real-time prediction of the effective tangent stiffness and performing inverse analysis to determine optimal geometric parameters.