homogenization techniques

To address the high computational cost arising from microscale repetitive calculations in simulating multiscale rate-dependent materials (e.g., viscoelastic solids), this work proposes a neural-operator-driven hybrid micromechanical constitutive model. The method innovatively embeds a neural operator into the evolution equations of microscale internal variables, synergistically integrating physical constraints with data-driven learning to explicitly capture microstructural effects. By coupling computational homogenization with physics-informed deep learning, the model preserves mechanistic interpretability while enabling generalization across material types and mesh resolutions. Experiments demonstrate homogenized stress prediction errors below 6% and approximately 100× speedup over conventional multiscale approaches. The core contribution is a physics-guided neural-operator framework for micromechanical modeling—achieving a balanced trade-off among accuracy, computational efficiency, and generalizability.

This work addresses the longstanding challenge of non-invasively inferring statistical characteristics of material microstructures from macroscopic mechanical responses. The authors propose a distributional inverse homogenization framework that, for the first time, integrates probabilistic modeling with homogenization theory to formulate a novel class of inverse problems. By leveraging large datasets of macroscopic measurements, the method learns global statistical properties of underlying microstructures. Built upon Voronoi tessellations and accelerated via surrogate models, the approach is theoretically grounded in the one-dimensional setting. Numerical experiments on two-dimensional Voronoi-based material systems demonstrate that the statistical distribution of microstructural features can be accurately recovered solely from macroscopic responses, enabling non-invasive, data-driven inference of microstructural variability.

Predicting effective constitutive tensors (e.g., elasticity, thermal conductivity, permeability) for heterogeneous materials faces key bottlenecks: high computational cost, non-differentiability, and violations of fundamental physical bounds—namely the Voigt (upper) and Reuss (lower) limits. To address these, this paper proposes a physics-constrained neural network incorporating spectral normalization. At the architectural level, the model intrinsically enforces Loewner order constraints, rigorously guaranteeing output tensor symmetry and adherence to Voigt–Reuss bounds—achieving feature-agnostic generality. By synergistically integrating Voigt–Reuss homogenization theory with surrogate modeling, the method attains superior accuracy and strong robustness on large-scale datasets, significantly outperforming conventional neural networks. Moreover, its fully differentiable formulation enables gradient-based microstructure inverse design—a capability absent in standard black-box surrogates.

Conventional homogenization methods for high-dimensional spatiotemporal physical systems with complex microstructures rely on ad hoc averaging assumptions, compromising both accuracy and physical interpretability. Method: This project establishes a rigorous asymptotic homogenization framework tailored to systems exhibiting finite-scale separation—deriving macroscopic multi-continuum (i.e., multi-microstructural) models directly from first-principles microscale physics, without presupposing any averaging hypotheses. The approach integrates asymptotic analysis, modern dynamical systems theory, and symbolic computation to automate, verify, and control the accuracy of model derivation. Contribution/Results: The resulting models are physically transparent and mathematically rigorous, significantly improving the accuracy and reliability of macroscopic dynamical predictions. This work introduces a novel paradigm for multiscale modeling of systems with intricate microstructures, enabling systematic, assumption-free upscaling from microscale physics to macroscopic continuum descriptions.

Traditional numerical homogenization methods suffer from high computational cost and poor generalizability in complex geometries, multiphase materials, and high-resolution settings. To address this, we propose the first foundational neural operator model for multiscale mechanical homogenization. Our method integrates Fourier neural operators, multiscale geometric embeddings, and triply periodic minimal surface (TPMS) structural modeling to enable end-to-end, millisecond-scale prediction of effective elastic tensors for arbitrary geometries, material parameters, and spatial resolutions. It overcomes the computational rigidity of finite element methods and breaks the cross-scale and cross-geometry generalization bottlenecks. On TPMS-based periodic materials, our model achieves an 80× speedup over conventional solvers, with a mean absolute error in predicted elastic modulus below 1.2%. Crucially, it supports practical scenarios where training and testing resolutions differ—enabling robust deployment across heterogeneous discretizations.

This work addresses the high computational cost of traditional numerical homogenization methods in predicting the effective mechanical properties of hyperelastic composites with Boolean microstructures. A supervised learning–based neural network surrogate model is proposed, integrating multiscale statistical descriptors—including area fraction, shape descriptor τ, two-point correlation function S₂(r), and lineal path function ℓ(z)—to enable rapid prediction of effective Lamé parameters for two-phase hyperelastic composites. Model extrapolation capability is enhanced through data augmentation with extreme loading scenarios, and generalization performance is rigorously evaluated via leave-one-grain-type-out cross-validation. The study demonstrates that combining τ and S₂(r) suffices for accurate and compact modeling, while inclusion of ℓ(z) further reduces sampling error, albeit requiring careful attention to physical plausibility within interpolation regions.

This work addresses the prohibitive computational cost of full-field loss evaluation in physics-informed operator learning for microstructure surrogate modeling. To overcome this challenge, the authors propose a novel framework that integrates Equivariant Neural Operators (EquiNO) with a QR-based Discrete Empirical Interpolation Method (Q-DEIM). By constructing reduced-order representations of displacement and stress fields using periodic, divergence-free basis functions and evaluating constitutive relations only at a small number of spatial points, the method—introducing Q-DEIM to this domain for the first time—dramatically reduces training expenses. It directly predicts homogenized stresses without reconstructing full fields and demonstrates strong interpolation and extrapolation capabilities even with very few training snapshots. Numerical experiments show a reduction of approximately three orders of magnitude in per-step training cost and acceleration of homogenization computations by factors of $10^3$–$10^4$ compared to full-field simulations, while accurately capturing both microscopic stress fields and macroscopic responses.

This study addresses the high computational cost associated with computational homogenization of representative volume elements (RVEs) composed of hyperelastic materials undergoing large deformations. To this end, the authors propose a highly reduced-order modeling approach formulated in strain space. The method leverages empirical material sampling and clustering during an offline stage to construct a piecewise linearized constitutive approximation. Combined with Proper Orthogonal Decomposition (POD) modes, it enables linear estimation of stress responses at each load increment, thereby circumventing online Newton iterations. Owing to its affine reduced representation and exact integration, the approach achieves substantial gains in online efficiency. Numerical experiments on porous hyperelastic RVEs demonstrate that the proposed method attains a Pareto improvement over existing strain-space reduction techniques, offering superior trade-offs between accuracy and computational time.

This study addresses the challenge of achieving efficient and highly accurate model order reduction for large-deformation solid mechanics problems involving parametrized materials and boundary conditions. It extends strain-space hyper-reduction methods—specifically ECM, E3C, and EMSL—to non-uniform large-deformation scenarios for the first time. By constructing, offline, compatible lifting fields that satisfy arbitrary Dirichlet boundary conditions, the approach rigorously enforces boundary consistency. In two hyperelastic test cases, the proposed strain-space methods significantly outperform the displacement-space ECSW method: EMSL achieves acceleration of approximately 10⁵-fold, while E3C delivers exceptional accuracy at only marginally higher computational cost. This work overcomes key limitations of conventional displacement-based reduced-order modeling in nonlinear large-deformation settings.

This work addresses the formidable computational challenges in high-dimensional, multiscale structural reliability analysis, where uncertainty propagation from microscale material randomness to macroscale responses renders conventional methods inefficient. The authors propose a novel framework that integrates a physics-informed Voigt–Reuss neural network (VRNN) with a deep inverse Rosenblatt transformation (DIRT). Under assumptions of scale separation and anisotropic linear elasticity, this approach uniquely combines the VRNN—enforcing physical consistency—with tensor-train-based high-dimensional importance sampling. The method enables efficient joint treatment of multiscale simulation and uncertainty quantification, achieving low-variance estimates of failure probabilities in benchmark three-dimensional heterogeneous material problems with up to 150 stochastic dimensions, thereby substantially enhancing computational efficiency and scalability.