Generative machine learning models have revolutionized material discovery by capturing complex structure–property relationships, yet extending these approaches to the inverse design of three-dimensional metamaterials remains limited by computational complexity and underexplored design spaces due to the lack of expressive representations. Here we present DiffuMeta, a generative framework integrating diffusion transformers with an algebraic language representation, encoding three-dimensional geometries as mathematical sentences. This compact, unified parameterization spans diverse topologies, enabling the direct application of transformers to structural design. DiffuMeta leverages diffusion models to generate new shell structures with precisely targeted stress–strain responses under large deformations, accounting for buckling and contact while addressing the inherent one-to-many mapping by producing diverse solutions. Uniquely, our approach enables simultaneous control over multiple mechanical objectives, including linear and nonlinear responses beyond training domains. Experimental validation of fabricated structures further confirms the efficacy of our approach for accelerated design of metamaterials and structures with tailored properties.

Constitutive evaluations often dominate the computational cost of finite element (FE) simulations whenever material models are complex. Neural constitutive models (NCMs), i.e., neural network-based constitutive models, offer a highly expressive and flexible framework for modeling complex material behavior in solid mechanics. However, their practical adoption in large-scale FE simulations remains limited due to significant computational costs, especially in repeatedly evaluating stress and stiffness. NCMs thus represent an extreme case: their large computational graphs make stress and stiffness evaluations prohibitively expensive, restricting their use to small-scale problems. In this work, we introduce COMMET, an open-source FE framework whose architecture has been redesigned from the ground up to accelerate high-cost constitutive updates. Our framework features a novel assembly algorithm that supports batched and vectorized constitutive evaluations, compute-graph-optimized derivatives that replace automatic differentiation, and distributed-memory parallelism via MPI. These advances dramatically reduce runtime, with speed-ups exceeding three orders of magnitude relative to traditional non-vectorized automatic differentiation-based implementations. While we demonstrate these gains primarily for NCMs, the same principles apply broadly wherever for-loop based assembly or constitutive updates limit performance, establishing a new standard for large-scale, high-fidelity simulations in computational mechanics.

We propose a computational framework, Hetero-EUCLID, for segmentation and parameter identification to characterize the full hyperelastic behavior of all constituents of a heterogeneous material. In this work, we leverage the Bayesian-EUCLID (Efficient Unsupervised Constitutive Law Identification and Discovery) framework to efficiently solve the heterogenized formulation through parsimonious model selection using sparsity-promoting priors and Monte Carlo Markov Chain sampling. We utilize experimentally observable 3D surface displacement and boundary-averaged force data generated from Finite Element simulations of non-equi-biaxial tension tests on heterogeneous specimens. The framework broadly consists of two steps– residual force-based segmentation and constitutive parameter identification. We validate and demonstrate the ability of the proposed framework to segment the domain and characterize the constituent materials on various types of thin square heterogeneous domains. We validate the framework's ability to segment and characterize materials with multiple levels of displacement noises and non-native mesh discretizations, i.e, using different meshes for the forward FE simulations and the inverse EUCLID problem. This demonstrates the applicability of the Hetero-EUCLID framework in Digital Image/Volume Correlation-based experimental scenarios. Furthermore, the proposed framework performs successful segmentation and material characterizations based on data from a single experiment, thereby making it viable for rapid, interpretable model discovery in domains such as aerospace and defense composites and for characterization of selective tissue stiffening in medical conditions such as fibroatheroma, atherosclerosis, or cancer.

Background: Determining accurate constitutive laws from experimental data remains a key challenge in mechanics, particularly when the material behavior is nonlinear and the dataset is limited or noisy. Traditional approaches rely on identifying parameters of preselected material models, which separates the model selection and the calibration tasks leading to a potentially long and tedious trial-and-error procedure. Objective: This work aims to assess the performance of EUCLID (Efficient Unsupervised Constitutive Law Identification and Discovery), a recently proposed framework for automated discovery of constitutive laws, when applied to experimental data. Methods: Mechanical tests are conducted on natural rubber specimens of varying geometrical complexity. Both global (force–elongation) and local (full-field displacement) data are collected. Constitutive laws are obtained via two routes: (i) conventional parameter identification in a priori selected models, and (ii) EUCLID, which integrates model selection and parameter identification in a unified model discovery pipeline. Results: The two approaches are compared in terms of predictive accuracy, generalization to unseen geometries, and robustness to experimental noise. The coverage of the material state space achieved by each dataset is quantified, and the relative performance of different datasets and models is analyzed. Conclusions: EUCLID enables automated and data-driven discovery of constitutive laws, offering improved flexibility compared to conventional identification methods, and showing strong potential for reliable material modeling based on experimental data.

Ions play a fundamental role in solid-liquid interface processes, whether as essential or undesirable components, highlighting the need for precise and quantitative real-time monitoring. Electrochemical sensors are identified as promising tools, particularly for field-deployable applications. However, conventional electrochemical sensing is inherently restricted to redox-active species and is often single use, constraining its scope. This study presents electrochemical impedance spectroscopy as an alternative for ion detection, utilizing physico-chemical interactions at the electrode-electrolyte interface. We introduce a first-principles model that describes the interfacial impedance behavior and shows how ion specific processes shape the impedance response. Based on this framework, an extensive dataset is compiled, and a machine learning model is trained to predict electrolyte composition with consistent accuracy, demonstrating detection limits at the parts-per-billion level. The findings indicate that this method has considerable potential as a real-time method for ion sensing, providing a perspective on selectivity and sensitivity beyond traditional electrochemical approaches. This work could serve as a foundation for advanced models of impedance behavior, and development of impedance-based sensors with applicability in complex environments, including biological fluids and industrial liquids.

The increasing availability of full-field displacement data from imaging techniques in experimental mechanics is determining a gradual shift in the paradigm of material model calibration and discovery, from using several simple-geometry tests towards a few, or even one single test with complicated geometry. The feasibility of such a “one-shot” calibration or discovery heavily relies upon the richness of the measured displacement data, i.e., their ability to probe the space of the state variables and the stress space (whereby the stresses depend on the constitutive law being sought) to an extent sufficient for an accurate and robust calibration or discovery process. The richness of the displacement data is in turn directly governed by the specimen geometry. In this paper, we propose a density-based topology optimisation framework to optimally design the geometry of the target specimen for calibration of an anisotropic elastic material model. To this end, we perform automatic, high-resolution specimen design by maximising the robustness of the solution of the inverse problem, i.e., the identified material parameters, given noisy displacement measurements from digital image correlation. We discuss the choice of the cost function and the design of the topology optimisation framework, and we analyse a range of optimised topologies generated for the identification of isotropic and anisotropic elastic responses.

Vitrimers represent an emerging class of sustainable polymers with self-healing capabilities enabled by dynamic covalent adaptive networks. However, their limited molecular diversity constrains their property space and potential applications. Recent developments in machine learning (ML) techniques accelerate polymer design by predicting properties and virtually screening candidates, yet the scarcity of available experimental vitrimer data poses challenges in training ML models. To address this, we leverage molecular dynamics (MD) data generated by our previous work to train and benchmark seven ML models covering six feature representations for glass transition temperature (Tg) prediction. By averaging predicted Tg from different models, the model ensemble approach outperforms individual models, allowing for accurate and efficient property prediction on unlabeled datasets. Two novel vitrimers are identified and synthesized, exhibiting experimentally validated higher Tg than existing bifunctional transesterification vitrimers, along with demonstrated healability. This work explores the possibility of using MD data to train ML models in the absence of sufficient experimental data, enabling the discovery of novel, synthesizable polymer chemistries with a wide range of desirable properties. The integrated MD–ML approach offers polymer chemists an efficient tool for accurate property prediction and designing polymers tailored to diverse applications.

The thermal conductivity of covalent organic frameworks (COFs), an emerging class of nanoporous polymeric materials, is crucial for many applications, yet the link between their structure and thermal properties remains poorly understood. Analysis of a dataset containing over 2400 COFs reveals that conventional features such as density, pore size, void fraction, and surface area do not reliably predict thermal conductivity. To address this, an attention-based machine learning model was trained, accurately predicting thermal conductivities even for structures outside the training set. The attention mechanism was then utilized to investigate the model's success. The analysis identified dangling molecular branches as a key predictor of thermal conductivity, leading us to define the dangling mass ratio (DMR), a descriptor that quantifies the fraction of atomic mass in dangling branches relative to the total COF mass. Feature importance assessments on regression models confirm the significance of DMR in predicting thermal conductivity. These findings indicate that COFs with dangling functional groups exhibit lower thermal transfer capabilities. Molecular dynamics simulations support this observation, revealing significant mismatches in the vibrational density of states due to the presence of dangling branches.

Traditional constitutive models rely on hand-crafted parametric forms with limited expressivity and generalizability, while neural network-based models can capture complex material behavior but often lack interpretability. To balance these trade-offs, we present monotonic Input-Convex Kolmogorov-Arnold Networks (ICKANs) for learning polyconvex hyperelastic constitutive laws. ICKANs leverage the Kolmogorov-Arnold representation, decomposing the model into compositions of trainable univariate spline-based activation functions for rich expressivity. We introduce trainable monotonic input-convex splines within the KAN architecture, ensuring physically admissible polyconvex models for isotropic compressible hyperelasticity. The resulting models are both compact and interpretable, enabling explicit extraction of analytical constitutive relationships through a monotonic input-convex symbolic regression technique. Through unsupervised training on full-field strain data and limited global force measurements, ICKANs accurately capture nonlinear stress–strain behavior across diverse strain states. Finite element simulations of unseen geometries with trained ICKAN hyperelastic constitutive models confirm the framework's robustness and generalization capability.

Spinodal metamaterials, with architectures inspired by natural phase-separation processes, have presented a significant alternative to periodic and symmetric morphologies when designing mechanical metamaterials with extreme performance. While their elastic mechanical properties have been systematically determined, their large-deformation, nonlinear responses have been challenging to predict and design, in part due to limited data sets and the need for complex nonlinear simulations. This work presents a novel physics-enhanced machine learning (ML) and optimization framework tailored to address the challenges of designing intricate spinodal metamaterials with customized mechanical properties in large-deformation scenarios where computational modeling is restrictive and experimental data is sparse. By utilizing large-deformation experimental data directly, this approach facilitates the inverse design of spinodal structures with precise finite-strain mechanical responses. The framework sheds light on instability-induced pattern formation in spinodal metamaterials—observed experimentally and in selected nonlinear simulations—leveraging physics-based inductive biases in the form of nonconvex energetic potentials. Altogether, this combined ML, experimental, and computational effort provides a route for efficient and accurate design of complex spinodal metamaterials for large-deformation scenarios where energy absorption and prediction of nonlinear failure mechanisms is essential.

We introduce HyperCAN, a machine learning framework that utilizes hypernetworks to construct adaptable constitutive artificial neural networks for a wide range of beam-based metamaterials exhibiting diverse mechanical behavior under finite deformations. HyperCAN integrates an input convex neural network that models the nonlinear stress–strain map of a truss lattice, while ensuring adherence to fundamental mechanics principles, along with a hypernetwork that dynamically adjusts the parameters of the convex network as a function of the lattice topology and geometry. This unified framework demonstrates robust generalization in predicting the mechanical behavior of previously unseen metamaterial designs and loading scenarios well beyond the training domain. We show how HyperCAN can be integrated into multiscale simulations to accurately capture the highly nonlinear responses of large-scale truss metamaterials, closely matching fully resolved simulations while significantly reducing computational costs. This offers new efficient opportunities for the multiscale design and optimization of truss metamaterials.

Vitrimer is a new, exciting class of sustainable polymers with healing abilities due to their dynamic covalent adaptive networks. However, a limited choice of constituent molecules restricts their property space and potential applications. To overcome this challenge, an innovative approach coupling molecular dynamics (MD) simulations and a novel graph variational autoencoder (VAE) model for inverse design of vitrimer chemistries with desired glass transition temperature (T_g) is presented. The first diverse vitrimer dataset of one million chemistries is curated and T_g for 8,424 of them is calculated by high-throughput MD simulations calibrated by a Gaussian process model. The proposed VAE employs dual graph encoders and a latent dimension overlapping scheme which allows for individual representation of multi-component vitrimers. High accuracy and efficiency of the framework are demonstrated by discovering novel vitrimers with desirable T_g beyond the training regime. To validate the effectiveness of the framework in experiments, vitrimer chemistries are generated with a target T_g = 323 K. By incorporating chemical intuition, a novel vitrimer with T_g of 311–317 K is synthesized, experimentally demonstrating healability and flowability. The proposed framework offers an exciting tool for polymer chemists to design and synthesize novel, sustainable polymers for various applications.

This study demonstrates that two- and three-dimensional spatially graded, truss-based polymeric-material metamaterials can be designed for beneficial impact mitigation and energy absorption capabilities. Through a combination of numerical and experimental techniques, we highlight the broad property space of periodic viscoelastic trusses, realized using 3D printing via selective laser sintering. Extending beyond periodic designs, we investigate the impact response of spatially variant viscoelastic lattices in both two and three dimensions. Our result reveal that introducing spatial variations in lattice topology allows for redirecting of the impact trajectory, opening new opportunities for engineering and tailoring lightweight materials with target impact functionality. This is achieved through the combined selection of base material and metamaterial design.

Smooth and curved microstructural topologies found in nature—from soap films to trabecular bone—have inspired several mimetic design spaces for architected metamaterials and bio-scaffolds. However, the design approaches so far are ad hoc, raising the challenge: how to systematically and efficiently inverse design such artificial microstructures with targeted topological features? Herein, surface curvature is explored as a design modality and a deep learning framework is presented to produce topologies with as-desired curvature profiles. The inverse design framework can generalize to diverse topological features such as tubular, membranous, and particulate features. Moreover, successful generalization beyond both the design and data space is demonstrated by inverse designing topologies that mimic the curvature profile of trabecular bone, spinodoid topologies, and periodic nodal surfaces for application in bio-scaffolds and implants. Lastly, curvature and mechanics are bridged by showing how topological curvature can be designed to promote mechanically beneficial stretching-dominated deformation over bending-dominated deformation.

Slender beams are often employed as constituents in engineering materials and structures. Prior experiments on lattices of slender beams have highlighted their complex failure response, where the interplay between buckling and fracture plays a critical role. In this paper, we introduce a novel computational approach for modeling fracture in slender beams subjected to large deformations. We adopt a state-of-the-art geometrically exact Kirchhoff beam formulation to describe the finite deformations of beams in three-dimensions. We develop a discontinuous Galerkin finite element discretization of the beam governing equations, incorporating discontinuities in the position and tangent degrees of freedom at the inter-element boundaries of the finite elements. Before fracture initiation, we enforce compatibility of nodal positions and tangents weakly, via the exchange of variationally-consistent forces and moments at the interfaces between adjacent elements. At the onset of fracture, these forces and moments transition to cohesive laws modeling interface failure. We conduct a series of numerical tests to verify our computational framework against a set of benchmarks and we demonstrate its ability to capture the tensile and bending fracture modes in beams exhibiting large deformations. Finally, we present the validation of our framework against fracture experiments of dry spaghetti rods subjected to sudden relaxation of curvature.

Natural porous materials have exceptional properties—for example, light weight, mechanical resilience, and multi-functionality. Efforts to imitate their properties in engineered structures have limited success. This, in part, is caused by the complexity of multi-phase materials composites and by the lack of quantified understanding of each component's role in overall hierarchy. This challenge is twofold: 1) computational. because non-periodicity and defects render constructing design guidelines between geometries and mechanical properties complex and expensive and 2) experimental. because the fabrication and characterization of complex, often hierarchical and non-periodic 3D architectures is non-trivial.

We extend EUCLID, a computational strategy for automated material model discovery and identification, to linear viscoelasticity. For this case, we perform a priori model selection by adopting a generalized Maxwell model expressed by a Prony series, and deploy EUCLID for identification. The methodology is based on four ingredients: i. full-field displacement and net force data; ii. a very wide material model library — in our case, a very large number of terms in the Prony series; iii. the linear momentum balance constraint; iv. the sparsity constraint. The devised strategy comprises two stages. Stage 1 relies on sparse regression; it enforces momentum balance on the data and exploits sparsity-promoting regularization to drastically reduce the number of terms in the Prony series and identify the material parameters. Stage 2 relies on k-means clustering; starting from the reduced set of terms from stage 1, it further reduces their number by grouping together Maxwell elements with very close relaxation times and summing the corresponding moduli. Automated procedures are proposed for the choice of the regularization parameter in stage 1 and of the number of clusters in stage 2. The overall strategy is demonstrated on artificial numerical data, both without and with the addition of noise, and shown to efficiently and accurately identify a linear viscoelastic model with five relaxation times across four orders of magnitude, out of a library with several hundreds of terms spanning relaxation times across seven orders of magnitude.

Advancements in machine learning have sparked significant interest in designing mechanical metamaterials, i.e., materials that derive their properties from their inherent microstructure rather than just their constituent material. We propose a data-driven exploration of the design space of growth-based cellular metamaterials based on star-shaped distances. These two-dimensional metamaterials are based on periodically-repeating unit cells consisting of material and void patterns with non-trivial geometries. Machine learning models exploiting large datasets are then employed to inverse design growth-based metamaterials for tailored anisotropic stiffness. Firstly, a forward model is created to bypass the growth and homogenization process and accurately predict the mechanical properties given a finite set of design parameters. Secondly, an inverse model is used to invert the structure–property maps and enable the accurate prediction of designs for a given anisotropic stiffness query. We successfully demonstrate the frameworks’ generalization capabilities by inverse designing for stiffness properties chosen from outside the domain of the design space.

The numerical simulation of additive manufacturing techniques promises the acceleration of costly experimental procedures to identify suitable process parameters. We recently proposed Floating Isogeometric Analysis (FLIGA), a new computational solid mechanics approach, which is mesh distortion-free in one characteristic spatial direction. FLIGA emanates from Isogeometric Analysis and its key novel aspect is the concept of deformation-dependent “floating” of individual B-spline basis functions along one parametric axis of the mesh. Our previous work showed that FLIGA not only overcomes the problem of mesh distortion associated to this direction, but is also ideally compatible with material point integration and enjoys a stability similar to that of conventional Lagrangian mesh-based methods. These features make the method applicable to the simulation of large deformation problems with history-dependent constitutive behavior, such as additive manufacturing based on polymer extrusion. In this work, we enhance the first version of FLIGA by (i) a novel quadrature scheme which further improves the robustness against mesh distortion, (ii) a procedure to automatically regulate floating of the basis functions (as opposed to the manual procedure of the first version), and (iii) an adaptive refinement strategy. We demonstrate the performance of enhanced FLIGA on relevant numerical examples including a selection of viscoelastic extrusion problems.

The rise of machine learning has fueled the discovery of new materials and, especially, metamaterials—truss lattices being their most prominent class. While their tailorable properties have been explored extensively, the design of truss-based metamaterials has remained highly limited and often heuristic, due to the vast, discrete design space and the lack of a comprehensive parameterization. We here present a graph-based deep learning generative framework, which combines a variational autoencoder and a property predictor, to construct a reduced, continuous latent representation covering an enormous range of trusses. This unified latent space allows for the fast generation of new designs through simple operations (e.g., traversing the latent space or interpolating between structures). We further demonstrate an optimization framework for the inverse design of trusses with customized mechanical properties in both the linear and nonlinear regimes, including designs exhibiting exceptionally stiff, auxetic, pentamode-like, and tailored nonlinear behaviors. This generative model can predict manufacturable (and counter-intuitive) designs with extreme target properties beyond the training domain.