Historically, buckling and large deformations are often viewed as the onset of failure, and much investigation has gone into improving materials' performance to avoid them. As interest in metamaterials has grown, so has interest in leveraging controlled buckling/deformation to achieve more exotic and controllable material properties such as stiffness, shape change, or acoustic properties. Understanding microstructural behaviour under loading is necessary for designing metamaterials for specific target applications or interpreting results from inverse design schemes. This work seeks to develop a proper understanding of the relationship between microstructural morphology, buckling-like patterning modes and resulting macroscopic stiffness. We aim to provide design principles for creating microstructures that enable passive and active control of buckling behaviour, achieved through microstructure design and magnetic actuation. A numerical study of microstructural morphology variations shows that small changes strongly influence the competition between local and global buckling modes. Furthermore, the derivation of an idealised analytical model representing the microstructure shows good qualitative agreement with numerical simulations. Using the insights gained from the idealised model, basic design principles are developed for actuated materials with switchable properties. Using these design principles, we successfully design and numerically validate magnetic patterns that demonstrate different buckling modes, depending on the choice of pattern and application of external magnetic fields. Having understood the effect of microstructure on deformation and buckling, new material microstructures are proposed to promote different buckling modes. Hence, the proposed analytical model serves as a versatile design guide to achieve target deformations and active control of metamaterials.

ABSTRACT Lattice systems are indispensable for modeling and analyzing physical phenomena in materials with discrete or heterogeneous micro‐ or meso‐structures. However, the computational requirements for practical engineering applications of lattice systems remain high. The quasicontinuum (QC) method addresses this by reducing the system of equations using a finite element mesh interpolation, rather than considering all nodes of a fully resolved lattice. Nevertheless, interfaces between separate phases in heterogeneous materials, such as concrete, require fine meshes throughout the domain, diminishing the effectiveness of QC. Enrichment strategies originally introduced in the extended finite element method (XFEM) can also account for material interfaces in discrete systems using nonconforming meshes, thereby resolving this issue. In a previous study, we applied Heaviside enrichment to investigate concrete mesostructures with the extended QC method, achieving a tenfold reduction in the number of unknowns while maintaining similar accuracy compared to discretizations with fully resolved interfaces. In the present study, we employ the meshless local maximum entropy (LME) interpolation, which transitions seamlessly from widespread meshfree to linear basis functions. Additionally, we combine LME interpolation with Heaviside enrichment and systematically investigate the role of the locality parameter and its optimization in heterogeneous lattices. This combination of optimized LME basis functions with Heaviside enrichment leads to an order‐of‐magnitude improvement in displacement accuracy while using the same number of degrees of freedom (DOF) compared to QC with linear interpolation. Moreover, we identify optimized distributions of the LME locality parameter and propose simple, non‐optimized rules that deliver comparable accuracy at a fraction of the computational cost. Results from three numerical examples show that the optimal locality‐parameter fields are nonuniform near interfaces and can be approximated by simple pattern‐based rules that retain much of the benefit of full optimization.

Lattice systems are effective for modeling heterogeneous materials, but their computational cost is often prohibitive. The QuasiContinuum (QC) method reduces this cost by interpolating the lattice response over a coarse finite-element mesh, yet material interfaces in heterogeneous systems still require fine discretizations. Enrichment strategies from the eXtended Finite Element Method (XFEM) address this by representing interfaces on nonconforming meshes. In this work, we combine Heaviside enrichment with meshless Local Maximum Entropy (LME) interpolation in the QC framework for heterogeneous lattice systems. We systematically investigate the role of the LME locality parameter and its optimization. The results show that optimized LME interpolation improves displacement accuracy by about one order of magnitude over QC with linear interpolation at the same number of degrees of freedom. In addition, the optimal locality-parameter fields are nonuniform near interfaces and exhibit systematic spatial structure. Based on these observations, we derive simple pattern-based rules that retain much of the benefit of full optimization at a fraction of the computational cost. The approach is demonstrated on three numerical examples.

Physics-informed neural networks (PINNs) have shown promise in solving partial differential equations (PDEs) relevant to multiscale modeling, but they often fail when applied to materials with discontinuous coefficients, such as media with piecewise constant properties. This paper introduces a dual formulation for the PINN framework to improve the reliability of the homogenization of periodic thermo-conductive composites, for both strong and variational (weak) formulations. The dual approach facilitates the derivation of guaranteed upper and lower error bounds, enabling more robust detection of PINN failure. We compare standard PINNs applied to smoothed material approximations with variational PINNs (VPINNs) using both spectral and neural network-based test functions. Our results indicate that while strong-form PINNs may outperform VPINNs in controlled settings, they are sensitive to material discontinuities and may fail without clear diagnostics. In contrast, VPINNs accommodate piecewise constant material parameters directly but require careful selection of test functions to avoid instability. Dual formulation serves as a reliable indicator of convergence quality, and its integration into PINN frameworks enhances their applicability to homogenization problems in micromechanics.

Honeycomb microstructures with circular voids are well known to undergo pattern transformations under macroscopic strain loading. Depending on the biaxiality of the applied strain, they deform into three different patterns. Here we demonstrate that the same three patterns can be triggered also by pneumatic actuation of the voids, with resulting patterns depending on an applied pressurization scheme, i.e., the ratio of suction pressures introduced in different voids within the microstructure. Our numerically obtained findings are experimentally validated on two finite size samples made of silicone rubber cast in a 3D printed mold. In numerical studies, we first showcase the evolution of homogenized stiffness and its anisotropy during the genesis of pneumatically-induced patterns. Our loading schemes result in macroscopic stiffness varying by a factor of two with loading direction, chosen by a reversible actuation independent of the load itself. Second, we study the effect of the pneumatic actuation on acoustic properties of the microstructure, finding the proposed method viable to trigger different acoustic bandgaps.

Pattern-forming metamaterials feature microstructures specifically designed to change the material's macroscopic properties due to internal instabilities. These can be triggered either by mechanical deformation or, in the case of active materials, by other external stimuli, such as pneumatic actuation. We study a two-dimensional rectangular lattice microstructure which is pneumatically actuated by non-uniform pressure patterns in its voids, and demonstrate that this actuation may lead to different instability patterns. The patterns are associated with a significant reduction in the macroscopic stiffness of the material. The magnitude of this reduction can be controlled by different arrangements of the pressure actuation, thus choosing the precise buckled shape of the microstructure. We develop an analytical model and complement it with computational tests on a two-dimensional plane-strain finite element model. We explain the phenomenon and discuss ways of further developing the concept to actively control the stiffness of materials and structures.

While modular structures offer great construction efficiency, scalability, safety, and reusability in engineering and architectural applications, their wide-spread adoption is hindered by the perceived material inefficiency and low design flexibility. Finding an optimal design within a modular system is a significant challenge, mostly because of associated computational complexity. Existing methods of accelerating combinatorial optimization with machine learning rely on heuristics and are often not transferrable between varying domain shapes, boundary conditions, and external loads.In this work, we present two key contributions to address this issue: (i) a deep neural network (DNN)-based surrogate model that accelerates the evaluation of mechanical responses by predicting reduced-order stiffness matrices, and (ii) a stochastic gradient optimization method that leverages the surrogate’s capability to compute sensitivities of the structure’s response to changes in module types. Our model combines convolutional layers with a physics-guided approach, ensuring that the output stiffness matrices are symmetric positive definite, consistent with the structure’s reduced-order representation via Schur’s complement.A distinguishing feature of our approach is its intrinsic independence from the specific domain shape, boundary conditions, and applied loads, allowing for broader applicability once the DNN-based surrogate is trained on a specific module set. We validate our method by optimizing multiple modular layout plans differing in size and loading conditions and demonstrate its efficacy by comparing its performance against the standard density-based topology optimization method. We achieve a computational speed-up of up to 1000x compared to the full-scale simulation, with a fast converging optimization for different domain sizes. This work lays the foundation for more flexible, efficient, and scalable modular design processes.

Bifurcation phenomena in nonlinear dynamical systems often lead to multiple coexisting stable solutions, particularly in the presence of symmetry breaking. Deterministic machine learning models struggle to capture this multiplicity, averaging over solutions and failing to represent lower-symmetry outcomes. In this work, we propose a generative framework based on flow matching to model the full probability distribution over bifurcation outcomes. Our method enables direct sampling of multiple valid solutions while preserving system symmetries through equivariant modeling. We introduce a symmetric matching strategy that aligns predicted and target outputs under group actions, allowing accurate learning in equivariant settings. We validate our approach on a range of systems, from toy models to complex physical problems such as buckling beams and the Allen-Cahn equation. Our results demonstrate that flow matching significantly outperforms non-probabilistic and variational methods in capturing multimodal distributions and symmetry-breaking bifurcations, offering a principled and scalable solution for modeling multistability in high-dimensional systems.

This paper introduces a deep learning approach for predicting time-dependent full-field damage in concrete. The study uses an auto-regressive U-Net model to predict the evolution of the scalar damage field in a unit cell given microstructural geometry and evolution of an imposed shrinkage profile. By sequentially using the predicted damage output as input for subsequent predictions, the model facilitates the continuous assessment of damage progression. Complementarily, a convolutional neural network (CNN) utilises the damage estimations to forecast key mechanical properties, including observed shrinkage and residual stiffness. The proposed dual-network architecture demonstrates high computational efficiency and robust predictive performance on the synthesised datasets. The approach reduces the computational load traditionally associated with full-field damage evaluations and is used to gain insights into the relationship between aggregate properties, such as shape, size, and distribution, and the effective shrinkage and reduction in stiffness. Ultimately, this can help to optimize concrete mix designs, leading to improved durability and reduced internal damage.

Soft, porous mechanical metamaterials exhibit pattern transformations that may have important applications in soft robotics, sound reduction and biomedicine. To design these innovative materials, it is important to be able to simulate them accurately and quickly, in order to tune their mechanical properties. Since conventional simulations using the finite element method entail a high computational cost, in this article we aim to develop a machine learning-based approach that scales favorably to serve as a surrogate model. To ensure that the model is also able to handle various microstructures, including those not encountered during training, we include the microstructure as part of the network input. Therefore, we introduce a graph neural network that predicts global quantities (energy, stress stiffness) as well as the pattern transformations that occur (the kinematics). To make our model as accurate and data-efficient as possible, various symmetries are incorporated into the model. The starting point is an E(n)-equivariant graph neural network (which respects translation, rotation and reflection) that has periodic boundary conditions (i.e., it is in-/equivariant with respect to the choice of RVE), is scale in-/equivariant, can simulate large deformations, and can predict scalars, vectors as well as second and fourth order tensors (specifically energy, stress and stiffness). The incorporation of scale equivariance makes the model equivariant with respect to the similarities group, of which the Euclidean group E(n) is a subgroup. We show that this network is more accurate and data-efficient than graph neural networks with fewer symmetries. To create an efficient graph representation of the finite element discretization, we use only the internal geometrical hole boundaries from the finite element mesh to achieve a better speed-up and scaling with the mesh size.

Mechanical metamaterials often exhibit pattern transformations through instabilities, enabling applications in, e.g., soft robotics, sound reduction, and biomedicine. These transformations and their resulting mechanical properties are closely tied to the symmetries in these metamaterials' microstructures, which remain under-explored. Designing such materials is challenging due to the unbounded design space, and while machine learning offers promising tools, they require extensive training data. Here, we present a large dataset of 2D microstructures and their macroscopic mechanical responses in the hyperelastic, finite-strain regime, including buckling. The microstructures are generated using a novel method, which covers all 17 wallpaper symmetry groups and employs Bézier curves for a rich parametric space. Mechanical responses are obtained through finite element-based computational homogenization. The dataset includes 1,020 distinct geometries, each subjected to 12 loading trajectories, totaling 12,240 trajectories. Our dataset supports the development and benchmarking of surrogate models, facilitates the study of symmetry-property relationships, and enables investigations into symmetry-breaking during pattern transformations, potentially revealing emergent behavior in mechanical metamaterials.

The structural properties of mechanical metamaterials are typically studied with two-scale methods based on computational homogenization. Because such materials have a complex microstructure, enriched schemes such as second-order computational homogenization are required to fully capture their non-linear behavior, which arises from non-local interactions due to the buckling or patterning of the microstructure. In the two-scale formulation, the effective behavior of the microstructure is captured with a representative volume element (RVE), and a homogenized effective continuum is considered on the macroscale. Although an effective continuum formulation is introduced, solving such two-scale models concurrently is still computationally demanding due to the many repeated solutions for each RVE at the microscale level. In this work, we propose a reduced-order model for the microscopic problem arising in second-order computational homogenization, using proper orthogonal decomposition and a novel hyperreduction method that is specifically tailored for this problem and inspired by the empirical cubature method. Two numerical examples are considered, in which the performance of the reduced-order model is carefully assessed by comparing its solutions with direct numerical simulations (entirely resolving the underlying microstructure) and the full second-order computational homogenization model. The reduced-order model is able to approximate the result of the full computational homogenization well, provided that the training data is representative for the problem at hand. Any remaining errors, when compared with the direct numerical simulation, can be attributed to the inherent approximation errors in the computational homogenization scheme. Regarding run times for one thread, speed-ups on the order of 100 are achieved with the reduced-order model as compared to direct numerical simulations.

This paper presents the first method to enable friction in the Third Medium Contact (TMC) method. TMC embeds a solid in a highly compliant medium, which becomes infinitely stiff under ultimate compression, thus allowing forces to be transferred between the solids when the medium between the solids is compressed. This approach is increasingly adopted for integrating internal contact in structural design processes, owing to its continuous, fully implicit characteristics, simplicity, and its stability when applying regularisation to the third medium regions. The lack of friction has previously restricted the use of the TMC method in simulating real-world contact conditions. Here, we address this issue by (1) integrating an anisotropic term into a Neo-Hookean material model to provide shear resistance, and (2) employing a framework inspired by crystal plasticity that includes a yield criterion specifically designed to replicate the effects of Coulomb friction.The effectiveness of the method is demonstrated through two examples: (1) a smooth sliding contact problem and (2) a non-smooth C-shaped structure. Results demonstrate a close agreement with reference solutions obtained by a conventional Lagrange multiplier approach. While the method, for now, requires user-defined slip directions, it represents a significant advancement by enabling the integration of friction into TMC, thereby broadening its applicability to problems involving realistic frictional contact. Future research should focus on restoring the fully implicit nature of TMC in the presence of friction, and on developing automated slip direction definitions to enhance usability and expand the method’s versatility.

In recent years, there has been a growing interest in understanding complex microstructures and their effect on macroscopic properties. In general, it is difficult to derive an effective constitutive law for such microstructures with reasonable accuracy and meaningful parameters. One numerical approach to bridge the scales is computational homogenization, in which a microscopic problem is solved at every macroscopic point, essentially replacing the effective constitutive model. Such approaches are, however, computationally expensive and typically infeasible in multi-query contexts such as optimization and material design. To render these analyses tractable, surrogate models that can accurately approximate and accelerate the microscopic problem over a large design space of shapes, material and loading parameters are required. In this work, we develop a reduced order model based on Proper Orthogonal Decomposition (POD), Empirical Cubature Method (ECM) and a geometrical transformation method with the following key features: (i) large shape variations of the microstructure are captured, (ii) only relatively small amounts of training data are necessary, and (iii) highly non-linear history-dependent behaviors are treated. The proposed framework is tested and examined in two numerical examples, involving two scales and large geometrical variations. In both cases, high speed-ups and accuracies are achieved while observing good extrapolation behavior.

Lattice networks are indispensable to study heterogeneous materials such as concrete or rock as well as textiles and woven fabrics. Due to the discrete character of lattices, they quickly become computationally intensive. The QuasiContinuum (QC) Method resolves this challenge by interpolating the displacement of the underlying lattice with a coarser finite element mesh and sampling strategies to accelerate the assembly of the resulting system of governing equations. In lattices with complex heterogeneous microstructures with a high number of randomly shaped inclusions the QC leads to an almost fully-resolved system due to the many interfaces. In the present study the QC Method is expanded with enrichment strategies from the eXtended Finite Element Method (XFEM) to resolve material interfaces using nonconforming meshes. The goal of this contribution is to bridge this gap and improve the computational efficiency of the method. To this end, four different enrichment strategies are compared in terms of their accuracy and convergence behavior. These include the Heaviside, absolute value, modified absolute value and the corrected XFEM enrichment. It is shown that the Heaviside enrichment is the most accurate and straightforward to implement. A first-order interaction based summation rule is applied and adapted for the extended QC for elements intersected by a material interface to complement the Heaviside enrichment. The developed methodology is demonstrated by three numerical examples in comparison with the standard QC and the full solution. The extended QC is also able to predict the results with 5 percent error compared to the full solution, while employing almost one order of magnitude fewer degrees of freedom than the standard QC and even more compared to the fully-resolved system.

Elastomeric mechanical metamaterials exhibit unconventional behaviour, emerging from their microstructures often deforming in a highly nonlinear and unstable manner. Such microstructural pattern transformations lead to non-local behaviour and induce abrupt changes in the effective properties, beneficial for engineering applications. To avoid expensive simulations fully resolving the underlying microstructure, homogenization methods are employed. In this contribution, a systematic comparative study is performed, assessing the predictive capability of several computational homogenization schemes in the realm of two-dimensional elastomeric metamaterials with a square stacking of circular holes. In particular, classical first-order and two enriched schemes of second-order and micromorphic computational homogenization type are compared with ensemble-averaged full direct numerical simulations on three examples: uniform compression and bending of an infinite specimen, and compression of a finite specimen. It is shown that although the second-order scheme provides good qualitative predictions, it fails in accurately capturing bifurcation strains and slightly over-predicts the homogenized response. The micromorphic method provides the most accurate prediction for tested examples, although soft boundary layers induce large errors at small scale ratios. The first-order scheme yields good predictions for high separations of scales, but suffers from convergence issues, especially when localization occurs.

A micromorphic computational homogenization framework has recently been developed to deal with materials showing long-range correlated interactions, i.e. displaying patterning modes. Typical examples of such materials are elastomeric mechanical metamaterials, in which patterning emerges from local buckling of the underlying microstructure. Because pattern transformations significantly influence the resulting effective behaviour, it is vital to distinguish them from the overall deformation. To this end, the following kinematic decomposition into three parts was introduced in the micromorphic scheme: (i) a smooth mean displacement field, corresponding to the slowly varying deformation at the macro-scale, (ii) a long-range correlated fluctuation field, related to the buckling pattern at the meso-scale, and (iii) the remaining uncorrelated local microfluctuation field at the micro-scale. The micromorphic framework has proven to be capable of predicting relevant mechanical behaviour, including size effects and spatial as well as temporal mixing of patterns in elastomeric metamaterials, making it a powerful tool to design metamaterials for engineering applications. The long-range correlated fluctuation fields need to be, however, provided a priori as input parameters. The main goal of this study is experimental identification of the decomposed kinematics in cellular metamaterials based on the three-part ansatz. To this end, a full-field micromorphic Integrated Digital Image Correlation (IDIC) technique has been developed. The methodology is formulated for finite-size cellular elastomeric metamaterial specimens deformed in (i) virtually generated images and (ii) experimental images attained during in-situ compression of specimens with millimetre sized microstructure using optical microscopy. The proposed IDIC method identifies the different kinematic fields, both before and after the microstructural buckling, and without any prior knowledge determines correctly the relevant patterning modes required by the homogenization scheme. It is further argued that patterning modes are independent of the unit cell size, the hole diameter to cell size ratio, as well as local material properties, allowing for modelling and design of (finite- and infinite-size) metamaterials and specimens with graded microstructures in terms of geometry and/or material properties. It is shown that the proposed methodology is also applicable to cellular metamaterials and structures with different microstructural designs.

Active mechanical metamaterials are artificially engineered microstructures that can be externally controlled to exhibit novel and switchable mechanical behavior on the macroscopic scale. Pneumatically actuated variants of these metamaterials can then change their mechanical, acoustic, or other types of effective behavior in response to applied pressure with possible applications ranging from soft robotic actuators to phononic crystals. Such materials often undergo large deformations, leading to self-contact, which makes their modeling and generative design challenging. We propose to use the third medium method to simulate both pneumatic actuation and contact behavior. In contrast to existing formulations, we (i) combine together contact with pneumatic actuation in a single material model for the third medium, (ii) include more permissive regularization scheme for our model, penalizing gradient of pseudo-rotation and gradient of volume change only, and (iii) introduce pneumatic actuation as an exact value of Cauchy stress. Our formulation is energetically consistent and admits advanced finite element solvers, such as the modified Cholesky decomposition to detect instabilities. We demonstrate the behavior of the proposed formulation on several examples of traditional contact benchmarks, including a standard patch test, and validate it with experimental measurement.

Micromechanical constitutive parameters are important for many engineering materials, typically in microelec-tronic applications and material design. Their accurate identification poses a three-fold experimental challenge: (i) deformation of the microstructure is observable only at small scales, requiring SEM or other microscopy techniques; (ii) external loadings are applied at a (larger) engineering or device scale; and (iii) material parameters typically depend on the applied manufacturing process, necessitating measurements on material produced with the same process. In this paper, micromechanical parameter identification in heterogeneous solids is addressed through multiscale experiments combined with Integrated Digital Image Correlation (IDIC) in conjunction with various possible computational homogenization schemes. To this end, some basic concepts underlying multiscale approaches available in the literature are first reviewed, discussing their respective advantages and disadvantages from the computational as well as experimental point of view. A link is made with recently introduced uncoupled methods, which allow for identification of material parameter ratios at the microscale, still lacking a proper normalization. Two multiscale methods are analysed, allowing to bridge the gap between microstructural kinematics and macroscopically measured forces, providing the required normalization. It is shown that an integrated experimental-computational scheme provides relaxed requirements on scale separation. The accuracy and performance of the discussed techniques are analysed by means of virtual experimentation under plane strain and large strain assumptions for unidirectional fibre-reinforced composites. The robustness against image noise is also assessed. The obtained results demonstrate that the expected accuracy is typically within 10% RMS error for all multiscale methods, but decreasing to 1% RMS error for the optimal method.