arXiv subjects

Martin Doškář

Publications and source records attributed to Martin Doškář.

18 recordsLinked to original sources

arXiv2026

Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problems

Bifurcation phenomena in nonlinear dynamical systems often lead to multiple coexisting stable solutions, particularly in the presence of symmetry breaking. Deterministic machine learning models are unable to capture this multiplicity, averaging over solutions and failing to represent lower-symmetry outcomes. In this work, we formalize the use of generative AI, specifically flow matching, as a principled way to model the full probability distribution over bifurcation outcomes. Our approach builds on existing techniques by combining flow matching with equivariant architectures and an optimal-transport-based coupling mechanism. We generalize equivariant flow matching to a symmetric coupling 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 simple conceptual systems to physical problems such as buckling beams and the Allen--Cahn equation. The results demonstrate that the approach accurately captures multimodal distributions and symmetry-breaking bifurcations. Moreover, our results demonstrate that flow matching significantly outperforms non-probabilistic and variational methods. This offers a principled and scalable solution for modeling multistability in high-dimensional systems.

cs.LG↗

arXiv2026

Auto-Regressive U-Net for Full-Field Prediction of Shrinkage-Induced Damage in Concrete

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.

cs.LG↗

arXiv2026

Homogenization with Guaranteed Bounds via Primal-Dual Physically Informed Neural Networks

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.

cs.LG↗

arXiv2025

Non-uniform pneumatic actuation switches macroscopic properties of elastomeric honeycombs

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.

physics.app-ph↗

arXiv2025

Wallpaper Group-Based Mechanical Metamaterials: Dataset Including Mechanical Responses

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.

cond-mat.soft↗

arXiv2025

Similarity Equivariant Graph Neural Networks for Homogenization of Metamaterials

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.

cond-mat.soft↗

arXiv2025

Towards active stiffness control in pattern-forming pneumatic metamaterials

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 toifferent 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.

physics.app-ph↗

arXiv2024

Third Medium Finite Element Contact Formulation for Pneumatically Actuated Systems

Mechanical metamaterials are artificially engineered microstructures that exhibit novel mechanical behavior on the macroscopic scale. Active metamaterials can be externally controlled. Pneumatically actuated metamaterials can 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. To facilitate the design of such pneumatically actuated metamaterials and structures by topology optimization, a robust way of their computational modeling, capturing both pneumatic actuation of internal voids and internal contact, is needed. Since voids in topology optimization are often modeled using a soft material model, the third medium contact formulation lends itself as a suitable stepping stone. We propose a single hyperelastic material model capable of maintaining a prescribed hydrostatic Cauchy stress within a void in the pre-contact phase while simultaneously acting as a third medium to enforce frictionless contact, contrasting existing third medium approaches focused solely on contact. We split the overall third-medium energy density into contact, regularization, and pneumatic pressure contributions, all of which can be individually controlled and tuned. To prevent distortions of the compliant third medium, we include curvature penalization in our model. This improves on existing formulations in terms of compliant third medium behavior, leading ultimately to better numerical stability of the solution. Since our formulation is energetically consistent, we are able to employ more advanced finite element solvers, such as the modified Cholesky algorithm 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.

cs.CE↗

arXiv2023

Wang tiles enable combinatorial design and robot-assisted manufacturing of modular mechanical metamaterials

In this paper, we introduce a novel design paradigm for modular architectured materials that allows for spatially nonuniform designs from a handful of building blocks, which can be robotically assembled for efficient and scalable production. The traditional, design-limiting periodicity in material design is overcome by utilizing Wang tiles to achieve compatibility among building blocks. We illustrate our approach with the design and manufacturing of an L-shaped domain inspired by a scissor-like soft gripper, whose internal module distribution was optimized to achieve an extreme tilt of a tip of the gripper's jaw when the handle part was uniformly compressed. The geometry of individual modules was built on a 3$\times$3 grid of elliptical holes with varying semi-axes ratios and alternating orientations. We optimized the distribution of the modules within the L-shaped domain using an enumeration approach combined with a factorial search strategy. To address the challenge of seamless interface connections in modular manufacturing, we produced the final designs by casting silicone rubber into modular molds automatically assembled by a robotic arm. The predicted performance was validated experimentally using a custom-built, open-hardware test rig, Thymos, supplemented with digital image correlation measurements. Our study demonstrates the potential for enhancing the mechanical performance of architectured materials by incorporating nonuniform modular designs and efficient robot-assisted manufacturing.

cond-mat.mtrl-sci↗

arXiv2022

Modular-topology optimization of structures and mechanisms with free material design and clustering

Topology optimization of modular structures and mechanisms enables balancing the performance of automatically-generated individualized designs, as required by Industry 4.0, with enhanced sustainability by means of component reuse. For optimal modular design, two key questions must be answered: (i) what should the topology of individual modules be like and (ii) how should modules be arranged at the product scale? We address these challenges by proposing a bi-level sequential strategy that combines free material design, clustering techniques, and topology optimization. First, using free material optimization enhanced with post-processing for checkerboard suppression, we determine the distribution of elasticity tensors at the product scale. To extract the sought-after modular arrangement, we partition the obtained elasticity tensors with a novel deterministic clustering algorithm and interpret its outputs within Wang tiling formalism. Finally, we design interiors of individual modules by solving a single-scale topology optimization problem with the design space reduced by modular mapping, conveniently starting from an initial guess provided by free material optimization. We illustrate these developments with three benchmarks first, covering compliance minimization of modular structures, and, for the first time, the design of non-periodic compliant modular mechanisms. Furthermore, we design a set of modules reusable in an inverter and in gripper mechanisms, which ultimately pave the way towards the rational design of modular architectured (meta)materials.

cond-mat.mtrl-sci↗

arXiv2021

Comparison of FETI-based domain decomposition methods for topology optimization problems

We critically assess the performance of several variants of dual and dual-primal domain decomposition strategies in problems with fixed subdomain partitioning and high heterogeneity in stiffness coefficients typically arising in topology optimization of modular structures. Our study considers Total FETI and FETI Dual-Primal methods along with three enhancements: k-scaling, full orthogonalization of the search directions, and considering multiple search-direction at once, which gives us twelve variants in total. We test these variants both on academic examples and snapshots of topology optimization iterations. Based on the results, we conclude that (i) the original methods exhibit very slow convergence in the presence of severe heterogeneity in stiffness coefficients, which makes them practically useless, (ii) the full orthogonalization enhancement helps only for mild heterogeneity, and (iii) the only robust method is FETI Dual-Primal with multiple search direction and k-scaling.

cs.CE↗

arXiv2021

Microstructure reconstruction via artificial neural networks: A combination of causal and non-causal approach

We investigate the applicability of artificial neural networks (ANNs) in reconstructing a sample image of a sponge-like microstructure. We propose to reconstruct the image by predicting the phase of the current pixel based on its causal neighbourhood, and subsequently, use a non-causal ANN model to smooth out the reconstructed image as a form of post-processing. We also consider the impacts of different configurations of the ANN model (e.g. number of densely connected layers, number of neurons in each layer, the size of both the causal and non-causal neighbourhood) on the models' predictive abilities quantified by the discrepancy between the spatial statistics of the reference and the reconstructed sample.

cond-mat.mtrl-sci↗

arXiv2020

Modular-topology optimization with Wang tilings: An application to truss structures

Modularity is appealing for solving many problems in optimization. It brings the benefits of manufacturability and reconfigurability to structural optimization, and enables a trade-off between the computational performance of a Periodic Unit Cell (PUC) and the efficacy of non-uniform designs in multi-scale material optimization. Here, we introduce a novel strategy for concurrent minimum-compliance design of truss modules topologies and their macroscopic assembly encoded using Wang tiling, a formalism providing independent control over the number of modules and their interfaces. We tackle the emerging bilevel optimization problem with a combination of meta-heuristics and mathematical programming. At the upper level, we employ a genetic algorithm to optimize module assemblies. For each assembly, we obtain optimal module topologies as a solution to a convex second-order conic program that exploits the underlying modularity, incorporating stress constraints, multiple load cases, and reuse of module(s) for various structures. Merits of the proposed strategy are illustrated with three representative examples, clearly demonstrating that the best designs obtained by our method exhibited decreased compliance: from 56% to 69% compared to the PUC designs.

cs.CE↗

arXiv2020

Reduced integration schemes in micromorphic computational homogenization of elastomeric mechanical metamaterials

Exotic behaviour of mechanical metamaterials often relies on an internal transformation of the underlying microstructure triggered by its local instabilities, rearrangements, and rotations. Depending on the presence and magnitude of such a transformation, effective properties of a metamaterial may change significantly. To capture this phenomenon accurately and efficiently, homogenization schemes are required that reflect microstructural as well as macro-structural instabilities, large deformations, and non-local effects. To this end, a micromorphic computational homogenization scheme has recently been developed, which employs the particular microstructural transformation as a non-local mechanism, magnitude of which is governed by an additional coupled partial differential equation. Upon discretizing the resulting problem it turns out that the macroscopic stiffness matrix requires integration of macro-element basis functions as well as their derivatives, thus calling for a higher-order integration rules. Because evaluation of constitutive law in multiscale schemes involves an expensive solution of a non-linear boundary value problem, computational efficiency can be improved by reducing the number of integration points. Therefore, the goal of this paper is to investigate reduced-order schemes in computational homogenization, with emphasis on the stability of the resulting elements. In particular, arguments for lowering the order of integration from the expensive mass-matrix to a cheaper stiffness-matrix equivalent are first outlined. An efficient one-point integration quadrilateral element is then introduced and proper hourglass stabilization discussed. Performance of the resulting set of elements is finally tested on a benchmark bending example, showing that we achieve accuracy comparable to the full quadrature rules.

cond-mat.soft↗

arXiv2020

Level-set based design of Wang tiles for modelling complex microstructures

Microstructural geometry plays a critical role in the response of heterogeneous materials. Consequently, methods for generating microstructural samples are increasingly crucial to advanced numerical analyses. We extend Sonon et al.'s unified framework, developed originally for generating particulate and foam-like microstructural geometries of Periodic Unit Cells, to non-periodic microstructural representations based on the formalism of Wang tiles. This formalism has been recently proposed in order to generalize the Periodic Unit Cell approach, enabling a fast synthesis of arbitrarily large, stochastic microstructural samples from a handful of domains with predefined compatibility constraints. However, a robust procedure capable of designing complex, three-dimensional, foam-like and cellular morphologies of Wang tiles has not yet been proposed. This contribution fills the gap by significantly broadening the applicability of the tiling concept. Since the original Sonon et al.'s framework builds on a random sequential addition of particles enhanced with an implicit representation of particle boundaries by the level-set field, we first devise an analysis based on a connectivity graph of a tile set, resolving the question where a particle should be copied when it intersects a tile boundary. Next, we introduce several modifications to the original algorithm that are necessary to ensure microstructural compatibility in the generalized periodicity setting of Wang tiles. Having established a universal procedure for generating tile morphologies, we compare strictly aperiodic and stochastic sets with the same cardinality in terms of reducing the artificial periodicity in reconstructed microstructural samples. We demonstrate the superiority of the vertex-defined tile sets for two-dimensional problems and illustrate the capabilities of the algorithm with two- and three-dimensional examples.

cs.CG↗

arXiv2018

Wang tiling aided statistical determination of the Representative Volume Element size of random heterogeneous materials

Wang tile based representation of a heterogeneous material facilitates fast synthesis of non-periodic microstructure realizations. In this paper, we apply the tiling approach in numerical homogenization to determine the Representative Volume Element size related to the user-defined significance level and the discrepancy between bounds on the apparent properties. First, the tiling concept is employed to efficiently generate arbitrarily large, statistically consistent realizations of investigated microstructures. Second, benefiting from the regular structure inherent to the tiling concept, the Partition theorem, and statistical sampling, we construct confidence intervals of the apparent properties related to the size of a microstructure specimen. Based on the interval width and the upper and lower bounds on the apparent properties, we adaptively generate additional microstructure realizations in order to arrive at an RVE satisfying the prescribed tolerance. The methodology is illustrated with the homogenization of thermo-mechanical properties of three two-dimensional microstructure models: a microstructure with mono-disperse elliptic inclusions, foam, and sandstone.

physics.app-ph↗

arXiv2016

A jigsaw puzzle framework for homogenization of high porosity foams

An approach to homogenization of high porosity metallic foams is explored. The emphasis is on the \Alporas{} foam and its representation by means of two-dimensional wire-frame models. The guaranteed upper and lower bounds on the effective properties are derived by the first-order homogenization with the uniform and minimal kinematic boundary conditions at heart. This is combined with the method of Wang tilings to generate sufficiently large material samples along with their finite element discretization. The obtained results are compared to experimental and numerical data available in literature and the suitability of the two-dimensional setting itself is discussed.

cond-mat.mtrl-sci↗

arXiv2014

Aperiodic compression and reconstruction of real world material systems based on Wang tiles

The paper presents a concept/technique to compress and synthesize complex material morphologies that is based on Wang tilings. Specifically, a microstructure is stored in a set of Wang tiles and its reconstruction is performed by means of a stochastic tiling algorithm. A substantial part of the study is devoted to the setup of optimal parameters of the automatic tile design by means of parametric studies with statistical descriptors at heart. The performance of the method is demonstrated on four two-dimensional two-phase target systems, monodisperse media with hard and soft discs, sandstone, and high porosity metallic foam.

cond-mat.mtrl-sci↗