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A recently introduced representation by a set of Wang tiles—a generalization of the traditional Periodic Unit Cell-based approach—serves as a reduced geometrical model for materials with stochastic heterogeneous microstructure, enabling an efficient synthesis of microstructural realizations. To facilitate macroscopic analyses with a fully resolved microstructure generated with Wang tiles, we develop a reduced order modelling scheme utilizing pre-computed characteristic features of the tiles. In the offline phase, inspired by computational homogenization, we extract continuous fluctuation fields from the compressed microstructural representation as responses to generalized loading represented by the first- and second-order macroscopic gradients. In the online phase, using the ansatz of the generalized finite element method, we combine these fields with a coarse finite element discretization to create microstructure-informed reduced modes specific for a given macroscopic problem. Considering a two-dimensional scalar elliptic problem, we demonstrate that our scheme delivers less than 3% error in both the relative $$L_2$$ and energy norms with only 0.01% of the unknowns when compared to the fully resolved problem. Accuracy can be further improved by locally refining the macroscopic discretization and/or employing more pre-computed fluctuation fields. Finally, unlike standard snapshot-based reduced-order approaches, our scheme handles significant changes in the macroscopic geometry or loading without the need for recalculating the offline phase, because the fluctuation fields are extracted without any prior knowledge of the macroscopic problem.

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., Comment: 17 pages, 16 figures

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., 7 pages, 5 figures; Submitted to Acta Polytechnica Proceedings; Based on a talk given at NMM2021 - Conference Nano and Macro Mechanics 2021

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., Comment: 30 pages

Mechanical metamaterials feature engineered microstructures designed to exhibit exotic, and often counter-intuitive, effective behaviour. Such a behaviour is often achieved through instability-induced transformations of the underlying periodic microstructure into one or multiple patterning modes. Due to a strong kinematic coupling of individual repeating microstructural cells, non-local behaviour and size effects emerge, which cannot easily be captured by classical homogenization schemes. In addition, the individual patterning modes can mutually interact in space as well as in time, while at the engineering scale the entire structure can buckle globally. For efficient numerical macroscale predictions, a micromorphic computational homogenization scheme has recently been developed. Although this framework is in principle capable of accounting for spatial and temporal interactions between individual patterning modes, its implementation relied on a gradient-based quasi-Newton solution technique. This solver is suboptimal because (i) it has sub-quadratic convergence, and (ii) the absence of Hessians does not allow for proper bifurcation analyses. Given that mechanical metamaterials often rely on controlled instabilities, these limitations are serious. To address them, a full Newton method is provided in detail in this paper. The construction of the macroscopic tangent operator is not straightforward due to specific model assumptions on the decomposition of the underlying displacement field pertinent to the micromorphic framework, involving orthogonality constraints. Analytical expressions for the first and second variation of the total potential energy are given, and the complete algorithm is listed. The developed methodology is demonstrated with two examples in which a competition between local and global buckling exists and where multiple patterning modes emerge., 34 pages, 17 figures, 1 table, 1 algorithm, abstract shortened to fulfill 1920 character limit

In this paper we present our recent work focused on the analysis of the abilities of Wang Tiles method and Automatic tile design method to synthesize the micro-structure of cellular materials, especially particular type of metal foam.Wang Tiles method stores and compress the micro-structure in a set of Wang Tiles and by the means of stochastic tiling algorithms the planar domain is reconstructed. The used tiles are created by the Automatic tile design method from respective number of small specimens extracted from the original micro-structure image. As an additional step the central areas of automatically designed tiles are patched to suppress the influence of repeating tile edges (and relevant tile quarters) on inducing artifacts. In the presented analysis the performance of raw and patched tiles of different sizes in conjunction of various tile sets is investigated.

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., 19 pages, 22 figures, post-print version

In this contribution, we present the concept of Wang Tiles as a surrogate of the periodic unit cell method (PUC) for modelling of materials with heterogeneous microstructures and for synthesis of micro-mechanical fields. The concept is based on a set of specifically designed cells that compresses the stochastic microstructure into a small set of statistical volume elements – tiles. Tiles are placed side by side according to matching edges like in a game of domino. Opposite to the repeating pattern of PUC the Wang Tiles method with the stochastic tiling algorithm preserves the randomness for reconstructed microstructures. The same process is applied to obtain the micro-mechanical response of domains where the evaluation as one piece would be time consuming. Therefore the micro-mechanical quantities are evaluated only on tiles (with surrounding layers of tiles of each addressed tile included into the evaluation) and then synthesized to the micro-mechanical field of whole domain.

The Extended Finite Element Method (XFEM) enhances the approximation space of the standard Finite Element Method (FEM) with functions reflecting local features in order to yield more accurate results with less degrees of freedom. XFEM performance is, thus, closely related to the quality of enrichment functions. Analogously to our previous works, in which we have employed the concept of Wang tiles to assembly microstructure geometries, in this contribution we use Wang tiles to assemble microstructure-informed enrichment functions. We compare two ways of generating the enrichments: (i) inspired by the first-order numerical homogenization and (ii) based on spectral analysis of the global stiffness matrix for the whole set. The methodology and performance of both approaches are illustrated through a linear diffusion problem in two dimensions

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., 16 pages, 16 figures. Abstract has been shortened to match the arXiv.org's limit of 1920 characters

Preliminary outcomes on a novel modular additive manufacturing technology are presentedin this contribution. The capabilities of the proposed system to synthesize materials with conventionalor meta properties are explored. Two limiting and a single mixed compositions are tested in com-pression in order to reveal both, global and local behaviours of the conglomerates. The experimentalmeasurements are supported by DIC (Digital Image Correlation) observations.

A concept of a planar modular mechanical metamaterial inspired by the nature's principle of local adaptivity is proposed. The metamaterial consists of identical pieces similar to jigsaw puzzle tiles. Their rotation within assembly provides a substantial flexibility in terms of structural behavior and mechanical interlocks enable reassembly. The tile design with a diagonal elliptical opening allows us to vary elastic properties--from stiff to compliant, with positive, zero, or negative Poisson's ratio. The outcomes of experimental testing on additively manufactured specimens confirm that the assembly properties can be accurately designed using optimization approaches with finite element analysis at heart., 9 pages, 8 figures

The paper deals with a modelling of random heterogeneous materials composed of circular solid particles within a matrix. The concept of Wang tiling is utilized to generate material patterns. Wang tiling principles enable to create random heterogeneous structures with high reduction of periodicity artefacts in comparison with the traditional approach of the Periodic Unit Cell (PUC). A packing algorithm based on molecular dynamics for Wang tiles generation is introduced. During the process of generation particles grow, move and collide with tile edges and with each other in order to achieve required or optimal particle placements without an overlapping. A set of Wang tiles include eight different tiles with specific boundary conditions which allow composing of random heterogeneous materials applying a stochastic tiling algorithm. An optimization technique based on the Particle Swarm Optimization (PSO) is utilized to get a material model with similar statistical information compared with a reference pattern. Due to the nature of investigated materials the Radial Distribution Function (RDF) was chosen as a tool for statistical description.

With the current contribution we continue in our long-term ambition to develop an efficient numerical method that incorporates detailed knowledge of material microstructure into upper-scale response. In particular, we investigate potential of the Wang tiling concept to provide microstructure-informed enrichment functions for the eXtended/Generalized Finite Element Method. Preliminary results for a one-dimensional linear elasticity problem are presented, covering two modes of loading and their combination. The performance of the proposed methodology is quantified by means of $L^2$ and $H^1$ norms of errors for sequentially refined macro-scale discretization.

This contribution presents a statistical evaluation of microstructures assembled by means of2D Wang tiles, morphology of which is designed employing an algorithm based on Discrete ElementMethod. Comparison between realization of Wang tiling and Periodic Unit Cell representation is providedin terms of spatial descriptors. Namely, radial distribution function g2 and two-point probabilityfunction S2 are used to assess the artefacts related to periodicity in synthesized microstructures. Wedemonstrate significant reduction in periodicity achieved with Wang tiling (compared to the periodicsynthesis) in both the microstructures and results of numerical simulations.

An algorithm generating morphology of 2D and 3D stochastic Wang tiles is presented in thiscontribution. The algorithm is based on the discrete element method (DEM) and is therefore applicableprimarily for matrix-based microstructures with separate inclusions with emphasis on producingmaximally random dense packings. An open-source free DEM code YADE was used for all the computations.The method is illustrated with a 2D exemplary realization of Wang tiling and comparisonagainst Periodic Unit Cell representation in terms of spatial statistics is provided.

Stochastic Wang tiling has been shown to bring unexpected insights to microstructure representation efforts as it generalizes the conventional unit-cell approach. It allows to reconstruct stochastic realizations of the compressed medium without prior periodic assumptions on microstructural patterns. Moreover, once the microstructure is compressed, its realizations of various sizes can be generated at almost negligible cost. In this paper, we follow the standard numerical homogenization procedure and make use of the realizations as domains over which local quantities are computed and averaged subsequently. In order to alleviate computational cost, a domain decomposition method is adopted such that it benefits from the fact that the computational domains are composed of limited number of repetitive patterns -- tiles.

The present study is on the concept of modeling of heterogeneous materials by means of Wang tilings. The central idea is to store a microstructural information within a finite set of Wang Tiles, which allow for reconstructing heterogeneity patterns of random media in planar domains of arbitrary sizes. The particular objective of presented work is our automatic tile set designer in conjunction with stochastic tiling synthesis algorithm. The proposed methodology is demonstrated on different examples. The proximity of synthesized microstructures to reference media is explored by statistical descriptors and discussed in terms of parasitic spatial orientation orders that may occur.

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., 11 pages, 7 figures, 3 tables

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., v2: title and PACS number change; 8 pages, 11 figures