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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.

Background/Aim: Patients with unresectable liver colorectal cancer metastases are treated with neoadjuvant chemotherapy often accompanied by biological therapy aimed at reducing the mass of metastases and thus increasing the chances of resectability. Bevacizumab comprises an anti-VEGF (vascular endothelial growth factor) humanized IgG monoclonal antibody that is used for biological therapy purposes. It acts to inhibit angiogenesis, thereby slowing down the growth of metastases. Due to its being administered systematically, bevacizumab also exerts an effect on the surrounding healthy liver parenchyma and potentially limits the process of neovascularization and thus regeneration of the liver. Since the remnant liver volume forms an important factor in postoperative morbidity and mortality following a major hepatectomy, we decided to study the effect of bevacizumab on vascular and biliary microarchitecture in healthy liver parenchyma and its ability to regenerate following major hepatectomy. Materials and Methods: We performed an experiment employing a large animal model where a total of 16 piglets were divided into two groups (8 piglets in the control group and 8 piglets in the experimental group with bevacizumab). All the animals were subjected to major hepatectomy and the experimental group was given bevacizumab prior to hepatectomy. All the animals were sacrificed after 4 weeks. We performed biochemical analyses at regular time intervals during the follow-up period. Histological examination of the liver tissue was performed following sacrifice of the animals. Results: No statistical difference was shown between groups in terms of the biochemical and immunohistochemical parameters. The histological examination of the regenerating liver tissue revealed the higher length density of sinusoids in the experimental group. Conclusion: Bevacizumab does not act to impair liver regeneration following hepatectomy.

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

The emerging field of passive macro-scale tile-based self-assembly (TBSA) shows promise in enabling effective manufacturing processes by harnessing TBSA's intrinsic parallelism. However, current TBSA methodologies still do not fulfill their potentials, largely because such assemblies are often prone to errors, and the size of an individual assembly is limited due to insufficient mechanical stability. Moreover, the instability issue worsens as assemblies grow in size. Using a novel type of magnetically-bonded tiles carried by bristle-bot drives, we propose here a framework that reverses this tendency; i.e., as an assembly grows, it becomes more stable. Stability is achieved by introducing two sets of tiles that move in opposite directions, thus zeroing the assembly net force. Using physics-based computational experiments, we compare the performance of the proposed approach with the common orbital shaking method, proving that the proposed system of tiles indeed possesses self-stabilizing characteristics. Our approach enables assemblies containing hundreds of tiles to be built, while the shaking approach is inherently limited to a few tens of tiles. Our results indicate that one of the primary limitations of mechanical, agitation-based TBSA approaches, instability, might be overcome by employing a swarm of free-running, sensorless mobile robots, herein represented by passive tiles at the macroscopic scale., 7 pages, 14 figures, 1 table; incorporates referees' and editor's comments

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

Tato práce se podrobně věnuje způsobu, jakým David Hilbert (1862–1943) pojal aritmetizaci geometrie v knize Grundlagen der Geometrie z roku 1899. Nejprve stručně představíme Hilbertovy předchůdce z téže doby, kteří buď po změnách v založení geometrie volali, nebo je již sami prostřednictvím axiomaticko-deduktivní metody zapracovali. Neopomeneme přitom, co dílu předcházelo v dřívějších Hilbertových přednáškách. Následně se pokusíme nastínit­ obsah prvních dvou kapitol knihy a vysvětlit dobové i věcné souvislosti, nutné k jejich pochopení. Představíme způsob implicitních definic základních pojmů a vztahů v axiomech, a dále Hilbertovo rozdělení axiomů do skupin, přičemž se zejména zaměříme na axiomy spojitosti v kontextu s otázkou o její bezespornosti. K tomu popíšeme konstrukci aritmetického modelu axiomů geometrie, který Hilbert pro důkaz bezespornosti používá. V závěru se pokusíme nastínit hlavní důvody, které Hilberta k napsání díla vedly, a některé klíčové důsledky jeho pojetí axiomatiky geometrie.

The design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is usually a polynomial function of design variables, allowing us to build an equivalent non-linear semidefinite programming formulation over a semi-algebraic feasible set. This formulation is subsequently solved using the Lasserre moment-sum-of-squares hierarchy, generating a sequence of outer convex approximations that monotonically converges from below to the optimum of the original problem. Globally optimal solutions can subsequently be extracted using the Curto-Fialkow flat extension theorem. Furthermore, we show that a simple correction to the solutions of the relaxed problems establishes a feasible upper bound, thereby deriving a simple sufficient condition of global $\varepsilon$-optimality. When the original problem possesses a unique minimum, we show that this solution is found with a zero optimality gap in the limit. These theoretical findings are illustrated on several examples of topology optimization of frames and shells, for which we observe that the hierarchy converges in a finite (rather small) number of steps., 16 pages, 7 figures

Peridynamics is a recently developed extension of continuum mechanics, which replaces the traditional concept of stress by force interactions between material points at a finite distance. The peridynamic continuum is thus intrinsically nonlocal. In this contribution, a bond-based peridynamic model with elastic-brittle interactions is considered and the critical strain is defined for each bond as a function of its length. Various forms of length functions are employed to achieve a variety of macroscopic responses. A detailed study of three different localization mechanisms is performed for a one-dimensional periodic unit cell. Furthermore, a convergence study of the adopted finite element discretization of the peridynamic model is provided and an effective event-driven numerical algorithm is described.

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

A phase-field description of brittle fracture is employed in the reported four-point bending analyses of monolithic and laminated glass plates. Our aims are: (i) to compare different phase-field fracture formulations applied to thin glass plates, (ii) to assess the consequences of the dimensional reduction of the problem and mesh density and refinement, and (iii) to validate for quasi-static loading the time-/temperature-dependent material properties we derived recently for two commonly used polymer foils made of polyvinyl butyral or ethylene-vinyl acetate. As the nonlinear response prior to fracture, typical of the widely used Bourdin&ndash, Francfort&ndash, Marigo model, can lead to a significant overestimation of the response of thin plates under bending, the numerical study investigates two additional phase-field fracture models providing the linear elastic phase of the stress-strain diagram. The typical values of the critical fracture energy and tensile strength of glass lead to a phase-field length-scale parameter that is challenging to resolve in the numerical simulations. Therefore, we show how to determine the fracture energy concerning the applied dimensional reduction and the value of the length-scale parameter relative to the thickness of the plate. The comparison shows that the phase-field models provide very good agreement with the measured stresses and resistance of laminated glass, despite the fact that only one/two cracks are localised using the quasi-static analysis, whereas multiple cracks evolve during the experiment. It was also observed that the stiffness and resistance of the partially fractured laminated glass can be well approximated using a 2D plane-stress model with initially predefined cracks, which provides a better estimation than the one-glass-layer limit.

A method of estimating all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in T. Gergelits, K.A. Mardal, B.F. Nielsen, Z. Strako\v{s}: Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator, SIAM Journal on Numerical Analysis 57(3) (2019), 1369-1394. Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite element method, preconditioned by the inverse of a matrix of the same operator with different data. Our results hold for mixed Dirichlet and Robin or periodic boundary conditions applied to the original and preconditioning problems. The bounds are two-sided, guaranteed, easily accessible, and depend solely on the material data., Comment: 20 pages, 5 figures

This paper focuses on the modal analysis of laminated glass beams. In these multilayer elements, the stiff glass plates are connected by compliant interlayers with frequency/temperature-dependent behavior. The aim of our study is (i) to assess whether approximate techniques can accurately predict the behavior of laminated glass structures and (ii) to propose an easy tool for modal analysis based on the enhanced effective thickness concept by Galuppi and Royer-Carfagni. To this purpose, we consider four approaches to the solution of the related nonlinear eigenvalue problem: a complex-eigenvalue solver based on the Newton method, the modal strain energy method, and two effective thickness concepts. A comparative study of free vibrating laminated glass beams is performed considering different geometries of cross-sections, boundary conditions, and material parameters for interlayers under two ambient temperatures. The viscoelastic response of polymer foils is represented by the generalized Maxwell model. We show that the simplified approaches predict natural frequencies with an acceptable accuracy for most of the examples. However, there is a considerable scatter in predicted loss factors. The enhanced effective thickness approach adjusted for modal analysis leads to lower errors in both quantities compared to the other two simplified procedures, reducing the extreme error in loss factors to one half compared to the modal strain energy method or to one quarter compared to the original dynamic effective thickness method., 31 pages, 12 figures, 10 tables

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

Computational micromechanics and homogenization require the solution of the mechanical equilibrium of a periodic cell that comprises a (generally complex) microstructure. Techniques that apply the Fast Fourier Transform have attracted much attention as they outperform other methods in terms of speed and memory footprint. Moreover, the Fast Fourier Transform is a natural companion of pixel-based digital images which often serve as input. In its original form, one of the biggest challenges for the method is the treatment of (geometrically) non-linear problems, partially due to the need for a uniform linear reference problem. In a geometrically linear setting, the problem has recently been treated in a variational form resulting in an unconditionally stable scheme that combines Newton iterations with an iterative linear solver, and therefore exhibits robust and quadratic convergence behavior. Through this approach, well-known key ingredients were recovered in terms of discretization, numerical quadrature, consistent linearization of the material model, and the iterative solution of the resulting linear system. As a result, the extension to finite strains, using arbitrary constitutive models, is at hand. Because of the application of the Fast Fourier Transform, the implementation is substantially easier than that of other (Finite Element) methods. Both claims are demonstrated in this paper and substantiated with a simple code in Python of just 59 lines (without comments). The aim is to render the method transparent and accessible, whereby researchers that are new to this method should be able to implement it efficiently. The potential of this method is demonstrated using two examples, each with a different material model. (C) 2016 Elsevier B.V. All rights reserved.

Lattice networks with dissipative interactions can be used to describe the mechanics of discrete meso-structures of materials such as 3D-printed structures and foams. This contribution deals with the crack initiation and propagation in such materials and focuses on an adaptive multiscale approach that captures the spatially evolving fracture. Lattice networks naturally incorporate non-locality, large deformations, and dissipative mechanisms taking place inside fracture zones. Because the physically relevant length scales are significantly larger than those of individual interactions, discrete models are computationally expensive. The Quasicontinuum (QC) method is a multiscale approach specifically constructed for discrete models. This method reduces the computational cost by fully resolving the underlying lattice only in regions of interest, while coarsening elsewhere. In this contribution, the (variational) QC is applied to damageable lattices for engineering-scale predictions. To deal with the spatially evolving fracture zone, an adaptive scheme is proposed. Implications induced by the adaptive procedure are discussed from the energy-consistency point of view, and theoretical considerations are demonstrated on two examples. The first one serves as a proof of concept, illustrates the consistency of the adaptive schemes, and presents errors in energies. The second one demonstrates the performance of the adaptive QC scheme for a more complex problem.

Fourier solvers have become efficient tools to establish structure-property relations in heterogeneous materials. Introduced as an alternative to the Finite Element (FE) method, they are based on fixed-point solutions of the Lippmann-Schwinger type integral equation. Their computational efficiency results from handling the kernel of this equation by the Fast Fourier Transform (FFT). However, the kernel is derived from an auxiliary homogeneous linear problem, which renders the extension of FFT-based schemes to non-linear problems conceptually difficult. This paper aims to establish a link between FE- and FFT-based methods, in order to develop a solver applicable to general history- and time-dependent material models. For this purpose, we follow the standard steps of the FE method, starting from the weak form, proceeding to the Galerkin discretization and the numerical quadrature, up to the solution of non-linear equilibrium equations by an iterative Newton-Krylov solver. No auxiliary linear problem is thus needed. By analyzing a two-phase laminate with non-linear elastic, elasto-plastic, and visco-plastic phases, and by elasto-plastic simulations of a dual-phase steel microstructure, we demonstrate that the solver exhibits robust convergence. These results are achieved by re-using the non-linear FE technology, with the potential of further extensions beyond small-strain inelasticity considered in this paper.

This paper investigates a time-stepping procedure of the Newmark type for dynamic analyses of viscoelastic structures characterized by a generalized Maxwell model. We depart from a scheme developed for a three-parameter model by Hatada et al. in 2000, which we extend to a generic Maxwell chain and demonstrate that the resulting algorithm can be derived from a suitably discretized Hamilton variational principle. This variational structure manifests itself in an excellent stability and a low artificial damping of the integrator, as we confirm with a mass-spring-dashpot example. After a straightforward generalization to distributed systems, the integrator may find use in, e.g., fracture simulations of laminated glass units, once combined with variationally-based fracture models., 9 pages, 4 figures, 1 table

Fueled by their excellent stiffness-to-weight ratio and the availability of mature manufacturing technologies, filament wound carbon fiber reinforced polymers represent ideal materials for thin-walled laminate structures. However, their strong anisotropy reduces structural resistance to wall instabilities under shear and buckling. Increasing laminate thickness degrades weight and structural efficiencies and the application of a dense internal core is often uneconomical and labor-intensive. In this contribution, we introduce a convex linear semidefinite programming formulation for truss topology optimization to design an efficient non-uniform lattice-like internal structure. The internal structure not only reduces the effect of wall instabilities, mirrored in the increase of the fundamental free-vibration eigenfrequency, but also keeps weight low, secures manufacturability using conventional three-dimensional printers, and withstands the loads induced during the production process. We showcase a fully-automatic procedure in detail for the design, prototype manufacturing, and verification of a simply-supported composite machine tool component, including validation with roving hammer tests. The results confirm that the 3D-printed optimized internal structure almost doubles the fundamental free-vibration eigenfrequency, allowing to increase working frequency of the machine tool, even though the ratio between elastic properties of the carbon composite and the ABS polymer used for 3D printing exceeds two orders of magnitude., 13 pages, 13 figures

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

Following publication of the original article [1], the authors reported the errors in the equations.

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