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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

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

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 presents a detailed analysis and extended formulation of a rate-independent regularized damage model proposed by Mielke and Roub\'i\v{c}ek (2006). Localization properties are studied in the context of a simple one-dimensional problem, but the results reveal the fundamental features of the basic model and of its modified versions. The initial bifurcation from a uniform solution is described analytically while the complete failure process is studied numerically. Modifications of the regularizing term and of the dissipation distance are introduced and their effect on the global response is investigated. It is shown that, with a proper combination of model parameters, a realistic shape of the load-displacement diagram can be achieved and pathological effects such as extremely brittle response or expansion of the damage zone accompanied by stress locking can be eliminated., Comment: 48 pages, 26 figures; v2: moderate revision after the first round of reviews

In this paper, we study a non-linear numerical scheme arising from the implicit time discretization of the Ba\v{z}ant-Thonguthai model for hygro-thermal behavior of concrete at high temperatures. Existence and uniqueness of the time-discrete solution in two dimensions is established using the theory of pseudomonotone operators in Banach spaces. Next, the spatial discretization is accomplished by the conforming finite element method. An illustrative numerical example shows that the numerical model reproduces well the rapid increase of pore pressure in wet concrete due to extreme heating. Such phenomenon is of particular interest for the safety assessment of concrete structures prone to thermally-induced spalling., Comment: 20 pages, 8 figures

The paper presents analytical or semi-analytical solutions for the formation and evolution of localized plastic zone in a uniaxially loaded bar with variable cross-sectional area. A variationally based formulation of explicit gradient plasticity with linear softening is used, and the ensuing jump conditions and boundary conditions are discussed. Three cases with different regularity of the stress distribution are considered, and the problem is converted to a dimensionless form. Relations linking the load level, size of the plastic zone, distribution of plastic strain and plastic elongation of the bar are derived and compared to another, previously analyzed gradient formulation., Comment: 42 pages, 11 figures

Lattice systems and discrete networks with dissipative interactions are successfully employed as meso-scale models of heterogeneous solids. As the application scale generally is much larger than that of the discrete links, physically relevant simulations are computationally expensive. The QuasiContinuum (QC) method is a multiscale approach that reduces the computational cost of direct numerical simulations by fully resolving complex phenomena only in regions of interest while coarsening elsewhere. In previous work (Beex et al., J. Mech. Phys. Solids 64, 154-169, 2014), the originally conservative QC methodology was generalized to a virtual-power-based QC approach that includes local dissipative mechanisms. In this contribution, the virtual-power-based QC method is reformulated from a variational point of view, by employing the energy-based variational framework for rate-independent processes (Mielke and Roub\'i\v{c}ek, Rate-Independent Systems: Theory and Application, Springer-Verlag, 2015). By construction it is shown that the QC method with dissipative interactions can be expressed as a minimization problem of a properly built energy potential, providing solutions equivalent to those of the virtual-power-based QC formulation. The theoretical considerations are demonstrated on three simple examples. For them we verify energy consistency, quantify relative errors in energies, and discuss errors in internal variables obtained for different meshes and two summation rules., Comment: 38 pages, 21 figures, 4 tables; moderate revision after review, one example in Section 5.3 added

Lattice networks with dissipative interactions are often employed to analyze materials with discrete micro- or meso-structures, or for a description of heterogeneous materials which can be modelled discretely. They are, however, computationally prohibitive for engineering-scale applications. The (variational) QuasiContinuum (QC) method is a concurrent multiscale approach that reduces their computational cost by fully resolving the (dissipative) lattice network in small regions of interest while coarsening elsewhere. When applied to damageable lattices, moving crack tips can be captured by adaptive mesh refinement schemes, whereas fully-resolved trails in crack wakes can be removed by mesh coarsening. In order to address crack propagation efficiently and accurately, we develop in this contribution the necessary generalizations of the variational QC methodology. First, a suitable definition of crack paths in discrete systems is introduced, which allows for their geometrical representation in terms of the signed distance function. Second, special function enrichments based on the partition of unity concept are adopted, in order to capture kinematics in the wakes of crack tips. Third, a summation rule that reflects the adopted enrichment functions with sufficient degree of accuracy is developed. Finally, as our standpoint is variational, we discuss implications of the mesh refinement and coarsening from an energy-consistency point of view. All theoretical considerations are demonstrated using two numerical examples for which the resulting reaction forces, energy evolutions, and crack paths are compared to those of the direct numerical simulations., Comment: 36 pages, 23 figures, 1 table, 2 algorithms; small changes after review, paper title changed

The purpose of this paper is to present a simple micromechanics-based model to estimate the effective thermal conductivity of real-world macroscopically isotropic materials of matrix-inclusion type. The methodology is based on the well-established Mori-Tanaka method for composite media reinforced with ellipsoidal inclusions, extended to account for imperfect thermal contact at the matrix-inclusion interface, random orientation of particles and particle size distribution. Using simple ensemble averaging arguments, we show that the original Mori-Tanaka relations are still applicable for these complex systems, provided that the inclusion conductivity is appropriately modified. Such conclusion is supported by the verification of the model against a detailed finite-element study as well as its validation against experimental data for a wide range of engineering material systems., Corrected Eqs. (23), (38) and (39); 21 pages, 11 figures

In this paper, we study the mechanics of statistically non-uniform two-phase elastic discrete structures. In particular, following the methodology proposed in (Luciano and Willis, Journal of the Mechanics and Physics of Solids 53, 1505-1522, 2005), energetic bounds and estimates of the Hashin-Shtrikman-Willis type are developed for discrete systems with a heterogeneity distribution quantified by second-order spatial statistics. As illustrated by three numerical case studies, the resulting expressions for the ensemble average of the potential energy are fully explicit, computationally feasible and free of adjustable parameters. Moreover, the comparison with reference Monte-Carlo simulations confirms a notable improvement in accuracy with respect to approaches based solely on the first-order statistics., Comment: 32 pages, 8 figures

In this short note, we present a new technique to accelerate the convergence of a FFT-based solver for numerical homogenization of complex periodic media proposed by Moulinec and Suquet in 1994. The approach proceeds from discretization of the governing integral equation by the trigonometric collocation method due to Vainikko (2000), to give a linear system which can be efficiently solved by conjugate gradient methods. Computational experiments confirm robustness of the algorithm with respect to its internal parameters and demonstrate significant increase of the convergence rate for problems with high-contrast coefficients at a low overhead per iteration., Comment: 13 pages, 4 figures

The purpose of this study is to explore three numerical approaches to the elastic homogenization of disordered masonry structures with moderate meso/macro-lengthscale ratio. The methods investigated include a representative of perturbation methods, the Karhunen-Lo\`{e}ve expansion technique coupled with Monte-Carlo simulations and a solver based on the Hashin-Shtrikman variational principles. In all cases, parameters of the underlying random field of material properties are directly derived from image analysis of a real-world structure. Added value as well as limitations of individual schemes are illustrated by a case study of an irregular masonry panel., Comment: 18 pages, 8 figures

This paper presents a comprehensive approach to the evaluation of macroscopic material parameters for natural stone and quarry masonry. To that end, a reliable non-linear material model on a meso-scale is developed to cover the random arrangement of stone blocks and quasi-brittle behaviour of both basic components, as well as the impaired cohesion and tensile strength on the interface between the blocks and mortar joints. The paper thus interrelates the following three problems: (i) definition of a suitable periodic unit cell (PUC) representing a particular masonry structure; (ii) derivation of material parameters of individual constituents either experimentally or running a mixed numerical-experimental problem; (iii) assessment of the macroscopic material parameters including the tensile and compressive strengths and fracture energy., Comment: 38 pages, 15 figures

An uncoupled multi-scale homogenization approach is used to estimate the effective thermal conductivities of plain weave C/C composites with a high degree of porosity. The geometrical complexity of the material system on individual scales is taken into account through the construction of a suitable representative volume element (RVE), a periodic unit cell, exploiting the information provided by the image analysis of a real composite system on every scale. Two different solution procedures are examined. The first one draws on the classical first order homogenization technique assuming steady state conditions and periodic distribution of the fluctuation part of the temperature field. The second approach is concerned with the solution of a transient flow problem. Although more complex, the latter approach allows for a detailed simulation of heat transfer in the porous system. Effective thermal conductivities of the laminate derived from both approaches through a consistent homogenization on individual scales are then compared with those obtained experimentally. A reasonably close agreement between individual results then promotes the use of the proposed multi-scale computational approach combined with the image analysis of real material systems., 17 pages, 7 figures

The purpose of this paper is to propose a continuum micromechanics model for the simulation of uniaxial compressive and tensile tests on lime-based mortars, in order to predict their stiffness, compressive and tensile strengths, and tensile fracture energy. In tension, we adopt an incremental strain-controlled form of the Mori-Tanaka scheme with a damageable matrix phase, while a simple $J_2$ yield criterion is employed in compression. To reproduce the behavior of lime-based mortars correctly, the scheme must take into account shrinkage cracking among aggregates. This phenomenon is introduced into the model via penny-shaped cracks, whose density is estimated on the basis of a particle size distribution combined with the results of finite element analyses of a single crack formation between two spherical inclusions. Our predictions show a good agreement with experimental data and explain the advantages of compliant crushed brick fragments, often encountered in ancient mortars, over stiff sand particles. The validated model provides a reliable tool for optimizing the composition of modern lime-based mortars with applications in conservation and restoration of architectural heritage., Comment: 21 pages, 12 figures, 1 table

Architectural conservation and repair are becoming increasingly important issues in many countries due to numerous prior improper interventions, including the use of inappropriate repair materials over time. As a result, the composition of repair masonry mortars is now being more frequently addressed in mortar research. Just recently, for example, it has become apparent that Portland cement mortars, extensively exploited as repair mortars over the past few decades, are not suitable for repair because of their chemical, physical, mechanical, and aesthetic incompatibilities with original materials. This paper focuses on the performance of various lime-based alternative materials intended for application in repairing historic structures when subjected to mechanical loading. Results of basic material tests indicate that the use of metakaolin as a pozzolanic additive produces mortars with superior strength and sufficiently low shrinkage. Moreover, mortar strength can be further enhanced by the addition of crushed brick fragments, which explains the longevity of Roman concretes rich in pozzolans and aggregates from crushed clay products such as tiles, pottery, or bricks. An integrated experimental-numerical approach was used to identify key mortar parameters influencing the load-bearing capacity of masonry piers subjected to a combination of compression and bending. The simulations indicate increased load-bearing capacities for masonry piers containing metakaolin-rich mortars with crushed brick fragments, as a result of their superior compressive strength., Comment: 22 pages, 19 figures, 2 tables

The purpose of this paper is to provide analytical and numerical solutions of the formation and evolution of the localized plastic zone in a uniaxially loaded bar with variable cross-sectional area. An energy-based variational approach is employed and the governing equations with appropriate physical boundary conditions, jump conditions, and regularity conditions at evolving elasto-plastic interface are derived for a fourth-order explicit gradient plasticity model with linear isotropic softening. Four examples that differ by regularity of the yield stress and stress distributions are presented. Results for the load level, size of the plastic zone, distribution of plastic strain and its spatial derivatives, plastic elongation, and energy balance are constructed and compared to another, previously discussed non-variational gradient formulation., Comment: 41 pages, 24 figures; moderate revision after the first round of review, Appendix A re-written completely

In this paper, we assess the performance of four iterative algorithms for solving non-symmetric rank-deficient linear systems arising in the FFT-based homogenization of heterogeneous materials defined by digital images. Our framework is based on the Fourier-Galerkin method with exact and approximate integrations that has recently been shown to generalize the Lippmann-Schwinger setting of the original work by Moulinec and Suquet from 1994. It follows from this variational format that the ensuing system of linear equations can be solved by general-purpose iterative algorithms for symmetric positive-definite systems, such as the Richardson, the Conjugate gradient, and the Chebyshev algorithms, that are compared here to the Eyre-Milton scheme - the most efficient specialized method currently available. Our numerical experiments, carried out for two-dimensional elliptic problems, reveal that the Conjugate gradient algorithm is the most efficient option, while the Eyre-Milton method performs comparably to the Chebyshev semi-iteration. The Richardson algorithm, equivalent to the still widely used original Moulinec-Suquet solver, exhibits the slowest convergence. Besides this, we hope that our study highlights the potential of the well-established techniques of numerical linear algebra to further increase the efficiency of FFT-based homogenization methods., Comment: 23 pages, 11 figures, 1 table; minor revision after the first review

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

In 1994, Moulinec and Suquet introduced an efficient technique for the numerical resolution of the cell problem arising in homogenization of periodic media. The scheme is based on a fixed-point iterative solution to an integral equation of the Lippmann-Schwinger type, with action of its kernel efficiently evaluated by the Fast Fourier Transform techniques. The aim of this work is to demonstrate that the Moulinec-Suquet setting is actually equivalent to a Galerkin discretization of the cell problem, based on approximation spaces spanned by trigonometric polynomials and a suitable numerical integration scheme. For the latter framework and scalar elliptic setting, we prove convergence of the approximate solution to the weak solution, including a-priori estimates for the rate of convergence for sufficiently regular data and the effects of numerical integration. Moreover, we also show that the variational structure implies that the resulting non-symmetric system of linear equations can be solved by the conjugate gradient method. Apart from providing a theoretical support to Fast Fourier Transform-based methods for numerical homogenization, these findings significantly improve on the performance of the original solver and pave the way to similar developments for its many generalizations proposed in the literature., 22 pages, 1 figure

This paper presents a stochastic Wang tiling based technique to compress or reconstruct disordered microstructures on the basis of given spatial statistics. Unlike the existing approaches based on a single unit cell, it utilizes a finite set of tiles assembled by a stochastic tiling algorithm, thereby allowing to accurately reproduce long-range orientation orders in a computationally efficient manner. Although the basic features of the method are demonstrated for a two-dimensional particulate suspension, the present framework is fully extensible to generic multi-dimensional media., 4 pages, 6 figures, v2: minor changes as suggested by reviewers, v3: corrected two typos in the revised version

The paper reviews several topics associated with the homogenization of transport processed in historical masonry structures. Since these often experience an irregular or random pattern, we open the subject by summarizing essential steps in the formulation of a suitable computational model in the form of Statistically Equivalent Periodic Unit Cell (SEPUC). Accepting SEPUC as a reliable representative volume element is supported by application of the Fast Fourier Transform to both the SEPUC and large binary sample of real masonry in search for effective thermal conductivities limited here to a steady state heat conduction problem. Fully coupled non-stationary heat and moisture transport is addressed next in the framework of two-scale first-order homogenization approach with emphases on the application of boundary and initial conditions on the meso-scale., 19 pages, 13 figures, 2 tables

In the analysis of composite materials with heterogeneous microstructures, full resolution of the heterogeneities using classical numerical approaches can be computationally prohibitive. This paper presents a micromechanics-enhanced finite element formulation that accurately captures the mechanical behaviour of heterogeneous materials in a computationally efficient manner. The strategy exploits analytical solutions derived by Eshelby for ellipsoidal inclusions in order to determine the mechanical perturbation fields as a result of the underlying heterogeneities. Approximation functions for these perturbation fields are then incorporated into a finite element formulation to augment those of the macroscopic fields. A significant feature of this approach is that the finite element mesh does not explicitly resolve the heterogeneities and that no additional degrees of freedom are introduced. In this paper, hybrid-Trefftz stress finite elements are utilised and performance of the proposed formulation is demonstrated with numerical examples. The method is restricted here to elastic particulate composites with ellipsoidal inclusions but it has been designed to be extensible to a wider class of materials comprising arbitrary shaped inclusions., 28 pages, 12 figures, 2 tables

This paper presents an approach to constructing microstructural enrichment functions to local fields in non-periodic heterogeneous materials with applications in Partition of Unity and Hybrid Finite Element schemes. It is based on a concept of aperiodic tilings by the Wang tiles, designed to produce microstructures morphologically similar to original media and enrichment functions that satisfy the underlying governing equations. An appealing feature of this approach is that the enrichment functions are defined only on a small set of square tiles and extended to larger domains by an inexpensive stochastic tiling algorithm in a non-periodic manner. Feasibility of the proposed methodology is demonstrated on constructions of stress enrichment functions for two-dimensional mono-disperse particulate media., Comment: 27 pages, 12 figures; v2: completely re-written after the first review

The present paper deals with mathematical models of heat and moisture transport in layered building envelopes. The study of such processes generates a system of two doubly nonlinear evolution partial differential equations with appropriate initial and boundary conditions. The existence of the strong solution in two dimensions on a (short) time interval is proven. The proof rests on regularity results for elliptic transmission problem for isotropic composite-like materials., 34 pages, 3 figures

A two-layer statistically equivalent periodic unit cell is offered to predict a macroscopic response of plain weave multilayer carbon-carbon textile composites. Falling-short in describing the most typical geometrical imperfections of these material systems the original formulation presented in (Zeman and \v{S}ejnoha, International Journal of Solids and Structures, 41 (2004), pp. 6549--6571) is substantially modified, now allowing for nesting and mutual shift of individual layers of textile fabric in all three directions. Yet, the most valuable asset of the present formulation is seen in the possibility of reflecting the influence of negligible meso-scale porosity through a system of oblate spheroidal voids introduced in between the two layers of the unit cell. Numerical predictions of both the effective thermal conductivities and elastic stiffnesses and their comparison with available laboratory data and the results derived using the Mori-Tanaka averaging scheme support credibility of the present approach, about as much as the reliability of local mechanical properties found from nanoindentation tests performed directly on the analyzed composite samples., Comment: 28 pages, 14 figures

Localization of plastic strain induced by softening can be objectively described by a regularized plasticity model that postulates a dependence of the current yield stress on a nonlocal softening variable defined by a differential (gradient) expression. This paper presents analytical solutions of the one-dimensional localization problem under certain special nonuniform stress distributions. The one-dimensional problem can be interpreted as describing either a tensile bar with variable cross section, or a beam subjected to a nonuniform bending moment. Explicit as well as implicit gradient formulations are considered. The evolution of the plastic strain profile and the shape of the load-displacement diagram are investigated. It is shown that even if the local constitutive law exhibits softening right from the onset of yielding, the global load-displacement diagram has a hardening part. The interplay between the internal length scales characterizing the material and the geometry is discussed., to appear in International Journal for Multiscale Computational Engineering

A well established framework of an uncoupled hierarchical modeling approach is adopted here for the prediction of macroscopic material parameters of the Generalized Leonov (GL) constitutive model intended for the analysis of flexible pavements at both moderate and elevated temperature regimes. To that end, a recently introduced concept of a statistically equivalent periodic unit cell (SEPUC) is addressed to reflect a real microstructure of Mastic Asphalt mixtures (MAm). While mastic properties are derived from an extensive experimental program, the macroscopic properties of MAm are fitted to virtual numerical experiments performed on the basis of first order homogenization scheme. To enhance feasibility of the solution of the underlying nonlinear problem a two-step homogenization procedure is proposed. Here, the effective material properties are first found for a mortar phase, a composite consisting of a mastic matrix and a fraction of small aggregates. These properties are then introduced in place of the matrix in actual unit cells to give estimates of the model parameters on macroscale. Comparison with the Mori-Tanaka predictions is also provided suggesting limitations of classical micromechanical models., Comment: to appear in International Journal for Multiscale Computational Engineering

An efficient approach to the evaluation of effective elastic properties of carbon-carbon plain weave textile composites using the Mori-Tanaka method is presented. The method proves its potential even if applied to real material systems with various types of imperfections including the non-uniform waviness of the fiber-tow paths, both along its longitudinal direction and through the laminate thickness. Influence of the remaining geometrical parameters is accounted for by optimal calibration of the shape of the equivalent ellipsoidal inclusion. An application of the method to a particular sample of the carbon-carbon composite laminate demonstrates not only its applicability but also its efficiency particularly when compared to finite element simulations., 18 pages, 6 figures

Functionally graded materials (FGMs) are two-phase composites with continuously changing microstructure adapted to performance requirements. Traditionally, the overall behavior of FGMs has been determined using local averaging techniques or a given smooth variation of material properties. Although these models are computationally efficient, their validity and accuracy remain questionable, since a link with the underlying microstructure (including its randomness) is not clear. In this paper, we propose a modeling strategy for the linear elastic analysis of FGMs systematically based on a realistic microstructural model. The overall response of FGMs is addressed in the framework of stochastic Hashin-Shtrikman variational principles. To allow for the analysis of finite bodies, recently introduced discretization schemes based on the Finite Element Method and the Boundary Element Method are employed to obtain statistics of local fields. Representative numerical examples are presented to compare the performance and accuracy of both schemes. To gain insight into similarities and differences between these methods and to minimize technicalities, the analysis is performed in the one-dimensional setting., Comment: 33 pages, 14 figures

The purpose of this paper is to propose an inexpensive micromechanics-based scheme for stiffness homogenization and strength estimation of mortars containing crushed bricks, known as cocciopesto. The model utilizes the Mori-Tanaka method to determine the effective stiffness, combined with estimates of quadratic invariants of the deviatoric stresses inside phases to predict the compressive strength. Special attention is paid to the representation of C-S-H gel layer around bricks and interfacial transition zone around sand aggregates, which renders the predictions sensitive to particle sizes. Several parametric studies are performed to demonstrate that the method correctly reproduces data and trends reported in available literature. Moreover, the model is based exclusively on parameters with clear physical or geometrical meaning and as such it provides a convenient framework for its further experimental validation., Comment: 12 pages, 6 figures, 1 table, v2: moderate revisions after the first round of review

Micromechanical homogenization is often carried out with Fourier-accelerated methods that are prone to ringing artifacts. We here generalize the compatibility projection introduced by Vond��ejc, Zeman & Marek [Comput. Math. Appl. 68, 156 (2014)] beyond the Fourier basis. In particular, we formulate the compatibility projection for linear finite elements while maintaining Fourier-acceleration and the fast convergence properties of the original method. We demonstrate that this eliminates ringing artifacts and yields an efficient computational homogenization scheme that is equivalent to canonical finite-element formulations on fully structured grids., 29 pages, 8 figures