Search

We discuss how Dirichlet boundary conditions can be directly imposed for the Moulinec–Suquet discretization on the boundary of rectangular domains in iterative schemes based on the fast Fourier transform (FFT) and computational homogenization problems in mechanics. Classically, computational homogenization methods based on the fast Fourier transform work with periodic boundary conditions. There are applications, however, when Dirichlet (or Neumann) boundary conditions are required. For thermal homogenization problems, it is straightforward to impose such boundary conditions by using discrete sine (and cosine) transforms instead of the FFT. This approach, however, is not readily extended to mechanical problems due to the appearance of mixed derivatives in the Lamé operator of elasticity. Thus, Dirichlet boundary conditions are typically imposed either by using Lagrange multipliers or a "buffer zone" with a high stiffness. Both strategies lead to formulations which do not share the computational advantages of the original FFT-based schemes. The work at hand introduces a technique for imposing Dirichlet boundary conditions directly without the need for indefinite systems. We use a formulation on the deformation gradient—also at small strains—and employ the Green's operator associated to the vector Laplacian. Then, we develop the Moulinec–Suquet discretization for Dirichlet boundary conditions—requiring carefully selected weights at boundary points—and discuss the seamless integration into existing FFT-based computational homogenization codes based on dedicated discrete sine/cosine transforms. The article culminates with a series of well-chosen numerical examples demonstrating the capabilities of the introduced technology. [ABSTRACT FROM AUTHOR]

Classical homogenization approaches applied to heterogeneous materials are suitable for the cases where a scale-separation is eminent. As the length-scale at the effective continuum reaches the length-scale of the microstructure of the material, classical homogenization approaches fail to be accurate. In such cases, higher-gradient theories may be stimulated for multi-scale material modeling of complex structures in terms of geometry and material. In this study, a multi-scale homogenization framework is presented for additively manufactured (3-D printed) composite parts with specific infill design. The overall framework consists of two major steps, namely micro-to-material and material-to-structure homogenization. In both steps, an asymptotic homogenization procedure is applied to determine constitutive parameters. In the micro-to-material homogenization, the constitutive parameters of the composite material are first determined regarding the material composition. Then, in the material-to-structure homogenization, the constitutive parameters are obtained regarding the infill design of the additively manufactured part. The developed two-step homogenization framework is applied for an off-the-shelf composite material commonly used in 3-D printers. Specifically, in this study, composite parts printed with grid infills are investigated numerically considering different infill ratios. [ABSTRACT FROM AUTHOR]

This paper presents a new method of dynamic homogenization of periodic elastic media under harmonic antiplane deformation based on energy equivalence. The dynamic homogenization is based on two main steps, (1) determining the dispersion relation and the detailed local response in a representative volume element (RVE) by analyzing the propagation of Bloch waves and (2) using energy equivalence between the periodic and effective media to determine the effective properties. The method is first applied to a one-dimensional periodic medium, from which the analytical solutions of the effective material properties are obtained. The results are compared with that from the original periodic medium and an excellent agreement is observed. The dynamic homogenization method is then applied to general two-dimensional periodic media to determine the effective properties and to predict wave fields under typical loading conditions. Illustrative examples are presented and compared with the results from the multiple scattering model. The method has also been applied to multiscale modeling of complicated inhomogeneous media containing multiple groups of periodic inhomogeneities. By treating each group as a homogeneous material with effective properties determined by the current homogenization method, the wave field is obtained using the boundary element method. The resulting wave fields from the current method show an excellent agreement with that from the multiple scattering model with matching local response details. [ABSTRACT FROM AUTHOR]

In this contribution, we derive a consistent variational formulation for computational homogenization methods and show that traditional FE 2$$ {}^2 $$ and IGA 2$$ {}^2 $$ approaches are special discretization and solution techniques of this most general framework. This allows us to enhance dramatically the numerical analysis as well as the solution of the arising algebraic system. In particular, we expand the dimension of the continuous system, discretize the higher dimensional problem consistently and apply afterwards a discrete null‐space matrix to remove the additional dimensions. A benchmark problem, for which we can obtain an analytical solution, demonstrates the superiority of the chosen approach aiming to reduce the immense computational costs of traditional FE 2$$ {}^2 $$ and IGA 2$$ {}^2 $$ formulations to a fraction of the original requirements. Finally, we demonstrate a further reduction of the computational costs for the solution of general nonlinear problems. [ABSTRACT FROM AUTHOR]

A computational homogenization framework is presented to study the dynamics of locally resonant acoustic metamaterial structures. Modelling the resonant units at the microscale as representative volume elements and building on well-established scale transition relations, the framework brings as a main novelty a reduced-order macroscopic homogenized continuum whose governing equations involve no additional variables to describe the microscale dynamics unlike micromorphic homogenized continua obtained by alternative computational homogenization approaches. This model-order reduction is obtained by formulating the governing equations of the micro- and macroscale problems in the frequency domain, introducing a finite-element discretization of the two problems and performing an exact dynamic condensation of all the degrees of freedom at the microscale. An appropriate inverse Fourier transform approach is implemented on the frequency-domain equations to capture transient dynamics as well; notably, the implementation involves the Exponential Window Method, here applied for the first time to calculate the time-domain response of undamped locally resonant acoustic metamaterial structures. The framework may handle arbitrary geometries of micro- and macro-structures, any transient excitations and any boundary conditions on the macroscopic domain. [ABSTRACT FROM AUTHOR]

A second-order multiscale theory based on the concept of a Representative Volume Element (RVE) is proposed to link a classical poromechanical model at the RVE scale to a high-order poromechanical model at the macro-scale in the context of finite-strain kinematics. The proposed theory is carefully derived from the Principle of Multiscale Virtual Power, which is a generalization of the Hill-Mandel Principle of Macrohomogeneity. The coupled governing equations of the low-scale and the homogenization rules for the flux and stress-like quantities are obtained by means of standard variational arguments. The main theoretical result is that the minimally constrained space for the pore pressure field allows for non-zero net fluid flow across the RVE boundaries, unlike first-order theories. The direct consequence of this finding is that the present theory can be consistently applied in cases where the low-scale (RVE level) exhibits substantial volume changes (swelling or shrinking) as a consequence of the evolution of the macro-scale kinematics. Details of formulation development and expression for the homogenized tangent operators are presented for those interested in the computational implementation of the model. [ABSTRACT FROM AUTHOR]

A computational homogenization framework is proposed for solving transient wave propagation in the linear regime in heterogeneous poroelastic media that may exhibit local resonances due to microstructural heterogeneities. The microscale fluid‐structure interaction problem and the macroscale are coupled through an extended version of the Hill‐Mandel principle, leading to a variationally consistent averaging scheme of the microscale fields. The effective macroscopic constitutive relations are obtained by expressing the microscale problem with a reduced‐order model that contains the longwave basis and the so‐called local resonance basis, yielding the closed‐form expressions for the homogenized material properties. The resulting macroscopic model is an enriched porous continuum with internal variables that represent the microscale dynamics at the macroscale, whereby the Biot model is recovered as a special case. Numerical examples demonstrate the framework's validity in modeling wave transmission through a porous layer. [ABSTRACT FROM AUTHOR]

Enhanced transformation field analysis (E‐TFA), recently proposed for reduced‐order modeling, is here formulated for and applied to multiscale analysis. The approach is able to reproduce a highly complex nonlinear macroscale behavior, resulting from a microstructure with cohesive interfaces embedded in an elasto‐plastic bulk. E‐TFA features a consistent tangent matrix in its solution procedure, which enables a straightforward definition of the upscaled tangent stiffness tensor. Numerical tests show that, compared to FE 2$$ {}^2 $$, the proposed approach yields accurate solutions at a lower computational cost. [ABSTRACT FROM AUTHOR]

For scientific software, especially those used for large-scale simulations, achieving good performance and efficiently using the available hardware resources is essential. It is important to regularly perform benchmarks to ensure the efficient use of hardware and software when systems are changing and the software evolves. However, this can become quickly very tedious when many options for parameters, solvers, and hardware architectures are available. We present a continuous benchmarking strategy that automates benchmarking new code changes on high-performance computing clusters. This makes it possible to track how each code change affects the performance and how it evolves. [ABSTRACT FROM AUTHOR]

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

This work deals with the composite voxel method, which—in its original form—furnishes voxels containing more than one material with a surrogate material law accounting for the heterogeneity in the voxel. We show that the laminate composite voxel technique naturally arises as an assumed strain method, that is, the general framework introduced by Simo‐Rifai, for a specific choice of enhanced strain field. As a consequence, laminate composite voxels may be regarded as a kinematic assumption within a discretization scheme rather than a part of material modeling, as suggested originally. We discuss how to seamlessly integrate composite voxels into the framework of a level‐set description of the microstructure, in particular the accurate and efficient computation of normals and cut‐volume fractions. In contrast to more traditional strategies based on subvoxelizations, the introduced method avoids systematic errors when computing composite voxel properties. We demonstrate the applicability of the developed technology for a number of relevant computational examples. [ABSTRACT FROM AUTHOR]

AbstractConcurrent multiscale damage analysis is still a challenge in woven composite structural level due to its extremely high computational cost. In this paper, a novel efficient concurrent multiscale framework for woven composite structures is proposed based on data-driven reduced order model, and the multiscale damage behavior of 4-H satin weave carbon/carbon composites is investigated using this framework. The elastic properties of yarn are predicted using the information from fibers and matrix. The mesoscale anisotropic damage behaviors of the open hole plate under uniaxial tension are simulated using this framework. The simulation results have good agreement with the experiment. [ABSTRACT FROM AUTHOR]

This article deals with the development of an elastic tetragonal model for the 2D auxetic rotating units structures in the framework of orthogonal transformations. The existing anisotropy in the structure was first determined by numerical simulations on the samples with different pattern orientation angles. A suitable representative volume element (RVE), which correctly represents the mechanical properties of the whole structure both in macroscale and in microscale, was then proposed by implementation of the kinematic periodic boundary conditions. In the next step, with the help of the orthogonal transformations relations, an anisotropic elastic model was developed, which correctly reflects the present tetragonal symmetry in the structure. Finally, the model parameters were identified and validated with the help of the corresponding experiments [ABSTRACT FROM AUTHOR]

To compute the effective properties of random heterogeneous materials, a number of different boundary conditions are used to define the apparent properties on cells of finite size. Typically, depending on the specific boundary condition, different numerical methods are used. The article at hand provides a unified framework for Lippmann–Schwinger solvers in thermal conductivity and Dirichlet (prescribed temperature), Neumann (prescribed normal heat flux) as well as periodic boundary conditions. We focus on the explicit jump finite‐difference discretization and discuss different techniques for computing the discrete Green's operator. These include Fourier‐type methods, that is, based on discrete sine, cosine and Fourier transformations, as well as low‐rank tensor methods, that is, the tensor‐train approximation, whose computational prowess was demonstrated recently. We use the developed computational technology to investigate a number of interesting random materials and assess different microstructure‐generation techniques in terms of their compatibility with the prescribed boundary conditions. [ABSTRACT FROM AUTHOR]

This study provides a methodology for exploring the microstructural and mechanical properties of the Haynes®282® alloy produced via the Powder Bed Fusion-Electron Beam (PBF-EB) process. Employing 2D Electron Backscatter Diffraction (EBSD) data, we have successfully generated 3D representations of columnar microstructures using the Representative Volume Element (RVE) method. This methodology allowed for the validation of elastic properties through Crystal Elasticity Finite Element (CEFE) computational homogenization, revealing critical insights into the material behavior. This study highlights the importance of accurately representing the grain morphology and crystallographic texture of the material. Our findings demonstrate that created virtual models can predict directional elastic properties with a high level of accuracy, showing a maximum error of only ~5% compared to the experimental results. This precision underscores the potential of our approach for predictive modeling in Additive Manufacturing (AM), specifically for materials with complex, non-homogeneous microstructures. It can be concluded that the results uncover the intricate link between microstructural features and mechanical properties, underscoring both the challenges encountered and the critical need for the accurate representation of grain data, as well as the significance of achieving a balance in EBSD area selection, including the presence of anomalies in strongly textured microstructures. [ABSTRACT FROM AUTHOR]

The computational efficiency of FFT‐based computational micromechanics is deeply rooted in the underlying regular, that is, Cartesian, discretization. The bottleneck for most industrial applications is evaluating the typically rather expensive constitutive law on the regular grid. In the work at hand, we exploit coarsening strategies to evaluate the material law with the intention of speeding up the overall computation time while retaining the level of achieved accuracy. Inspired by wavelet‐compression techniques, we form aggregates of voxels where the local strain tensors are close, and compute the stresses on these coarsened elements. If done naively, such a strategy will lead to intrinsic instabilities whose origin is apparent from a mathematical perspective. As a remedy, we introduce a stabilization technique which is inspired by hourglass control well‐known for underintegrated finite elements. We introduce octree as well as sandwich coarsening, discuss the handling of internal variables, report on the efficient implementation of the concepts and demonstrate the effectiveness of the developed technology on simple as well as industrial examples. [ABSTRACT FROM AUTHOR]

Locking effects can be a major concern during the numerical modelling of elastic materials, especially for large strains. Those effects arise from volumetric constraints such as incompressibility or anisotropic effects of the underlying material class. One particular solution strategy is to employ mixed formulations, which provide solutions tailored to the specific locking phenomena at hand. While being in general powerful, one drawback of such solution strategies is that the provided strategy to overcome locking is often tied or limited to some specific topological type of finite elements and thus forfeiting generality. Contrary the Virtual Element Method (VEM) benefits by definition from allowing arbitrary element shapes and number of nodes at element level. A variety of approaches for the treatment of locking phenomena for hyperelastic material is formulated in this contribution for the Virtual Element Method with a low order ansatz. Key ingredient to the implementation of multi-field mixed principles in VEM is the consideration of only one constant variable per field and one corresponding Lagrange multiplier over the entire virtual element. Hereby the stabilization contribution utilizes the mixed formulation but shares the element-wise constant variable with the projection part of the virtual element. A direct consequence of this rather simple implementation strategy is the combination of powerful mixed formulations with a computational approach that is able to treat general element shapes. The proposed formulations are tested with regard to structured mesh at standard examples in computational mechanics as well as at specific computational engineering applications where also unstructured meshes are utilized. [ABSTRACT FROM AUTHOR]

This study provides a methodology for exploring the microstructural and mechanical properties of the Haynes®282® alloy produced via the Powder Bed Fusion-Electron Beam (PBF-EB) process. Employing 2D Electron Backscatter Diffraction (EBSD) data, we have successfully generated 3D representations of columnar microstructures using the Representative Volume Element (RVE) method. This methodology allowed for the validation of elastic properties through Crystal Elasticity Finite Element (CEFE) computational homogenization, revealing critical insights into the material behavior. This study highlights the importance of accurately representing the grain morphology and crystallographic texture of the material. Our findings demonstrate that created virtual models can predict directional elastic properties with a high level of accuracy, showing a maximum error of only ~5% compared to the experimental results. This precision underscores the potential of our approach for predictive modeling in Additive Manufacturing (AM), specifically for materials with complex, non-homogeneous microstructures. It can be concluded that the results uncover the intricate link between microstructural features and mechanical properties, underscoring both the challenges encountered and the critical need for the accurate representation of grain data, as well as the significance of achieving a balance in EBSD area selection, including the presence of anomalies in strongly textured microstructures.

Moulinec and Suquet introduced a method for computational homogenization based on the fast Fourier transform which turned out to be rather computationally efficient. The underlying discretization scheme was subsequently identified as an approach based on trigonometric polynomials, coupled to the trapezoidal rule to substitute full integration. For problems with smooth solutions, the power of spectral methods is well‐known. However, for heterogeneous microstructures, there are jumps in the coefficients, and the solution fields are not smooth enough due to discontinuities across material interfaces. Previous convergence results only provided convergence of the discretization per se, that is, without explicit rates, and could not explain the effectiveness of the discretization observed in practice. In this work, we provide such explicit convergence rates for the local strain as well as the stress field and the effective stresses based on more refined techniques. More precisely, we consider a class of industrially relevant, discontinuous elastic moduli separated by sufficiently smooth interfaces and show rates which are known to be sharp from numerical experiments. The applied techniques are of independent interest, that is, we employ a local smoothing strategy, utilize Féjer means as well as Bernstein estimates and rely upon recently established superconvergence results for the effective elastic energy in the Galerkin setting. The presented results shed theoretical light on the effectiveness of the Moulinec–Suquet discretization in practice. Indeed, the obtained convergence rates coincide with those obtained for voxel finite element methods, which typically require higher computational effort. [ABSTRACT FROM AUTHOR]

In this manuscript, a reduced order homogenization model is developed for polycrystalline microstructures with microstructurally small cracks. The proposed approach employs and advances the eigendeformation‐based homogenization method to account for the plastic deformation within the microstructure and the presence of cracks. A novel approach to construct the reduced order basis for the separation field is proposed for approximating crack opening profiles of kinked cracks. To capture the variable stress fields around the crack tips, a domain partitioning strategy that automatically refines the reduced order parts in these regions is proposed. The model performance is evaluated against reference crystal plasticity finite element (CPFE) simulations under various loading conditions and crack configurations. Both the overall and local response predictions show reasonable accuracy with only a fraction of the computational cost of the reference simulations. [ABSTRACT FROM AUTHOR]

This study aimed to propose a new approach for predicting the warm deformation behaviour of AA2060-T8 sheets by coupling computational homogenization (CH) with crystal plasticity (CP) modeling. Firstly, to reveal the warm deformation behaviour of the AA2060-T8 sheet, isothermal warm tensile testing was accomplished using a Gleeble-3800 thermomechanical simulator at the temperatures and strain rates that varied from 373 to 573 K and 0.001 to 0.1 s−1. Then, a novel crystal plasticity model was proposed for describing the grains' behaviour and reflecting the crystals' actual deformation mechanism under warm forming conditions. Afterward, to clarify the in-grain deformation and link the mechanical behaviour of AA2060-T8 with its microstructural state, RVE elements were created to represent the microstructure of AA2060-T8, where several finite elements discretized every grain. A remarkable accordance was observed between the predicted results and their experimental counterparts for all testing conditions. This signifies that coupling CH with CP modelling can successfully determine the warm deformation behaviour of AA2060-T8 (polycrystalline metals) under different working conditions. [ABSTRACT FROM AUTHOR]

Herein, we present a new data-driven multiscale framework called FE ANN which is based on two main keystones: the usage of physics-constrained artificial neural networks (ANNs) as macroscopic surrogate models and an autonomous data mining process. Our approach allows the efficient simulation of materials with complex underlying microstructures which reveal an overall anisotropic and nonlinear behavior on the macroscale. Thereby, we restrict ourselves to finite strain hyperelasticity problems for now. By using a set of problem specific invariants as the input of the ANN and the Helmholtz free energy density as the output, several physical principles, e. g., objectivity, material symmetry, compatibility with the balance of angular momentum and thermodynamic consistency are fulfilled a priori. The necessary data for the training of the ANN-based surrogate model, i. e., macroscopic deformations and corresponding stresses, are collected via computational homogenization of representative volume elements (RVEs). Thereby, the core feature of the approach is given by a completely autonomous mining of the required data set within an overall loop. In each iteration of the loop, new data are generated by gathering the macroscopic deformation states from the macroscopic finite element simulation and a subsequently sorting by using the anisotropy class of the considered material. Finally, all unknown deformations are prescribed in the RVE simulation to get the corresponding stresses and thus to extend the data set. The proposed framework consequently allows to reduce the number of time-consuming microscale simulations to a minimum. It is exemplarily applied to several descriptive examples, where a fiber reinforced composite with a highly nonlinear Ogden-type behavior of the individual components is considered. Thereby, a rather high accuracy could be proved by a validation of the approach. [ABSTRACT FROM AUTHOR]

This work develops a novel homogenization-based computational strategy for predicting the mechanical properties of fused filament fabricated parts by characterizing the component according to the different printing patterns used. The anisotropic constitutive material models obtained for each printing pattern allows the accurate prediction of the behavior of the entire component. The model is validated against experiments with the samples manufactured according to corresponding printing patterns. Various materials including Polycarbonate/Acrylonitrile Butadiene Styrene and ULTEM 9085 are tested. The good accuracy of the predictions obtained using present approach indicates that material characterization can be successfully performed in a fully numerical way. [ABSTRACT FROM AUTHOR]

In this work, a method is proposed for modeling RTM process and the associated mechanical behavior of composites filled with mono-sized spherical Alumina particles. This method combines (i) a numerical model (RTM model) that allows the simulation of the RTM process during the injection of particle filled resins and (ii) a computational strategy of mechanical properties based on the homogenization methods. These proposed models have already been validated with experimental results. The RTM model is based on 3 sub-models: the first one to describe the suspension flow, the second one to simulate the advance of the flow front, and the last one to model the particles filtration by the fibrous medium. The distribution result of the concentration of particles in the fibrous medium obtained at the end of the simulation of the injection is used as input data for mechanical models of homogenization. The homogenization numerical model was constructed from a representative volume element of the microstructures using the Poisson process. The idea here is to couple these two steps (RTM simulation + mechanical property computation) in a complete model which allows at the same time and in a single operation: to simulate the process of the manufactured composites loaded with particles and to deduce their induced mechanical properties. The pertinence of the proposed method is confirmed by the simulation of nine elastic properties of composites with the finite element method. The influence of post-filling on the induced mechanical properties has been studied. [ABSTRACT FROM AUTHOR]

This paper studies the convergences of several FFT-based discretization schemes that are widely applied in computational micromechanics for deriving effective coefficients, and “convergence” here means the limiting behaviors as spatial resolutions tending to infinity. Those schemes include Moulinec–Suquet’s scheme, Willot’s scheme and the FEM scheme. Under some reasonable assumptions, we prove that the effective coefficients obtained by those schemes all converge to the theoretical ones. Moreover, for the FEM scheme, we can present several convergence rate estimates under additional regularity assumptions. [ABSTRACT FROM AUTHOR]

Data-driven thermal and percolation analyses are conducted to elucidate the effects of various characteristics on the effective thermal conductivity of complex 3D composite structures. These characteristics include the thermal and geometric properties of the composite constituents, the interface resistance, and the existence of percolation paths. A series of voxel-wise microstructure samples with various characteristics are generated. Their effective thermal conductivities are evaluated using a diffuse-interface-based computational homogenization method. A voxel-based algorithm is employed to identify the potential percolation paths in the structures. The homogenization results show particularly significant effects of the percolation path in composite samples with higher aspect ratios and interface resistances. The importance of different thermal and geometric features to the effective thermal conductivity is analyzed using a data-driven sensitivity study. The analysis also demonstrates that the particle volume fraction and interface thermal resistance are the most influential characteristics for determining the effective thermal conductivity. Finally, employing a surrogate-based classification model, microstructures with and without percolation can be distinguished with an accuracy of 93%.

We provide theoretical investigations and empirical evidence that the effective stresses in computational homogenization of inelastic materials converge with a higher rate than the local solution fields. Due to the complexity of industrial‐scale microstructures, computational homogenization methods often utilize a rather crude approximation of the microstructure, favoring regular grids over accurate boundary representations. As the accuracy of such an approach has been under continuous verification for decades, it appears astonishing that this strategy is successful in homogenization, but is seldom used on component scale. A part of the puzzle has been solved recently, as it was shown that the effective elastic properties converge with twice the rate of the local strain and stress fields. Thus, although the local mechanical fields may be inaccurate, the averaging process leads to a cancellation of errors and improves the accuracy of the effective properties significantly. Unfortunately, the original argument is based on energetic considerations. The straightforward extension to the inelastic setting provides superconvergence of (pseudoelastic) potentials, but does not cover the primary quantity of interest: the effective stress tensor. The purpose of the work at hand is twofold. On the one hand, we provide extensive numerical experiments on the convergence rate of local and effective quantities for computational homogenization methods based on the fast Fourier transform. These indicate the superconvergence effect to be valid for effective stresses, as well. Moreover, we provide theoretical justification for such a superconvergence based on an argument that avoids energetic reasoning. [ABSTRACT FROM AUTHOR]

We propose a new multiscale method that combines the radial basis function (RBF)-based surrogate homogenization for elastoplastic composites and numerical material test (NMT) that allows for nonperiodic distributions of constituent materials within a representative volume element (RVE). In the offline process, we construct a constitutive database of the macroscopic stress-strain responses by conducting NMTs with a special constraint such that the volume integral of gradient of fluctuation displacement over the RVE domain is zero. Then, a surrogate model is created by using the constitutive database. In the online process, we conduct a numerical simulation of a macrostructure with the created surrogate model, followed by localization analysis using arbitrary strain histories. The validity of the proposed multiscale analysis method using RBF-based surrogate modeling is confirmed through these offline and online processes. In numerical examples, to demonstrate the modeling capability of nonperiodic microstructures, cubic and circular RVEs are targeted. [ABSTRACT FROM AUTHOR]

The power of fast Fourier transform (FFT)‐based methods in computational micromechanics critically depends on a seamless integration of discretization scheme and solution method. In contrast to solution methods, where options are available that are fast, robust and memory‐efficient at the same time, choosing the underlying discretization scheme still requires the user to make compromises. Discretizations with trigonometric polynomials suffer from spurious oscillations in the solution fields and lead to ill‐conditioned systems for complex porous materials, but come with rather accurate effective properties for finitely contrasted materials. The staggered grid discretization, a finite‐volume scheme, is devoid of bulk artifacts in the solution fields and works robustly for porous materials, but does not handle anisotropic materials in a natural way. Fully integrated finite‐element discretizations share the advantages of the staggered grid, but involve a higher memory footprint, require a higher computational effort due to the increased number of integration points and typically overestimate the effective properties. Most widely used is the rotated staggered grid discretization, which may also be viewed as an underintegrated trilinear finite element discretization, which does not impose restrictions on the constitutive law, has fewer artifacts than Fourier‐type discretizations and leads to rather accurate effective properties. However, this discretization comes with two downsides. For a start, checkerboard artifacts are still present. Second, convergence problems occur for complex porous microstructures. The work at hand introduces FFT‐based solution techniques for underintegrated trilinear finite elements with hourglass control. The latter approach permits to suppress local hourglass modes, which stabilizes the convergence behavior of the solvers for porous materials and removes the checkerboards from the local solution field. Moreover, the hourglass‐control parameter can be adjusted to "soften" the material response compared to fully integrated elements, using only a single integration point for nonlinear analyses at the same time. To be effective, the introduced technology requires a displacement‐based implementation. The article exposes an efficient way for doing so, providing minimal interfaces to the most commonly used solution techniques and the appropriate convergence criterion. [ABSTRACT FROM AUTHOR]

A model for the macroscopic mechanical behavior of rank-1 laminates including two shape memory alloy (SMA) phases is presented. The model expresses the general behavior of the composite with phases undergoing rate-independent elastic and inelastic deformations. Homogenization techniques (including the rank-1 laminate model) are used to establish the effective behavior of the SMA laminated composite (SLC) based on the information about the mechanical response of the individual phases and their volume concentrations. A stress-control algorithm is put forward to implement the model. With the aid of the stress-control algorithm, an implicit expression for the effective tangent stiffness and an evolution equation for the effective inelastic strain are derived. Results are compared with the outcomes of an FE-based computational homogenization and a very good agreement is seen. By using a constitutive model with internal variables for the dense SMA, the overall response of the SLC under different mechanical loadings is evaluated. The effective response of the SLC for various volume concentrations of the phases is assessed and exclusive comparisons are illustrated. Furthermore, the influence of different temperatures on the effective superelastic behavior of the SLCs is studied. The findings have implications for the analysis and the design of more complex shape-memory-alloy laminated composites for high-end applications. • An implicit time-independent model for the rank-1 SMA laminated composites is offered. • Tangent stiffness, strain evolution and a homogenization algorithm are provided. • The model provides an effective transversely isotropic behavior for SMA laminates. • Results are confronted with full-field 3D computational homogenization results. • Analysis and design of more complex SMA laminated composites can be performed. [ABSTRACT FROM AUTHOR]

Although additive manufacturing processes have matured in many areas, difficulties with regard to printing accuracy persist. Possible defects are, e.g. , the generation of unwanted internal pores or a lack of fusion between layers. In general, defects result in a deviation between as-planned and as-built geometries. Assessing the influence of such deviations on effective (mechanical) properties requires either experimental testing or numerical investigation, both of which are often too complex for being used in time-critical production settings. Previous work with the Finite Cell Method (FCM) has shown that image-based computational homogenization can assist in principle in quality monitoring of produced parts by providing effective mechanical properties of as-built geometries. However, conducting FCM computations online for each and every part of a series production demands a prohibitive amount of computational resources. The paper at hand suggests a remedy to this problem by using generative adversarial networks. This manuscript presents an integrated computational workflow for outlier detection and property monitoring, which in principle consists of four steps: (i) Carry out experiments under varying process conditions and collect respective (micro)-structural images. (ii) Compute effective properties for these images using FCM. (iii) Compose a generative adversarial network (GAN) training dataset from those images that meet desired effective properties. (iv) Use the GAN discriminator to detect outliers in series production. To this end, the discriminator is used as a classifier on as-built parts to judge whether an as-built structure is acceptable or defective. The viability of the approach is demonstrated on additively manufactured lattice structures whose geometry is acquired after production via computed tomography. The methodology is not only applicable for automated property monitoring but potentially also for reliability estimates of neural network-based property predictors. • Presentation of an integrated computational workflow for quality monitoring of additively manufactured structures. • Generative adversarial network-based (micro)-structure generation for additive manufacturing • Discriminator as a reliability indicator for image-based structural property monitoring. [ABSTRACT FROM AUTHOR]

This study addresses the challenge of estimating the elastic properties of heterogeneous frozen clay soils by introducing a comprehensive approach that combines analytical and numerical models. The frozen clay soil is treated as a mixture composed of frozen clay-water composites and nonclay mineral inclusions. An inversion algorithm is employed to deduce the elastic properties of the matrix (clay-water composites) of two artificially frozen sandy clay samples with known temperature-dependent elastic properties. Subsequently, a two-dimensional numerical simulation using the eXtended Finite Element Method (XFEM) is conducted to carry out numerical homogenization by considering the imperfect bond among frozen clay-water composites and nonclay minerals. The numerical homogenization model offers insights into the temperature-dependent behavior of the interface stiffness parameter. The numerical homogenization results are compared with conventional numerical homogenization approaches like the FEM, which rigidly defines the bonding between inclusions and the matrix. The comparison indicates that the neglect of imperfect bonds among clay-water composites and nonclay minerals will lead to unrealistic outcomes in cases with a high fraction of inclusions. This integrated approach advances the understanding and prediction of elastic properties of frozen clay soils by considering their heterogeneous nature. • A novel approach for estimating elastic properties of frozen clay soils. • An XFEM-based numerical homogenization considers imperfect interfacial bonds. • Temperature-dependent interface stiffness parameters are quantified. • Comparative study highlights the importance of considering imperfect bonds. [ABSTRACT FROM AUTHOR]

• A direct FE2 method was proposed for concurrent multilevel modeling of Joule heating effect in multiscale materials and structures. • The proposed methods can accurately capture the coupled thermoelectric response of Joule heating phenomenon in heterogeneous materials and structures. • Computational efficiency of the proposed method is 15 times higher than traditional FE simulations. Coupled thermoelectric problem significantly affects the performance of electric devices, while there seems to be no method for concurrent multiscale modeling of such problems, which significantly limits the development of advanced electric devices consisting of multiscale structures. To this end, a Direct FE2 (D-FE2) method is proposed for concurrent multiscale modeling of a typical type of coupled thermoelectric problems, i.e., Joule heating effect, in heterogeneous materials and structures. To enforce energy equilibrium and realize information transfer between the macro- and meso‑scales, the Hill-Mandel homogenization condition was generalized to account for both thermal and electric energy, while periodic boundary conditions derived from kinematic constraints were prescribed to both electric potential and temperature to link meso‑scale RVEs to macro-elements in the D -FE2 model. The proposed D -FE2 method can be easily implemented using the available feature "Multiple Points Constraint (MPC)" in many commercial FE software. A series of numerical examples including steady-state analysis and transient analysis of fiber composites, porous materials and lattice structures validate the favorable accuracy and efficiency of the proposed D-FE2 method in modeling the Joule heating phenomenon in multiscale materials and structures. This also implies the promising potential of the proposed D -FE2 method in modeling and design of large-scale electric devices with multiscale structures. [ABSTRACT FROM AUTHOR]

2D nanoparticles, such as graphene or graphite nanoplatelets, are used as additives in polymer matrices to improve their stiffness and electrical conductivity. In this paper, a finite element-based model for homogenized macrolevel stiffness is developed to understand the increase in stiffness of the epoxy matrix induced by graphene nanoplatelets. The model uses image segmentation of regular SEM micrographs to account for the morphology of the graphene platelet network. Here, it is essential to include a fluctuation field in computational homogenization to describe microstructural relaxation. Platelets of the microstructure are modeled as embedded membranes, assuming perfect bonding to the polymer. To estimate the stiffness of the membrane, we used molecular dynamics simulations from a related paper on layered graphene platelets. A novel feature is the identified anisotropic and isotropic elastic surrogate models obtained by least-squares fits of homogenized microstructural responses. Surrogate models serve as a basis for the evaluation of the stiffness of the nanocomposites, and these models are validated through the Halpin–Tsai and Mori–Tanaka models. According to the experimental investigation, the results show that the samples exhibit an increase in stiffness of up to 10 % to 30 % for GNP contents ranging from 1 to 5 wt. %, respectively, obtained from the morphological properties and the weight fraction of the carbon filler. [Display omitted] [ABSTRACT FROM AUTHOR]

The high computational demands of multiscale modeling necessitate advanced parallel and adaptive strategies. To address this challenge, we introduce an adaptive method that utilizes two microscale models based on an offline database for multiscale modeling of curved interfaces (e.g., adhesive layers). This database employs nonlinear classifiers, developed using Support Vector Machines from microscale sampling data, as a preprocessing step for multiscale simulations. Next, we develop a new parallel network library that enables seamless model selection with customized communication layers, ensuring scalability in parallel computing environments. The correctness and effectiveness of the hierarchically parallel solver are verified on a crack propagation problem within the curved adhesive layer. Finally, we predict the ultimate bending moment and adhesive layer failure of a wind turbine blade and validate the solver on a difficult large-scale engineering problem. [ABSTRACT FROM AUTHOR]

The lack of interest in Mindlin's second strain-gradient theory, despite its success in characterizing various phenomena for which there is no explanation in the classical elasticity, is predominantly due to the numerous higher-order elastic constants involved therein. As an attempt to overcome this shortcoming, a computational homogenization method for determining such elastic constants of polycrystalline face-centered cubic (fcc) metals is proposed in the present study. In this homogenization method, using the Voigt-type averaging scheme, a polycrystalline material is modeled by an isotropic aggregate of randomly oriented single crystals without accounting for the grain size effect and the complexities due to grain boundaries and junctions. Subsequently, analytical expressions for the strain energy due to certain modes of loading are determined, and molecular simulations of a few fcc metals under such modes of loading are performed. Then, by fitting the corresponding analytical expressions to the results obtained from these simulations, the effective elastic constants and consequently the effective characteristic lengths of the fcc metals in their polycrystalline form are determined. Moreover, the free-surface-induced reconstruction in a thin layer of a solid is addressed both analytically and by molecular simulations, as a result of which the effective moduli of cohesion of the fcc metals in their polycrystalline form are calculated. In addition, the complete set of conditions for the positive-definiteness of the strain-energy-density function of isotropic materials within the adopted theory is derived, and subsequently a discussion of whether or not the numerical values obtained for the effective elastic moduli of the polycrystalline fcc metals satisfy such conditions is provided. • Proposing a computational-based homogenization for polycrystalline fcc metals. • Determination of the elastic moduli of fcc metals in second strain-gradient theory. • Addressing the free-surface-induced reconstruction of thin layers of fcc metals. • Determination of the moduli of cohesion of fcc metals. • Derivation of conditions for the positive-definiteness of the strain-energy density. [ABSTRACT FROM AUTHOR]

• A 3D first-order NMM-based multiscale scheme is presented for hydro-mechanics of the heterogeneous porous media. • Micro-scale dynamics is fully considered based on the extended Hill-Mandel lemma. • Simple, closed and efficient algorithms are presented for extracting macro-scale quantities and Jacobian. A two-scale computational homogenization model is presented for modeling three-dimensional (3D) heterogeneous poro-elastic media in the framework of the Numerical Manifold Method (NMM). The micro-dynamics is fully incorporated using the extended first-order Hill-Mandel principle for simulation of the hydro-dynamic response. The micro- and macro-scale Initial Boundary-Value Problems (IBVPs) are simultaneously solved with the NMM by exchanging quantities between different scales. The micro-scale IBVPs are solved under both Linear Boundary Conditions (LBCs) and Periodic Boundary Conditions (PBCs). The macro-scale IBVP is solved using the Newton's algorithm iteratively. For both micro-scale LBCs and PBCs, highly efficient algorithms for extracting macro-scale quantities and Jacobian matrix from the micro-scale are established merely by simple matrix manipulations of the micro-scale Jacobian matrix without solving any IBVPs. By conducting benchmark simulations of the heterogeneous porous media under uniform, partial and impact loads, the accuracy, stability and versatility of the presented multiscale model are verified. [ABSTRACT FROM AUTHOR]

Topology optimization (TO) is an optimal design method to obtain an efficient structure with minimal usage of material by satisfying two conflicting objectives of weight reduction and structural safety. Owing to the recent advances in additive manufacturing technology, TO has been developed in connection with the use of microscale lattices, of which complicated geometries require considerable computational loads to verify their structural performance. This study aims to develop an efficient computational method to analyze a complex TO model. Computational homogenization was then developed for efficient computation of the TO model that contains a number of microscale lattices. The proposed homogenization scheme was then applied to perform three-dimensional (3D) finite element analysis (FEA) on various TO models with three scales (i.e., macroscale, microscale, andmultiscale TOs). The homogenized FEAs were conducted to verify the static and dynamic deformation behaviors of three optimized meta-sandwich beams, and their results and computational efficiency were compared with those from full solid FEAs. Experimental verification revealed that the proposed homogenized FEA provided more reliable results and better computational efficiency for the microscale and multiscale TO models, whereas the conventional solid FEA was advantageous for the macroscale TO model. To apply the proposed simulation strategy to a more complex 3D geometry, three TO models were calculated for a 3D block under a compression load. The simulation strategy combining the full solid and homogenized FEAs was then applied to analyze the static and dynamic deformation behaviors of various TO models, which provided reliable predictions of the experimentally observed behaviors within an acceptable computational time. [ABSTRACT FROM AUTHOR]

With the complexity of modern microstructured materials, computational homogenization methods have been shown to provide accurate estimates of their effective mechanical properties, reducing the involved experimental effort considerably. After solving the balance of linear momentum on the microscale, the effective stress is traditionally computed through volume averaging the microscopic stress field. In the work at hand, we exploit the idea that averaging the elastic energy may lead to much more accurate effective elastic properties than through stress averaging. We show that the accuracy is roughly doubled when using energy equivalence instead of strain equivalence for compatible iterates of iterative schemes. Thus, to achieve a prescribed accuracy, the necessary effort is roughly reduced by a factor of two. In addition to the theory, we provide a handbook for utilizing these ideas for modern solvers prominent in FFT‐based micromechanics. We demonstrate the superiority of energy averaging through computational examples, discuss the peculiarities of polarization methods with their non‐compatible iterates and expose a superaccuracy phenomenon occurring for the linear conjugate gradient method. [ABSTRACT FROM AUTHOR]

Deep material networks (DMNs) are a recent multiscale technology which enable running concurrent multiscale simulations on industrial scale with the help of powerful surrogate models for the micromechanical problem. Classically, the parameters of the DMNs are identified based on linear elastic precomputations. Once the parameters are identified, DMNs may process inelastic material models and were shown to reproduce micromechanical full-field simulations with the original microstructure to high accuracy. The work at hand was motivated by creep loading of thermoplastic components with fiber reinforcement. In this context, multiple scales appear, both in space (due to the reinforcements) and in time (short- and long-term effects). We demonstrate by computational examples that the classical training strategy based on linear elastic precomputations is not guaranteed to produce DMNs whose long-term creep response accurately matches high-fidelity computations. As a remedy, we propose an inelastically informed early stopping strategy for the offline training of the DMNs. Moreover, we introduce a novel strategy based on a surrogate material model, which shares the principal nonlinear effects with the true model but is significantly less expensive to evaluate. For the problem at hand, this strategy enables saving significant time during the parameter identification process. We demonstrate that the novel strategy provides DMNs which reliably generalize to creep loading. [ABSTRACT FROM AUTHOR]