Computational Homogenization Research Articles

Surrogate modeling for the homogenization of elastoplastic composites based on RBF interpolation

We propose a new framework for creating a surrogate model of computational homogenization for elastoplastic composite materials that serves as a homogenized constitutive law for decoupled two-scale analysis. Two key ingredients of the proposed surrogate modeling are the introduction of a variable representing the macroscopic strain history along with the macroscopic strain and the application of radial basis function (RBF) interpolation. Not only the coefficients of the RBF but also the type of its function form are considered hyperparameters and are determined by applying an optimization algorithm. In the offline process, numerical material tests (NMTs) are performed on a unit cell consisting of multiple elastoplastic materials subjected to various patterns of macroscopic strain to create a database that stores the discrete relationships between variables representing macroscopic deformation states with the macroscopic stresses. Then, RBF interpolation is applied to represent the macroscopic stress in a continuous function with these input data as independent variables. In the online process, this function is used as a macroscopic constitutive law in the macroscopic analysis, which can be followed by localization analysis using an arbitrary macroscopic strain history if necessary.

Computational homogenization of histological microstructures in human prostate tissue: Heterogeneity, anisotropy and tension-compression asymmetry.

Human prostatic tissue exhibits complex mechanical behaviour due to its multiphasic, heterogeneous nature, with hierarchical microstructures involving epithelial compartments, acinar lumens and stromal tissue all interconnected in complex networks. This study aims to establish a computational homogenization framework for quantifying the mechanical behaviour of prostate tissue, considering its multiphasic heterogeneous microstructures and the mechanical characteristics of tissue constituents. Representative tissue microstructure models were reconstructed from high-resolution histology images. Parametric studies on the mechanical properties of the tissue constituents, particularly the fibre-reinforced hyper-elasticity of the stromal tissue, were carried out to investigate their effects on the apparent tissue properties. These were then benchmarked against the experimental data reported in literature. Results showed significant anisotropy, both structural and mechanical, and tension-compression asymmetry of the apparent behaviours of the prostatic tissue. Strong correlation with the key microstructural indices such as area fractions of tissue constituents and the tissue fabric tensor was also observed. The correlation between the stromal tissue orientation and the principal directions of the apparent properties suggested an essential role of stromal tissue in determining the directions of anisotropy and the compression-tension asymmetry characteristics in normal human prostatic tissue. This work presented a homogenization and histology-based computational approach to characterize the apparent mechanical behaviours of human prostatic or other similar glandular tissues, with the ultimate aim of assessing how pathological conditions (e.g., prostate cancer and benign prostatic hyperplasia) could affect the tissue mechanical properties in a future study.

An explicit dynamic FFT method for homogenizing heterogeneous solids under large deformations

We propose an explicit displacement-based FFT method for computational homogenization and multi-scale modeling of heterogeneous solids. Unlike existing FFT methods that solve the static equilibrium equation using implicit iterative techniques, our proposed explicit method solves dynamic equations with the Central Difference method. The voxel-like discretizations of the periodic domain and the direct calculation of gradient and divergence operations using Discrete Fourier Transform and its inverse result in a straightforward algorithm that avoids convergence issues and does not require material consistent tangents. The stability condition of this explicit method is also addressed. Four numerical examples (2D particle-reinforced composites of hyperelasticity and viscoelasticity, a 3D fiber-reinforced composite, and a 3D polycrystalline metal) illustrate that the proposed method calculates as accurately and efficiently as implicit FFT methods for quasi-static problems. Furthermore, the proposed explicit displacement-based FFT method outperforms implicit FFT methods when dealing with microstructures comprising phases with very high contrast. Moreover, since the method does not require material consistent tangents, it extends the applicability of FFT methods to the homogenization of materials with complex or black-boxed constitutive models, where consistent tangents are challenging to compute.

Upscaling of chemo-mechanical properties of battery electrode material

A variationally consistent model-based computational homogenization approach for transient chemo- mechanically coupled problems is developed based on the classical assumption of first order prolongation of the displacement and chemical potential fields within a Representative Volume Element (RVE). An upscaling procedure is introduced that is based on the assumption of micro-stationarity for the RVE problem. This is motivated by sufficient separation of time-scales. Periodic boundary conditions on the pertinent fields provide the general variational setting for the uniquely solvable RVE-problems. Due to the assumed linearity and micro-stationary, it is possible to use the pertinent sensitivity analysis for the RVE in order to derive effective macro-scale properties in closed form: the elastic stiffness, the insertion strain tensor and the mobility tensor. Statistically representative results are obtained by considering a sufficient number of RVE-realizations with varying volume fractions of the constituents. With the application to structural batteries in mind, a numerical study is conducted for a three-phase RVE microstructure representing a battery electrode material.

Machine learning aided multiscale magnetostatics

Computational material modeling using advanced numerical techniques speeds up the design process and reduces the costs of developing new engineering products. In the field of multiscale modeling, huge computation efforts are expected for modeling heterogeneous materials while trying to reach high accuracy levels. In this work, a machine learning approach, namely the convolutional neural network (CNN), is developed as a solution providing a high level of accuracy while being computationally efficient. The input for the CNN model consists of two/three-dimensional images of artificial periodic and biphasic microstructures in the form of nonoverlapping and overlapping, mono- and polydisperse circular/spherical inclusion systems, which are generated by a random sequential inhibition process. These correspond to Statistical Volume Elements (SVE). Considering linear magnetostatics at the microscale, the output is the apparent permeability of the SVE. Training and testing data for the apparent properties is produced with finite element method-based two-scale asymptotic homogenization. The model efficiency is revealed by employing some representative examples in two and three-dimensional settings. In this regard, the performance of the CNN model is assessed with the applied computational homogenization method relating to the accuracy and computational efficiency. The results with the CNN model show high accuracy in predicting the homogenized permeability and a significant decrease in computation time.

A two-scale framework for coupled mechanics-diffusion-reaction processes

There is a wide range of industrially-relevant problems where mechanical stresses directly affect kinetics of chemical reactions. For example, this includes formation of oxide layers on parts of micro-electro-mechanical systems (MEMS) and lithiation of Si in Li-ion batteries. Detailed understanding of these processes requires thermodynamically-consistent theories describing the coupled thermo-chemo-mechanical behaviour of those systems. Furthermore, as the majority of materials used in those systems have complex microstructures, multiscale modelling techniques are required for efficient simulation of their behaviour. Hence, the purpose of the present paper is two-fold: (1) to derive a thermodynamically-consistent thermo-chemo-mechanical theory; and (2) to propose a two-scale modelling approach based on the concept of computational homogenisation for the considered theory. The theory and the two-scale computational approach are implemented and tested using a number of computational examples, including the case of the reaction locking due to mechanical stresses.

Material database construction for data-driven computing via a continuous path-following method

Data-driven computational homogenization has been proposed recently for the analyses of composite structures. Its basic idea is to construct an equivalent stress–strain database of composites via offline homogenization on the representative volume element and conduct online macroscopic simulation through distance-minimizing data-driven computing. Thanks to the scale separation of concurrent multiscale systems, this framework allows for improving online computational efficiency. However, high-density database construction in the offline stage remains a burdensome and time-consuming task. To this end, this work proposed an efficient approach that associates computational homogenization with the Asymptotic Numerical Method (ANM) to construct a high-density database. Being a reliable and efficient perturbation technique, the ANM allows for accurate tracking of the displacement–load paths and easily generates abundant equivalent stress–strain data on the paths. A fiber reinforced composite material with fiber buckling has been considered to demonstrate the accuracy and efficiency of the proposed method for the database construction of composites.

Coupling Computational Homogenization with Crystal Plasticity Modelling for Predicting the Warm Deformation Behaviour of AA2060-T8 Al-Li Alloy.

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.

Effect of phase contrast and inclusion shape on the effective response of viscoelastic composites using peridynamic computational homogenization theory

Viscoelastic composites are widely used in engineering applications, necessitating a comprehensive understanding of their effective response to external loads. This study employs peridynamic computational homogenization theory to investigate the influence of phase contrast and inclusion shape on the effective properties of these composites. By focusing on a two-phase matrix-inclusion composite material and considering circular and elliptical inclusion shapes, this study explores the relationship between these factors and the resulting effective behavior. The simulation results demonstrate that the use of circular inclusions result in isotropic effective response of the composites for all values of phase contrasts, while elliptical inclusions lead to an increase in anisotropy as phase contrast increases. These findings highlight the significant impact of inclusion shape and phase contrast on the effective behavior of composites.

A multiscale framework for atomistic–continuum transition in nano-powder compaction process using a cap plasticity model

In this paper, a computational atomistic-continuum multiscale framework is presented for the simulation of the nano-powder compaction process using a cone-cap plasticity model. The atomistic representative volume element (RVE) comprises of nano-powder particles that is used to perform the molecular dynamics analysis in order to capture the mechanical behavior of nano-powder material. The periodic boundary condition is applied over the atomistic RVE to satisfy the inter-scale kinematic compatibility, and the stress tensor is derived according to the inter-scale energetic consistency principals from the nanoscale at each material point of the powder component. A multiscale framework is performed using the constitutive law of the coarse-scale domain enhanced by the molecular dynamics results of the fine-scale domain. The coarse-scale constitutive law is conformed to the cone-cap plasticity model as a robust and capable constitutive relation in modeling the powder compaction process, in which the cap plasticity parameters are determined using the data derived from the molecular dynamics simulations. The mechanical response of nano-powder particles within the RVE is obtained from the confining pressure test and true-triaxial test, where a sensitivity analysis is accomplished on the computed parameters of the cone-cap plasticity model. The enhanced cap plasticity model is then employed to simulate various industrial nano-powder components that illustrates the performance and efficiency of the proposed multiscale computational homogenization method.

A nonlocal method to compute effective properties of viscoelastic composite materials based on peridynamic computational homogenization theory

This article presents a computational homogenization framework for viscoelastic composites within the framework of non-ordinary state-based peridynamic theory. The motivation to develop this framework stems from the desire to develop a homogenization scheme that can model processes and phenomena that are driven by nonlocal behaviour such as size effect and fracture for which frameworks based on the classical continuum theory lack the capability of modelling. The proposed framework was used to calculate the effective properties in both time and frequency domains of two viscoelastic matrix-inclusion composite systems, one with an elastic inclusion and viscoelastic matrix, and the other with viscoelastic inclusion and matrix. The results of calculations were found to compare well with results from the literature. A parametric study was also conducted to investigate the influence of nonlocal interaction on the effective properties by varying the horizon size. Results showed that increasing the degree of nonlocality reduces the stiffness of the composite system as well as increase its rate of creep. The capability to account for nonlocal interaction highlights the potential of this proposed scheme to provide a more comprehensive understanding of the behaviour of viscoelastic composite materials over a wide range of material behaviour.

A hybrid direct FE2 method for modeling of multiscale materials and structures with strain localization

Strain localization is a common phenomenon existing in various multiscale materials or structures, e.g., the bulking bands of thin-walled structures, the local collapse of porous materials, and the crack in solid materials, etc. However, this phenomenon cannot be captured by the conventional homogenization methods, such as mean-field homogenization (e.g., Mori–Tanaka method), first-order or high-order computational homogenization (e.g., the Finite Element Square method), etc., due to the high strain gradient associated with strain localization. Aiming at this, a hybrid Direct FE2 method is proposed by combining the D-FE2 (Direct FE2) and the traditional FE methods, while the multiscale structure is modeled using the D-FE2 method in the region exhibiting low deformation gradient, and the other region displaying high deformation gradient is modeled using the traditional FE method. Moreover, a node displacement constraint and an overall node displacement constraint derived from the multilevel equilibrium equations using the Gauss–Ostrogradsky theorem are respectively prescribed to the interface between the D-FE2 model and the FE model of the multiscale structure, to enforce the energy equilibrium and deformation continuity. The proposed hybrid D-FE2 method is then applied to predict the strain localization behavior of multiscale materials or structures, including local bulking of honeycomb structures, in-situ crack propagation, and localized plastic deformation in fiber reinforced composites, etc. Comparison of the simulation results obtained from the hybrid D-FE2 method and the traditional FE method validates the accuracy, efficiency and ease of numerical implementation of the proposed hybrid D-FE2 method.

A multiscale homogenization iterative algorithm of 2D orthogonal prepreg carbon/aramid/epoxy woven-ply

An orthogonal mixture woven composites with aramid yarns and carbon yarns are investigated in this work. A dual-scale computational homogenization algorithm is also proposed to address the cross-scale numerical problems. Models and methods for mesoscale and macroscale are illustrated separately. Two types of meso-structures are selected for comparison, which are termed as representative unit cell (RUC) and equivalent cross-ply laminate (ECPL), and the corresponding macro-structures are referred to as RUCs and equivalent cross-ply laminate plain-woven composites (ECPL-PWC). The key finds clarified that the ECPL-PWC takes more computational expenses compared with the RUCs, but the RUCs is in acceptable agreement with testing results, as well as the evaluation results of the ECPL-PWC would obtain very satisfactory accuracy through minor numerical adjustments. The final finite element damage results reveals that the RUCs is more accurate for local damage assessment, while the ECPL-PWC is better for global damage analysis.

Predicting the influence of geometric imperfections on the mechanical response of 2D and 3D periodic trusses

Although architected materials based on truss networks have been shown to possess advantageous or extreme mechanical properties, those can be highly affected by tolerances and uncertainties in the manufacturing process, which are usually neglected during the design phase. Deterministic computational tools typically design structures with the assumption of perfect, defect-free architectures, while experiments have confirmed the inevitable presence of imperfections and their possibly detrimental impact on the effective properties. Information about the nature and expected magnitude of geometric defects that emerge from the additive manufacturing processes would allow for new designs that aim to mitigate (or at least account for) the effects of defects and to reduce the uncertainty in the effective properties. To this end, we here investigate the effects of four most commonly found types of geometric imperfections in trusses, applied to eleven representative truss topologies in two and three dimensions. Through our study, we (i) quantify the impact of imperfections on the effective stiffness through computational homogenization, (ii) examine the sensitivity of the various truss topologies with respect to those imperfections, (iii) demonstrate the applicability of the model through experiments on 3D-printed trusses, and (iv) present a machine learning framework to predict the presence of defects in a given truss architecture based merely on its mechanical response.

Multiscale modelling of sandwich structured composites using direct FE2

In finite element analyses, composite laminates are often homogenised and modelled using beam and plate elements for computational efficiency. Aside from difficulties in deriving homogenised properties for nonlinear materials and the loss in fine-scale deformation details, such structural elements are notoriously inaccurate for laminates comprising plies of distinctly different in-plane stiffnesses, notably, sandwich structured composites. A multiscale layerwise beam model is proposed to address these shortcomings. The model is based on the Timoshenko-Ehrenfest beam theory with layerwise kinematics incorporated to capture the key deformation modes of bending and transverse shear, and implemented using Direct FE2 – a versatile computational homogenisation scheme. Beyond homogenisation for the structural elements, Direct FE2 also captures deformation mechanisms at the mesoscale. More importantly, it is readily implemented using available functions in commercial FE codes, allowing for ease of adoption. A foam core sandwich structure is modelled to demonstrate that the proposed approach leads to macroscale responses and resolution to the mesoscale deformations that are almost identical to that of finely meshed solid element models, while taking only 23% of the computation time.

Challenges in two‐scale computational homogenization of mechanical metamaterials

AbstractThanks to the advancement of additive manufacturing technologies, mechanical metamaterials have attracted a great deal of attention in recent years. With the employment of such technologies, materials with exceptional or tailored mechanical properties can be easily manufactured mainly by 3D printing of different microstructures rather than by changing the material composition. These lattice materials can provide remarkable material properties in spite of being significantly lighter than typical bulk materials. Due to the large number of degrees of freedom for engineering structures, single‐scale numerical simulation of such cellular materials is computationally demanding. Therefore, two‐scale computational homogenization approaches, such as FE2 and FE‐FFT, can perform a key role in the cost‐effective numerical modeling of metamaterials. Two‐scale computational homogenization methods rely on solving a boundary value problem (BVP) for each of the macroscopic and microscopic scales in a nested procedure. Although representative homogenization techniques have been widely used to study materials with heterogeneous microstructures, there still exist some challenges in their employment for lattice materials. This study addresses main challenges in two‐scale‐based computational homogenization methods for numerical modeling of mechanical metamaterials. High dependence of convergence rate and accuracy on phase contrast for fast Fourier transform (FFT) solvers and comparable macro and micro characteristic lengths in metamaterials (i.e. the applicability of the principle of scale separation) are some examples of such challenges.

High‐performance computational homogenization of Stokes–Brinkman flow with an Anderson‐accelerated FFT method

AbstractFluid flow problems in bi‐porous media have strong implications in a variety of industrial applications, such as nuclear waste disposal and composites manufacturing. Despite various numerical solutions proposed in the literature, the prediction of dual‐scale flow remains a challenging task due to the requirement of fine meshes to resolve the complex microstructures of bi‐porous media. This paper introduces a new numerical solver based on fast Fourier transform (FFT) for the Stokes–Brinkman problem to predict fluid flow in bi‐porous media with local anisotropy. The novel FFT solver is derived from a velocity‐form of the problem, in contrast to the stress‐form proposed by a recent work. The Anderson acceleration technique is applied for the first time to the Stokes–Brinkman problem, leading to a substantial (orders of magnitude) improvement of the convergence rate. A range of detailed numerical examples are provided to validate the method against the analytical and literature results. Parametric studies are also demonstrated to aid in the selection of model parameters to achieve optimum numerical performance. With a 3D unit cell of a woven textile fabric, we demonstrate that the proposed method is capable of handling high‐resolution simulations with strongly heterogeneous microstructures. Combined with a parallelized implementation over high‐performance computing systems, our method demonstrates a new state‐of‐the‐art in numerical solvers for the Stokes‐Brinkman problem, in terms of computation capacity.

Isotropic yield surfaces for porous ductile materials: complete geometric representation by a computational homogenisation procedure

PurposeThe present paper aims to explore a computational homogenisation procedure to investigate the full geometric representation of yield surfaces for isotropic porous ductile media. The effects of cell morphology and imposed boundary conditions are assessed. The sensitivity of the yield surfaces to the Lode angle is also investigated in detail.Design/methodology/approachThe microscale of the material is modelled by the concept of Representative Volume Element (RVE) or unit cell, which is numerically simulated through three-dimensional finite element analyses. Numerous loading conditions are considered to create complete yield surfaces encompassing high, intermediate and low triaxialities. The influence of cell morphology on the yield surfaces is assessed considering a spherical cell with spherical void and a cubic RVE with spherical void, both under uniform strain boundary condition. The use of spherical cell is interesting as preferential directions in the effective behaviour are avoided. The periodic boundary condition, which favours strain localization, is imposed on the cubic RVE to compare the results. Small strains are assumed and the cell matrix is considered as a perfect elasto-plastic material following the von Mises yield criterion.FindingsDifferent morphologies for the cell imply in different yield conditions for the same load situations. The yield surfaces in correspondence to periodic boundary condition show significant differences compared to those obtained by imposing uniform strain boundary condition. The stress Lode angle has a strong influence on the geometry of the yield surfaces considering low and intermediate triaxialities.Originality/valueThe exhaustive computational study of the effects of cell morphologies and imposed boundary conditions fills a gap in the full representation of the flow surfaces. The homogenisation-based strategy allows us to further investigate the influence of the Lode angle on the yield surfaces.

A Green’s function-based approach to the concentration tensor fields in arbitrary elastic microstructures

Computational homogenization based on FEM models is the gold standard when it comes to homogenization over a representative volume element (RVE), of so-called complex material microstructures, i.e., such which cannot be satisfactorily represented by an assemblage of homogeneous subdomains called phases. As a complement to the aforementioned models, which depend on the boundary conditions applied to the representative volume element and which, as a rule, do not give direct access to the macro-micro-relations in terms of concentration tensors, we here introduce a Green’s function-based homogenization method for arbitrary inhomogeneous microstructures: Inspired by the ideas underlying traditional phase-based homogenization schemes, such as the Mori-Tanaka or the self-consistent model, the new method rests on mapping, through the strain average rule, the microscopic strain fields associated with an auxiliary problem to the macroscopic strains subjected to the RVE. Thereby, the auxiliary problem is defined on a homogeneous infinite matrix subjected to homogeneous auxiliary strains and to inhomogeneous (fluctuating) polarization stresses representing the fluctuations of the microstiffness field, i.e., the complex microstructure within the RVE. The corresponding microscopic strains appear as the solution of a Fredholm integral equation, delivering a multilinear operator linking the homogeneous auxiliary strains to the microscopic strains. This operator, together with the aforementioned mapping, eventually allows for completing the model in terms of concentration tensor and homogenized stiffness quantification. This is illustrated by example of a sinusoidally fluctuating microstructure, whereby the corresponding singular convolution integrals are analytically evaluated from the solution of the Poisson’s equation, and this evaluation strategy is then analytically verified through a Cauchy principal value analysis, and numerically validated by a state-of-the-art FFT homogenization procedure. For the given example, the novel analytical method is several thousand times faster than an FTT-based computational homogenization procedure.

Reduced order homogenization of thermoelastic materials with strong temperature dependence and comparison to a machine-learned model

In this work, an approach for strongly temperature-dependent thermoelastic homogenization is presented. It is based on computational homogenization paired with reduced order models (ROMs) that allow for full temperature dependence of material parameters in all phases. In order to keep the model accurate and computationally efficient at the same time, we suggest the use of different ROMs at few discrete temperatures. Then, for intermediate temperatures, we derive an energy optimal basis emerging from the available ones. The resulting reduced homogenization problem can be solved in real time. Unlike classical homogenization where only the effective behavior, i.e., the effective stiffness and the effective thermal expansion, of the microscopic reference volume element are of interest, our ROM delivers also accurate full-field reconstructions of all mechanical fields within the microstructure. We show that the proposed method referred to as optimal field interpolation is computationally as efficient as simplistic linear interpolation. However, our method yields an accuracy that matches direct numerical simulation in many cases, i.e., very accurate real-time predictions are achieved. Additionally, we propose a greedy sampling procedure yielding a minimal number of direct numerical simulations as inputs (two to six discrete temperatures are used over a range of around 1000 K). Further, we pick up a black box machine-learned model as an alternative route and show its limitations in view of the limited amount of training data. Using our new method to generate an abundance of data, we demonstrate that a highly accurate tabular interpolator can be gained easily.