Modeling Of Heterogeneous Materials Research Articles

A statistical high-order reduced model for nonlinear random heterogeneous materials with three-scale micro-configurations

An effective statistical higher-order three-scale reduced homogenization (SHTRH) method is established to analyze the nonlinear random heterogeneous materials with multiple micro-configurations. Firstly, the various unit cell functions based on the microscale and mescoscale regions are given, and two expected homogenization coefficients are computed through Kolmogorov's strong laws of large number. Further, the nonlinear homogenized equations are formulated, and the corresponding reduced-order multiscale systems for displacement and stress solutions are derived by using the high-order unit cell solutions and homogenized solutions. The key features of the new statistical multiscale methods are (i) the novel reduced models established to solve the inelastic problems of random composites at a fraction of cost, (ii) the high-order homogenized solutions which do not need high-order continuity for the macro solutions of the random problems and (iii) the statistical high-order multiscale algorithms developed for analyzing the nonlinear random composites with three-scale structures. Finally, the effectiveness and correctness of the algorithm are confirmed according to several hyperelastic, plasticity and damage periodic/random composites with multiple-scale configurations. The computation shows that the proposed SHTRH methods are useful for analyzing the macroscopic nonlinear performance, and can efficiently catch the microscopic and mesoscopic information for the random heterogeneous composites.

Multi-Objective Topology Optimization of Conjugate Heat Transfer Using Level Sets and Anisotropic Mesh Adaptation

This study proposes a new computational framework for the multi-objective topology optimization of conjugate heat transfer systems using a continuous adjoint approach. It relies on a monolithic solver for the coupled steady-state Navier–Stokes and heat equations, which combines finite elements stabilized by the variational multi-scale method, level set representations of the fluid–solid interfaces and immersed modeling of heterogeneous materials (fluid–solid) to ensure that the proper amount of heat is exchanged to the ambient fluid by solid objects in arbitrary geometry. At each optimization iteration, anisotropic mesh adaptation is applied in near-wall regions automatically captured by the level set. This considerably cuts the computational effort associated with calling the finite element solver, in comparison to traditional topology optimization algorithms operating on isotropic grids with a comparable refinement level. Given that we operate within the constraint of a specified number of nodes in the mesh, this allows not only to improve the accuracy of interface representation and motion but also to retain the high fidelity of the numerical solutions at the grid points just adjacent to the interface. Finally, the remeshing and resolution steps both run within a highly parallel environment, which makes it possible for the proposed algorithm to tackle large-scale problems in three dimensions with several tens of millions of state degrees of freedom. The developed solver is validated first by minimizing dissipation in a flow splitter device, for which the method delivers relevant optimal designs over a wide range of volume constraints and flow rate distributions over the multiple outlet orifices but yields better accuracy compared to reference data from literature obtained using uniform meshes (in the sense that the layouts are more smooth, and the solutions are better resolved). The scheme is then applied to a two-dimensional heat transfer problem, using bi-objective cost functionals combining flow resistance and thermal recoverable power. A comprehensive parametric study reveals a complex arrangement of optimal solutions on the Pareto front, with multiple branches of symmetric and asymmetric designs, some of them previously unreported. Finally, the algorithmic developments are substantiated with several three-dimensional numerical examples tackled under fixed weights for heat transfer and flow resistance, for which we show that the optimal layouts computed at low Reynolds number, that are intrinsically relevant to a broad range of microfluidic application, can also serve as smooth solutions to high-Reynolds-number engineering problems of practical interest.

X-ray scattering tensor tomography based finite element modelling of heterogeneous materials

Among micro-scale imaging technologies of materials, X-ray micro-computed tomography has evolved as most popular choice, even though it is restricted to limited field-of-views and long acquisition times. With recent progress in small-angle X-ray scattering these downsides of conventional absorption-based computed tomography have been overcome, allowing complete analysis of the micro-architecture for samples in the dimension of centimetres in a matter of minutes. These advances have been triggered through improved X-ray optical elements and acquisition methods. However, it has not yet been shown how to effectively transfer this small-angle X-ray scattering data into a numerical model capable of accurately predicting the actual material properties. Here, a method is presented to numerically predict mechanical properties of a carbon fibre-reinforced polymer based on imaging data with a voxel-size of 100 μm corresponding to approximately fifteen times the fibre diameter. This extremely low resolution requires a completely new way of constructing the material’s constitutive law based on the fibre orientation, the X-ray scattering anisotropy, and the X-ray scattering intensity. The proposed method combining the advances in X-ray imaging and the presented material model opens for an accurate tensile modulus prediction for volumes of interest between three to six orders of magnitude larger than those conventional carbon fibre orientation image-based models can cover.

On micropolar elastic foundations

The modelling of heterogeneous and architected materials poses a significant challenge, demanding advanced homogenisation techniques. However, the complexity of this task can be considerably simplified through the application of micropolar elasticity. Conversely, elastic foundation theory is widely employed in fracture mechanics and the analysis of delamination propagation in composite materials. This study aims to amalgamate these two frameworks, enhancing the elastic foundation theory to accommodate materials exhibiting micropolar behaviour. Specifically, we present a novel theory of elastic foundation for micropolar materials, employing stress potentials formulation and a unique normalisation approach. Closed-form solutions are derived for stress and couple stress reactions inherent in such materials, along with the associated restoring stiffness. The validity of the proposed theory is established through verification using the double cantilever beam configuration. Concluding our study, we elucidate the benefits and limitations of the developed theory by quantifying the derived parameters for materials known to exhibit micropolar behaviour. This integration of micropolar elasticity into the elastic foundation theory not only enhances our understanding of material responses but also provides a versatile framework for the analysis of heterogeneous materials in various engineering applications.

The fully coupled thermo-mechanical dual-horizon peridynamic correspondence damage model for homogeneous and heterogeneous materials

To accurately address the thermally induced dynamic and steady-state crack propagation problems for homogeneous and heterogeneous materials involving crack branching, interfacial de-bonding and crack kinking, we propose the fully coupled thermo-mechanical dual-horizon peridynamic correspondence damage model (TM-DHPD). To this end, the integral coupled equations for TM-DHPD are firstly derived within the framework of thermodynamics. And then, the alternative dual-horizon peridynamic correspondence principle is used to derive the constitutive bond force state, heat flow state and their general linearizations. Moreover, the unified criterion for bond damage is proposed to characterize the internal bond damage in a single material and the interface bond damage in dissimilar materials. To ensure convergence and accuracy, the coupled equations are solved using the standard implicit method without the use of artificial damping. In both homogeneous and heterogeneous materials, some representative and challenging numerical cases are examined, such as dynamic crack branching in a centrally heated disk and multiple interface failure of thermal barrier coating. The numerical results are in good agreement with the available experimental results or the previous predictions, which shows the great potential of the proposed TM-DHPD in addressing the physics of numerous thermally induced fractures in the real-world engineering problems.

Micromechanics-informed parametric deep material network for physics behavior prediction of heterogeneous materials with a varying morphology

Deep Material Network (DMN) has recently emerged as a data-driven surrogate model for heterogeneous materials. Given a particular microstructural morphology, the effective linear and nonlinear behaviors can be successfully approximated by such physics-based neural-network like architecture. In this work, a novel micromechanics-informed parametric DMN (MIpDMN) architecture is proposed for multiscale materials with a varying microstructure characterized by several parameters. A single-layer feedforward neural network is used to account for the dependence of DMN fitting parameters on the microstructural ones. Micromechanical constraints are prescribed both on the architecture and the outputs of this new neural network. The proposed MIpDMN is also recast in a multiple physics setting, where physical properties other than the mechanical ones can also be predicted. In the numerical simulations conducted on three parameterized microstructures, MIpDMN demonstrates satisfying generalization capabilities when morphology varies. The effective behaviors of such parametric multiscale materials can thus be predicted and encoded by MIpDMN with high accuracy and efficiency.

Self-optimization wavelet-learning method for predicting nonlinear thermal conductivity of highly heterogeneous materials with randomly hierarchical configurations

In the present work, we propose a self-optimization wavelet-learning method (SO-W-LM) with high accuracy and efficiency to compute the equivalent nonlinear thermal conductivity of highly heterogeneous materials with randomly hierarchical configurations. The randomly structural heterogeneity, temperature-dependent nonlinearity and material property uncertainty of heterogeneous materials are considered within the proposed self-optimization wavelet-learning framework. Firstly, meso- and micro-structural modeling of random heterogeneous materials are achieved by the proposed computer representation method, whose simulated hierarchical configurations have relatively high volume ratio of material inclusions. Moreover, temperature-dependent nonlinearity and material property uncertainties of random heterogeneous materials are modeled by a polynomial nonlinear model and Weibull probabilistic model, which can closely resemble actual material properties of heterogeneous materials. Secondly, an innovative stochastic three-scale homogenized method (STSHM) is developed to compute the macroscopic nonlinear thermal conductivity of random heterogeneous materials. Background meshing and filling techniques are devised to extract geometry and material features of random heterogeneous materials for establishing material databases. Thirdly, high-dimensional and highly nonlinear material features of material databases are preprocessed and reduced by wavelet decomposition technique. The neural networks are further employed to excavate the predictive models from dimension-reduced low-dimensional data. At the same time, advanced intelligent optimization algorithms are utilized to self-search the optimal network structure and learning rate for obtaining the optimal predictive models. Finally, the computational accuracy and efficiency of the presented approach are validated via various numerical experiments on realistic random composites.

A quasi-conforming embedded reproducing kernel particle method for heterogeneous materials

We present a quasi-conforming embedded reproducing kernel particle method (QCE-RKPM) for modeling heterogeneous materials that makes use of techniques not available to mesh-based methods such as the finite element method (FEM) and avoids many of the drawbacks in current embedded and immersed formulations which are based on meshed methods. The different material domains are discretized independently thus avoiding time-consuming, conformal meshing. In this approach, the superposition of foreground (inclusion) and background (matrix) domain integration smoothing cells are corrected by a quasi-conforming quadtree subdivision on the background integration smoothing cells. Due to the non-conforming nature of the background integration smoothing cells near the material interfaces, a variationally consistent (VC) correction for domain integration is introduced to restore integration constraints and thus optimal convergence rates at a minor computational cost. Additional interface integration smoothing cells with area (volume) correction, while non-conforming, can be easily introduced to further enhance the accuracy and stability of the Galerkin solution using VC integration on non-conforming cells. To properly approximate the weak discontinuity across the material interface by a penalty-free Nitsche’s method with enhanced coercivity, the interface nodes on the surface of the foreground discretization are also shared with the background discretization. As such, there are no tunable parameters, such as those involved in the penalty type method, to enforce interface compatibility in this approach. The advantage of this meshfree formulation is that it avoids many of the instabilities in mesh-based immersed and embedded methods. The effectiveness of QCE-RKPM is illustrated with several examples.

Bayesian Nonlocal Operator Regression: A Data-Driven Learning Framework of Nonlocal Models with Uncertainty Quantification

We consider the problem of modeling heterogeneous materials where microscale dynamics and interactions affect global behavior. In the presence of heterogeneities in material microstructure it is often impractical, if not impossible, to provide quantitative characterization of material response. The goal of this work is to develop a Bayesian framework for uncertainty quantification (UQ) in material response prediction when using nonlocal models. Our approach combines the nonlocal operator regression (NOR) technique and Bayesian inference. Specifically, additive independent identically distributed Gaussian noise is employed to model the discrepancy between the nonlocal model and the data. Then, we use a Markov chain Monte Carlo (MCMC) method to sample the posterior probability distribution on parameters involved in the nonlocal constitutive law and associated modeling discrepancies relative to higher-fidelity computations. As an application, we consider the propagation of stress waves through a one-dimensional heterogeneous bar with randomly generated microstructure. Several numerical tests illustrate the construction, enabling UQ in nonlocal model predictions. Although nonlocal models have become popular means for homogenization, their statistical calibration with respect to high-fidelity models has not been presented before. This work is a first step in this direction, focused on Bayesian parameter calibration.

Implementation of a phase field damage model with a nonlinear evolution equation in an FFT-based solver

This paper focuses on the numerical implementation of phase-field models of fracture using the Fast Fourier Transform (FFT) based numerical method. Recent studies on phase-field models focus on the discussions of the choice of the value of regularization length proposed to smear the discontinuity of the sharp crack. Some studies argue that it should be considered as a material property because it has a significant impact on the mechanical behavior of a material in some phase-field models, for instance, in the model proposed by Miehe. However, our results in this study for heterogeneous materials have shown that the choice of regularization length not only affects the macroscopic mechanical behavior but also the local crack propagation patterns. As a result, it can be challenging to select an appropriate value that produces both accurate macroscopic responses and local crack patterns for certain phase-field models, such as Miehe’s model. Thus, the phase-field model proposed by Wu, which has been proven to reduce the length sensitivity for homogeneous material, has been successfully implemented in an FFT-based solver with the application of Newton–Krylov algorithm.The length sensitivity of Wu’s phase-field model for heterogeneous materials is also investigated in this paper. Our tests show that Wu’s phase-field model has partial length sensitivity for heterogeneous materials. However, the main reason for this sensitivity is that one phase enters the damage zone of other phase, which is different from Miehe’s model. As a result, a set of criteria for accurately choosing the regularization value has been established for Wu’s model in this work. Meanwhile, it has also been found that Wu’s model may be more suitable than Miehe’s model for brittle failure due to the introduction of an elastic stage.

PiNNwall: Heterogeneous Electrode Models from Integrating Machine Learning and Atomistic Simulation.

Electrochemical energy storage always involves the capacitive process. The prevailing electrode model used in the molecular simulation of polarizable electrode-electrolyte systems is the Siepmann-Sprik model developed for perfect metal electrodes. This model has been recently extended to study the metallicity in the electrode by including the Thomas-Fermi screening length. Nevertheless, a further extension to heterogeneous electrode models requires introducing chemical specificity, which does not have any analytical recipes. Here, we address this challenge by integrating the atomistic machine learning code (PiNN) for generating the base charge and response kernel and the classical molecular dynamics code (MetalWalls) dedicated to the modeling of electrochemical systems, and this leads to the development of the PiNNwall interface. Apart from the cases of chemically doped graphene and graphene oxide electrodes as shown in this study, the PiNNwall interface also allows us to probe polarized oxide surfaces in which both the proton charge and the electronic charge can coexist. Therefore, this work opens the door for modeling heterogeneous and complex electrode materials often found in energy storage systems.

Uncertainty quantification and propagation in the microstructure-sensitive prediction of the stress-strain response of woven ceramic matrix composites

Hierarchical multiscale modeling of heterogeneous materials has traditionally relied upon a deterministic estimation of constitutive properties when making microstructure-sensitive predictions of effective response at each subsequent length-scale. Such an approach is wholly unsuitable for a variety of material classes, such as ceramic matrix composites, which exhibit large variability at multiple length-scales. This work demonstrates a framework for approaching two open problems towards improved microstructure-sensitive predictions, namely, (i) probabilistically calibrating complex constitutive models at the mesoscale to sparsely observed macroscale experimental data, and (ii) propagating this stochastic constituent behavior at the mesoscale towards low-cost homogenized predictions for unseen microstructures. The proposed stochastic scale-bridging framework displays a continuity of information flow where no portion of the experimental data is neglected out of convenience, facilitating the greatest information gain from oftentimes costly experiments. In this paper, suitable protocols were developed to address the challenges described above. The protocols were subsequently demonstrated on ceramic matrix composite’s uniaxial tensile stress–strain response, where constituent behavior at the mesoscale was described using continuum damage mechanics, and predictions encapsulating constitutive model parameter uncertainty were made for novel microstructures. The methodology presented in this work is broadly applicable to various material classes and constitutive models with high-dimensional parameter sets.

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.

Multiscale modelling of strongly heterogeneous materials using geometry informed clustering

Computationally efficient and numerically accurate modelling of heterogeneous materials with complex internal architectures at the macroscale is a current problem. For instance, in engineering materials such as 3D woven composites, retaining the description of material architectures is important to obtain an accurate prediction of stiffness. However, computational cost increases in proportion to the level of mesoscale details modelled. To enable efficient and accurate calculations on the structural scale, a multiscale method that can identify repeatable patterns in the mesoscale and represent them efficiently at the macroscale is proposed. This method has two important novelties, (i) a method to identify and store repeatable patterns during offline stage in the form of 3D Voronoi cells using a data compression algorithm, k-means clustering. This improves the identification with a minimum number of data clusters and minimises the effects of mesh sizes, and (ii) a method to select most similar Voronoi cells from the mesoscale database during online stage using image registration and k-d tree data structure. This enables computations being performed without explicitly modelling mesoscale details and reduces the computational cost that are otherwise required. The proposed method is validated against 3D woven unit-cell and three-point bending examples. Furthermore, the ability to find repeatable patterns is tested by analysing a woven architecture previously not stored in the database.

Efficient multiscale modeling of heterogeneous materials using deep neural networks

Material modeling using modern numerical methods accelerates the design process and reduces the costs of developing new products. However, for multiscale modeling of heterogeneous materials, the well-established homogenization techniques remain computationally expensive for high accuracy levels. In this contribution, a machine learning approach, convolutional neural networks (CNNs), is proposed as a computationally efficient solution method that is capable of providing a high level of accuracy. In this work, the data-set used for the training process, as well as the numerical tests, consists of artificial/real microstructural images (“input”). Whereas, the output is the homogenized stress of a given representative volume element mathcal {RVE}. The model performance is demonstrated by means of examples and compared with traditional homogenization methods. As the examples illustrate, high accuracy in predicting the homogenized stresses, along with a significant reduction in the computation time, were achieved using the developed CNN model.

Introduction to the Special Issue on Modeling of Heterogeneous Materials

Cite This Article Liu, L., Chu, X., Yang, X., Chen, J., Huang, Q. (2023). Introduction to the Special Issue on Modeling of Heterogeneous Materials. CMES-Computer Modeling in Engineering & Sciences, 134(1), 1–3.

Contact modeling of heterogeneous materials of the machine tool bed–foundation interface based on the gradient of contact stress distribution

Contact modeling of heterogeneous materials of the machine tool bed–foundation interface based on the gradient of contact stress distribution

Plasto-elastohydrodynamic lubrication of heterogeneous materials in impact motion

The analysis of plastic evolution and material inhomogeneity is required for an accurate description of lubrication under impact loading. This work develops a semi-analytical model for heterogeneous materials in impact motion under plasto-elastohydrodynamic lubrication (PEHL) with limited inlet oil. A plasticity loop is proposed to obtain the accumulative plastic strain iteratively based on the analysis of the stress state. The inhomogeneous inclusion within the materials is homogenized with unknown eigenstrains according to the equivalent inclusion method. The surface displacements induced by the plastic strains and eigenstrains are introduced into the gap between the contact bodies to update the lubrication film thickness until the convergence is achieved. The consideration of plastic strains and eigenstrains makes the model more realistic for PEHL under impact loading. The responses including the pressure, film thickness, starvation parameter, residual deformation, temperature, and subsurface elastoplastic fields during the impact–rebound process are discussed. After the model validation against the reference data, the effects of material inhomogeneity and strain hardening on the lubrication response, as well as the typical features of elastic perfectly-plastic materials, are investigated to provide guidance for material reliability analysis.

An analysis of embedded weak discontinuity approaches for the finite element modelling of heterogeneous materials

This paper analyses in detail the use of the Embedded Finite Element Method (E-FEM) to simulate local material heterogeneities. The work starts by a short review on the evolution of weak discontinuity models within the E-FEM framework to discuss how they account for the presence of multiple materials within a single element structure. A theoretical basis is introduced through some mathematical weak discontinuity definitions and the Hu-Washizu variational principle, for then establishing a set of requirements for retaining variational and kinematic consistency for any weak discontinuity enhancement proposal. From a general definition of a displacement enhancement field, two particular enhancement functions are derived by considering different consistency requirements: one which has been typically used in previous works and other which truly possesses variational consistency. A discussion is held on enhancement stability properties and the impact to global finite element solution processes. In the end, numerical simulations are made to assess the performance of each of these enhancements on the task of modelling a classical bi-material layered 3D tension problem and a more realistic heterogeneous sample having spherical inclusions of different radii. The final discussion evaluates both model performance and ease of implementation.

Preface to the Special Issue on Multiscale Modelling of Heterogeneous Materials

Preface to the Special Issue on Multiscale Modelling of Heterogeneous Materials