Computational Homogenization Approach Research Articles

A reduced-order computational homogenization framework for locally resonant metamaterial structures

AbstractA 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.

DATA-PHYSICS DRIVEN REDUCED ORDER HOMOGENIZATION FOR CONTINUUM DAMAGE MECHANICS AT MULTIPLE SCALES

The paper introduces a novel approach, referred to as data-physics driven reduced order homogenization, for continuum damage mechanics. The proposed method combines the benefits of the physics-based reduced order homogenization and data-driven surrogate modeling by striking a balance between accuracy, computational efficiency, and physical interpretability. The primary objective of this hybrid approach is to minimize computational cost associated with online predictions at the macroscopic scale while preserving accuracy and physical interpretability. This is achieved by leveraging a surrogate-based Bayesian inference to extract crucial information at a representative volume element (RVE) level. With the inferred data, online predictions are performed using a data-enhanced reduced order homogenization. Consequently, the computational time required at a macroscopic scale is significantly reduced compared to both the conventional direct numerical simulation and the computational homogenization approach.

FAST FOURIER TRANSFORM METHOD FOR PERIDYNAMIC BAR OF PERIODIC STRUCTURE

The basic feature of the peridynamics [introduced by Silling (2000)] considered is a continuum description of material behavior as the integrated nonlocal force interactions between infinitesimal material points. A heterogeneous bar of the periodic structure of constituents with peridynamic mechanical properties is analyzed. One introduces the volumetric periodic boundary conditions (PBCs) at the interaction boundary of a representative unit cell (UC), whose local limit implies the known locally elastic PBCs. This permits us to generalize the classical computational homogenization approach to its counterpart in peridynamic micromechanics (PM). Alternative to the finite element methods (FEM) for solving computational homogenization problems are the fast Fourier transforms (FFTs) methods developed in local micromechanics (LM). The Lippmann-Schwinger (L-S) equation-based approach of the FFT method in the LM is generalized to the PM counterpart. Instead of one convolution kernel in the L-S equation, we use three convolution kernels corresponding to the properties of the matrix, inclusions, and interaction interface. The Eshelby tensor in LM depending on the inclusion shape is replaced by PM counterparts depending on the inclusion size and interaction interface (although the Eshelby concept of homogeneous eigenfields does no work in PM). The mentioned tensors are estimated one time (as in LM) in a frequency domain (also by the FFT method). Numerical examples for 1-D peridynamic inhomogeneous bar are considered. Computational complexities O (N <i>log</i><sub>2</sub> N) of the FFT methods are the same in both LM and PM.

A 3D RVE-based computational homogenization approach for predicting the effective fourth-order elasticity tensor of periodic porous materials

A 3D RVE-based computational homogenization approach for predicting the effective fourth-order elasticity tensor of periodic porous materials

Overall elastic characterization of equivalent FE models for aluminum foams through computational homogenization approach and genetic algorithm optimization

Aluminum foams are gaining increasing interest because of their widespread set of functional attributes combined with highly specific structural properties and affordable manufacturing processes. The finite element analysis of foam materials is typically performed with complex and time-consuming 3D models. Here, a beam FE model strategy is utilized to obtain a precise and time-saving simulation instrument for open-cell aluminum foams. The beam FE modeling is based on the Kelvin cell as a fundamental unit and provides a simplified geometrical description of the intersection node between ligaments whose elastic properties need to be adjusted through a calibration procedure. The beam FE model elastic properties are measured through a homogenization procedure that evaluates the elastic constants of an equivalent material. The issue of model calibration is formulated as an optimization problem exploiting the NSGA-II genetic algorithm. This procedure allows obtaining the full elastic calibration of the beam FE model, reproducing the orthotropic elastic behavior of the aluminum foams with reduced computational efforts.

A multiscale approach in modeling of chemically reactive porous media

Chemical reactions occur in geotechnical, biological and synthetic porous media. Swelling and shrinkage phenomena can be observed in clays, shales and gels, when they are in close contact with water. In petroleum engineering, wellbore-stability problems are caused by swelling or shrinking of the wellbore. Thus, it is significant to model the Chemo-Hydro-Mechanical (CHM) processes in porous media that include different physical and geometrical heterogeneities. In this paper, a multiscale computational homogenization approach is developed for the analysis of CHM problems. In this manner, the first-order homogenization technique is adopted for the two-scale formulation. The heterogeneities are assumed in the microscale level and a homogeneous domain is considered for the macroscale level, where the constitutive behavior is derived from the microscopic level. The governing equations and constitutive equations of the CHM processes are presented in the coupled manner. The primary variables are taken as the displacement, pore pressure, and solute concentration fields. Moreover, the transient terms are included in the microscale level to increase the accuracy of computations. Consequently, a generalized form of Hill-Mandel principle of macro-homogeneity is defined between the two scales. Appropriate microscopic boundary conditions, i.e. the linear and periodic boundary conditions, are employed to satisfy the averaging constraints. Finally, the accuracy and efficiency of the proposed computational multiscale method are illustrated through several numerical examples.

A representative volume element model to evaluate the effective properties of flexoelectric nanocomposite

In this paper, a computational homogenization approach is developed to investigate the effective material properties of flexoelectric nanocomposites. The analysis incorporates the isogeometric analysis (IGA) to solve an electromechanically coupled system using a 3D representative volume element (RVE) model with periodic boundary conditions. In particular, a micromechanical analysis is carried out to represent the microstructure of a heterogeneous domain composed of an elastic matrix and cylindrical flexoelectric fibers. The numerical simulations indicated that while the effective tensor of piezoelectricity is isotropic, the flexoelectricity tensor is transversely anisotropic concerning the geometry of the composite’s phases. Furthermore, the flexoelectricity coefficients, μ¯3113 and μ¯3333, which couple the polarization to the strain gradients with respect to the longitudinal axis, improved remarkably with increasing the inclusion’s volume fraction. The presented computational methodology and the findings of this study is expected to contribute valuably towards the design of flexoelectric nanostructures and devices.

Absolute nodal coordinate formulation – Multilevel finite element framework for the nonlinear multi-scale multibody dynamic analysis of composite structures

In this paper, a multi-scale modeling approach for the dynamic response of flexible multibody system made of composite material is suggested using the absolute nodal coordinate formulation (ANCF) – multilevel finite element (FE2) method. On many occasions, the analysis of composite structures requires a multi-scale modeling approach. FE2 method is one of the famous multi-scale modeling methods. And it is a versatile computational homogenization approach that is commonly applicable to many different materials. Because the present FE2 is capable of computing the mechanical response at two different scales simultaneously, ANCF element will be used at the macroscopic scale. And the microscopic RVE at each macroscopic integration point will be obtained by the finite element method. Since ANCF element uses only the absolute nodes and gradients, the mass matrix will be constant. Due to such characteristics, when combined with FE2, the linearization process will become easier compared against the existing formulation. And Coriolis or centrifugal force will not need to be considered in the dynamic formulation. Therefore, computational cost will be reduced, and the relevant algebraic manipulation will become simplified. Several numerical examples are presented to demonstrate the improved accuracy and diminished computational cost of the present suggestion.

Continuous self‐adversarial training of recurrent neural network–based constitutive description

AbstractData‐driven methods yield advantages in computational homogenization approaches due to the ability to capture complex material behaviour without the necessity to assume specific constitutive models. Neural network–based constitutive descriptions are one of the most widely used data‐driven approaches in the context of computational mechanics. The accuracy of this method strongly depends on the available data. Additionally, when considering inelastic materials, whose constitutive responses depend on the loading history, the accuracy and robustness of the approximation are influenced by the training algorithm. The applied recurrent neural networks exhibit reduced robustness in the presence of errors in the input. When capturing the history dependency using previously predicted material responses, occurrences of prediction errors accumulate over several time steps. An approach for achieving enhanced robustness of the predictions is based on extending the initial training dataset by iteratively generating adversarial examples, subjected to perturbations, based on the current prediction errors. In this contribution, a continuous self‐adversarial training approach yielding robust recurrent neural network constitutive descriptions for inelastic materials is presented. Compared to the iterative method it is based on, it exhibits significantly improved training efficiency. In order to demonstrate the capabilities of the proposed methods, numerical examples with datasets obtained by numerical material tests on representative volume elements are carried out. Validation of the results is performed using both test load cases from the numerical dataset, as well as application as a constitutive model in the finite element method.

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.

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.

Limit state analysis of asymmetrical microstructures based on yield design computational homogenization approach

This paper presents a novel formulation for the computational homogenization analysis of materials at the limit state. The polynomial interpolations are employed to impose the periodic boundary conditions for the fluctuating term of the displacement field when using arbitrary finite element meshes. Second-order cone programming provides an efficient solution to solve the resulting optimization problems, and accurate load multipliers can be obtained with the minimum computational cost. Several asymmetrical material models are investigated to perform the efficiency of the proposed method. The collapse mechanisms of the representative volume elements are also presented.

A Review of the Accuracy of Primal Assembly Model Order Reduction Techniques

Model order reduction (MOR) techniques have been used in structural dynamics for over sixty years as a quick and effective way to examine the dynamic behavior of complex finite element (FE) models. Various macroscopic phenomena arise from the physics and mechanics of the underlying microstructure, and MOR approaches can be employed in conjunction with the multiscale computational homogenization approach, which is deriving local macroscopic constitutive responses from underlying microstructure; to predict the dynamic behavior of heterogeneous macrostructures. The accuracy and efficiency of the most representative approaches from dynamic condensation, component mode synthesis (CMS), and general model reduction (GMR) methods are compared in this paper. The techniques considered are Guyan and the improved reduced system (IRS) from classical dynamic condensation, Craig-Bampton (CB) and enhanced Craig-Bampton (ECB) from CMS, IRS-based substructuring, GMR, and GMRPlus. All the substructuring methods employ primal assembly at the boundary interface of the substructures. The accuracy and computing efficiency of the techniques are compared in-depth using real-world engineering problems, and the effect of the number of substructures on the accuracy and computational cost is also examined.

Complete geometric representation of yield surfaces for porous ductile media by a 3D computational homogenization approach: an assessment of the Gurson yield criterion

Complete geometric representation of yield surfaces for porous ductile media by a 3D computational homogenization approach: an assessment of the Gurson yield criterion

Strength and fatigue analysis of periodic perforated materials using the computational homogenization and ES-FEM approaches

This paper presents a computational homogenization shakedown analysis of periodic perforated materials with von Mises matrices. The plastic behaviors of the perforated materials under cyclic macroscopic loads are studied by means of kinematic shakedown theorem and computational homogenization method. The kinematic micro-fields are approximated by the edge-based smoothed finite element method. The resulting large-scale optimization problem is efficiently solved by using conic solver, enabling a large number of points on the macroscopic strength and fatigue surface to be calculated rapidly. The effects of the hole's shape and size on the overall strength and fatigue domains are also investigated.

A computational homogenization framework with enhanced localization criterion for macroscopic cohesive failure in heterogeneous materials

Computational homogenization allows to let the macroscopic constitutive behavior of materials emerge from microscale simulations without loss of generality with respect to microstructure and microscale constitutive response. Although computationally demanding, computational homogenization works very well for the hardening response of materials where the macroscopic stress and strain fields are smooth. However, in case of softening materials, when localization of deformation takes place, special care is needed to ensure objectivity of the method. In this paper, a generic multiscale computational homogenization approach for modeling onset and propagation of cracks in heterogeneous materials that is capable of considering various microscale mechanisms is presented. The common acoustic tensor bifurcation criterion is reinforced by an additional condition to help detect the localization mode more robustly. After the onset of macroscale localization, a key scale transition parameter is needed to translate the macroscopic displacement jump to an averaged strain over the micromodel domain. Then the macroscale crack is governed by a homogenized traction-separation relation evaluated from the underlying micromodel in which micro-failure accumulates. The scale transition parameter is studied for a range of different scenarios and endowed with a geometrical interpretation. Various numerical tests have been performed to confirm the objectivity and validity of the framework. The framework is generic in the sense that no assumptions on the microscale constitutive or kinematic representation of material failure are made in the scale transition. The framework is also highly compatible with the first order computational homogenization, which minimizes the additional complexity of adding macroscopic crack growth to the computational implementation.

An FE-based orientation averaging model for elasto-plastic behavior of short fiber composites

A new micro-mechanical model for predicting the non-linear elasto-plastic behavior of short fiber reinforced composites is presented. The model is developed based on a two-step Orientation Averaging method, and is capable of accommodating a wide variety of micro-structural parameters. In the first step, Finite Element Analyses are performed on a unit cell (single fiber surrounded by matrix). Then, the unit cell response is up-scaled by calibrating its response to an elasto-plastic surrogate constitutive model. In a subsequent second homogenization step, a self-consistent interaction scheme is proposed. The predictive capability of the resulting two-step homogenization scheme is then evaluated, next to versions that adopt more traditional averaging schemes (Voigt and Reuss, providing upper and lower bounds, respectively), through comparisons to experiments and direct numerical simulations of realistic Representative Volume Elements. Results show that the model gives fairly good predictions. It is emphasized that the model is capable of accommodating any desired fiber orientation distribution, very large ranges of fiber aspect ratios and fiber volume fractions. Also, the model is computationally very efficient compared to the RVE computational homogenization approach, and thus, it could be conveniently used in applications possessing different fiber orientations at different points of a component.

Computational modeling and data‐driven homogenization of knitted membranes

AbstractKnitting is an effective technique for producing complex three‐dimensional surfaces owing to the inherent flexibility of interlooped yarns and recent advances in manufacturing, providing better control of local stitch patterns. Fully yarn‐level modeling of large‐scale knitted membranes is not feasible. Therefore, we use a two‐scale homogenization approach and model the membrane as a Kirchhoff‐Love shell on the macroscale and as Euler‐Bernoulli rods on the microscale. The governing equations for both the shell and the rod are discretized with cubic B‐spline basis functions. For homogenization, we consider only the in‐plane response of the membrane. The solution of the nonlinear microscale problem requires a significant amount of time due to the large deformations and the enforcement of contact constraints, rendering conventional online computational homogenization approaches infeasible. To sidestep this problem, we use a pretrained statistical Gaussian process regression (GPR) model to map the macroscale deformations to macroscale stresses. During the offline learning phase, the GPR model is trained by solving the microscale problem for a sufficiently rich set of deformation states obtained by either uniform or Sobol sampling. The trained GPR model encodes the nonlinearities and anisotropies present in the microscale and serves as a material model for the membrane response of the macroscale shell. The bending response can be chosen in dependence of the mesh size to penalize the fine out‐of‐plane wrinkling of the membrane. After verifying and validating the different components of the proposed approach, we introduce several examples involving membranes subjected to tension and shear to demonstrate its versatility and good performance.

Variationally consistent computational homogenization of chemomechanical problems with stabilized weakly periodic boundary conditions

AbstractA variationally consistent model‐based computational homogenization approach for transient chemomechanically coupled problems is developed based on the classical assumption of first‐order prolongation of the displacement, chemical potential, and (ion) concentration fields within a representative volume element (RVE). The presence of the chemical potential and the concentration as primary global fields represents a mixed formulation, which has definite advantages. Nonstandard diffusion, governed by a Cahn–Hilliard type of gradient model, is considered under the restriction of miscibility. Weakly periodic boundary conditions on the pertinent fields provide the general variational setting for the uniquely solvable RVE‐problem(s). These boundary conditions are introduced with a novel approach in order to control the stability of the boundary discretization, thereby circumventing the need to satisfy the LBB‐condition: the penalty stabilized Lagrange multiplier formulation, which enforces stability at the cost of an additional Lagrange multiplier for each weakly periodic field (three fields for the current problem). In particular, a neat result is that the classical Neumann boundary condition is obtained when the penalty becomes very large. In the numerical examples, we investigate the following characteristics: the mesh convergence for different boundary approximations, the sensitivity for the choice of penalty parameter, and the influence of RVE‐size on the macroscopic response.

Exploring the Role of Manufacturing Parameters on Microstructure and Mechanical Properties in Fused Deposition Modeling (FDM) Using PETG

Additive manufacturing develops rapidly, especially, fused deposition modeling (FDM) is one of the economical methods with moderate tolerances and high design flexibility. Ample studies are being undertaken for modeling the mechanical characteristics of FDM by using the Finite Element Method (FEM). Even in use of amorphous materials, FDM creates anisotropic structures effected by the chosen manufacturing parameters. In order to investigate these process-related characteristics and tailored properties of FDM structures, we prepare FDM-printed poly(ethylene terephthalate) glycol (PETG) samples with different process parameters. Mechanical and optical characterizations are carried out. We develop 2D-digital-image-correlation code with machine learning algorithm, namely K-means cluster, to analyze microstructures (contact surfaces, the changes in fiber shapes) and calculate porosity. By incorporating these characteristics, we draw CAD images. A digital twin of mechanical laboratory tests are realized by the FEM. We use computational homogenization approach for obtaining the effective properties of the FDM-related anisotropic structure. These simulations are validated by experimental characterizations. In this regard, a systematic methodology is presented for acquiring the anisotropy from the process related inner substructure (microscale) to the material response at the homogenized length scale (macroscale). We found out that the layer thickness and overlap ratio parameters significantly alter the microstructures and thereby, stiffness of the macroscale properties.Graphical