Computational Homogenization Method Research Articles

Computational homogenization of tensile deformation behaviors of a third generation Al-Li alloy 2060-T8 using crystal plasticity finite element method

A computational homogenization procedure based on the crystal plasticity model was proposed herein to predict the tensile deformation behavior and investigate the anisotropic response of a novel third generation Al-Li alloy (AA2060-T8) at room temperature and different deformation conditions. To elucidate the in-grain deformation features, a representative volume element was constructed to reveal the microstructure of the real AA2060-T8 (polycrystalline material) in which each grain was discretized by many finite elements. Afterward, numerical results were assigned to each grain to explore the pre-texture formed by previous thermomechanical processes. A dislocation density-based crystal plasticity model was developed to infer the constitutive equation of each grain and simulate the plastic deformation of AA2060-T8 alloy. The material parameters used in the dislocation density-based crystal plasticity model were calibrated against a tensile stress-strain curve deformed at 30° with respect to rolling direction. The results obtained from the proposed computational homogenization method are in line with those obtained from experimentation. This indicates that the proposed computational homogenization method can definitely predict the tensile deformation behavior and capture the anisotropic responses of AA2060-T8 (polycrystalline materials) originating from deformation induced texture as well as the initially anisotropic texture.

Identification of viscoelastic properties from numerical model reduction of pressure diffusion in fluid-saturated porous rock with fractures

This paper deals with the computational homogenization and numerical model reduction of deformation driven pressure diffusion in fractured porous rock. Exposed to seismic waves, the heterogeneity of the material leads to local fluid pressure gradients which are equilibrated via pressure diffusion. However, a macroscopic observer is not able to measure the diffusion process directly but senses the intrinsic attenuation of an apparently monophasic viscoelastic solid. The aim of this paper is to establish a reliable, yet numerically efficient, computational homogenization method to identify the viscoelastic properties of the macroscopic substitute model. Inspired by the Nonuniform Transformation Field Analysis, we incorporate a Numerical Model Reduction procedure. The proposed method is validated for several scenarios ranging from pressure diffusion in an unfractured poroelastic matrix, via localized pressure diffusion in interconnected fractures embedded in an impermeable matrix, to the fully coupled pressure diffusion both in fractures and the embedding poroelastic matrix.

Effective elastic properties of periodic irregular open-cell foams

This paper presents a micromechanical modeling for predicting the effective linear elastic properties of irregular open-cell foams using the periodic computational homogenization method. The tomography images are analyzed to obtain the morphological description of the irregular open-cell structure. An approach based on Voronoi diagram is used to generate realistic periodic foam structures with the morphological parameters. Hill’s lemma computational homogenization approach is performed to predict the effective elastic properties. In this paper, we focus especially on the determination of the Representative Volume Element (RVE) and the importance of choosing the RVE parameters, i.e. the number of realizations and the volume of RVE, on the accuracy of models. The efficiencies of different approaches are discussed. Some recommendations and improvements are proposed.

A hierarchical multiscale finite element approach for modelling the heat affected zone of welded joints

AbstractIn the present work a hierarchical multiscale approach, based on finite element method, is applied in order to characterize the heat affected zone (HAZ) of a welded joint with parent material S460M. The metallurgical constituents of particular regions of the HAZ was determined experimentally and, regarding the multiscale approach, a first‐order computational homogenization method was implemented. In this context, effective stress‐strain‐relations in particular spots of the HAZ are determined applying prescribed displacement boundary conditions. The obtained relations were considered in subsequent finite element calculations of butt‐welded joints under tensile loading, allowing a comparison of the numerical strain states to experimental results. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Cascade Continuum Micromechanics model for the effective permeability of solids with distributed microcracks: Comparison with numerical homogenization

The effective permeability of microcracked heterogeneous materials such as rocks, ceramics and concrete can be determined using analytical and computational homogenization methods. While in the companion paper (Timothy and Meschke, 2017) a semi-analytical Cascade Continuum Micromechanics (CCM) model is proposed to predict the effective permeability and the percolation threshold for porous materials with microcracks, the focus of this paper is to validate the CCM model by means of direct numerical meso-scale simulations of representative elementary volumes concerned with water flow through porous materials. Algorithms are developed to generate overlapping and non-overlapping microcrack morphologies within the framework of the finite element method to analyse the effective permeability as a function of the microcrack volume fraction. The numerical results confirm the predictions from the CCM model for different scenarios, including a low and high permeable matrix and different microcrack morphologies, both qualitatively as well as quantitatively. It is also shown, that in computational homogenization procedures, the effective permeability around the percolation threshold is strongly dependent on the discretization of the numerical REV.

Computational homogenization for thermal conduction in heterogeneous concrete after mechanical stress

Concrete structures are subjected to various loadings during their service life and their internal structures will be changed, resulting variations in mechanical and thermo-physical properties. To investigate the effect of mechanical stress on thermal conduction, a computational homogenization method from a mesoscopic perspective was proposed in the present work. In the simulations, concrete was considered as a three-phase composite material consisting of aggregate, mortar matrix and the interfacial transition zones (ITZs) between them. The mechanical analysis was conducted firstly to study the damage distribution within concrete. The outcomes of mechanical computation were then used as the initial input data in the thermal conduction computation. The equivalent thermal conductivity of damaged element was homogenized by a composite mechanical method based on damage and the initial thermal conductivity of sound material. Accordingly, a meso-scale model in which the mechanical and thermal behavior were one-way coupled was built. The method was calibrated by comparing the numerical results with the available experimentally measured ones. Based on the verified simulation method, effective thermal conductivity (ETC) and temperature field of concrete subjected to different loading levels were calculated. Besides, the effects of loading type (compressive and tensile loadings) on ETC and temperature field of concrete were studied. It is found that ETC of concrete decreases with an increasing loading level. In addition, the effect of tensile loading on thermal behavior depends on whether the direction of thermal conduction is parallel or perpendicular to loadings.

An FFT-based fast gradient method for elastic and inelastic unit cell homogenization problems

Building upon the previously established equivalence of the basic scheme of Moulinec–Suquet’s FFT-based computational homogenization method with a gradient descent method, this work concerns the impact of the fast gradient method of Nesterov in the context of computational homogenization. Nesterov’s method leads to a significant speed up compared to the basic scheme for linear problems with moderate contrast, and compares favorably to the (Newton-)conjugate gradient (CG) method for problems in digital rock physics and (small strain) elastoplasticity. We present an efficient implementation requiring twice the storage of the basic scheme, but only half of the storage of the CG method.

Unified treatment of microscopic boundary conditions and efficient algorithms for estimating tangent operators of the homogenized behavior in the computational homogenization method

This work provides a unified treatment of arbitrary kinds of microscopic boundary conditions usually considered in the multi-scale computational homogenization method for nonlinear multi-physics problems. An efficient procedure is developed to enforce the multi-point linear constraints arising from the microscopic boundary condition either by the direct constraint elimination or by the Lagrange multiplier elimination methods. The macroscopic tangent operators are computed in an efficient way from a multiple right hand sides linear system whose left hand side matrix is the stiffness matrix of the microscopic linearized system at the converged solution. The number of vectors at the right hand side is equal to the number of the macroscopic kinematic variables used to formulate the microscopic boundary condition. As the resolution of the microscopic linearized system often follows a direct factorization procedure, the computation of the macroscopic tangent operators is then performed using this factorized matrix at a reduced computational time.

Multiscale modelling of hydro-mechanical couplings in quasi-brittle materials

A multiscale computational homogenization method for the modeling of hydro-mechanical coupling problem for quasi-brittle materials is developed. The present method is based on an asymptotic expansion homogenization combined with the semi-concurrent finite element modelling approach. Modified periodic boundary conditions and a molecular dynamics (MD) based inclusion or filler generation procedure are devised for the hydro-mechanical coupling problem. A modified elastic damage constitutive model and a damage induced permeability law have been developed for the hydraulic fracturing. The statistical convergence of the microscale representative volume element (RVE) model regarding the RVE characteristic size is studied. It was found that the RVE characteristic size is determined by both the mechanical and hydraulic properties of the RVE simultaneously. The present method is validated by the experimental results for brittle material. The damage zone and crack propagation path captured by the present method is compared with the experimental results (Chitrala et al. in J Pet Sci Eng 108:151–161, 2013). The results show that the present method is an effective for the modelling of hydro-mechanical coupling for brittle materials.

Microstructure characterization and homogenization of acoustic polyurethane foams: Measurements and simulations

The sound absorption ability of porous materials is strongly related to the underlying microstructure. In this paper, acoustic properties of a polyurethane (PU) foam are determined from its microstructure with a computational homogenization method. The foam is analyzed using X-ray computed tomography (CT) and scanning electron microscopy (SEM). Based on the obtained microstructure information, a parallel model of a fully-open and a partially-open Kelvin cell with thin membranes is built to represent the foam. The corresponding effective material parameters, including the dynamic density and the stiffness tensor, are obtained by applying a computational homogenization approach. Numerical simulations of an impedance tube test based on Biot’s equations with parameters obtained from the homogenization are compared with the measured sound absorption coefficients. Considering the limitations of the simplified microscopic model, a good agreement between the measurements and the simulation results for the PU foam is found.

Perturbation-based stochastic multi-scale computational homogenization method for the determination of the effective properties of composite materials with random properties

Quantifying uncertainty in the overall elastic properties of composite materials arising from randomness in the material properties and geometry of composites at microscopic level is crucial in the stochastic analysis of composites. In this paper, a stochastic multi-scale finite element method, which couples the multi-scale computational homogenization method with the second-order perturbation technique, is proposed to calculate the statistics of the overall elasticity properties of composite materials in terms of the mean value and standard deviation. The uncertainties associated with the material properties of the constituents are considered. Performance of the proposed method is evaluated by comparing mean values and coefficients of variation for components of the effective elastic tensor against corresponding values calculated using Monte Carlo simulation for three numerical examples. Results demonstrate that the proposed method has sufficient accuracy to capture the variability in effective elastic properties of the composite induced by randomness in the constituent material properties.

Perturbation-based stochastic multi-scale computational homogenization method for woven textile composites

In this paper, a stochastic homogenization method that couples the state-of-the-art computational multi-scale homogenization method with the stochastic finite element method, is proposed to predict the statistics of the effective elastic properties of textile composite materials. Uncertainties associated with the elastic properties of the constituents are considered. Accurately modeling the fabric reinforcement plays an important role in the prediction of the effective elastic properties of textile composites due to their complex structure. The p-version finite element method is adopted to refine the analysis. Performance of the proposed method is assessed by comparing the mean values and coefficients of variation for components of the effective elastic tensor obtained from the present method against corresponding results calculated by using Monte Carlo simulation method for a plain-weave textile composite. Results show that the proposed method has sufficient accuracy to capture the variability in effective elastic properties of the composite induced by the variation of the material properties of the constituents.

Topology optimization of multiscale elastoviscoplastic structures

SummaryThis paper extends current concepts of topology optimization to the design of structures made of nonlinear microheterogeneous materials. The objective is to maximize the macroscopic structural stiffness for a prescribed material volume usage while accounting for the nonlinearity and the microstructure of the material. The resulting design problem considers two scales: the macroscopic scale at which the optimization is performed and the microscopic scale at which the material heterogeneities and the nonlinearities are observed. The topology optimization at the macroscopic scale is performed by means of the bi‐directional evolutionary structural optimization method. The solution of the macroscopic boundary value problem requires as inputs the effective constitutive response with full consideration of the microstructure. While computational homogenization methods such as the FE2 method could be used to solve the nonlinear multiscale problem, the associated numerical expense (CPU time and memory) is highly unacceptable. In order to regain the computational feasibility of the computational scale transition, a recent model reduction technique of the authors is employed: the potential‐based reduced basis model order reduction with graphics processing unit acceleration. Numerical examples show the efficiency of the resulting nonlinear two‐scale designs. The impact of different load amplitudes on the design is examined. Copyright © 2015 John Wiley & Sons, Ltd.

Stochastic analysis of the interphase effects on the mechanical properties of clay/epoxy nanocomposites

The polymeric clay nanocomposites are a new class of materials which recently have become the center of attention because of their superior mechanical and physical properties. Several studies have been performed on the mechanical characterization of these nanocomposites; however most of those studies have neglected the effect of the interphase and interfacial region between the clays and the matrix despite of its significance on the mechanical performance of the nanocomposites. Based on a hierarchical multiscale approach on the mesoscopic level, in this study we use the stochastic analysis and the computational homogenization method to analyse the effect of thickness and stiffness of the interphase region on the overall elastic properties of the clay/epoxy nanocomposites. The results show that considering interphase layer reduces the stiffness of clay/epoxy nanocomposites and this decrease becomes significant in higher clay contents. Then, the significance of the stiffness and thickness of the interphase layer on the elastic modulus of clay/epoxy nanocomposites are studied using a sensitivity analysis. Finally, the results of this study are validated with available experimental results.

Computational homogenization of nonlinear elastic materials using neural networks

SummaryIn this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the effective potential is represented as a response surface parameterized by the macroscopic strains and some microstructural parameters. The discrete values of the effective potential are computed by finite element method through random sampling in the parameter space, and neural networks are used to approximate the surface response and to derive the macroscopic stress and tangent tensor components. We show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10, allowing to consider three‐dimensional representative volume elements and an explicit dependence of the effective behavior on microstructural parameters like volume fraction. We present several applications of this technique to the homogenization of nonlinear elastic composites, involving a two‐scale example of heterogeneous structure with graded nonlinear properties. Copyright © 2015 John Wiley & Sons, Ltd.

Effect of cell geometry and material properties on wood rigidity

Rigidity of complex cellular materials, such as wood, depends on multiple geometrical and mechanical parameters of the cell and cell structure. Computational homogenization method gives the effective elastic properties of an infinitely large collection of cells out of a calculation on the representative cell. If the geometric and material properties of the cells are considered constants so that the cell structure can be though to be built from translating and copying the representative cell. Cell shape, cell wall thickness, and anisotropic elastic properties of the cell wall are accounted for in the application, whereas the cell structure is considered regular. The model gives the effective rigidity of a sample of earlywood, transition-/latewood. The effect of the cell shape and cell wall properties on the effective rigidity is illustrated.

Computational homogenization of rope-like technical textiles

This contribution investigates a computational homogenization method for one-dimensional technical textiles i.e. ropes or cables. On the macroscopic level rope-like textiles are characterized by a large length-to-thickness ratio, such that a discretization with structural elements, i.e. beam elements, is numerically efficient. The material behavior is, however, strongly influenced by the heterogeneous micro structure. Here, the fibers at the micro structure are modeled explicitly via a representative volume element whereby the contact interactions between fibers are captured. To transfer the microscopic response to the macro level a beam specific homogenization scheme is introduced. Theoretical aspects are discussed, e.g. the advocated beam specific power averaging theorem and the corresponding scale transition, and numerical examples for periodic rope-like structures are given.

Boundary value driven approach to computational homogenization

AbstractA novel approach to computational homogenization methods for the analysis of heterogeneous structures with large length scale differences between their micro‐ and macroscale is presented. The boundary value driven approach to homogenization (BVDH) is efficient and reliable compared to existing multi‐scale frameworks and the limit cases of homogenization, respectively. Furthermore, the proposed approach is applicable to multi‐physical problems such as thermomechanical analysis. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A hierarchical multi-scale approach to mechanical characterization of heat affected zone in welded connections

During the welding process, the microstructure of the base material changes locally that leads to altering mechanical properties in the weld and the neighboring areas, Heat Affected Zone (HAZ). The constitution of the HAZs also varies spatially as a result of different temperature gradients during heating and cooling cycles. Consequently, the macroscopic material behavior of the HAZ also varies spatially as well and therefore the determination of the mechanical behavior of the welded connection is very challenging. In this study, a hierarchical multiscale approach is presented and applied in order to characterize the material behavior of the Heat Affected Zone (HAZ) in welded connections. First, the metallurgical constituents in particular areas of HAZ have been identified experimentally. Next, several representative volume elements (RVEs) have been constructed using microscopic images.Then employing the computational homogenization methods, the stress–strain curves representing the macroscopic material behavior in respective points of the HAZ have been calculated. Meanwhile, some miniature tensile tests have been performed to experimentally identify the stress–strain behavior in the HAZ. Finally, the numerically calculated stress–strain curves are compared with the measured experimental data and they show a good agreement. The comparison of the results depicts that the proposed multiscale approach is suitable for the characterization of the material properties in welded connections or in general for materials with specially varying microstructures.

A Multi-Scale Scheme for Modelling Fracture under Dynamic Loading Conditions

Fracture in heterogeneous materials under dynamic loading is modelled using a multi-scale method. Computational homogenization is considered, in which the overall properties at the global-scale are obtained by solving a boundary value problem for a representative volume element (RVE) assigned to each material point of the global-scale model. In order to overcome the problems with upscaling of localized deformations, a non-standard failure zone averaging scheme is used. Discontinuous cohesive macro-cracking is modelled using the XFEM and a gradient-enhanced damage model is used to model diffuse damage at the local-scale. A continuous-discontinuous computational homogenization method is employed to obtain the traction-separation law for macro-cracks using averaged properties calculated over the damaged zone in the RVE. In the multi-scale model, a dynamic analysis is performed for the global-scale model and the local-scale model is solved as a quasi-static problem. Dispersion effects are then captured by accounting for the inertia forces at the local-scale model via a so-called dispersion tensor which depends on the heterogeneity of the RVE. Numerical examples are presented and the multi-scale model results are compared to direct numerical simulation results. Objectivity of the multi-scale scheme with respect to the RVE size is examined.