Reduced Order Homogenization Research Articles

Multiscale crack band model for eigenstrain based reduced order homogenization

A multiscale crack band model is proposed to alleviate spurious mesh size dependence in the solution for a strain softening constitutive model. A continuum damage mechanics-based material model for the constituent phases of a carbon fiber reinforced polymer (CFRP) based composite material is applied to the Eigenstrain based Reduced-order Homogenization framework for multiscale modeling. This paper presents the formulation and demonstrates dissipated energy regularization within the multiscale modeling framework. A key contribution is computationally efficient implementation of multiscale crack band model as user defined subroutine for the commercial software Abaqus. The accuracy of the multiscale crack band model is demonstrated by critical evaluation of the numerical results for CFRP laminates.

Computational model for oxidation-assisted rupture of ceramic matrix composites

The manuscript presents a novel reduced order homogenization model for oxidation-assisted rupture in SiC/SiC composites at intermediate temperatures. Starting with a one-dimensional model based on the slow crack growth model originally proposed by Iyengar and Curtin, which is subsequently generalized to three dimensions focusing on the woven composite architecture. The proposed oxidation-assisted rupture model is developed within the framework of continuum damage mechanics where the constitutive equation in the axial tow direction is governed by the continuum damage mechanics variant of the slow crack-growth model and the availability of oxygen to fibers, which in turn depends on the initial matrix pores and subsequent matrix cracking. The constitutive model in the transverse tow directions as well as in the matrix phase outside the tows is assumed to obey a time-independent orthotropic continuum damage mechanics model, but the time-dependent nature of the axial tow model, inevitably causes the time-dependent redistribution of damage in all directions and phases. The reduced order homogenization approach that provides an efficient framework for computing overall composite behavior is employed for model reduction. Numerical validation studies are conducted on the eight-harness weave architecture.

Enriched continuum for multi-scale transient diffusion coupled to mechanics

In this article, we present a computationally efficient homogenization technique for linear coupled diffusion–mechanics problems. It considers a linear chemo-mechanical material model at the fine scale, and relies on a full separation of scales between the time scales governing diffusion and mechanical phenomena, and a relaxed separation of scales for diffusion between the matrix and the inclusion. When the characteristic time scales associated with mass diffusion are large compared to those linked to the deformation, the mechanical problem can be considered to be quasi-static, and a full separation of scales can be assumed, whereas the diffusion problem remains transient. Using equivalence of the sum of virtual powers of internal and transient forces between the microscale and the macroscale, a homogenization framework is derived for the mass diffusion, while for the mechanical case, considering its quasi-static nature, the classical equivalence of the virtual work of internal forces is used instead. Model reduction is then applied at the microscale. Assuming a relaxed separation of scales for diffusion phenomena, the microscopic fields are split into steady-state and transient parts, for which distinct reduced bases are extracted, using static condensation for the steady-state part and the solution of an eigenvalue problem for the transient part. The model reduction at the microscale results in emergent macroscopic enriched field variables, evolution of which is described with a set of ordinary differential equations which are inexpensive to solve. The net result is a coupled diffusion–mechanics enriched continuum at the macroscale. Numerical examples are conducted for the cathode–electrolyte system characteristic of a lithium ion battery. The proposed reduced order homogenization method is shown to be able to capture the coupled behavior of this system, whereby high computational gains are obtained relative to a full computational homogenization method.

Dislocation density informed eigenstrain based reduced order homogenization modeling: verification and application on a titanium alloy structure subjected to cyclic loading

This manuscript presents a dislocation density informed eigenstrain based reduced order homogenization model (DD-EHM), and its application on a titanium alloy structure subjected to cyclic loading. The eigenstrain based reduced order homogenization (EHM) approach has been extended to account for the presence of HCP (primary α phase) and BCC (β phase) grains, within which the deformation process is modeled using a dislocation density based crystal plasticity formulation. DD-EHM has been thoroughly verified to assess the accuracy of the reduced order model in capturing local and global behavior compared with direct crystal plasticity finite element method simulations. A structural scale study of titanium alloy Ti-6242S is performed using DD-EHM to quantify and characterize the spatial distribution and evolution of the dislocation pile-ups subjected to cyclic loading. The evolution of pileups at two spatial scales are tracked using a nonlocal parameter based on dislocation density discrepancy across neighboring grains. The effect of non-uniform texture on the response of the structural component has been investigated.

Discrete eigenseparation-based reduced order homogenization method for failure modeling of composite materials

This manuscript presents a new reduced order computational homogenization method for failure analysis of composite materials. The proposed approach relies on the discrete representation of the fracture process at the fine scale that consistently bridges to a continuum representation of damage at the coarse scale. In order to achieve very high computational efficiency, the proposed approach builds on the ideas of the eigendeformation-based reduced order homogenization method, but tracks microscale failure evolution over a set of discrete cohesive failure surfaces. The formulation concurrently achieves mesh size objectivity and multiscale consistency by establishing analytical relationships between the microstructure size and the macroscale element size. The performance of the proposed formulation is demonstrated in the context of three-dimensional, laminated fiber reinforced composite configurations. The numerical experiments demonstrate mesh size objectivity in the presence of multiscale failure mechanisms including fiber fracture and transverse matrix cracking.

Notch-strength prediction of ceramic matrix composites using multi-scale continuum damage model

In-plane failure behavior of SiC/SiC Ceramic Matrix Composites (CMCs) containing notches of different shapes is studied using experiments and associated computational model. The experimental work consisted of room-temperature tensile tests on an 8-harness satin woven CMC material with three different notch geometries: (1) a through-thickness center hole, (2) a semi-circular notch on both edges, and (3) a rectangular slit on both edges. A multi-scale continuum damage mechanics model is used in conjunction with Finite Element Analysis (FEA) to predict the overall failure response of the notched specimens. The continuum damage model is based on reduced order homogenization theory with damage variables defined at the meso-scale (i.e., at the scale of tows and matrix in the representative volume element of the CMC), and furthermore, the model considers failure in each of the constituents via softening stress-strain response with appropriate fracture energy-based rescaling to eliminate mesh-dependent behavior. The model parameters are derived from unnotched tension tests in [0/90] and [+/−45] orientations. It is shown that the model is able to successfully predict the overall stress-strain response as well as the ultimate strength of all three notch configurations. In addition, the model predicts the notch size effect (notch-sensitivity) in a manner consistent with existing data on CMCs.

Multiscale analysis of nonlinear composites via a mixed reduced order formulation

In this paper a new multiscale approach is presented for the analysis of structures made of composite material characterized by elastoplastic or viscoplastic nonlinear response. Scale separation is assumed so that the homogenization theory can be applied: at the structural scale (macroscale) the material appears as homogeneous, while at the microscale it is characterized by a heterogeneous microstructure that affects the global behavior of the structure. The resolution of the micromechanical problem is performed developing a reduced order homogenization technique based on a mixed variational formulation, named Mixed Transformation Field Analysis (MxTFA). The microscopic reduced internal variables are the stress and plastic multiplier parameters of RVE subsets, whose evolution is computed enforcing the weak form of the governing equations for every subset. Some numerical tests are performed to show the accuracy and efficiency of the proposed multiscale technique.

Comparison of reduced order homogenization techniques: pRBMOR, NUTFA and MxTFA

The introduction of composites in engineering applications led to a need for tools that can predict the mechanical response with account for the heterogeneities in the materials in order to safely design complex structures. These predictions are required to be sufficiently accurate yet computationally inexpensive, especially when dealing with nonlinear materials: for these the amount of internal variables and the computing times can both become prohibitively high. Here, three model order reduction homogenization methods are compared in detail: the potential-based Reduced Basis Model Order Reduction (pRBMOR), the NonUniform Transformation Field Analysis and the Mixed Transformation Field Analysis (MxTFA). All methods are developed in the framework of the Nonuniform TFA, initially introduced by Michel and Suquet. Two of the illustrated methods, the pRBMOR and the MxTFA, deviate from the TFA of Michel and Suquet in that they are based on mixed variational formulations. The comparison will investigate differences and similarities of the techniques in terms of the reduced degrees of freedom and their evolution law, the storage requirements and the accuracy of the effective constitutive behavior. Numerical tests on three-dimensional periodic composites under complex normal and shear loading paths will show the performances of the techniques.

Towards practical multiscale approach for analysis of reinforced concrete structures

We present a novel multiscale approach for analysis of reinforced concrete structural elements that overcomes two major hurdles in utilization of multiscale technologies in practice: (1) coupling between material and structural scales due to consideration of large representative volume elements (RVE), and (2) computational complexity of solving complex nonlinear multiscale problems. The former is accomplished using a variant of computational continua framework that accounts for sizeable reinforced concrete RVEs by adjusting the location of quadrature points. The latter is accomplished by means of reduced order homogenization customized for structural elements. The proposed multiscale approach has been verified against direct numerical simulations and validated against experimental results.

Sparse and scalable eigenstrain-based reduced order homogenization models for polycrystal plasticity

In this manuscript, accelerated, sparse and scalable eigenstrain-based reduced order homogenization models have been developed for computationally efficient multiscale analysis of polycrystalline materials. The proposed model is based on the eigenstrain-based reduced order homogenization (EHM) approach, and provides significant efficiency in analysis of structures composed of complex polycrystalline microstructures. The acceleration is achieved by introducing sparsity into the linearized reduced order system through selectively considering the interactions between grains based on the idea of grain clustering. The proposed approach results in a hierarchy of reduced models that recovers eigenstrain-based homogenization, when full range of interactions are considered, and degrades to the Taylor model, when all inter-grain interactions are neglected. The resulting sparse system is solved efficiently using both direct and iterative sparse solvers, both of which show significant efficiency improvements compared to the full EHM. A layer-by-layer neighbor grain clustering scheme is proposed and implemented to define ranges of grain interactions. Performance of the proposed approach is evaluated by comparing the results against the full EHM and crystal plasticity finite element (CPFE) simulations.

Multiscale modeling of femur fracture

SummaryWe present two multiscale approaches for fracture analysis of full scale femur. The two approaches are the reduced order homogenization (ROH) previously developed by the first author and his associates and a novel accelerated ROH (AROH). The AROH is based on utilizing ROH calibrated to limited experimental data as a training tool to calibrate a simpler, single‐scale anisotropic damage model. For bone tissue orientation, we take advantage of so‐called Wolff's law, which states that bone tissue orientation is well correlated with principal strain direction in a stance position. The meso‐phase properties are identified by minimizing error between the overall cortical and trabecular bone properties resulting from the quantitative computed tomography (QCT) scans and those predicted by the two‐scale homogenization. The overall elastic and inelastic properties of the cortical and trabecular bone microstructure are derived from bone density that can be estimated from the Hounsfield units, which represent the measured grey levels in the QCT scans. For model validation, we conduct ROH and AROH simulations of full scale finite element model of femur created from the QCT and compare the simulation results with available experimental data. Copyright © 2016 John Wiley & Sons, Ltd.

Reduced order homogenization for viscoplastic composite materials including dissipative imperfect interfaces

The computational effort of homogenization schemes based on full-field simulations is prohibitively high for many applications and motivates the use of model order reduction techniques. A reduced order homogenization scheme is presented which allows for a coupled treatment of dissipative mechanisms both in the bulk and on imperfect interfaces within the microstructure. Extending the key idea of the nonuniform transformation field analysis (NTFA), reduced bases are introduced for all fields of internal variables and also for the displacement discontinuities at the interfaces. In the online phase, an initial value problem is solved, which is derived from a mixed incremental variational principle. The variational formulation requires that the underlying constitutive models are given by two potentials, as known from the framework of generalized standard materials. As an example, a unidirectional fiber composite with a viscoplastic matrix and a viscoelastic interface is studied. It is demonstrated that the reduced order model can be used to analyze the rate-dependence of the overall behavior as well as the interface-induced size effect and the impact of varying interface parameters. At a significantly reduced computational effort, good quantitative agreement with finite element reference solutions is found for predictions of global quantities such as the effective stress and the effective mechanical work. For this is a rather simple training strategy sufficient. It is found that the reduced model provides good approximations of the local stress field and of the internal variables within the microstructure, which can benefit from a more comprehensive training.

Two accuracy improvements on nonuniform transformation field analysis for plasticity coupled to softening

AbstractA reduced order homogenization scheme for the case of plasticity coupled with softening effects is proposed. This is based on a straightforward extension of the so‐called nonuniform transformation field analysis (NTFA, [2]). Two related new methods, denoted as uneven NTFA and adaptive NTFA accounting for accuracy improvements, are also presented, which are based on the ideas of parameter identification and adaptive modeling, respectively. A complementary numerical study is given. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Prediction of progressive fatigue damage and failure behavior of IM7/977-3 composites using the reduced-order multiple space-time homogenization approach

This manuscript presents the blind prediction of fatigue life performance in three laminated carbon fiber reinforced polymer composite layups using a reduced-order space-time homogenization model. To bridge the spatial scales, the modeling approach relies on the Eigendeformation-based reduced order homogenization method. To bridge the time scales associated with a single load cycle and the overall life of the composite, a homogenization-based accelerated multiple-time-scale integrator with adaptive time stepping capability is employed. The proposed multiscale modeling approach was used to predict the evolution of composite stiffness and progressive damage accumulation as a function of loading cycles, as well as residual strength after fatigue in tension and compression, for three layups ([0,45,90,−45]2 s, [30,60,90,−60,−30]2 s, and [60,0,−60]3 s). Following blind prediction, the experimental data from the blind prediction specimens were employed to better understand the failure mechanisms and recalibrate the model. This study was performed as a part of the Air Force Research Laboratory's “Damage Tolerant Design Principles” Program.

Are the cohesive zone models necessary for delamination analysis?

We develop a dual-purpose damage model (DPDM) that can simultaneously model intralayer damage (ply failure) and interlayer damage (delamination) as an alternative to conventional practices that models ply failure by continuum damage mechanics (CDM) and delamination by cohesive elements. From purely computational point of view, if successful, the proposed approach will significantly reduce computational cost by eliminating the need for having double nodes at ply interfaces. At the core, DPDM is based on the regularized continuum damage mechanics approach with vectorial representation of damage and ellipsoidal damage surface. Shear correction factors are introduced to match the mixed mode fracture toughness of an analytical cohesive zone model. A predictor–corrector local–nonlocal regularization scheme, which treats intralayer portion of damage as nonlocal and interlayer damage as local, is developed and verified. Two variants of the DPDM are studied: a single- and two-scale DPDM. For the two-scale DPDM, reduced-order-homogenization (ROH) framework is employed with matrix phase modeled by the DPDP while the inclusion phase modeled by the CDM. The proposed DPDM is verified on several multi-layer laminates with various ply orientations including double-cantilever beam (DCB), end-notch-flexure (ENF), mixed-mode-bending (MMB), and three-point-bending (TPB).

Prediction of progressive damage and strength of IM7/977-3 composites using the Eigendeformation-based homogenization approach: Static loading

This paper presents the results from the authors’ participation in the Air Force Research Laboratory’s Damage Tolerance Design Principles Program. The Eigendeformation-based reduced order homogenization method was employed to predict the mechanical response of a suite of open hole and unnotched IM7/977-3 composite laminates under static tension and compression. Damage accumulation, effective stiffness, and ultimate strength blind predictions are included in addition to the results of the recalibration study. In blind predictions, the proposed multiscale model produced predictions with an average error of 13.1% compared to the experiments for static ultimate strength and 13.6% for stiffness. After recalibration, the average prediction error was improved to 8.7% for static ultimate strength and 4.4% for stiffness. Details of the blind predictions and the recalibration are discussed.

Numerical identification of a viscoelastic substitute model for heterogeneous poroelastic media by a reduced order homogenization approach

The paper deals with the computational homogenization of pressure diffusion processes in a poroelastic medium. The underlying physical phenomena are of interest for the interpretation of seismic data with applications in hydrocarbon production and geothermal energy. Pressure diffusion is assumed to take place on a length scale much smaller than the observer scale. Thus, the macroscopic observer is not able to measure the properties of the poroelastic medium directly but notices an intrinsic viscous attenuation. Under these circumstances, the macro-scale can be interpreted as a single-phase solid with (apparent) viscoelastic properties. In this paper, we establish a numerical upscaling procedure based on a volume averaging concept. This enables us to identify the material properties of the viscoelastic substitute model in a numerically efficient manner. For this purpose, the poroelastic medium on the small scale is modelled in terms of the momentum balance of the biphasic mixture and a coupled diffusion equation. We approximate the poroelastic pressure field on the small scale by a linear combination of pressure modes forming a reduced orthogonal basis and being identified by a Proper Orthogonal Decomposition (POD) technique. From the superposition principle, the evaluation of the poroelastic continuity equation results in a proper identification of the evolution equations defining the apparent viscoelastic model. In comparison to the nested FE2 solution schemes, the reduced order approach only requires a small amount of off-line precomputations and, therefore, causes very low numerical costs. The proposed method is validated for the simple setup of a layered porous rock with alternating water-and gas-saturated zones.

Eigenstrain based reduced order homogenization for polycrystalline materials

Abstract In this manuscript, an eigenstrain based reduced order homogenization method is developed for polycrystalline materials. A two-scale asymptotic analysis is used to decompose the original equations of polycrystal plasticity into micro- and macroscale problems. Eigenstrain based representation of the inelastic response field is employed to approximate the microscale boundary value problem using an approximation basis of much smaller order. The reduced order model takes into account the grain-to-grain interactions through influence functions that are numerically computed over the polycrystalline microstructure. The proposed approach is also endowed with a hierarchical model improvement capability that allows accurate representation of stress and deformation state within subgrains. The proposed approach was implemented and its performance was assessed against crystal plasticity finite element simulations. Numerical studies point to the capability to efficiently compute the mechanical response of the polycrystal RVEs with good accuracy and the ability to capture stress risers near grain boundaries.

Nonlinear reduced order homogenization of materials including cohesive interfaces

The mechanical response of composite materials is strongly influenced by the nonlinear behavior of the interface between the constituents. In order to make reliable yet computationally efficient predictions for such materials, a reduced order model is developed. Conceptual ideas of the NTFA (Michel and Suquet, Int J Solids Struct 40:6937---6955, 2003, Comput Methods Appl Mech Eng 193:5477---5502, 2004) and of the pRBMOR (Fritzen, Hodapp and Leuschner Comput Methods Appl Mech Eng 260:143---154, 2013, Fritzen et al., Comput Methods Appl Mech Eng 278:186---217, 2014) are adopted. The key idea is to parameterize the displacement jumps on the cohesive interfaces by a reduced basis of global ansatz functions. Micromechanical considerations and the potential structure of the constitutive models lead to a variational formulation and reduced equilibrium conditions. The effect of the preanalysis phase on the accuracy is investigated using geometrically optimal training directions. The reduced model is tested for three-dimensional microstructures. Besides the effective stress response, the tension---compression asymmetry and the distribution of the separation of the interface are investigated. Memory savings on the order of $$10^5$$105 are realized. The computing time is reduced considerably.

IDENTIFICATION OF OPTIMAL REDUCED ORDER HOMOGENIZATION MODELS FOR FAILURE OF HETEROGENEOUS MATERIALS

This manuscript presents a new methodology for the identification of optimal reduced order models for the inelastic and failure response of heterogeneous materials. The proposed methodology employs the eigendeformation-based reduced order homogenization approach. The identification of the optimal reduced order model is posed as an integer optimization problem and the genetic algorithm method is used to evaluate the optimization problem. A second optimization problem is posed to ensure that the errors associated with the optimal reduced order model are minimized through scaling of the failure parameters. The performance and capabilities of the optimal reduced order models identified based on the proposed approach are demonstrated by comparing model predictions with the computational homogenization method with full resolution of the material microstructure. Numerical simulations conducted using unidirectional reinforced matrix microstructures reveal that the reduced order models accurately describe the response characteristics of the composite material for a wide range of loading regimes.