Nonuniform Transformation Field Analysis Research Articles

Efficient nonlinear homogenization of bones using a cluster-based model order reduction technique.

We present a reduced order model for efficient nonlinear homogenization of bones, accounting for strength difference effects and containing some well-known plasticity models (like von Mises or Drucker-Prager) as special cases. The reduced order homogenization is done by using a cluster-based model order reduction technique, called cluster-based nonuniform transformation field analysis. For an offline phase, a space-time decomposition is performed on the mesoscopic plastic strain fields, while a clustering analysis is employed for a spatial decomposition of the mesoscale RVE model. A volumetric-deviatoric split is additionally introduced to capture the enriched characteristics of the mesoscopic plastic strain fields. For an online analysis, the reduced order model is formulated in a unified minimization problem, which is compatible with a large variety of material models. Both cortical and trabecular bones are considered for numerical experiments. Compared to conventional FE-based RVE computations, the developed reduced order model renders a considerable acceleration rate beyond , while maintaining a sufficient accuracy level.

Cluster-based Nonuniform Transformation Field Analysis of Gra-phene nanocomposites

        Graphene nanocomposites have attracted much attention in materials science due to their superior me-chanical properties. It is difficult for conventional mul-tiscale methods to provide substantial assistance to the research of such materials due to their huge computa-tional costs. Nonuniform transformation field analysis is a very effective reduced order homogenization meth-od for elastoplastic multiscale analysis. However, the reduced order model derived from this method has the shortcoming of low universality and high application threshold. Therefore, an improved reduced order model is proposed by combining the nonuniform transfor-mation field analysis with the k-means clustering algo-rithm. One can embed the required microscopic consti-tutive model into the reduced order homogenization framework without the need to derive a new reduced order model. Based on the cluster-based nonuniform transformation field analysis, the influence of the mi-croscopic plastic strain field evolution on the macro-scopic response of the material under consideration is revealed, while the mechanical properties of graphene nanocomposites are predicted. The numerical results show that the new reduced order model can accurately predict the macroscopic mechanical properties of com-posite materials, and its acceleration rate compared to the traditional finite element computations reaches103-104 .
 Keywords:graphene nanocomposites; reduced order model; multiscale methods; clustering; nonuniform transfomation field analysis

From Domain Decomposition to Model Reduction for Large Nonlinear Structures

The numerical simulation of multiscale and multiphysics problems requires efficient tools for spatial localization and model reduction. A general strategy combining Domain Decomposition and Nonuniform Transformation Field Analysis (NTFA) is proposed herein for the simulation of nuclear fuel assemblies at the scale of a full nuclear reactor. The model at subdomain level solves the full elastic problem but with a reduced nonlinear loading, based on simplified boundary conditions, reduced creep flow rules, projected sign preserving contact conditions, and a NTFA like reduced friction law to get the evolution of each slipping mode. With this loading reduction, the local solution can be explicitly obtained from a small set of precomputed elementary elastic solutions. The numerical tests indicate that considerable cost reduction (a factor of 50 to 1000) can be achieved while preserving engineering accuracy.

Multiscale identification and NTFA reduction of the elasto-viscoplastic polycrystalline behaviour of uranium dioxide (UO2) for wide range of loading conditions

This numerical study presents a micromechanical modelling of the behaviour of uranium dioxide fuel (UO2), a polycrystalline ceramic used in nuclear power plants, for loading condition typical of a reactivity initiated accident (RIA). Above a specific temperature, UO2 has an elasto-viscoplastic behaviour with strain hardening depending on both temperature and macroscopic strain rate (T,ε¯˙).In a first step, via a full-field approach, an inverse identification of the local single crystal evolution law is performed based on experimental data obtained at the macroscopic scale [4,5]. Beyond the macroscopic behaviour of the fuel pellet, it is sometimes necessary to go down in scale to study local phenomena. A first approach would be to perform a Finite Element Method squared to two (FEM2) resolution [37]. It consists to solve a thermomechanical problem on a first mesh defined at the pellet scale where at each integration point is solved another thermomechanical problem on a Representative Volume Element. The full-field approach guarantees access to the local field at the expense of relatively high computational time. It is not conceivable within industrial codes.In a second step it appeared necessary to replace the second full-field resolution by a model reduction method based on the Nonuniform Transformation Field Analysis (NTFA) approach. This method allows access to macroscopic quantities and local fields while drastically reducing the computational time. The NTFA model [21] with tangent second order (TSO) approximation [24,21] is developed and applied to the problem at hand. Two decompositions of the hardening variable are studied. It is taken as in the reference [21], as uniform per grain or taken as decomposed, like modes associated with the viscoplastic strain tensor. Then, work is done to integrate the specific loading conditions, i.e. temperatures and strain rates of the RIA, into the NTFA model while keeping the number of internal variables as low as possible. Finally, the results from the experimental, full-field and NTFA TSO models are compared. It is checked that the numerical results are not deteriorated either macroscopically or locally at the macroscopic and local scales for uniaxial and triaxial loading.

Multiscale computational homogenization for woven composites using the cluster-based nonuniform transformation field analysis

Multiscale computational homogenization for woven composites using the cluster-based nonuniform transformation field analysis

NTFA-enabled goal-oriented adaptive space–time finite elements for micro-heterogeneous elastoplasticity problems

In this work, we establish a goal-oriented space–time finite element method for a class of dissipative heterogeneous materials. Those materials are modeled on both micro- and macroscale, with a scale transition of volume averaging type satisfying the Hill–Mandel condition. A nonuniform transformation field analysis is performed on the microscopic inelastic strain fields for a model reduction. Reduced variables are deduced from a space–time decomposition of those inelastic strain fields. Closed-form constitutive relations are derived from some dissipative considerations, thus resulting into a reduced order homogenization problem. The resulting model error is sufficiently small for the considered class of materials, thus leaving the discretization error of the finite element method as a main error source. For ease of error estimate, we rewrite the reduced order problem in a multifield formulation. Based on duality techniques, a backward-in-time dual problem is derived from a Lagrange method, rendering error representations of a user-defined quantity of interest. Combining a patch recovery technique, a computable error estimator is developed to quantify both spatial and temporal discretization errors. By means of a localization technique, local error estimators are used to drive a greedy adaptive refinement algorithm in space and time. The effectiveness of the resulting algorithm is confirmed by several numerical examples w.r.t. a prototype model.

Multiscale analysis of composite structures with goal-oriented mesh adaptivity and reduced order homogenization

We present a multiscale finite element approach for composite structure analysis, applying reduced order homogenization to obtain an accurate surrogate model for microscale RVE computations and establishing macroscale mesh adaptivity towards an error control of a user-defined quantity of interest. By means of the so-called nonuniform transformation field analysis, reduced variables are deduced from a space–time decomposition of inelastic strain fields. Considering the macroscopic dissipation power as a volume-averaged microscopic dissipation power, closed-form macroscale constitutive relations are established, thus resulting into a reduced order homogenization problem. For ease of error estimate, we propose a multifield formulation for the reduced order problem. Based on duality techniques, a backward-in-time dual problem is derived from a Lagrange method, leading to error representations aiming at the quantity of interest. Accordingly, an efficient error estimator combined with a patch recovery technique avoiding nonlinear computations is proposed. The effectiveness of the resulting adaptive algorithm is illustrated by several numerical examples w.r.t. a prototype model.

Numerical Model Reduction with error estimation for computational homogenization of non-linear consolidation

Numerical Model Reduction (NMR) is adopted for solving the non-linear microscale problem that arises from computational homogenization of a model problem of porous media with displacement and pressure as unknown fields. A reduced basis is obtained for the pressure field using Proper Orthogonal Decomposition and the pertinent displacement basis is obtained using Nonuniform Transformation Field Analysis. An explicit, fully computable, a posteriori error estimator is derived based on the linearized error equation for quantification of the NMR error in terms of a suitably chosen energy norm. The performance of the error estimates is demonstrated via a set of numerical examples with varying load amplitudes.

Nonuniform transformation field analysis based reduced-order model of high-fidelity generalized method of cells

The accurate prediction of a composite’s nonlinear behavior using multiscale modeling techniques relies heavily on the description of microscale features and requires many internal variables, leading to high computational costs. This paper presents a reduced-order model (ROM) for composite materials based on a novel adaptation of nonuniform transformation field analysis (NTFA) in conjunction with the high-fidelity generalized method of cells (HFGMC) micromechanics theory accounting for micro constituent damage. First, the composite’s nonlinear behavior is simulated using the HFGMC technique with constituent damage represented as damage-induced inelastic strains. The nonuniformity in the inelastic strains is then approximated using a set of tensorial deformation modes. Next, a macroscopic continuum damage model is formulated based on the inelastic modes to have the same evolution function as their microscopic counterpart. The NTFA procedure is applied in a three-dimensional setting, followed by a detailed description of the localization operators necessary for linearizing the homogenized constitutive equations. Last, the ROM’s capability to capture damage effects using only a finite set of internal variables is demonstrated by comparing the predicted global and local responses through HFGMC simulations for different loading paths. The presented model is shown to decrease the computational cost of micromechanics-based unit cell homogenization significantly.

Cluster based nonuniform transformation field analysis: An efficient homogenization for inelastic heterogeneous materials

AbstractIn this article, the authors propose an efficient numerical homogenization method, the so‐called cluster based nonuniform transformation field analysis (NTFA), for inelastic heterogeneous material. The method enables to predict the effective properties directly from the local constitutive relations for microstructural phases of heterogeneous material, thus no requiring preliminary assumptions and any derivations for the effective constitutive relation as in the conventional NTFA. Evolution of plastic strain field is expressed by combination of nonuniform plastic strain modes evaluated from the nonlinear finite element analysis for training directions previously given and snapshot proper orthogonal decomposition. Eventually, microscopic stress and strain field are represented by scalar reduced variables and plastic strain modes. A simple maximization of minimum distance is proposed in order to choose uniformly distributed training directions on a unit sphere in the load space. Also, scalar reduced variables are directly determined from the integration of local constitutive relation, where material points are compressed into a number of clusters by clustering scheme. Thus, evolution of reduced variables is evaluated by the integration of local constitutive relation only in clusters. Numerical examples that consider material models followed J2 plasticity and pressure‐dependent non‐associated flow rule show that the current approach may predict the effective properties of inelastic heterogeneous material accurately and efficiently. It should be noted that the current approach can be available for material followed not only associated flow rule but also non‐associated flow one.

Multiscale analysis of elastic-viscoplastic composite using a cluster-based reduced-order model

In this paper, a cluster-based nonuniform transformation field analysis (CNTFA) is suggested for the multicale analysis of elastic-viscoplastic composite. The current method estimates the effective behavior of composite directly from the local constitutive relation of phase materials, so that the effective behavior of composite can be predicted only by the integration of local constitutive equation. Furthermore, the number of integration points of the local constitutive relation necessary for the evolution of reduced variables is significantly decreased by dividing the representative volume element (RVE) into a number of material clusters using the data compression technology by clustering. This plays a decisive role in reducing the amount of information necessary for the prediction of effective behavior, and enhancing the computational efficiency. Numerical examples for the aluminium matrix composite with boron particles, the plate-fin structure under high temperature and short fiber reinforced epoxy matrix composite explain that the current method can be applied efficiently for the prediction of effective behavior in the elastic-viscoplastic composite.

A nonuniform transformation field analysis for composites with strength difference effects in elastoplasticity

Many composites and their constituents exhibit strength difference (SD) effects. To reduce the computational cost associated to twoscale simulations, the theory of nonuniform transformation field analysis (NTFA, originally proposed by Michel and Suquet, 2003) is extended to consider a general case of SD effects in elastoplasticity, containing several well-known plasticity models like von Mises or Drucker and Prager as special cases. A space-time decomposition is done separately for volumetric and deviatoric inelastic strain fields, thus leading to two sets of plastic modes with different charateristics. Localization rules are deduced from superposition principles for the microscopic strain and stress fields, which are then homogenized to obtain the effective response. A coupled model governing the evolution of the resulting reduced variables is proposed based on some dissipative considerations. A convenient matrix formulation for an FE-based implementation is presented in detail. The proposed coupled model is shown to be exact for homogeneous cases and sufficiently accurate for three-dimensional heterogeneous microstructures. Finally, an efficient structural analysis is performed by using corresponding algorithmic consistent tangent operators.

Reduced order modeling of viscoelastic properties of asphalt concrete

AbstractBiphasic asphalt concrete consists of mineral aggregates and a surrounding bituminous binding agent, of those material properties are known. To obtain the overall material behavior, we generate Statistical Volume Elements (SVE) of the asphalt concrete structure artificially. Afterwards, the Nonuniform Transformation Field Analysis (NTFA) is applied to homogenize them. This paper focuses on the stochastical differences of the microstructures of the SVEs and their influences on the macroscopic material behavior. Therefore, an indirect tensile test is simulated.

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.

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.

Model‐reduction in micromechanics of polycrystalline materials

AbstractThe present study is devoted to the generalization of the Nonuniform Transformation Field Analysis (NTFA), a model‐reduction approach introduced by the authors. First, the local fields of internal variables are decomposed on a reduced basis of modes. Second, the effective (average) dissipation potential of the phases is replaced by accurate approximations. The reduced evolution equations of the models, in other words the homogenized constitutive relations, can be entirely expressed explicitly in terms of quantities which are pre‐computed “off‐line”. The example of creep of polycrystalline ice is used to assess the accuracy of the models. Their predictions, both the overall response and the local response, are shown to be in good agreement with full‐field simulations with a significant speed‐up. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Effective potentials in nonlinear polycrystals and quadrature formulae.

This study presents a family of estimates for effective potentials in nonlinear polycrystals. Noting that these potentials are given as averages, several quadrature formulae are investigated to express these integrals of nonlinear functions of local fields in terms of the moments of these fields. Two of these quadrature formulae reduce to known schemes, including a recent proposition (Ponte Castañeda 2015 Proc. R. Soc. A471, 20150665 (doi:10.1098/rspa.2015.0665)) obtained by completely different means. Other formulae are also reviewed that make use of statistical information on the fields beyond their first and second moments. These quadrature formulae are applied to the estimation of effective potentials in polycrystals governed by two potentials, by means of a reduced-order model proposed by the authors (non-uniform transformation field analysis). It is shown how the quadrature formulae improve on the tangent second-order approximation in porous crystals at high stress triaxiality. It is found that, in order to retrieve a satisfactory accuracy for highly nonlinear porous crystals under high stress triaxiality, a quadrature formula of higher order is required.

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)

A model-reduction approach in micromechanics of materials preserving the variational structure of constitutive relations

In 2003 the authors proposed a model-reduction technique, called the Nonuniform Transformation Field Analysis (NTFA), based on a decomposition of the local fields of internal variables on a reduced basis of modes, to analyze the effective response of composite materials. The present study extends and improves on this approach in different directions. It is first shown that when the constitutive relations of the constituents derive from two potentials, this structure is passed to the NTFA model. Another structure-preserving model, the hybrid NTFA model of Fritzen and Leuschner, is analyzed and found to differ (slightly) from the primal NTFA model (it does not exhibit the same variational upper bound character). To avoid the “on-line” computation of local fields required by the hybrid model, new reduced evolution equations for the reduced variables are proposed, based on an expansion to second order (TSO) of the potential of the hybrid model. The coarse dynamics can then be entirely expressed in terms of quantities which can be pre-computed once for all. Roughly speaking, these pre-computed quantities depend only on the average and fluctuations per phase of the modes and of the associated stress fields. The accuracy of the new NTFA-TSO model is assessed by comparison with full-field simulations. The acceleration provided by the new coarse dynamics over the full-field computations (and over the hybrid model) is then spectacular, larger by three orders of magnitude than the acceleration due to the sole reduction of unknowns.