Transformation Field Analysis Research Articles

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.

Fully coupled nonlinear thermomechanical modeling of composites using mean-field Mori–Tanaka scheme combined with TFA theory

This article aims at proposing a new mean-field homogenization framework for the study of composites undergoing fully coupled thermomechanical processes. Strongly dissipative phenomena during high or moderate cyclic loading conditions in a structural component made of a composite material cause significant interplay between mechanical and thermal fields. The proposed framework attempts to address such effect by combining the Mori–Tanaka scheme and the Transformation Field Analysis (TFA) theory and by developing a multiscale framework capable of taking into account thermomechanically coupled processes. The numerical simulations performed in the examples section and validations with computations using periodic homogenization and full-structure analysis demonstrate the proposed strategy’s accuracy and robustness. The numerical simulation of a tube shows the model’s ability to simulate cyclic loading conditions with significantly less computational cost than the alternative FE2 computation strategies. This drastic computational time reduction is due to the semi-analytical formalism of the micromechanics methodology.

Enhanced transformation field analysis for reduced-order modeling of problems with cohesive interfaces

We propose a novel transformation field analysis (TFA) technique designed to solve problems with cohesive interfaces. The TFA approach, originally proposed for homogenization purposes, is (i) extended to account for arbitrary boundary conditions (i.e. not only those suitable for homogenization), (ii) endowed with an iterative Newton–Raphson solution scheme involving the consistent tangent matrix and (iii) provided with an arc-length strategy to follow possible snap-back branches. The extension to arbitrary boundary conditions leads to an efficient reduced-order modeling approach for solving complex non-linear problems. Numerical tests demonstrate that the approach performs well in challenging scenarios with several interfaces, and that sharp and multiple snap-back branches can be followed. The comparison with finite element results shows that the proposed technique yields fast yet accurate solutions.

Formula omitted]-TFA based multiscale analysis of failure in elasto-plastic composites

This paper describes a novel homogenization methodology for analyzing the failure of elastoplastic composite materials based on elastic and eigen influence tensors-driven transformation field analysis (E2-TFA). The proposed technique considers the microscopic eigenstrain field accounting for intra-phase damage and inelastic strains. This results in realistic computations by alleviating the post-damage stiffness response, which is a drawback of TFA-based methods. We attain computational efficiency by identifying the preprocessing data solely from the elastic and eigen transformation functions and adopting a reduced order modeling technique with a piecewise constant eigenstrain field throughout the subdomains. The performance of the model is assessed by simulating the response for (a) the representative volume element (RVE) as a homogenized continuum and (b) the various composites under complex load histories with intricate macroscale morphologies. Furthermore, the nonlinear shear stress–strain response of a glass fiber composite is calculated and compared to experimentally measured fracture initiation parameters, failure plane orientation, and strain histories. Finally, we show that E2-TFA can accurately and efficiently capture damage and inelastic deformations in order to estimate the mechanical response of composite materials in a better way.

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.

An iterative plane stress integrated Transformation Field Analysis for Equivalent Homogeneous Medium characterization and localization — Application to structural holes

Full-scale structure models of high-energy situations exhibited insufficient localization when connector elements were employed to model the assemblies. An alternative approach using a super-element featuring a hole was proposed and successful in mechanical fields localization. However this approach was restrained to linear problems. An extension to materially non-linear problems was thus looked for, and in particular perforated plate materially non-linear fields.An iterative plane stress integrated TFA method is thus developed to relocate the micro-scale elastic–plastic mechanical fields of a plate containing an extremely soft medium/a strong discontinuity: a hole. To this end, a RVE is defined and discretized, the accuracy of the localization relation is assessed, and a radial return algorithm is set-up to integrate the constitutive equation of the micro-scale problem. The results of the proposed TFA approach are compared to an analytical solution and to non-linear FEA. The results show that the localization of elastic–plastic perforated plate mechanical fields is accurate, provided that the method is iterative and that the hole is discretized in the RVE. It is foreseen that both the sequential and integrated TFA approaches will be combined to evaluate criteria in full-scale structure computations.

Modelling of damage and plasticity phenomena in 3D printed materials via a multiscale approach

A multiscale technique able to describe the overall mechanical response of 3D printed materials taking into account the plastic behaviour and the occurrence of failure is presented. Reference is made to elements produced by extrusion-based additive manufacturing. At the microscale, the de-cohesion between filaments that constitute the printed material is described introducing a cohesive damage interface model that also takes into account the unilateral contact. The micromechanical problem is solved with a homogenization procedure based on the PieceWise Uniform Transformation Field Analysis properly extended to the case of interfaces. In detail, the plastic strains in the filaments and the inelastic relative displacements along the interfaces are approximated as piecewise constant functions and then are computed solving the evolutive problem. Some numerical applications are carried out, comparing the numerical results with the experimental ones and with results obtained from non-linear finite element analyses, to show the potentiality of the proposed multiscale technique.

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

Transformation field analysis as a background of clustering discretization methods in micromechanics of composites

A linear composite material (CM) consisting of a homogeneous matrix with either the periodic or random set of heterogeneities is considered. One of the first reduced-order models (ROMs), Transformation Field Analysis (TFA) by Dvorak for the pair strain-eigenstrain (or stress-eigenstress) is modified in terms of the “mixed” pair strain-eigenstress (or stress-eigenstrain) for implementation to clustering-based ROMs initiated by the self-consistent clustering analysis (SCA) by WK Liu. A set of consistency conditions are obtained for both the effective properties and modified eigenfield concentration factors. It is found that modified TFA represents a unified background of all three central clustering-based ROMs: the SCA, virtual clustering analysis (VCA), and finite element analysis–clustering analysis (FCA). One presents the scheme of estimation of inhomogeneous strain concentration factors in some prescribed clusters analyzed by the SCA (nonlinear problem). For statistically homogeneous CMs, a localized version of the modified TFA is proposed for clustering of the matrix (coating) in the vicinity of inclusions. VCA used for the analysis of a finite size inclusion can be modified for the implementation of a combination of both the modified TFA and functional gradient theory.

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.

Non-Uniform TFA reduced multiscale procedure for shell-3D modeling of periodic masonry structures

This work proposes a reduced-order multiscale model for the analysis of masonry elements subjected to in-plane and out-of-plane loading conditions. The Transformation Field Analysis (TFA) is adopted to link the homogeneous shell model at the macroscale with a three-dimensional (3D) representative unit cell (UC) of the masonry material defined at the microscale, accounting for the regular arrangement of bricks/blocks and mortar joints. The UC is modeled considering linear elastic bricks joined by interfaces subjected to possible damage and frictional plasticity mechanisms. An enhanced TFA procedure is proposed, discretizing the interfaces in subsets, where non-uniform distribution of the inelastic quantities is considered and the plastic-damage evolution problem solved. Numerical simulations are developed to assess the advantages and drawbacks of the non-uniform TFA approach compared to previously proposed piece-wise uniform procedure. The results obtained through the proposed numerical approach are compared with both micromechanical and experimental outcomes.

Evaluation of macroscopic yield surfaces of periodic porous microstructures employing transformation field analysis and high order finite element method

Evaluation of macroscopic yield surfaces of periodic porous microstructures employing transformation field analysis and high order finite element method

A second-order reduced homogenization for nonlinear structures with periodic configurations in cylindrical coordinates

A high-efficiency second-order reduced homogenization (SORH) method is established for the inelastic problem of periodic heterogeneous structures in cylindrical coordinate. The composite structures investigated in this paper are periodical in radial, axial and circumferential directions. The new second-order linear/nonlinear local solutions at microscale are given by introducing various multiscale auxiliary functions. And, the nonlinear high-order homogenization solutions are obtained at macroscale using a modified asymptotic expansion method. The significant novelties of the work are (i) an effective reduced model on the basis of transformation field analysis given to solve the nonlinear multiscale problems in cylindrical coordinates with less computational amount and (ii) a new second-order nonlinear multiscale algorithm established using asymptotic expansion technique for simulating the heterogeneous cylindrical structure. Finally, by three typical numerical examples including nonlinear elastic and elasto-plastic periodic problems, the validity and accuracy of the proposed algorithms are verified.

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

Mechanical properties of 3D printed polylactic acid elements: Experimental and numerical insights

The mechanical behaviour of polylactic acid (PLA) samples printed via material extrusion is experimentally and numerically addressed. An experimental program comprising tensile and three-point bending tests is carried out. Some filament printing orientations and different values of flow rate percentage are considered and their influence on the mechanical performance is investigated. From a numerical point of view, a new two-level model for the multiscale analysis is considered. The macroscopic structural behaviour of the 3D printed component is described with a laminate finite element model based on the first-order shear deformation theory. Each layer of the laminate is described with an elasto-plastic constitutive law and the geometrical and mechanical properties are derived from the experimental results. The micromechanical analysis is conducted only when inelastic strain occurs performing a non-linear analytical homogenization technique based on the Transformation Field Analysis. The obtained numerical results are compared with the experimental results highlighting the effectiveness of the proposed modelling approach.

Electric field equivalent method for AC/DC hybrid electric equipment adapted to digital twin with low latency

• The equivalence of electric field is mainly related to the excitation source. • The 2D axisymmetric model under various excitations is constructed and verified. • This method can ensure the real-time and high fidelity in electric field digital twin technology. • A novel isolated power supply transformer in flexible DC transmission is introduced and the method is applied. Real-time synchronization and high fidelity are the most important features of the digital twin technology. When constructing the digital twin of power equipment in flexible DC transmission technology, the simulation analysis based on transient electric field is accurate, but the real-time is difficult to guarantee due to complex structure and various working conditions. In order to reduce the delay caused by the solution as much as possible, starting from the type of excitation source, an electric field equivalent method for AC/DC hybrid electric equipment adapted to digital twin with low latency is proposed: electrostatic field equivalent is used under power frequency AC or impulse voltage excitation, and constant electric field equivalent is used under typical DC voltage at the converter station. The method has been theoretically deduced in detail and verified by using a RC voltage divider model. The results show that the method proposed in this paper can shorten the solution time to 6.4% of the original, and ensure that the error after equivalence is within 5‱ . Finally the method is actually applied to the digital twin electric field analysis of a power supply transformer for 535 kV DC circuit breaker. The time required for solving is 40 min and 14 s. It can fully meet the demand of the dispatching department to update and calibrate the electric field once every hour and ensure safe and stable operation of equipment without corona.

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.

Piecewise-uniform homogenization of heterogeneous composites using a spatial decomposition based on inelastic micromechanics

The homogenized mechanical response of heterogeneous, elasto-plastic composite materials is investigated by the use of the transformation field analysis (TFA), a two-scale algorithm relying on microscopically piece-wise uniform fields of internal variables. Not optimized spatial subdomain decompositions of the microscopic domain cause over-stiff composite material responses modeled by the TFA since the main characteristics of the inelastic field interactions are not well-represented. To improve mechanical predictions using the TFA approach, emerging inelastic fields were used to achieve enhanced spatial decompositions. The numerical estimation of the interaction functions between the subdomains allows the use of this TFA approach for the numerical modeling of a wide variety of composite materials without the need of any pre-determined reference stiffnesses. The new TFA approach was tested for materials with isotropic and anisotropic microstructures and various material systems, with a particular emphasis on the complex case of perfectly plastic material phases. Comparisons are drawn between the TFA modeling using elasticity-based and inelasticity-based spatial divisions and to reference full-field computations. The achieved results prove that more accurate predictions for the mechanical responses of composite materials can be found when inelastic fields are considered as the foundation of the spatial division into subdomains.

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.

SHELL-3D MULTISCALE MODELING OF MASONRY VAULTS BASED ON THE TFA PROCEDURE

This paper investigates the response of masonry curved structural elements with periodic texture by adopting the modeling strategy recently developed by the authors in [1]. Relying on a multiscale approach, the Mindlin-Reissner shell formulation is assumed at the macroscale and the classical 3D Cauchy continuum is adopted at the microscopic level to model the representative masonry unit cell (UC). The procedure for linking the two scales involves a Transformation Field Analysis (TFA) homogenization technique based on piece-wise uniform distributions of the damaging and frictional mechanisms over the mortar joints. Advantages and limitations of this assumption are explored considering a masonry vault with stack bond texture, that is assuming a UC made of one linear elastic brick surrounded by one head and one bed joint modeled as nonlinear interfaces. The more appropriate interface discretizations required by the TFA model are identified in relation to the activated deformation modes and the resulting computational effort is evaluated. Hence, an enriched TFA procedure is proposed to limit the computational cost and the time of analysis associated to the denser joint partitions, while still preserving the model accuracy. Finally, the efficiency of the model is investigated at the structural level by analyzing the response of the vault to differential settlements. The comparison between results obtained with the proposed multiscale formulation and those recovered by detailed micromechanical analysis confirms that the TFA approach is a very reliable and effective method for resorting to a fast reduced order model also for curved geometries.