Simulation of the multi-scale convergence in computational homogenization approaches

Terada, Kenjiro; Hori, Muneo; Kyoya, Takashi; Kikuchi, Noboru

doi:10.1016/s0020-7683(98)00341-2

Cited by 500 publications

(256 citation statements)

References 17 publications

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“…Kaczmarczyk et al [225] made similar conclusions in the context of second-order computational homogenization. However, this does not imply that the results obtained under periodic boundary conditions are always the closest ones to the exact solutions as clearly stated by Terada et al [221] "there is no guarantee that periodic boundary conditions are the best among a class of possible boundary conditions. Nonetheless, the periodic boundary conditions provide the reasonable estimates on the effective moduli in the sense that they are always bounded by the other."…”

Section: Computational Homogenizationmentioning

confidence: 87%

“…Many authors, e.g., Refs. [215][216][217][218][219][220][221][222][223][224], have shown that in pure mechanical linear and nonlinear problems, the effective behavior derived under periodic boundary conditions is bounded by linear displacement boundary conditions from above and constant traction boundary conditions from below for a finite size of the RVE. Kaczmarczyk et al [225] made similar conclusions in the context of second-order computational homogenization.…”

Section: Computational Homogenizationmentioning

confidence: 99%

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Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

190

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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Section: Computational Homogenizationmentioning

confidence: 87%

Section: Computational Homogenizationmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

190

138

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“…An effective remedy, which is known as the computational homogenization, has been developed to link up straightforwardly the responses of the large scale problems, also called the macroscopic problems, to the behavior of the smaller scale problems, also called the microscopic problems, where the presence of heterogeneities is considered. The basic ideas of the computational homogenization approach have been presented in papers by Michel et al [1], Terada et al [2], Miehe et al [3,4], Kouznetsova et al [5,6,7], Kaczmarczyk et al [8], Peric et al [9], Geers et al [10] and references therein, as a non-exhaustive list. By this technique, two boundary value problems are defined at two separate scales, Figure 1: Illustration of first-order and second-order multiscale computational homogenization schemes.…”

Section: Introductionmentioning

confidence: 99%

“…The classical multiscale computational homogenization approach (so-called the first order multiscale computational homogenization approach -FMCH) provides a versatile tool to model the micro-macro transitions and is based on the standard continuum theory [1,2,3,4,5,9,10]. For a given macroscopic deformation gradient tensor, the stress and the associated material tangent are estimated from the response of the micro-structure, see Fig.…”

Section: Introductionmentioning

confidence: 99%

Multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous Galerkin formulation

Nguyễn

Becker

Noels

2013

Computer Methods in Applied Mechanics and Engineering

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When considering problems of dimensions close to the characteristic length of the material, the size effects can not be neglected and the classical (so-called first-order) multiscale computational homogenization scheme (FMCH) looses accuracy, motivating the use of a second-order multiscale computational homogenization (SMCH) scheme. This second-order scheme uses the classical continuum at the micro-scale while considering second-order continuum at the macro-scale. Although the theoretical background of the second-order continuum is increasing, the implementation into a finite element code is not straightforward because of the lack of high-order continuity of the shape functions. In this work, we propose a SMCH scheme relying on the discontinuous Galerkin (DG) method at the macro-scale, which simplifies the implementation of the method. Indeed, the DG method is a generalization of weak formulations allowing for inter-element discontinuities either at the C 0 level or at the C 1 level, and it can thus be used to constrain weakly the C 1 continuity at the macro-scale. The C 0 continuity can be either weakly constrained by using the DG method or strongly constrained by using usual C 0 displacement-based finite elements. Therefore, two formulations can be used at the macro-scale: (i) the full-discontinuous Galerkin formulation (FDG) with weak C 0 and C 1 continuity enforcements, and (ii) the enriched discontinuous Galerkin formulation (EDG) with high-order term enrichment into the conventional C 0 finite element framework. The micro-problem is formulated in terms of standard equilibrium and periodic boundary conditions. A parallel implementation in three dimensions for non-linear finite deformation problems is developed, showing that the proposed method can be integrated into conventional finite element codes in a straightforward and efficient way.

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“…In practice, the heterogeneities within a composite are not periodic as in the case of fiberreinforced matrices . In order to adapt to general heterogeneous materials, the size of RVE must be sufficiently large to contain enough microscopic heterogeneous information [3,54], thus increasing the corresponding computational cost. Furthermore, in an elasto-plastic problem, periodicity on the RVEs also dictates periodicity on the damage induced which could result in erroneous results.…”

mentioning

confidence: 99%

A hysteretic multiscale formulation for nonlinear dynamic analysis of composite materials

Triantafyllou

Chatzi

2014

Comput Mech

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A new multiscale finite element formulation is presented for nonlinear dynamic analysis of heterogeneous structures. The proposed multiscale approach utilizes the hysteretic finite element method to model the microstructure. Using the proposed computational scheme, the micro-basis functions, that are used to map the microdisplacement components to the coarse mesh, are only evaluated once and remain constant throughout the analysis procedure. This is accomplished by treating inelasticity at the micro-elemental level through properly defined hysteretic evolution equations. Two types of imposed boundary conditions are considered for the derivation of the multiscale basis functions, namely the linear and periodic boundary conditions. The validity of the proposed formulation as well as its computational efficiency are verified through illustrative numerical experiments.

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Simulation of the multi-scale convergence in computational homogenization approaches

Cited by 500 publications

References 17 publications

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous Galerkin formulation

A hysteretic multiscale formulation for nonlinear dynamic analysis of composite materials

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