Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy

Kouznetsova, V Varvara; Geers, Mgd Marc; Brekelmans, Wam Marcel

doi:10.1016/j.cma.2003.12.073

Cited by 588 publications

(448 citation statements)

References 49 publications

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“…as for example pointed out in [34,43,44]. In this context, the symbols "+" and "−" in the latter condition denote opposing edges of the underlying RVE.…”

Section: Finite Element Discretisationmentioning

confidence: 99%

Partially relaxed energy potentials for the modelling of microstructures – application to shape memory alloys

Bartel

Menzel

2012

GAMM-Mitteilungen

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Energy relaxation is well-established by several researchers -especially in the field of the modelling of solid-solid phase transformations. Nevertheless, critics still counter this concept by considering it as a purely mathematical tool with poor physical significance. In this contribution we aim at emphasising the significance of energy relaxation methods for the modelling of dissipative solids and especially microstructure formation and its further evolution. In particular, we shall point out aspects and advantages of this concept which are not straight forward to achieve within alternative schemes.

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“…as for example pointed out in [34,43,44]. In this context, the symbols "+" and "−" in the latter condition denote opposing edges of the underlying RVE.…”

Section: Finite Element Discretisationmentioning

confidence: 99%

Partially relaxed energy potentials for the modelling of microstructures – application to shape memory alloys

Bartel

Menzel

2012

GAMM-Mitteilungen

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“…Equation (35) is satisfied automatically if the periodic constraints (24) and (27) are satisfied either by using the constraint elimination method or by using the Lagrange multipliers method, in which case the multipliers represent the boundary forces. In a finite element analysis, the discrete constraints created from these periodic boundary conditions are easily obtained when using conformal meshes [4,7,8,9]. In a more general setting, the conformity of mesh distributions on opposite boundaries of the representative volume element cannot always be guaranteed, leading to a non-periodic mesh.…”

Section: Problem At the Microscopic Scalementioning

confidence: 99%

“…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%

“…For the multiscale problems, the generalized continuum can potentially be used at both the macroscopic and the microscopic scales. Recent extensions of the FMCH scheme to the second-order continuum, as for the so-called second-order multiscale computational homogenization (SMCH) [6,7,8], provide a systematic way to couple the strain-gradient continuum at the macro-scale with the classical continuum at the micro-scale, see Fig. 1.…”

Section: Introductionmentioning

confidence: 99%

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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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“…This assumption is particularly valid when macrogradients remain small and material failure does not occur. The second-order computational homogenization partly alleviates the assumption of scale separation by taking the gradient of the macrodeformation gradient tensor into account [246][247][248][249][250]. Furthermore, second-order computational homogenization introduces a physical length to the microscale that is missing in the first-order homogenization.…”

Section: 22mentioning

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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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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Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy

Cited by 588 publications

References 49 publications

Partially relaxed energy potentials for the modelling of microstructures – application to shape memory alloys

Partially relaxed energy potentials for the modelling of microstructures – application to shape memory alloys

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

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

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