Computational homogenization

Geers, M.G.D.; Kouznetsova, V Varvara; Brekelmans, W.A.M.

doi:10.1007/978-3-7091-0283-1_7

Cited by 17 publications

(21 citation statements)

References 73 publications

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“…The FE 2 method has its origins in solid mechanics, [31], [46], [47], [48], [28], [27], [55], and has found considerable interest in academia and industry; as a versatile method FE 2 has been used in non-linear problems of elasticity and inelasticity. For recent, comprehensive overviews of the FE 2 method we refer to [30], [61] and [56]. In order to account for size-dependency observed in materials science, Kouznetsova et al [41], [42] have introduced a second-order homogenization into FE 2 .…”

Section: Introductionmentioning

confidence: 99%

The heterogeneous multiscale finite element method FE‐HMM for the homogenization of linear elastic solids

Eidel

Fischer

2016

Proc Appl Math and Mech

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The objective of the present paper is a formulation of the Heterogeneous Multiscale Finite Element Method (FE-HMM) for the homogenization of linear elastic solids in a geometrical linear frame, and doing so, for the first time, of a vector-valued field problem. The macro stiffness is estimated by stiffness sampling on heterogeneous microdomains in terms of a modified quadrature formula, which implies an equivalence of energy densities of the microscale with the macroscale. Existing a-priori estimates are assessed and used for optimal micro-macro refinement strategies in the H 1 -norm and the L 2 -norm.

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Section: Introductionmentioning

confidence: 99%

The heterogeneous multiscale finite element method FE‐HMM for the homogenization of linear elastic solids

Eidel

Fischer

2016

Proc Appl Math and Mech

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“…microstructure evolution has also been taken into account within the framework of finite inelasticity with 'internal variables' [15], and also to extend the results within the framework of nonsimple materials, various models particularly addressing nonlocal and higher order materials have also been proposed [16][17][18]. Such formulations become significant in problems in which macroscopic length scales are comparable to material internal length scales and the field equations for the classical continuum become ill-posed.…”

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confidence: 98%

“… Voigt ([46], p. 294) 17. This is the linearized representation of the Cauchy-Born rule (see footnote 2).…”

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confidence: 98%

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Genesis of the multiscale approach for materials with microstructure

2008

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This paper presents an overview of the origin of multiscale approaches in mechanics. While the pioneer molecular models of linear elastic bodies by Navier, Cauchy and Poisson were contradicted by experiments, the phenomenological energetic approach by Green still seems suitable for simple materials only. Voigt's molecular model, here reinterpreted in the light of contemporary mechanics, reconciled the two approaches providing a conceptual guideline for developing constitutive models based on a direct link between continuum and discrete solid mechanics. Such a theoretical background proves to be especially suitable for new complex materials. An example referred to masonry-like materials is given.

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“…As a result, F m will be decomposed into a constant part F M (the superscript M means Macroscopic) and a periodic part with zero average over Ω m 0 . These assumptions correspond to first order homogenization theory [Moulinec & Suquet, 1998, Geers et al, 2010. Data science is used in the mechanical science of materials to predict either u m or F m as a function of F M .…”

Section: Data Science Formulationmentioning

confidence: 99%

Data science for finite strain mechanical science of ductile materials

et al. 2018

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A mechanical science of materials, based on data science, is formulated to predict process-structure-propertyperformance relationships. Sampling techniques are used to build a training database, which is then compressed using unsupervised learning methods, and finally used to generate predictions by means of supervised learning methods or mechanistic equations. The method presented in this paper relies on an a priori deterministic sampling of the solution space, a K-means clustering method, and a mechanistic Lippmann-Schwinger equation solved using a self-consistent scheme. This method is formulated in a finite strain setting in order to model the large plastic strains that develop during metal forming processes. An efficient implementation of an inclusion fragmentation model is introduced in order to model this micromechanism in a clustered discretization. With the addition of a fatigue strength prediction method also based on data science, process-structureproperty-performance relationships can be predicted in the case of cold-drawn NiTi tubes.

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Computational homogenization

Cited by 17 publications

References 73 publications

The heterogeneous multiscale finite element method FE‐HMM for the homogenization of linear elastic solids

The heterogeneous multiscale finite element method FE‐HMM for the homogenization of linear elastic solids

Genesis of the multiscale approach for materials with microstructure

Data science for finite strain mechanical science of ductile materials

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