A class of general algorithms for multi-scale analyses of heterogeneous media

Terada, Kenjiro; Kikuchi, Noboru

doi:10.1016/s0045-7825(01)00179-7

Cited by 418 publications

(254 citation statements)

References 35 publications

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“…Furthermore, most discoveries in this paper actually exist elsewhere in various previous publications [12][13][14][15][16]; however, it is useful to provide a more systematic explanation for mechanics-based homogenization. Another important point is that theoretically the mathematical asymptotic homogenization process requires the micro-cell to be small or infinitely small to assume convergence of the process, but this is not necessary for mechanics-based homogenization.…”

Section: Introductionmentioning

confidence: 85%

Homogenization Method for Designing Novel Architectured Cellular Materials

Ma¹

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. Two types of novel architectured cellular materials have been developed [1,2]. To estimate the effective material properties and to optimally design such materials, a suitable homogenization method was needed. An existing asymptotic homogenization process has provided an effective means for the multiscale modeling of continuum solids; however, that method was not easily adapted to the aforementioned materials, which are naturally discrete systems. First, we developed a strain-based homogenization method equivalent to the existing asymptotic homogenization method, but it was developed based on an engineering approach rather than on a mathematical approach such as the one used in asymptotic homogenization. The new approach separates the strain field into a homogenized strain field and a strain variation field in the local cellular domain superposed on the homogenized strain field. The Principle of Virtual Displacements for the relationship between the strain variation field and the homogenized strain field is then used to condense the strain variation field onto the homogenized strain field, and the homogenization process becomes a coordinate reduction process comparable to the Guyan Reduction used in structural dynamics analyses. The characteristic modes and the stress recovery process are also discussed. The new method is then extended to a stress-based homogenization process based on the Principle of Virtual Forces, and it is further applied to address the discrete systems of the aforementioned architectured cellular materials.

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Section: Introductionmentioning

confidence: 85%

Homogenization Method for Designing Novel Architectured Cellular Materials

Ma¹

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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“…Following similar approaches [58,16,61,14,57,41,42,38,8,9,62,15,28], the computational homogenization is performed through a nested solution scheme for the coupled multiscale numerical analysis. A numerical computation of the representative volume element is carried out simultaneously in order to obtain constitutive equations at the macroscopic scale.…”

Section: The Reduced Model Multiscale Methods (R3m)mentioning

confidence: 99%

“…In techniques of this type, e.g. [58,16,61,14,57,41,42,38,8,9,62,15,28], among others, the macroscopic deformation (gradient) tensor is calculated for every material point of the macrostructure and is next used to formulate kinematic boundary conditions to be applied on the associated microstructural representative volume element (RVE). After the solution of the microstructural boundary value problem, the macroscopic stress tensor is obtained by averaging the resulting microstructural stress field over the volume of the microstructural cell.…”

Section: Introductionmentioning

confidence: 99%

The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains

Yvonnet¹,

He²

2007

Journal of Computational Physics

263

164

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To cite this version:Julien Yvonnet, Qi-Chang He. The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. AbstractThis paper presents a new multi-scale method for the homogenization analysis of hyperelastic solids undergoing finite strains. The key contribution is to use an incremental nonlinear homogenisation technique in tandem with a model reduction method, in order to alleviate the complexity of multiscale procedures, which usually involve a large number of nonlinear nested problems to be solved. The problem associated with the Representative Volume Element (RVE) is solved via a model reduction method (Proper Orthogonal Decomposition). The reduced basis is obtained through pre-computations on the RVE. The technique, coined as Reduced Model Multiscale Method (R3M), allows reducing significantly the computation times, as no large matrix needs to be inverted, and as the convergence of both macro and micro problems is enhanced. Furthermore, the R3M drastically reduces the size of the data base describing the history of the micro problems. In order to validate the technique in the context of porous elastomers at finite strains, a comparison between a full and a reduced multiscale analysis is performed through numerical examples, involving different micro and macro structures, as well as different nonlinear models (Neo-Hookean, Mooney-Rivlin). It is shown that the R3M gives good agreement with the full simulations, at lower computational and data storage requirements.

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“…This idea has been further developed in Refs. [222] and [352][353][354][355][356][357][358][359][360][361][362][363]. Mo€ es et al [364] presented an extended version of the classical finite element method, referred to as XFEM, to solve microproblems involving complex geometries [365].…”

Section: Analysis At the Rve Levelmentioning

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

185

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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A class of general algorithms for multi-scale analyses of heterogeneous media

Cited by 418 publications

References 35 publications

Homogenization Method for Designing Novel Architectured Cellular Materials

Homogenization Method for Designing Novel Architectured Cellular Materials

The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains

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

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