Hierarchical modeling of heterogeneous solids

Oden, J. Tinsley; Vemaganti, Kumar; Moës, Nicolas

doi:10.1016/s0045-7825(98)00224-2

Cited by 124 publications

(68 citation statements)

References 10 publications

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“…Taking into account the periodicity of the boundary displacement fluctuations leading to u þ ¼ u À þ M F Á X þ À X À ½ and antiperiodicity of the boundary tractions, that is, t þ ¼ Àt À , Eq. (A2) 2 can be formulated as 5 …”

Section: B0mentioning

confidence: 99%

“…Multiscale models are traditionally categorized into the homogenization method, where the length scales of micro-and macroproblems are sufficiently separate, and the concurrent method, see, e.g., Refs. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], which considers strong coupling between the scales. 2 This contribution details on the former one.…”

Section: Introductionmentioning

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

183

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: B0mentioning

confidence: 99%

Section: Introductionmentioning

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

183

138

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“…In the first stage of the adaptation (discretization error) the elements are refined to the size of unit cell with the purpose of replacing the refined and flagged elements in this second stage with an adjusted composite micro-structure. As in many studies [2,4,7] the composite micro-structure is chosen as a particle-matrix structure. In order to have an inexpensive Section 8: Multiscales and homogenization Fig.…”

Section: Modeling Error and Resolving The Microstructurementioning

confidence: 99%

“…For a better analysis of the composite substructure in nonlinear regimes and near highly stressed regions, the mesh refinement needs to be complemented by a model adaptivity method. Though the concept of modeling error is developed in some researches [2,[4][5][6], there are still only few researches who used a physical parameter, like strain-gradient or damage, to specify the adaptation zone [1,7]. Combining mathematical and physical error estimation together with using strain-gradient to identify the designated sub-domains are the main focuses of this work.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Multi‐scale Modeling Error Estimation for Composites

Hajibeik

Temizer

Löhnert

et al. 2011

Proc Appl Math and Mech

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Analysing composite materials through efficient higher-order homogenization techniques requires accurate simulations of highly heterogeneous domains. However the validity of the homogenization response in regions with highly localized variations is still questionable. A scale adaptation strategy is developed for a more accurate analysis of generally inelastic macrostructures, where a transition from a homogenized description to an explicit micro-structural resolution is pursued. The designated zones of interest are indicated through two different methods of error estimation. First for mesh refinement approaches, the discretization error is estimated using an indicator developed by Zienkiewicz-Zhu and studying L2 norm. Furthermore a second adaptation zone is identified based on a post-processing step on the homogenized solution and corresponds to regions with high strain-gradients, the gradient of the deformation gradient. In the designated sub-domains a micro-structure representation will replace the homogenized area. The introduced scale-adaptivity method returns an inexpensive and more accurate analysis of composite materials.

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“…One of these strategies uses the theory of the homogenization of periodic media [4][5][6]. Other developments and the associated computational approaches can be found in [7][8][9][10][11][12][13][14]. Besides periodicity, these strategies rely on the fundamental assumption that the ratio between the two scales is small.…”

Section: Introductionmentioning

confidence: 99%

On a mixed and multiscale domain decomposition method

Ladevèze

Néron

Gosselet

2007

Computer Methods in Applied Mechanics and Engineering

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This paper presents a reexamination of a multiscale computational strategy with homogenization in space and time for the resolution of highly heterogeneous structural problems, focusing on its suitability for parallel computing. Spatially, this strategy can be viewed as a mixed, multilevel domain decomposition method (or, more accurately, as a "structure decomposition" method). Regarding time, a "parallel" property is also described. We also draw bridges between this and other current approaches.Key words: domain decomposition, multiscale, computational mechanics IntroductionIn the structural mechanics field, one can observe a surge of interest in the multiscale analysis of structures with complex microstructural geometry and/or complex behavior. When accurate solutions are required, calculations must be performed on a finely discretized model of the structure (defined on what is called the "micro" level) consistent with short lengths of variation, both in space and in time. A major application is the analysis of composite structures described on the microscale or on the mesoscale [1]; there are also other applications, such as in [2,3]. These types of situations lead to problems with very large numbers of degrees of freedom and computation costs which are prohibitive if one uses classical finite element codes. One of the main objectives * Corresponding author. E-mail: pierre.ladeveze@lmt.ens-cachan.fr Preprint of the last few decades has been to develop efficient and robust computational strategies suitable for these types of problems. One of these strategies uses the theory of the homogenization of periodic media [4][5][6]. Other developments and the associated computational approaches can be found in [7][8][9][10][11][12][13][14]. Besides periodicity, these strategies rely on the fundamental assumption that the ratio between the two scales is small. Thus, the boundary zones require specific treatment because in these zones the microstructure cannot be homogenized. Here, we follow a recently introduced multiscale computational strategy for nonlinear evolution problems. This strategy involves an automatic homogenization technique in space as well as in time [15,16] which is an extension of previous works limited to space alone [17,18]. This strategy, developed in a general framework, makes no a priori assumption regarding the form of the solution and, therefore, does not suffer from the limitations of standard homogenization techniques; moreover, it relies on an iterative algorithm. Until now, this strategy has been developed in the framework of small displacements of (visco)plastic structures under possible contact with or without friction. This paper is a reconsideration of this computational strategy in order to assess its adaptability to parallel computing. Therefore, we examine its three main characteristics. The first characteristic is that it can be viewed as a mixed, multilevel domain decomposition method or, more precisely as will explained further on, as a "structure decomposition" method. Ind...

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Hierarchical modeling of heterogeneous solids

Cited by 124 publications

References 10 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

Adaptive Multi‐scale Modeling Error Estimation for Composites

On a mixed and multiscale domain decomposition method

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