Asymptotic expansion homogenization for heterogeneous media: computational issues and applications

Chung, Peter W.; Tamma, Kumar K.; Namburu, Raju R.

doi:10.1016/s1359-835x(01)00100-2

Cited by 139 publications

(74 citation statements)

References 8 publications

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“…In periodic structures with two scales, AEH consists of uncoupling those scales into a microscale and a macroscale. The general procedure for AEH applying the Finite Elements Method (FEM) comprises the following steps, as stated by Chung et al (2001): (1) definition of a global body X in a coordinate system x i , consisting of the structure without microstructural details, and a local body Y in a coordinate system y i , consisting of one microstructure period; (2) meshing of X and Y in finite elements; (3) primary variable approximation by an asymptotic series in scale parameter ∈ , which relates the two coordinate systems x i and y i ; (4) derivation of hierarchical equations, specific for the treated problem; (5) definition of a homogenized elastic tensor in the microscale Y; (6) resolution of the homogenized problem in the macroscale X. The homogenization deals with partial differential equations related to heterogeneous materials with periodical structure, considering the assumption that the amount of periodic cells tends to infinity.…”

Section: General Aspectsmentioning

confidence: 99%

See 1 more Smart Citation

Multiscale modeling of the elastic moduli of lightweight aggregate concretes: numerical estimation and experimental validation

Farage

Beaucour

Barra

et al. 2009

Rem: Rev. Esc. Minas

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Section: General Aspectsmentioning

confidence: 99%

“…According to Chung et al (2001), by considering a material whose microstructure is composed of multiple phases, periodically distributed over the body (Sanchez-Palencia, 1980;Farage et al, 2005), the periodic elastic material properties are defined by the following relations:…”

Section: Aeh Applied To Linear Elasticitymentioning

confidence: 99%

Multiscale modeling of the elastic moduli of lightweight aggregate concretes: numerical estimation and experimental validation

Farage

Beaucour

Barra

et al. 2009

Rem: Rev. Esc. Minas

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“…An extensive body of literature is devoted to study this technique among which we refer to Refs. [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. Reviews of the different multiscale approaches can be found in Refs.…”

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

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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“…Analysis is the most popular one, due to its flexibility in handling different type of problems, being computer programming friendly and the availability of commercial FE software [8,15]. Extensive studies have been carried out about FE homogenization techniques.…”

Section: Literature Reviewmentioning

confidence: 99%

“…Extensive studies have been carried out about FE homogenization techniques. Their computational issues and applications were overviewed and a relationship between macro and micro-scale properties were developed [8,41]. Mathematical expansions for modeling physical phenomena on inhomogeneous materials with periodic microstructure and also the explicit mathematical equations which describes the local stress and strain fields associated with a given global domain were derived [50].…”

Section: Literature Reviewmentioning

confidence: 99%

Characterization of Homogenized Mechanical Properties of Porous Ceramic Materials Based on Their Realistic Microstructure

Rastkar¹

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The recent advances in the Materials Engineering have led to the development of new materials with customized microstructure in which the properties of its constituents and their geometric distribution have a considerable effect on determination of the macroscopic properties of the substance. Direct inclusion of the material microstructure in the analysis on a macro level is challenging since spatial meshes created for the analysis should have enough resolution to be able to accurately capture the geometry of the microstructure. In most cases this leads to a huge finite element model which requires a substantial amount of computational resources.To circumvent this limitation a number of homogenization techniques were developed. By considering a small element of the material, referred to as Representative Volume Element (RVE), homogenization methods make it possible to include the effects of a materials microstructure on the overall properties at the macro level.However, complexity of the microstructure geometry and the necessity of satisfying periodic boundary conditions introduce additional difficulties into the analysis procedure.In this dissertation we propose a hybrid homogenization method that combines Asymptotic homogenization with MeshFree Solution Structures Method (SSM). Our vi approach allows realistic inclusion of complex geometry of the microstructure that can be captured from micrographs or micro CT scans. In addition to unprecedented flexibility in handling complex geometries, this method also provides a completely automatic analysis procedure. Using meshfree solution structures simplifies meshing to creating a simple cartesian grid which only needs to contain the domain. This also eliminates manual modifications which usually needs to be performed on meshes created from image data.A computational platform is developed in C++ based on meshfree/asymptotic method. In this platform also a novel meshfree solution structure is designed to provide exact satisfaction of periodic boundary conditions for boundary value problems such as homogenization. Performance of the developed platform is tested over 2D and 3D domains against previously published data and/or conventional finite element methods. After getting satisfactory results, homogenized properties are used to compute localized stress and strain distributions over inhomogeneous structures.Furthermore, effects of geometric features of pores/inclusions on homogenized mechanical properties is investigated and it is demonstrated that the developed platform could provide an automated quantitative analysis tool for studying effects of different design parameters on homogenized properties.

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Asymptotic expansion homogenization for heterogeneous media: computational issues and applications

Cited by 139 publications

References 8 publications

Multiscale modeling of the elastic moduli of lightweight aggregate concretes: numerical estimation and experimental validation

Multiscale modeling of the elastic moduli of lightweight aggregate concretes: numerical estimation and experimental validation

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

Characterization of Homogenized Mechanical Properties of Porous Ceramic Materials Based on Their Realistic Microstructure

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