Boundary element method for calculation of effective elastic moduli in 3D linear elasticity

Grzhibovskis, Richards; Rjasanow, Sergej; Andrä, Heiko; Zemitis, A.

doi:10.1002/mma.1233

Cited by 6 publications

(4 citation statements)

References 21 publications

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“…3), the variational identity (19) turns into (17). The transformation of the righthand side is given in (10). Next, taking the test function (v 1 ,v):=(1,1), (19) (19) is well-posed by means of the Lax-Milgram theorem.…”

Section: Stabilized Formulationmentioning

confidence: 99%

“…As a result, an FMM-BEM can be applied to the RVE including a large amount of periodic cells. [7][8][9] A different BEM method is proposed in Grzhibovskis et al, 10 where the RVE is just one periodic cell, and the homogeneous coefficients are computed by the energy method with prescribed constant elastic strains.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we propose a BEM different from Grzhibovskis et al 10 as we make use of the periodic structure in the spirit of the previous studies. 2,3,11 Besides numerical experiments, we provide an existence and convergence analysis of our boundary integral formulation and its boundary element counterparts, respectively.…”

Section: Introductionmentioning

confidence: 99%

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A boundary element method for homogenization of periodic structures

Lukáš

Zapletal

Bouchala

2019

Math Methods in App Sciences

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Homogenized coefficients of periodic structures are calculated via an auxiliary partial differential equation in the periodic cell. Typically, a volume finite element discretization is employed for the numerical solution. In this paper, we reformulate the problem as a boundary integral equation using Steklov-Poincaré operators. The resulting boundary element method only discretizes the boundary of the periodic cell and the interface between the materials within the cell. We prove that the homogenized coefficients converge super-linearly with the mesh size, and we support the theory with examples in two and three dimensions. KEYWORDS boundary element method, homogenization MSC CLASSIFICATION 65N38; 65N12; 35B27 | INTRODUCTIONSolving a boundary value problem, which involves materials with composite microstructure, is computationally demanding. Therefore, we look for homogeneous (constant) material coefficients imitating the original microstructure so that the solution to the original problem with a highly oscillating material function is in a sense close to the solution of a related problem with a constant material function. There are two well-known approaches to homogenization. In both approaches, auxiliary boundary value problems in the so-called representative volume element (RVE) are solved. In the first approach, the geometry of the microstructure does not need to be periodic, and a large portion of the domain has to be covered in the auxiliary problems. In such cases, energy methods 1 are often employed. The methods provide effective coefficients that preserve the energy of the solution for a fixed type of boundary conditions. The second approach, which our paper actually deals with, assumes a periodic microstructure. As stated in Allaire, 2 energy methods do not take full advantage of the periodicity. The homogenized coefficients are instead determined by solutions to auxiliary problems with periodic boundary conditions. For a presentation of mathematical theory and methods for the periodic case, we refer to Cioranescu and Donato. 3 As far as numerical methods for the solution to the auxiliary problems are concerned, volume discretization techniques such as finite element methods prevail in literature, cf Rohan and Lukeš. 4 In this paper, we consider boundary

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Section: Stabilized Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A boundary element method for homogenization of periodic structures

Lukáš

Zapletal

Bouchala

2019

Math Methods in App Sciences

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“…Torquato [1997] obtained exact series expansions for the effective stiffness tensor of macroscopically anisotropic, two-phase composite media in terms of the powers of the "elastic polarizabilities". Numerical estimates are also available, for example, with the finite element method in [Segurado and Llorca 2002;2006;Zohdi and Wriggers 2005] and with boundary element method in [Grzhibovskis et al 2010]. In addition, various effective medium theories and variational bounds have been generalized to estimate the overall three-dimensional anisotropic elastic properties (for example, [Willis 1977;Benveniste 1987;Ponte Castañeda and Willis 1995;Milton 2002;Torquato 2002]).…”

mentioning

confidence: 99%

Evaluation of the effective elastic moduli of particulate composites based on Maxwell’s concept of equivalent inhomogeneity: microstructure-induced anisotropy

Кущ¹,

Mogilevskaya²,

Stolarski³

et al. 2013

J. Mech. Mater. Struct.

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Maxwell's concept of equivalent inhomogeneity is employed for evaluating the effective elastic properties of macroscopically anisotropic particulate composites with isotropic phases. The effective anisotropic elastic properties of the material are obtained by comparing the far-field solutions for the problem of a finite cluster of isotropic particles embedded in an infinite isotropic matrix with those for the problem of a single anisotropic equivalent inhomogeneity embedded in the same matrix. The former solutions precisely account for the interactions between all particles in the cluster and for their geometrical arrangement. Illustrative examples involving periodic (simple cubic) and random composites suggest that the approach provides accurate estimates of their effective elastic moduli.

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Combining Maxwell’s methodology with the BEM for evaluating the two-dimensional effective properties of composite and micro-cracked materials

Mogilevskaya

Crouch

2012

Comput Mech

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Boundary element method for calculation of effective elastic moduli in 3D linear elasticity

Cited by 6 publications

References 21 publications

A boundary element method for homogenization of periodic structures

A boundary element method for homogenization of periodic structures

Evaluation of the effective elastic moduli of particulate composites based on Maxwell’s concept of equivalent inhomogeneity: microstructure-induced anisotropy

Combining Maxwell’s methodology with the BEM for evaluating the two-dimensional effective properties of composite and micro-cracked materials

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