Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites

International audienceWe report on the mathematical analysis of two different, FFT-based, numerical schemes for the homogenization of composite media within the framework of linear elasticity: the basic scheme of Moulinec and Suquet (1994, 1998) [9] and [10], and the energy-based scheme of Brisard and Dormieux (2010) [13]. Casting these two schemes as Galerkin approximations of the same variational problem allows us to assert their well-posedness and convergence. More importantly, we extend in this work their domains of application, by relieving some stringent conditions on the reference material which were previously thought necessary. The origins of the flaws of each scheme are identified, and a third scheme is proposed, which seems to combine the strengths of the basic and energy-based schemes, while leaving out their weaknesses. Finally, a rule is proposed for handling heterogeneous pixels/voxels, a situation frequently met when images of real materials are used as input to these schemes

FFT-based methods for the mechanics of composites: A general variational framework

International audienceFor more than a decade, numerical methods for periodic elasticity, based on the fast Fourier transform, have been used successfully as alternatives to more conventional (fem, bem) numerical techniques for composites. These methods are based on the direct, point-wise, discretization of the Lippmann-Schwinger equation, and a subsequent truncation of underlying Fourier series required for the use of the fast Fourier transform. The basic FFT scheme is very attractive, because of its simplicity of implementation and use. However, it cannot handle pores or rigid inclusions, for which a specific (and significantly more involved) treatment is required. In the present paper, we propose a new FFT-based scheme which is as simple as the basic scheme, while remaining valid for infinite contrasts. Since we adopted an energy principle as an alternative to the Lippmann-Schwinger equation, our scheme is derived within a variational framework. As a by-product, it provides an energetically consistent rule for the homogenization of boundary voxels, a question which has been pending since the introduction of Fourier-based methods

A Galerkin approach to FFT-based homogenization methods

International audienceSince their introduction by Moulinec and Suquet [1, 2], FFT-based full-field simulations of the mechanical properties of composites have become increasingly popular, with applications ranging from the linear elastic behaviour of cementitious materials [3], to the plasticity of polycrystals [4]. Recently, the authors have proposed [5] a new formulation of these numerical schemes, based on the energy principle of Hashin and Shtrikman [6]. While similar in principle to the original scheme of Moulinec and Suquet, the new scheme was shown to be much better-behaved. Indeed, convergence of the scheme is guaranteed for any contrast, without having to resort to augmented Lagrangian approaches [7]. Besides, convergence of the new scheme is generally much faster. However, the new scheme has two drawbacks. First, the reference material must be stiffer (or softer) than all constituants of the composite; this is not always possible, for example when the composite contains both pores and rigid inclusions. Second, the scheme requires the preliminary computation of the so-called consistent Green operator, which turned out to be a difficult task in three dimensions. In order to relax these requirements, an in-depth mathematical analysis of these schemes was carried out by the authors [8]. In this paper, the Lippmann-Schwinger equation and its variational form will briefly be recalled. The Galerkin approach will then be adopted for the discretization of this equation, and it will be shown that the basic scheme of [1] as well as the energy scheme proposed in [5] can both be viewed as well-posed Galerkin approximations of the Lippmann-Schwinger equation. Contrary to what was previously believed [7, 5] these approximations are convergent, regardless of the reference material (provided that its stiffness is positive definite). Comparison of these two approximations leads to the derivation of the so-called filtered, non-consistent approach, which combines the assets of the two former methods. Finally, some applications will be shown. In particular, the important problem of heterogeneous voxels will be addressed

Small-angle scattering of dense, polydisperse granular porous media: Computation free of size effects

19 pagesInternational audienceSmall-angle x-ray and neutrons scattering is a widespread experimental tool for the investigation of the microstructure of random heterogeneous materials. Validation of (computer-generated) model microstructures often requires the numerical computation of the scattering intensity, which must be carried out with great care due to finite size effects. In this paper, a new method for this computation is presented. It is superior to previously existing methods for three reasons: First, it applies to any type of microstructure (not necessarily granular). Second, closed-form expressions of the size effects inherent to the proposed method can be rigorously derived and removed (in this sense, our method is free of size effects). Third, the complexity of the new algorithm is linear and the computation can easily be updated to account for local changes of the microstructure, while most existing algorithms are quadratic and any change of the microstructure requires a full recomputation. The present paper provides full derivation and validation of this method. Application to the computation of the scattering intensity of dense, polydisperse assemblies of spheres is then presented. A new, simple algorithm for the generation of these dense configurations is introduced. Finally, the results are critically reviewed in the perspective of hardened cement pastes

Corrigendum to “Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites” [Comput. Methods Appl. Mech. Engrg. 217–220 (2012) 197–212]

International audienceAssumption 1 in our original paper can be replaced with the following, less stringent assumption

Hashin-Shtrikman bounds on the bulk modulus of a nanocomposite with spherical inclusions and interface effects

International audienceNanocomposites are becoming more and more popular and mechanical models are needed to help with their design and optimization. One of the key issues to be addressed by such models is the surface-stresses arising at the inclusion-matrix boundary, due to its high curvature. In this paper, we show that, contrary to what has previously been suggested, polarization techniques can be employed in the context of composites with interface effects. This requires a specific mathematical treatment of the interface, which must be regarded as a thin elastic layer. We then apply the proposed general methods to the specific case of nanocomposites with monodisperse spherical inclusions, for which a lower bound on the bulk modulus is derived. When interface effects are disregarded, this bound coincides with the classical Hashin-Shtrikman bound. In the presence of interface effects, we show that the existing Mori-Tanaka estimate is in fact a lower bound on the effective bulk modulus. Finally, lower bounds on the effective bulk modulus of nanocomposites with polydisperse spherical inclusions are proposed. Although this result can be considered as a by-product of the previous one, it is new, and has no published Mori-Tanaka counterpart

Towards improved Hashin--Shtrikman bounds on the effective moduli of random composites

Les bornes de Hashin et Shtrikman sur les propriétés effectives de composites sont valides pour une classe très large de matériaux. Néanmoins, elle ne prennent en compte qu’une information très limitée sur la microstructure (fraction volumique de chaque phase dans le cas isotrope). De ce fait, ces bornes ne sont en général pas très serrées. Dans ce travail, on présente une tentative d’amélioration de ces bornes par addition explicite de la fraction volumique locale au jeu des descripteurs locaux de la microstructure. On montre que, de façon inattendue, cette approche échoue, au sens où elle conduit aux bornes classiques. On montre ensuite que ce résultat négatif s’applique à tous descripteurs locaux de la microstructure faiblement isotropes (en un sens qui est précisé dans cet article). Cela suggère que des bornes améliorées pourraient être obtenues en considérant des descripteurs anisotropes

Overview of FFT-based homogenization techniques from the Galerkin point of view (slides)

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A variational form of the equivalent inclusion method for numerical homogenization

International audienceDue to its relatively low computational cost, the equivalent inclusion method is an attractive alternative to traditional full-field computations of heterogeneous materials formed of simple inhomogeneities (spherical, ellipsoidal) embedded in a homogeneous matrix. The method can be seen as the discretization of the Lippmann--Schwinger equation with piecewise polynomials. Contrary to the original approach of Moschovidis and Mura, who discretized the strong form of the Lippmann--Schwinger equation through Taylor expansions, we propose in the present paper a Galerkin discretization of the weak form of this equation. Combined with the new, mixed boundary conditions recently introduced by the authors, the resulting method is particularly well-suited to homogenization. It is shown that this new, variational approach has a number of benefits: (i) the resulting linear system is well-posed, (ii) the numerical solution converges to the exact solution as the maximum degree of the polynomials tends to infinity and (iii) the method can provide rigorous bounds on the apparent properties of the statistical volume element, provided that the matrix is stiffer (or softer) than all inhomogeneities. This paper presents the formulation and implementation of the new, variational form of the equivalent inclusion method. Its efficiency is investigated through numerical applications in 2D and 3D elasticity

New boundary conditions for the computation of the apparent stiffness of statistical volume elements

International audienceWe present a new auxiliary problem for the determination of the apparent stiffness of a Statistical Volume Element (SVE). The SVE is embedded in an infinite, homogeneous reference medium, subjected to a uniform strain at infinity, while tractions are applied to the boundary of the SVE to ensure that the imposed strain at infinity coincides with the average strain over the SVE. The main asset of this new auxiliary problem resides in the fact that the associated Lippmann-Schwinger equation involves without approximation the Green operator for strains of the infinite body, which is translation-invariant and has very simple, closed-form expressions. Besides, an energy principle of the Hashin and Shtrikman type can be derived from this modified Lippmann-Schwinger equation, allowing for the computation of rigorous bounds on the apparent stiffness. The new auxiliary problem requires a cautious mathematical analysis, because it is formulated in an unbounded domain. Observing that the displacement is irrelevant for homogenization purposes, we show that selecting the strain as main unknown greatly eases this analysis. Finally, it is shown that the apparent stiffness defined through these new boundary conditions "interpolates" between the apparent stiffnesses defined through static and kinematic uniform boundary conditions, which casts a new light on these two types of boundary conditions