Convergence of iterative methods based on Neumann series for composite materials: theory and practice

Iterative Fast Fourier Transform methods are useful for calculating the fields in composite materials and their macroscopic response. By iterating back and forth until convergence, the differential constraints are satisfied in Fourier space, and the constitutive law in real space. The methods correspond to series expansions of appropriate operators and to series expansions for the effective tensor as a function of the component moduli. It is shown that the singularity structure of this function can shed much light on the convergence properties of the iterative Fast Fourier Transform methods. We look at a model example of a square array of conducting square inclusions for which there is an exact formula for the effective conductivity (Obnosov). Theoretically some of the methods converge when the inclusions have zero or even negative conductivity. However, the numerics do not always confirm this extended range of convergence and show that accuracy is lost after relatively few iterations. There is little point in iterating beyond this. Accuracy improves when the grid size is reduced, showing that the discrepancy is linked to the discretization. Finally, it is shown that none of the three iterative schemes investigated over-performs the others for all possible microstructures and all contrasts.Comment: 41 pages, 14 figures, 1 tabl

ON THE CONVERGENCE OF THREE ITERATIVE FFT-BASED METHODS FOR COMPUTING THE MECHANICAL RESPONSE OF COMPOSITE MATERIALS

International audienceThe last decade has witnessed a growing interest for the so-called " FFT-based methods " for computing the overall and local properties of heterogeneous materials submitted to mechanical solicitations. Since the original method was introduced by Moulinec and Suquet [1], several authors have proposed different algorithms to better deal with non-linear materials or with materials whith highly contrasted mechanical properties between their constituents. The present paper aims to compare three methods of this family of algorithms which were designed to accelerate the convergence of the scheme. The study concerns a linear elastic material-although the methods involved can be extended into the case of non-linear behavior-submitted to a prescribed overall strain E. The stiffness tensor c(x) of the material varies with the position x. The numerical method proposed by Moulinec & Suquet lies on the iterative resolution of the Lippmann-Schwinger equation and can be summarized by the following relation between two successive iterates ε i and ε i+1 of the strain field: ε i+1 (x) = −Γ 0 * (c(x) − c 0) : ε i (x) + E (1

Comparaison de 3 méthodes à base de transformées de Fourier pour le calcul des propriétés mécaniques de matériaux hétérogènes

International audienceLes simulations numériques du comportement mécanique des matériaux hétérogènes doivent tenir compte de la complexité de leurs microstructures. Au cours des deux dernières décennies, les méth-odes dites " à base de transformées de Fourier " ont retenu l'attention par leur efficacité et la simplicité de leur mise en oeuvre. On se propose dans cette étude de comparer 3 méthodes numériques de cette famile, proposant une convergence accélérée en comparaison de celle du schéma itératif proposé par Moulinec & Suquet ([1], [2]). On s'intéressera tout particulièrement aux conditions de convergence et au choix des paramètres permettant une convergence optimale de ces schémas. Mots clés — transformée de Fourier, matériaux hétérogène

Comparison of different FFT-based methods for computing the mechanical response of heteregoneous materials

International audienceThe last decade has witnessed a growing interest for the so-called " FFT-based methods " for computing the overall and local properties of heterogeneous materials submitted to mechanical solicita-tions. Since the original method was introduced by Moulinec and Suquet [1], several authors have proposed different algorithms to better deal with non-linear materials or with materials with highly contrasted mechanical properties between their constituents. The study concerns a linear elastic material-although the methods involved can be extended into the case of non-linear behavior-submitted to a prescribed overall strain E. The stiffness tensor c(x) of the material varies with the position x. The numerical method proposed by Moulinec & Suquet lies on the iterative resolution of the Lippmann-Schwinger equation and can be summarized by the following relation between two successive iterates ε i and ε i+1 of the strain field: ε i+1 (x) = −Γ 0 * (c(x) − c 0) : ε i (x) + E , where c 0 is the stiffness tensor of a reference medium supposed to be linear elastic, where Γ 0 is a Green operator associated to c 0 and where * denotes the convolution operator. Eyre & Milton [2], Michel et al. [3] and Monchiet & Bonnet [4] proposed different schemes to accelerate the convergence of the initial scheme. It has been recently demonstrated in [5] that the two first schemes are particular cases of the last one. On the other hand, Zeman et al. [6] proposed to use a conjugate gradient method for solving the Lippmann-Schwinger equation. The present paper aims to compare these different methods with a special attention paid to their relative efficiency and their rates of convergence

Analysis of inhomogeneous materials at large strains using fast Fourier transform

invited lecture at the IUTAM Symposium - Stuttgart 2001International audienceThis paper focuses on a numerical method which has been recently developed to analyze the response of highly inhomogeneous materials, often with complex microstructure. This numerical method is based on Fast Fourier Transforms and allows to make direct use of digital images of the ”real” microstructure in the numerical simulation. The case of elastic nonhomogeneous phases is reduced to an integral equation (Lippman- Schwinger equation) which is solved iteratively. A nice feature of the method is that it involves a multiplication in Fourier space, a multiplication in real space, a FFT and an inverse FFT. The two first operations can be easily parallelized. It has been extended to various nonlinear behaviours (elastoplasticity, phase transformations, 3d analysis of texture evolution in polycrystals). Our work has been mainly concentrated on providing reference results to assess the accuracy of theoretical estimates for nonlinear compos- ites with simple behaviours (ideally plastic or power-law materials)

Effective flow surface of porous materials with two populations of voids under internal pressure: II. full-field simulations

International audienceThis study is devoted to the effective plastic flow surface of a bi-porous material saturated by a fluid. Highly irradiated uranium dioxide is a typical example of such a material. In part I of this study, a GTN-type approximation of the effective plastic flow surface has been derived. In this second part, the predictions of this new model are compared with full-field numerical simulations performed with a numerical method based on Fast Fourier Transforms. This method is successfully applied to voided materials with a Gurson matrix where the voids are subjected to internal pressure. Different microstructures containing a large number of spherical or ellipsoidal voids are investigated. The deviation from isotropy of their mechanical response is measured by a new criterion

Surface de charge de plasticité effective pour un matériau biporeux saturé : modélisation analytique et simulations numériques

Ce travail concerne la modélisation de la surface de charge de plasticité effective pour un matériau biporeux saturé. L’application visée concerne le combustible nucléaire UO2 fortement irradié, étudié par l’IRSN afin d’estimer la tenue des crayons combustibles lors d’un accident de réactivité. Ce matériau a la particularité de posséder, en première approximation, deux populations de cavités saturées : des bulles intragranulaires sphériques, de l’ordre de quelques nanomètres et des bulles intergranulaires, plutôt lenticulaires et de l’ordre du micromètre. A haute température, ce matériau est principalement ductile et les deux populations de cavités sont soumises à des pressions internes dues aux gaz de fission qu’elles contiennent. Des modèles pour déterminer la surface de charge de plasticité effective pour un tel matériau biporeux saturé ont été proposés dans [Vincent, Monerie, Suquet, 2009] et sont brièvement rappelés. Une procédure d’homogénéisation en deux étapes a été retenue : en homogénéisant le comportement du milieu à la plus petite échelle contenant les bulles intragranulaires sphériques, puis en homogénéisant le comportement du milieu à l’échelle supérieure constitué du milieu précédent dans lequel sont placées les bulles intergranulaires ellipsoïdales aplaties. Le modèle principal est basé sur l’approche variationnelle de Ponte Castañeda (ou méthode sécante modifiée) appliquée à une matrice de type Gurson contenant des cavités ellipsoïdales aplaties. Un deuxième modèle, généralisant l’approche de [Gologanu et al., 1994] aux matériaux à matrice compressible, également proposé dans l’article de [Vincent et al., 2009], a récemment été étendu au cas des cavités saturées. La pertinence des modèles est vérifiée par comparaison avec des simulations numériques par transformées de Fourier rapides (FFT, [Michel et al., 2000]). Des simulations sur des volumes contenant plus de 500 cavités ellipsoïdales dans une matrice de Gurson ont notamment été réalisées. L’effet des pressions dans les cavités est correctement reproduit par les modèles

Porous polycrystal plasticity modeling of neutron-irradiated austenitic stainless steels

A micromechanical model for quantifying the simultaneous influence of irradiation hardening and swelling on the mechanical stiffness and strength of neutron-irradiated austenitic stainless steels is proposed. The material is regarded as an aggregate of equiaxed crystalline grains containing a random dispersion of pores (large voids due to large irradiation levels) and exhibiting elastic isotropy but viscoplastic anisotropy. The overall properties are obtained via a judicious combination of various bounds and estimates for the elastic energy and viscoplastic dissipation of voided crystals and polycrystals. Reference results are generated with full-field numerical simulations for dense and voided polycrystals with periodic microstructures and crystal plasticity laws accounting for the evolution of dislocation and Frank loop densities. These results are calibrated with experimental data available from the literature and are employed to assess the capabilities of the proposed model to describe the evolution of mechanical properties of highly irradiated Solution Annealed 304L steels at 330°C. The agreement between model predictions and simulations is seen to be quite satisfactory over the entire range of porosities and loadings investigated. The expected decrease of overall elastic properties and strength for porosities observed at large irradiation levels is reported. The mathematical simplicity of the proposed model makes it particularly apt for implementation into finite-element codes for structural safety analyses.Centro Tecnológico Aeroespacia

Experimental characterization of the intragranular strain field in columnar ice during transient creep

A digital image correlation (DIC) technique has been adapted to polycrystalline ice specimens in order to characterize the development of strain heterogeneities at an intragranular scale during transient creep deformation (compression tests). Specimens exhibit a columnar microstructure so that plastic deformation is essentially two-dimensional, with few in-depth gradients, and therefore surface DIC analyses are representative of the whole specimen volume. Local misorientations at the intragranular scale were also extracted from microstructure analyses carried out with an automatic texture analyzer before and after deformation. Highly localized strain patterns are evidenced by the DIC technique. Local equivalent strain can reach values as much as an order of magnitude larger than the macroscopic average. The structure of the strain pattern does not evolve with strain in the transient creep regime. Almost no correlation between the measured local strain and the Schmid factor of the slip plane of the underlying grain is observed, highlighting the importance of the mechanical interactions between neighboring grains resulting from the very large viscoplastic anisotropy of ice crystals. Finally, the experimental microstructure was introduced in a full-field fast Fourier transform polycrystal model; simulated strain fields are a good match with experimental ones

Estimation du comportement thermo-viscoélastique effectif des pièces composites obtenues par impression 3D-FDM

peer reviewedDans le but d’estimer le comportement effectif des pièces obtenues par le procédé de fabrication FDM pour le cas des matériaux composites à fibres courtes on a visé une méthodologie permettant la mise en place d’une procédure d'homogénéisation analytique en thermo-viscoélasticité de façon analogue à celle des matériaux élastiques linéaires ; la prise en compte de la variation des paramètres qui déterminent l’état particulier des fibres dans le filament est achevé grâce à l’introduction des fonctions de distribution obtenues via l’analyse statistique de la microstructure. La procédure d'homogénéisation a été évaluée en comparant ses prédictions aux calculs basés sur la FFT en champ complet et des résultats des essais pour des échantillons traités en autoclave, pour enlever les porosités à l'échelle des couches du filament imprimé