François Willot

Fourier-based schemes for computing the mechanical response of composites with accurate local fields

Abstract We modify the Green operator involved in Fourier-based computational schemes in elasticity, in 2D and 3D. The new operator is derived by expressing continuum mechanics in terms of centered differences on a rotated grid. Using the modified Green operator leads, in all systems investigated, to more accurate strain and stress fields than using the discretizations proposed by Moulinec and Suquet (1994) [1] or Willot and Pellegrini (2008) [2] . Moreover, we compared the convergence rates of the “direct” and “accelerated” FFT schemes with the different discretizations. The discretization method proposed in this work allows for much faster FFT schemes with respect to two criteria: stress equilibrium and effective elastic moduli.

Stokes Flow Through a Boolean Model of Spheres: Representative Volume Element

The Stokes flow is numerically computed in porous media based on 3D Boolean random sets of spheres. Two configurations are investigated in which the fluid flows inside the spheres or in the complementary set of the spheres. Full-field computations are carried out using the Fourier method of Wiegmann (2007). The latter is applied to large system sizes representative of the microstructure. The overall permeability of the two models as well as the representative volume element are estimated as a function of the pore volume fraction. We give numerical estimates for the asymptotic behavior of the permeability in the dilute limit for the solid phase, and close to the percolation threshold of the pores. FFT maps of the velocity field are presented, for increasing values of the pore volume fraction. The patterns of the local velocity field is analyzed using various morphological criteria. The tortuosity of the streamlines is found to be much higher than the geometrical tortuosity, for both models. The histograms of the velocity field are computed at increasing pore volume fraction. The covariance of orientation is used to characterize the spatial correlation of the velocity field.

Modelling mesoporous alumina microstructure with 3D random models of platelets

This work focuses on a mesoporous material made up of nanometric alumina ‘platelets’ of unknown shape. We develope a 3D random microstructure to model the porous material, based on 2D transmission electron microscopy (TEM) images, without prior knowledge on the spatial distribution of alumina inside the material. The TEM images, acquired on samples with thickness 300 nm, a scale much larger than the plateletss size, are too blurry and noisy to allow one to distinguish platelets or platelets aggregates individually. In a first step, the TEM images correlation function and integral range are estimated. The presence of long‐range fluctuations, due to the TEM inhomogeneous detection, is detected and corrected by filtering. The corrected correlation function is used as a morphological descriptor for the model. After testing a Boolean model of platelets, a two‐scale model of microstructure is introduced to replicate the statistical dispersion of platelets observed on TEM images. Accordingly, a set of two‐scale Boolean models with varying physically admissible platelets shapes is proposed. Upon optimization, the model takes into account the dispersion of platelets in the microstructure as observed on TEM images. Comparing it to X‐ray diffraction and nitrogen porosimetry data, the model is found to be in good agreement with the material in terms of specific surface area.

Prediction of Effective Properties of Porous Carbon Electrodes from a Parametric 3D Random Morphological Model

Pore structures have a major impact on the transport and electrical properties of electrochemical devices, such as batteries and electric double-layer capacitors (EDLCs). In this work we are concerned with the prediction of the electrical conductivity, ion diffusivity and volumetric capacitance of EDLC electrodes, manufactured from hierarchically porous carbons. To investigate the dependence of the effective properties on the pore structures, we use a structurally resolved parametric model of a random medium. Our approach starts from 3D FIB-SEM imaging, combined with automatic segmentation. Then, a random set model is fitted to the segmented structures and the effective transport properties are predicted using full field simulations by iterations of FFT on 3D pore space images and calculations based on the geometric properties of the structure model. A parameter study of the model is used to investigate the sensitivity of the effective conductivity and diffusivity to changes in the model parameters. Finally, we investigate the volumetric capacitance of the EDLC electrodes with a geometric model, make a comparison with experimental measurements and do a parameter study to suggest improved microstructures.

Morphological modelling of three‐phase microstructures of anode layers using SEM images

A general method is proposed to model 3D microstructures representative of three‐phases anode layers used in fuel cells. The models are based on SEM images of cells with varying morphologies. The materials are first characterized using three morphological measurements: (cross‐)covariances, granulometry and linear erosion. They are measured on segmented SEM images, for each of the three phases. Second, a generic model for three‐phases materials is proposed. The model is based on two independent underlying random sets which are otherwise arbitrary. The validity of this model is verified using the cross‐covariance functions of the various phases. In a third step, several types of Boolean random sets and plurigaussian models are considered for the unknown underlying random sets. Overall, good agreement is found between the SEM images and three‐phases models based on plurigaussian random sets, for all morphological measurements considered in the present work: covariances, granulometry and linear erosion. The spatial distribution and shapes of the phases produced by the plurigaussian model are visually very close to the real material. Furthermore, the proposed models require no numerical optimization and are straightforward to generate using the covariance functions measured on the SEM images.

The Power Laws of Geodesics in Some Random Sets with Dilute Concentration of Inclusions

A method for computing upper-bounds on the length of geodesics spanning random sets in 2D and 3D is proposed, with emphasis on Boolean models containing a vanishingly small surface or volume fraction of inclusions f ≪ 1. The distance function is zero inside the grains and equal to the Euclidean distance outside of them, and the geodesics are shortest paths connecting two points far from each other. The asymptotic behavior of the upper-bounds is derived in the limit f → 0. The scalings involve powerlaws with fractional exponents ~f 2/3 for Boolean sets of disks or aligned squares and ~f 1/2 for the Boolean set of spheres. These results are extended to models of hyperspheres in arbitrary dimension and, in 2D and 3D, to a more general problem where the distance function is non-zero in the inclusions. Finally, other fractional exponents are derived for the geodesics spanning multiscale Boolean sets, based on inhomogeneous Poisson point processes, in 2D and 3D.

Mean covariogram of cylinders and applications to Boolean random sets

This work focuses on the variance properties of isotropic Boolean random sets containing randomly-oriented cylinders with circular cross-section. Emphasis is put on cylinders with large aspect ratios, of the oblate and prolate types. A link is established between the power law decay of the covariance function and the variance of the estimates of the volume fraction of cylinders. The covariance and integral range of the Boolean mixtures are expressed in terms of the orientation-averaged covariogram of cylinders, for which exact analytical formulas and approximate expressions are provided.

Three-dimensional morphological modelling of concrete using multiscale Poisson polyhedra: 3D MORPHOLOGICAL MODELLING OF CONCRETE

This paper aims at developing a random morphological model for concrete microstructures. A 3D image of concrete is obtained by microtomography and is used in conjunction with the concrete formulation to build and validate the model through morphological measurements. The morphological model is made up of two phases, corresponding to the matrix, or cement paste and to the aggregates. The set of aggregates in the sample is modelled as a combination of Poisson polyhedra of different scales. An algorithm is introduced to generate polyhedra packings in the continuum space. The latter is validated with morphological measurements.