Application of Hierarchical Matrix Techniques To The Homogenization of Composite Materials

In this paper, we study numerical homogenization methods based on integral equations. Our work is motivated by materials such as concrete, modeled as composites structured as randomly distributed inclusions imbedded in a matrix. We investigate two integral reformulations of the corrector problem to be solved, namely the equivalent inclusion method based on the Lippmann-Schwinger equation, and a method based on boundary integral equations. The fully populated matrices obtained by the discretization of the integral operators are successfully dealt with using the H-matrix format

Homogenization of a Multiscale Viscoelastic Model with Nonlocal Damping, Application to the Human Lungs

International audienceWe are interested in the mathematical modeling of the deformation of the human lung tissue, called the lung parenchyma, during the respiration process. The parenchyma is a foam–like elastic material containing millions of air–filled alveoli connected by a tree– shaped network of airways. In this study, the parenchyma is governed by the linearized elasticity equations and the air movement in the tree by the Poiseuille law in each airway. The geometric arrangement of the alveoli is assumed to be periodic with a small period ε > 0. We use the two–scale convergence theory to study the asymptotic behavior as ε goes to zero. The effect of the network of airways is described by a nonlocal operator and we propose a simple geometrical setting for which we show that this operator converges as ε goes to zero. We identify in the limit the equations modeling the homogenized behavior under an abstract convergence condition on this nonlocal operator. We derive some mechanical properties of the limit material by studying the homogenized equations: the limit model is nonlocal both in space and time if the parenchyma material is considered compressible, but only in space if it is incompressible. Finally, we propose a numerical method to solve the homogenized equations and we study numerically a few properties of the homogenized parenchyma model

Projective multiscale time-integration for electrostatic particle-in-cell methods

The simulation of problems in kinetic plasma physics is often challenging due to strongly coupled phenomena across multiple scales. In this work, we propose a wavelet-based coarse-grained numerical scheme, based on the framework of Equation-Free Projective Integration, for a kinetic plasma system modeled by the Vlasov–Poisson equations. A kinetic particle-in-cell (PIC) code is used to simulate the meso scale dynamics for short time intervals. This allows the extrapolation over long time-steps of the behavior of a coarse wavelet-based discretization of the system. To validate the approach and the underlying concepts, we perform two 1D1V numerical experiments: nonlinear propagation and steepening of an ion wave, and the expansion of a plasma slab in vacuum. The direct comparisons to resolved PIC simulations show good agreement. We show that the speedup of the projective integration scheme over the full particle scheme scales linearly with the system size, demonstrating efficiency while taking into account fully kinetic, non-Maxwellian effects. This suggests that the approach is potentially interesting for kinetic plasma problems with a large separation of scales

Multiscale modeling of sound propagation through the lung parenchyma

In this paper we develop and study numerically a model to describe some aspects of sound propagation in the human lung, considered as a deformable and viscoelastic porous medium (the parenchyma) with millions of alveoli filled with air. Transmission of sound through the lung above 1 kHz is known to be highly frequency-dependent. We pursue the key idea that the viscoelastic parenchyma structure is highly heterogeneous on the small scale ε and use two-scale homogenization techniques to derive effective acoustic equations for asymptotically small ε. This process turns out to introduce new memory effects. The effective material parameters are determined from the solution of frequency-dependent micro-structure cell problems. We propose a numerical approach to investigate the sound propagation in the homogenized parenchyma using a Discontinuous Galerkin formulation. Numerical examples are presented

Relaxation and domain wall structure of bilayer moire systems

Moire patterns result from setting a 2D material such as graphene on another 2D material with a small twist angle or from the lattice mismatch of 2D heterostructures. We present a continuum model for the elastic energy of these bilayer moire structures that includes an intralayer elastic energy and an interlayer misfit energy that is minimized at two stackings (disregistries). We show by theory and computation that the displacement field that minimizes the global elastic energy subject to a global boundary constraint gives large alternating regions of one of the two energy-minimizing stackings separated by domain walls. We derive a model for the domain wall structure from the continuum bilayer energy and give a rigorous asymptotic estimate for the structure. We also give an improved estimate for the L2-norm of the gradient on the moire unit cell for twisted bilayers that scales at most inversely linearly with the twist angle, a result which is consistent with the formation of one-dimensional domain walls with a fixed width around triangular domains at very small twist angles.Comment: 20 pages, 14 figure

Quantum plasmons with optical-range frequencies in doped few-layer graphene

Although plasmon modes exist in doped graphene, the limited range of doping achieved by gating restricts the plasmon frequencies to a range that does not include the visible and infrared. Here we show, through the use of first-principles calculations, that the high levels of doping achieved by lithium intercalation in bilayer and trilayer graphene shift the plasmon frequencies into the visible range. To obtain physically meaningful results, we introduce a correction of the effect of plasmon interaction across the vacuum separating periodic images of the doped graphene layers, consisting of transparent boundary conditions in the direction perpendicular to the layers; this represents a significant improvement over the exact Coulomb cutoff technique employed in earlier works. The resulting plasmon modes are due to local field effects and the nonlocal response of the material to external electromagnetic fields, requiring a fully quantum mechanical treatment. We describe the features of these quantum plasmons, including the dispersion relation, losses, and field localization. Our findings point to a strategy for fine-tuning the plasmon frequencies in graphene and other two-dimensional materials.MIT/Army Institute for Soldier Nanotechnologies (Contract W911NF-13-D-0001

Semaine d'Etude Mathématiques et Entreprises 5 : Propriétés asymptotiques de processus à volatilité stochastique

Cet article est une synthèse de notre travail de recherche durant la cinquième SEME (Semaine d'Etudes pour les Mathématiques en Entreprise) à l'Ecole des Mines de Nancy. Durant cette semaine, nous avons étudié une alternative au modèle de Black-Scholes: le modèle GARCH(1,1). Ce modèle est suffisamment simple pour être implémenté sur un ordinateur de bureau classique, et suffisamment proche du modèle de Black-Scholes pour ne pas dérouter les personnes habituées à ce dernier. Dans un premier temps, nous passons en revue différents résultats connus pour le modèle GARCH. En particulier, nous montrons que, sous certaines hypothèses, et contrairement au modèle de Black-Scholes, dans le modèle GARCH la distribution des rendements est à queue épaisse, similaire à celle d'une loi de puissance. Ensuite, nous illustrons à l'aide de simulations le comportement asymptotique de la distribution des rendements et nous proposons une méthode pour estimer le paramètre de la loi de puissance associée. Enfin, nous nous intéressons au phénomène de volatility clustering et à la durée des périodes de forte volatilité