Radiative transfer of acoustic waves in continuous complex media: Beyond the Helmholtz equation

Heterogeneity can be accounted for by a random potential in the wave equation. For acoustic waves in a fluid with fluctuations of both density and compressibility (as well as for electromagnetic waves in a medium with fluctuation of both permittivity and permeability) the random potential entails a scalar and an operator contribution. For simplicity, the latter is usually overlooked in multiple scattering theory: whatever the type of waves, this simplification amounts to considering the Helmholtz equation with a sound speed $c$ depending on position $\mathbf{r}$. In this work, a radiative transfer equation is derived from the wave equation, in order to study energy transport through a multiple scattering medium. In particular, the influence of the operator term on various transport parameters is studied, based on the diagrammatic approach of multiple scattering. Analytical results are obtained for fundamental quantities of transport theory such as the transport mean-free path $\ell^*$, scattering phase function $f$ and anisotropy factor $g$. Discarding the operator term in the wave equation is shown to have a significant impact on $f$ and $g$, yet limited to the low-frequency regime i.e., when the correlation length of the disorder $\ell_c$ is smaller than or comparable to the wavelength $\lambda$. More surprisingly, discarding the operator part has a significant impact on the transport mean-free path $\ell^*$ whatever the frequency regime. When the scalar and operator terms have identical amplitudes, the discrepancy on the transport mean-free path is around $300\,\%$ in the low-frequency regime, and still above $30\,\%$ for $\ell_c/\lambda=10^3$ no matter how weak fluctuations of the disorder are. Analytical results are supported by numerical simulations of the wave equation and Monte Carlo simulations

Transport laplacien, problème inverse et opérateurs de Dirichlet-Neumann

Le travail de ma thèse est basé sur ces 4 points : i) Transport laplacien d'une cellule absorbante : Soit un certain espèce (cellule) de concentration C(x), qui diffuse dans un milieu homogène et isotrope à partir d'une lointaine source localisée sur la frontière fermée ∂Ω₀ vers une interface compact semi-perméable ∂Ω (membrane de la "cellule") à laquelle elle disparaisse à un taux d'absorption donné : W≥0. La concentration C (transport laplacien avec un coefficient de diffusion D) satisfaite le problème (P1) (voir la thèse). On s'intéresse à résoudre le problème (P1) en dimension dim = 2; 3 et à calculer les courants local et total à travers les frontières des ∂Ω et ∂Ω₀ qui seront utiles pour résoudre le problèmeinverse de localisation. Pour faciliter les calculs et les rendre explicites, on prend ∂Ω et ∂Ω₀ avec des formes géométriquement régulières, précisément des boules, en distinguant les deux cas : Ω et Ω₀ sont concentriques ou non-concentriques. Pour le cas non-concentriques , on utilise la technique de transformation conforme et le développement orthogonal en série de Fourier pour résoudre le problème (P1) en cas bidimensionnel. Tandis que en cas tridimensionnel, on résout le problème (P1) en utilisant le développement orthogonal suivant les fonctions sphériques harmoniques. ii) Problème inverse de localisationOn s'intéresse dans cette partie à résoudre le problème inverse de localisation associé au problème (P1) où les domaines Ω et Ω₀ sont considérés avec des formes géométriques régulières (précisément des boules) . Ce problème consiste à trouver les conditions de Dirichlet-Neumann sur ∂Ω₀ (courant local, courant total) suffisantes pour déterminer la position de la cellule ∂ (par rapport à Ω₀), dont ces conditions sont disponibles par une suite des mesures expérimentales. iii) Problème invesre géomètrique : Dans cette partie on traite un autre type de problème inverse qui consiste à trouver la forme géométrique de la cellule en sachant les conditions de Dirichlet-Neumann au bord extérieur(∂Ω₀) qui sont mésurables par une suite d'expérience. Ce type du problème, on l'appelle le problème inverse géométrique. On résout ce problème en utilisant des techniques concernant les fonctions harmoniques et les transformations conformes. iv) Opérateur de Dirichlet-Neumann. On étudie l'opérateur de Dirichlet-Neumann relatif au problème (P1) dans les dimension deux et trois en distinguant les deux cas concentriques et non-concentriques. Ensuite, on montre que cet opérateur de Dirichlet-Neumann engendre certain semi-groupe qu'on l'appelle semi-groupe de Lax. Enfin, on construit ce semi-groupe de Lax associé à cet opérateur en cas tridimensionnel concentriques afin de vérifier que ce semi-groupe admet les mêmes propriétés que celui dans le cas général.The outline of my thesisi) Let some "species" of concentration C(p), x 2 Rd, diuse stationary in the isotropic bulk from a (distant) source localised on the closed boundary ∂Ω₀ towards a semipermeable compact interface ∂Ω of the cell Ω in Ω₀ where they disappear at a given rate W≥0. Then the steady field of concentrations C satisfy the problem (P1). (see the Thesis). We interest to solve (P1) in Twodimensional and Tridimensional cases and to calculate the local and total flux in order to solving the localisation inverse problem. In order to make easy the calculations, we take Ω and Ω₀ with a regularly geometricals forms by distinguishing the two cases : Concentrics and non-concentrics case. For the non-cncentrics case, we use the conformal mapping technique for resolving the problem (P1) in the twodimensional case. whereas in the tridimensional case, we use the development according to the spherical harmonics functions.ii) Localisation inverse problemThe aim of the localisation inverse problem is to find the necessary Dirichlet-to-Neumann conditions in order to determine the position of thecell Ω, where these conditions are measurable. iii) Geometrical inverse problem. Our main results concerns a formal solution of the geometrical inverse problem for the form of absorbing domains. We restrict this study to two dimensions and we study it by the conformal mapping technique and harmonic functions. iv) Dirichlet-to-Neumann operator. We study the Dirichlet-to-Neumann operatot relative to problem (P1) in the twodimensional and tridimensionnal cases by distinguishing the two cases : Concentrics and non-concentrics case. We prove that the Dirichlet-to-Neumann operator generates some semi-group, we call it the Lax semi-group. Finally we construct this semi group and verify that this demi-group satisfies the generals properties of a operator

Diffusion and Laplacian Transport for Absorbing Domains

International audienceWe study (stationary) Laplacian transport by the Dirichlet-to-Neumann formalism. Our results concern a formal solution of the geometrically inverse problem for localisation and reconstruction of the form of absorbing domains. Here, we restrict our analysis to the one- and two-dimensional cases. We show that the last case can be studied by the conformal mapping technique. To illustrate this, we scrutinize the constant boundary conditions and analyze a numeric example

1 2 DIFFUSION AND LAPLACIAN TRANSPORT FOR ABSORBING DOMAINS

We study (stationary) Laplacian transport by the Dirichlet-to-Neumann formalism. Our results concerns a formal solution of the geometrical inverse problem for localisation and reconstruction of the form of absorbing domains. Here we restrict our analysis to the one- and two-dimension cases. We show that the last case can be studied by the conformal mapping technique. To illustrate it we scrutinize constant boundary conditions and analyse a numeric example

Determining the initial configuration and characterizing mechanical properties of 3d interlock woven fabrics using finite element simulation

International audienceA finite element simulation approach, aimed at determining the equilibrium configuration of general fiber assemblies undergoing large displacements, and based on an implicit solver is used to predict the initial configuration of 3D interlock fabrics. Instead of simulating the weaving process, this approach offers to start from an arbitrary configuration where all tows lie in the same plane, interpenetrating each other, and to use frictional contact reactions generated by the model to gradually separate fibers from different tows, until achieving the stacking order between components defined by the selected weaving pattern. By this way, the initial configuration of the interlock woven fabric is characterized as an equilibrium configuration, and defined with only few parameters, without requiring any geometrical preprocessor. Once the initial configuration has been determined, various loading tests can be simulated in order to characterize the nonlinear behavior of the dry fabric

Anisotropic transport and diffusion of elastic waves in random media

International audienceWe discuss the influence of material anisotropy on the possible depolarization and diffusion of elastic waves in randomly heterogeneous media. Anisotropy is considered at two levels. The first one is related to the constitutive law of random materials, which may be handled by a random matrix theory for the elasticity tensor. The second level is related to the correlation structure of these random materials. Since the propagation of waves in such complex media cannot be described by deterministic models, a probabilistic framework shall be adopted based on a radiative transfer model in the so-called mesoscopic regime, when the wavelength and the correlation length are comparable