The accurate modelling of two-phase flows is a very active research field with significant implications for space applications, defense technologies, and geophysical phenomena. The derivation of models capable of describing the wide range of scales or the non-equilibrium effects that naturally arise in such flows, especially in the case of compressible high-velocity flows involving shocks, is still an open question.
The main goal of this presentation is to present a modelling strategy that allows for a systematic derivation of two-phase models within a variational framework. The strategy relies on two stages. First, the non-dissipative part of the model is derived by means of the Stationary Action Principle. A generic Eulerian framework, compatible with an underlying Lagrangian description, will be presented. The framework ensures that the resulting model admits a supplementary conservation law thus providing a coherent thermodynamic structure. The second stage introduces dissipative effects in accordance with the Second Principle of thermodynamics.
We will illustrate the application of this framework through several examples. After an introductory case to build familiarity, we will discuss the derivation of multi-fluid models which include a coupling between the relative velocity and velocity fluctuations. We will present a two-scale model which allows for the separated-to-disperse phase transition based on an interfacial energy budget that occurs in atomization processes. Finally, as a follow-up on the presentation of disperse two-phase flows, we will show how different levels of descriptions for the spray can naturally be integrated in the two-scale model.
The modeling and simulation of multiphase reacting flows covers a large spectrum of applications ranging from combustion in automobile and aeronautical engines to atmospheric pollution as well as biomedical engineering or pyroclastic flows. In the framework of this seminar, we will mainly focus on a disperse liquid phase carried by a gaseous flow field which can be either laminar or turbulent; however, this spray can be polydisperse, that is constituted of droplets with a large size spectrum; we have to deal with coalescence, break-up, droplet trajectory crossing and reduced order model for turbulent flows. Thus, such flows involve a large range of temporal and spatial scales, which have to be resolved in order to capture the dynamics of the phenomena and provide reliable and eventually predictive simulation tools. Even if the power of the computer resources regularly increases, such very stiff problems can lead to serious numerical difficulties and prevent efficient multi-dimensional simulations. The purpose of the presentation is to introduce to the Eulerian modeling of polydisperse evaporating spray for various applications, that is the disperse liquid phase carried by a gaseous flow field is modeled by "fluid" conservation equations. Such an approach is very competitive for real applications since it has strong ability for optimization on parallel architectures and leads to an easy coupling with the gaseous flow field resolution. We will show that all the necessary steps in order to develop a new generation of computational code have to be designed at the same time with a high level of coherence: mathematical modeling through Eulerian moment methods, development of new dedicated stable and accurate numerical methods, implementation of optimized algorithms as well as verification and validations of both model and methods using other codes and experimental measurements.
The Aronson-Benilan inequality, well known for the porous media equation $\partial_t \rho - \Delta \rho^m = 0$ provides a lower bound on the Laplacian of pressure: Delta \rho^{m-1} \geq C. In this presentation, I will show that this estimate remains valid for another equation: the Keller-Segel system, which is a porous medium equation to which we add an aggregation term. Among other things, this provides a new demonstration of global existence for this system. I will focus on the case in dimension 2, with a linear diffusion and with a small initial mass but the result can be extended for any dimension, with the critical diffusion exponent and up to and including the critical mass. This work is in collaboration with Alejandro Jimenez-Fernandez (Oxford) and Filippo Santambrogio (Lyon).
The migration of various cell populations partially relies on chemotaxis meaning that cells direct their movement in response to the gradient of a chemical signal. Widely used macroscopic models for chemotaxis are Keller-Segel systems consisting of partial differential equations. As one feature they can describe the spontaneous formation of cell aggregation. One analytical challenge related to that feature is the detection of unbounded solutions, the so-called blow-up. We will present several blow-up results for Keller-Segel systems and outline the main stategies of their proofs.
Cette thèse porte sur la stabilisation de problèmes de transmission d’interface impliquant les équations onde-plaque et les modèles de poutre/onde d'Euler-Bernoulli avec des contrôles frontières dynamiques, en utilisant la théorie des semi-groupes et l'analyse numérique. Tout d'abord, nous étudions un modèle de transmission bidimensionnel composé d'une équation des ondes et d'une équation de plaque de Kirchhoff, couplées par des conditions de transmission le long d'une interface fixe et contrôlées par trois contrôles frontières dynamiques sous certaines conditions géométriques. Bien que ce système ne soit pas exponentiellement stable, nous établissons une décroissance polynomiale de l’énergie de type 1/t pour des données initiales régulières en utilisant une approche fréquentielle combinant un raisonnement par contradiction avec la technique des multiplicateurs. Cette méthode impose des conditions géométriques particulières sur les domaines de l’onde et de la plaque. Ensuite, nous généralisons ces résultats à un modèle onde-plaque avec seulement deux contrôles frontières dynamiques au lieu des trois habituels. Notre analyse montre que cette réduction ne modifie pas le taux de décroissance polynomiale de l’énergie, qui reste de l’ordre de 1/t. Cette approche impose également des conditions géométriques spécifiques au domaine. Enfin, nous étudions la solution numérique d’un problème de transmission onde/poutre d’Euler-Bernoulli en une dimension avec contrôles frontières dynamiques. Le couplage s’effectue via des connexions aux frontières, où les équations d’onde et de poutre évoluent sur des intervalles adjacents. En utilisant la méthode des volumes finis, nous obtenons des estimations de stabilité et prouvons la convergence de la solution numérique vers la solution faible du système continu. De plus, nous présentons une expérimentation numérique validant l’étude théorique, notamment en ce qui concerne le taux de décroissance de la solution sous l'effet d’un unique contrôle frontière dynamique.
We consider a system of N Brownian particles interacting through a long-range smooth potential. It is known that "propagation of chaos" holds in the mean-field scaling. Assume indeed that the initial distribution of the particles is chaotic, i.e. that the particles are independent and identically distributed. Then, for any given time, and as N becomes large, the distribution of particles remains chaotic. Moreover, the distribution of a typical particle is given by the solution of a Vlasov-Fokker-Planck equation.
In this talk, we will investigate the creation of chaos phenomenon. Starting from an initial distribution of particles which is only exchangeable, we prove that in some weak norm, propagation of chaos holds up to an error stemming from initial correlations, exponentially damped over time. This is a joint work with Armand Bernou and Mitia Duerinckx.
Shape optimization is a central field in applied mathematics and engineering, with applications ranging from aerodynamics to material design. The objective is to find the optimal shape of a domain that minimizes a given criterion, such as energy or resistance, while satisfying geometric constraints like fixed volume, surface area, or minimal height. Classically, solving these optimization problems requires numerical methods that rely on shape derivatives and adjoint calculations, which are computationally expensive and difficult to adapt to complex geometries. To overcome these challenges and the inherent limitations in parallelization found in classical approaches, our objective is to find a good methodology that relies on neural networks and benefits on their specific properties (easy parallelization, automatic differentiation, and a meshless approach). This requires suitable representations for both the solution of the state equation and the shape of the domain in which the equation is defined. For this purpose, we introduce the DeepRitz method to approximate solutions to the state equation and symplectic neural networks to model the domain shape effectively. We illustrate this approach through an example: minimizing the Dirichlet energy of a domain under a volume constraint. We will then propose an extention of this work for odd dimensions.
We present the study of the non-linear stability of a class of travelling-wave solutions to the compressible pressureless Navier-Stokes system with a singular viscosity. These solutions encode the effect of congestion by connecting a congested left state to an uncongested right state. By using carefully weighted energy estimates we are able to prove the non-linear stability of viscous shock waves to this system under a small zero integral perturbation, which in particular extends previous results that do not handle the case where the viscosity is singular. This is a joint work with Muhammed Ali Mehmood from Imperial College, London.
Cette présentation vise à introduire un schéma monolithique GD/VF (Galerkin Discontinu/Volumes Finis) préservant les propriétés convexes (principes du maximum, positivité, entropie, ...) pour la résolution des systèmes de lois de conservation sur maillages non-structurés. Il est bien connu que les méthodes Galerkin discontinue (GD) nécessitent une limitation non linéaire pour éviter les oscillations parasites ou les instabilités non linéaires, susceptibles de provoquer l´arrêt anticipé du code de simulation. L'idée principale de ce travail est d'améliorer la robustesse des schémas GD tout en préservant autant que possible leur grande précision et leur résolution de sous-maille. Pour ce faire, une combinaison convexe entre un schéma GD d'ordre élevé et un schéma volumes finis (VF) d'ordre un sera localement effectué, à l'échelle des sous-mailles, où cela sera nécessaire. À cette fin, nous prouverons tout d'abord qu'il est possible de réécrire un schéma GD comme un schéma VF défini sur un sous-maillage, en introduisant des flux numériques spécifiques, qu'on appellera flux reconstruits GD. Le schéma monolithique GD/VF sera alors défini de la manière suivante : à chaque face de chaque sous-cellule seront assignés deux flux, un flux VF d'ordre un et un flux reconstruit d'ordre élevé, qui seront finalement combinés de manière convexe. L'objectif est alors de déterminer, par analyse, les coefficients de combinaison optimaux pour atteindre les propriétés souhaitées (par exemple, positivité, absence d´oscillations, inégalités d´entropie) tout en préservant la grande précision du schéma. Des résultats numériques sur divers types de problèmes hyperboliques seront présentés pour évaluer les performances de la méthode proposée.
Grâce à ce formalisme monolithique, nous tenterons de répondre à certaines questions : est-il possible d'assurer une stabilité entropique ? de quelle stabilité entropique parlons-nous (discrète, semi-discrète, de maille, de sous-maille, pour quelle entropie, ...) ? quels sont les coûts de telles stabilités (en terme de précision ou de perte d'autres propriétés) ? À quel point est-ce essentiel pour les problèmes qui nous intéressent ? Nous présenterons différents résultats numériques pour tenter de partiellement répondre à ces questions
In this seminar we present a mathematical model to describe the evolution of a city, which is determined by the interaction of two large populations of agents, workers and firms. The map of the city is described by a network with the edges representing at the same time residential areas and communication routes. The two populations compete for space while interacting through the labour market. The resulting model is described by a two population Mean-Field Game system coupled with an Optimal Transport problem. We prove existence and uniqueness of the solution and we provide some numerical tools to develop several numerical simulations. This is a joint work with Fabio Camilli (Sapienza Roma) and Luciano Marzufero (Libera Università di Bolzano).
The goal of this talk is to give an overview of two new results in the development of hybrid methods for elliptic and hyperbolic partial differential equations (PDEs). A hybrid method combines classical numerical analysis techniques (finite element method (FEM), discontinuous Galerkin (DG), ...) with tools from machine learning (ML). The first part of this talk is dedicated to a broad presentation of such ML tools, including a common framework to represent PDE approximators, be they classical or ML-based. Then, in a second part, we explain how to use a physics-informed prior to lower the error constant of the FEM while keeping the same order of accuracy. Thanks to the FEM framework applied to elliptic PDEs, we rigorously prove that our correction improves the FEM error constant by a factor depending on the prior quality. If time permits, in a third part, we discuss how to enhance the DG basis with physics-informed priors, to increase the resolution of near-equilibrium solutions to hyperbolic systems of balance laws. Once again, we rigorously prove that the error constant is improved. Numerical illustrations will be present throughout the presentation, to validate our results.
Le modèle Euler compressible bifluide ne présente pas de difficultés théoriques supplémentaires comparé au cas monofluide. Mais sa résolution numérique est notoirement plus difficile à cause du phénomène d'oscillations de pression à l'interface entre fluides. Nous présentons une approche basée sur un échantillonnage aléatoire "à la Glimm" à l'interface, qui permet de s'affranchir de ce défaut. Le schéma obtenu est applicable à des maillages non structurés, il a d'excellentes propriétés de robustesse et de convergence. Nous l'appliquons à des cas de déferlement.
Pendant l'élongation de l’axe de l'embryon de vertébré, on observe, grâce à l’imagerie live, un phénomène de turbulence cellulaire dans les différents tissus embryonnaires. Nous proposons un modèle mécanique en 2D pour modéliser la croissance des tissus pendant l'élongation de l'embryon, qui permet de retrouver ces flux turbulents à travers un rotationnel non trivial pour les vitesses des tissus. Une autre spécificité de ce modèle est que la ségrégation entre les tissus est assurée par une pression de ségrégation. Après avoir déterminé (formellement) la limite incompressible, nous étudions le comportement qualitatif à la limite et discutons d'un effet fantôme.
Plusieurs modèles physiques permettent de comprendre la dynamique des mélanges de fluides, parmi lesquels figurent les modèles dits de Baer-Nunziato. Les équations aux dérivées partielles associées à ces modèles ressemblent à celles de Navier-Stokes, avec, en plus, de nouveaux termes de relaxation.
Une stratégie pour obtenir ces modèles est l'homogénéisation : à partir d'un mélange mésoscopique, où deux fluides purs satisfaisant les équations de Navier-Stokes compressibles se répartissent l'espace, on effectue un changement d'échelle pour obtenir un mélange macroscopique, où, en chaque point de l'espace, les deux fluides peuvent coexister.
Ce problème relève de l'étude des équations de Navier-Stokes avec des données initiales fortement oscillantes. On commencera donc par expliquer certains résultats dans ce cadre de travail, en dimension un d'espace et sur le tore, d'abord pour des fluides barotropes, puis pour des gaz parfaits non barotropes. On détaillera ensuite les différentes étapes de la démonstration de l'homogénéisation.
Dynamic processes in continuum physics are modeled using time-dependent partial differential equations (PDE), which are based on the conservation of some physical quantities, such as mass, momentum and energy. Depending on the physical phenomenon under consideration, the governing equations can exhibit some mathematical structures like differential constraints, algebraic relations, physical admissible states as well as asymptotic limits and thermodynamics compatibility. An interesting class of mathematical models is provided by symmetric hyperbolic systems that intrinsically imply all the structures listed above. When passing at the discrete level, the exact satisfaction of these structural properties is not automatically guaranteed, thus Structure Preserving numerical schemes have recently emerged with the aim of exactly discretizing at least a subset of these constraints. We will investigate and present some of our research activity carried out in the framework of the development of Structure Preserving schemes, focusing on recent contributions delivered in the last three years. In particular, we will address asymptotic preserving schemes for low Mach flows, div-curl and curl-grad preserving operators for discontinuous Galerkin methods, and a novel geometric and thermodynamically compatible finite volume method for continuum mechanics.
The energy of saline gradients is a very promising source of non-intermittent renewable energy, the exploitation of which is hampered by the lack of economically viable technology. The most investigated harvesting methods rely on selective transport of ions or water molecules through semi-permeable or ion-selective membranes, which demonstrate limited power densities of the order of a few W/m2. While in the last decade single nanofluidic objects such as nanopores of nanotubes have opened up very promising prospects with power density capabilities in the kW or even MW/m2, scale-up efforts face serious issues, as concentration polarization phenomena result in a massive loss of performance.
At the LiPhy we work on a concept of nanofluidic exchanger for power generation from saline gradients, focused on designing a nanoscale flow able to harvest the power at the output of the nanopores. We will present the study of a simple exchanger made of a selective nanoslit fed by a nanofluidic assembly. One specific feature of such an exchanger relies on the non-linear ion fluxes through the nanoslit, according to the so-called 1D Poisson-Nernt Planck equations. Such an elemental brick could be massively parallelized in stackable electricity-generating layers using standard technologies of the semi-conductors industry. We demonstrate here a scheme for rationalizing the choice of the exchanger parameters, taking into account the transport properties at all scales. The simplified numerical resolution of the three-dimensional device shows that net power densities of 300 W/m2 and more can be achieved.