The hybrid method Volume of Fluid–Machine Learning and its three-dimensional extension

• Moreno Pintore (LJLL Sorbonne Université)

Room: amphithéâtre Hermite In this talk, we present the Volume of Fluid–Machine Learning (VOF-ML) method. It is a hybrid approach, belonging to the emerging Scientific Machine Learning framework, that combines the accuracy and robustness of the classical Volume of Fluid (VOF) method with the nonlinearity of neural networks (NNs). The VOF method is a numerical technique commonly used to advect sharp interfaces between two or more materials, and it is mainly used in combination with classical Finite Volume schemes when exact conservation is required. The basic idea consists in computing the percentage of each material in each cell of a computational mesh, and then modifying these percentages (called volume fractions) to simulate the movement of the materials. In most cases, to compute the update of the volume fractions, one approximates the interface with a simple parametrized surface and then moves the surface exactly according to the governing equation. However, when the interface is highly nonlinear or nonsmooth, the initial approximation may be inaccurate and may introduce artificial artifacts or excessive diffusion. To avoid this problem, we propose skipping the interface reconstruction step and directly computing the flux needed to update the volume fractions by means of a suitably trained neural network. The method was proposed in [1] and extended to the multi-material case in [2]. Here, instead, after an initial description of the method, we focus on its three-dimensional extension and the main challenges of its implementation. In particular, we describe how to train a neural network using only geometrical information and without the need for any expensive numerical simulations, and how to enforce certain physical properties in the neural network’s architecture. Numerical tests are provided to illustrate the performance of VOF-ML and to show numerically that the optimal convergence rate with respect to mesh refinement is achieved. [1] Bruno Després and Hervé Jourdren. Machine learning design of volume of fluid schemes for compressible flows. Journal of Computational Physics, 408:109275, 2020. [2] Matthieu Ancellin, Bruno Després, and Stéphane Jaouen. Extension of generic two-component vof interface advection schemes to an arbitrary number of components. Journal of Computational Physics, 473:111721, 2023.

Explicit T‑coercivity for the Stokes problem: A coercive finite element discretization / T‑coercivité explicite pour le problème de Stokes : discrétisation éléments finis coercive

• Erell Jamelot (CEA Saclay)

Room: amphithéâtre Hermite En utilisant la théorie de la T-coercivité telle que préconisée dans Chesnel et Ciarlet (2013), nous proposons une nouvelle formulation variationnelle du problème de Stokes. Avec cette nouvelle formulation, les paires d'éléments finis instables sont stabilisées. De plus, le schéma numérique est facile à mettre en œuvre, et une meilleure approximation de la vitesse et de la pression est observée numériquement lorsque la viscosité est petite.

A unified compressible two-phase, two-scale model with capillarity for separated-to-disperse transitions in atomizing flows

• Giuseppe Orlando (Ecole Polytechnique/CMAP)

Room: amphithéâtre Hermite We present a two-phase, two-scale compressible flow model with surface tension, enabling a unified description of both separated-interface and disperse-phase regimes. The formulation is derived in the barotropic setting and builds upon Hamilton’s Stationary Action Principle. Inter-scale mass-transfer terms, activated when local curvatures exceed a grid-independent cutoff length scale, allow for the transition from the large-scale representation to the small-scale disperse regime. This mechanism ensures the regularization of the large-scale interface while simultaneously modeling atomization through the formation of a spray of droplets at the small scale. The proposed process is dissipative for the extended thermodynamics which includes surface energies. We also propose robust numerical strategies that guarantee the preservation of admissible physical states. The capabilities of the model and the properties of the numerical scheme are demonstrated on a relevant test case. Perspectives and on-going work for extending the methodology to full thermodynamics will also be addressed.

Modélisation de la convection naturelle turbulente par simulation haute fidélité et apprentissage automatique informé par la physique

• Anne Sergent (LISN, Université Paris Saclay ; Institut Jean le Rond d´Alembert, Sorbonne Université)

Room: amphithéâtre Hermite La convection naturelle turbulente est un processus physique spontané présent dans de nombreux systèmes naturels ou d’ingénierie, comme le refroidissement passif. Malgré la constante amélioration des supercalculateurs, la simulation numérique directe (DNS) d’écoulements turbulents reste un défi au vu des besoins en parallélisme, capacité de calcul et de stockage, impliquant bien souvent de fortes restrictions spatio-temporelles sur les données stockées, ce qui entrave leur analyse. Cette question renvoie aussi à la complexité de la mesure expérimentale du champ de température en régime turbulent. L’introduction de lois physiques dans l'apprentissage semble être une voie prometteuse pour déduire des champs de vitesse et de température à partir de données partielles bruitées (images, faible densité de labels), en dépit des difficultés inhérentes aux fonctions de perte multi-objectifs. A partir de notre base de données DNS, nous cherchons à construire des modèles réduits par apprentissage automatique informé par la physique (PINN), explorant des configurations proches de situations expérimentales.

Session posters / poster session Room: amphithéâtre Hermite

A New Computational Approach for Fluid-Structure Interaction of Slender Bodies Immersed In Three-Dimensional Flows

• Fabien Lespagnol (IMAG Université de Montpellier)

Room: amphithéâtre Hermite Summary: The term ”slender” refers to structures with a very high ratio between their longitudinal length and their transverse dimensions, typically a cylinder with a height significantly larger than its radius. Because of this particular geometry, many models have been developed to provide a simplified description of the kinematics and dynamics of the structure. A standard approach in this context is to account for the distribution of forces and deformations only along the centerline. Consequently, the velocity fields and equilibrium equations of the structure are described in a one-dimensional (1D) setting. However, when a slender structure is immersed in a three-dimensional (3D) fluid, enforcing kinematic and dynamic coupling conditions on a 1D domain requires the introduction of a double-trace operator (codimension 2) which demands regularity of the solution within the fluid domain, a condition which is generally not satisfied a priori. In this talk, I will introduce and analyse a new mathematically sound approach for modelling and solving 3D–1D fluid–structure interaction problems. The main idea is to combine a fictitious domain approach with the projection of the kinematic constraint onto a Fourier based finite-dimensional space defined along the structure’s centerline. The discrete formulation is based on the finite element method and a semi-implicit treatment of the Dirichlet-Neumann coupling conditions, employing a partitioned procedure for the resolution of the fluid-structure interaction problem.

Characterization of turbulence-driven power fluctuations in a homogeneous unmoderated molten salt reactor

• Ayoub Ouazzani (CEA - Saclay)

Room: amphithéâtre Hermite Molten salt reactors are a promising type of innovative nuclear reactors characterized by a liquid nuclear fuel made of a molten salt in which fissile matter is dissolved. One challenge they face is the possibility of power and temperature fluctuations related to the turbulent character of the flow of the liquid fuel. We present a new computational framework for the analysis of the turbulent dynamics in the core which couples a large eddy simulation model with a deterministic neutronics model. These are implemented in the high-performance computing codes TrioCFD and TRUST-NK. We use the framework to study a simplified reactor core inspired by reference reactor concepts. In particular, we characterize the typical behavior of the core in terms of average fields and in terms of fluctuations of local and global observables. We explain the spatial profile of power density fluctuations in terms of the flow structure. Finally, we evaluate how the amplitude of total power fluctuations scale with power level and flow rate.

Sharp cartesian methods for multifluid flows and other applications

• Lisl WEYNANS (IMB - Univ. Bordeaux)

Room: amphithéâtre Hermite I will present numerical methods in the family of immersed boundary methods, designed to solve with elliptic problems with discontinuities across interfaces with complex shape. The convergence of these methods will be studied using the framework of a discrete maximum principle and discrete Green functions, allowing to account for the various orders of the truncation errors. These methods are applied to the simulation of flows with strong density ratios, as air-water interfaces.

Une méthode lattice Boltzmann vectorielle préservant la positivité pour les équations d'Euler compressibles

• Gauthier Wissocq (CEA - CESTA)

Room: amphithéâtre Hermite Ces travaux présentent un nouveau schéma cinétique positif basé sur l’algorithme efficace collide-and-stream de la méthode lattice Boltzmann (LBM) pour la résolution de lois de conservation hyperboliques, avec application aux équations d'Euler compressibles. À partir des modèles à relaxation de type BGK, nous montrons comment la discrétisation de la LBM conduit à des schémas cinétiques d'ordre un et deux. Le schéma d'ordre un a l'avantage de préserver le domaine invariant (positivité de la densité et de l'énergie interne) sous une condition CFL donnée. Le schéma d'ordre 2 est lui très peu dissipatif, mais oscillant. Nous proposons une stratégie originale de combinaison de ces schémas basée sur des limiteurs convexes, préservant la simplicité de la méthode. Des cas tests sévères appliqués aux équations d'Euler valident la précision, la robustesse et la capacité du schéma à capturer des discontinuités fines sans oscillations numériques.

Compressible single- and two-phase flows in rigid and elastic pipes: modeling, mathematical analysis and numerical schemes

• Frédéric Daude (EDF)

Room: amphithéâtre Hermite Compressible flows in pipelines are encountered in many industrial applications, i.e. oil/gas transportation, blood flows, and nuclear piping system. The mathematical modeling of such flows as well as the analysis of the corresponding governing equations are thus of great interest in particular for the design of dedicated numerical methods. The common approach for modeling compressible flows in pipelines is based on the cross-sectional averaging of the 3-D equations. Most of the research encountered in the literature deals with the complete Euler equations (with energy equation) on rigid pipes or with the barotropic Euler equations in elastic pipes. The two main examples of the latter case are the water-hammer and blood flows. In both cases, i.e. rigid and elastic pipes, non-conservative terms appear due to the pressure effect induced by the pipe cross-section variations. These non-conservative terms introduce an additional linear degenerate field with a zero-speed eigenvalue which are known to imply additional difficulties such as resonance, loss of uniqueness. In the context of rigid pipes, the pipe cross-section is temporally invariant, which is not the case for elastic pipes. For elastically deformable pipes, the elastic radial pipe deformation has a direct influence on the internal fluid which has to be considered in the models. Furthermore, an additional closure law named as tube law is required to represent the elastic radial pipe deformation as a function of the internal fluid pressure. In [1], the mathematical analysis of the cross-sectional averaged Euler equations with energy equations in elastic pipes is presented demonstrating hyperbolicity, the structure of the waves, the expression of the Riemann invariants and the existence of a mathematical entropy. This analysis has also been extended to the Kapila five-equation and the Baer-Nunziato seven-equation two-phase flow models. The influence of the pipe hoop elasticity on the characteristic fields is shown and the corresponding difficulties discussed. Instead of directly discretized the cross-sectional averaged equations, a Finite-Volume approach based on the integral form of the equations has been proposed for the Euler equations in elastic pipes [2]. In addition, a Finite-Volume junction methodology is also proposed for dealing with junction of several pipes at the same location [2]. This approach has also been extended to the Baer-Nunziato models [3]. Numerical results will be shown. Work in collaboration with Ray A. Berry and Pascal Galon. [1] F. Daude, R. A. Berry, F. Crouzet, P. Galon, 2025, Entropy consistent and hyperbolic formulations for compressible single- and two-phase flows modeling in both rigid and elastically deformable pipes: Application to Euler, Kapila and Baer-Nunziato equations, Applied Mathematical Modelling , Vol. 144, 116096. [2] F. Daude, P. Galon, 2018, A Finite-Volume approach for compressible single- and two-phase flows in flexible pipelines with fluid-structure interaction, Journal of Computational Physics , Vol. 362, pp. 375-408. [3] F. Daude, R. A. Berry, P. Galon, 2019, A Finite-Volume method for compressible non-equilibrium two-phase flows in networks of elastic pipelines using the Baer-Nunziato model, Computer Methods in Applied Mechanics and Engineering , Vol. 354, pp. 820-849.

Quasi-resonant collisions in kinetic theory and bi-temperature systems

• Thomas Borsoni (ENPC/CERMICS)

Room: amphithéâtre Hermite Abstract : Some molecules exhibit a peculiar behavior during collisions, called resonant: they separately exchange kinetic and internal energies. If the molecules of a gas undergo only resonant collisions, the equilibrium distribution exhibits two distinct temperatures, a kinetic and an internal one. To account for more realistic scenarios, we consider ‘’quasi’’-resonant collisions, where a very tiny exchange between kinetic and internal energies is allowed. We propose a mathematical framework for the notion of quasi-resonance, which leads to a Boltzmann model where the distribution is known at all times, a two-temperature Maxwellian, and converges towards a one-temperature Maxwellian. With this feature at hand, we derive so-called Landau-Teller equations, allowing us to replace the complicated Boltzmann equation by a simple ODE system of two equations.

In the shadow of hyperbolic models

• Martin Parisot (Inria)

Room: amphithéâtre Hermite Abstract: This presentation invites the audience to consider a class of models with high-order derivatives as hyperbolic models whose solutions are sought in a linear subspace. I will first justify this framework by presenting a few models from this family, in particular the approximate model derived from the water waves problem, but not only. I will then detail the mathematical structure based on an Helmholtz decomposition similar to that of incompressible flows, and we will highlight how this structure can aid in the analysis of the models, particularly with regard to boundary conditions. Finally, I will show how to preserve this structure at the discrete level in order to ensure robust, entropy-satisfying scheme, even in complex contexts such as dry front or bathymetry discontinuity.

Méthodes neuronales pour les problèmes de mécanique des fluides

• Emmanuel Franck (INRIA)

Room: amphithéâtre Hermite Depuis l’essor de l’apprentissage profond, un certain nombre de méthodes ont été proposées pour appliquer les réseaux de neurones aux problèmes d’EDP. L’une des plus populaires est la famille des méthodes PINNs, qui consiste à intégrer la contrainte d’EDP directement dans la fonction de coût à minimiser. Dans un premier temps, nous montrerons que ces méthodes peuvent être comprises comme des généralisations des approches traditionnelles de Galerkin ou Petrov–Galerkin. Nous discuterons ensuite les principales forces et faiblesses de ces approches par rapport aux méthodes numériques classiques, avec un accent particulier sur les aspects liés à l’optimisation. Dans un second temps, nous illustrerons, à travers plusieurs exemples issus de la mécanique des fluides, qu’une approche entièrement basée sur les PINNs n’est pas nécessairement la plus adaptée, et qu’il est souvent plus efficace de combiner schémas traditionnels et modèles neuronaux. Pour finir, nous présenterons des approches hybrides permettant de retrouver des garanties d’erreur.

Calcul exact des moments de l'intersection cuboïde/ellipsoïde, application à l'initialisation dans la méthode Moment Of Fluids

• Thomas Milcent (I2M - UMR CNRS 5295 - Bordeaux University)

Room: amphithéâtre Hermite Simuler numériquement de manière précise l'évolution des interfaces séparant différents milieux est un enjeu crucial dans de nombreuses applications (multi-fluides, fluide-structure, etc). La méthode MOF (Moment-Of-Fluid) est une extension récente de la méthode VOF (Volume-Of-Fluid) qui permet de suivre plusieurs matériaux évoluant au cours du temps. Elle utilise une reconstruction affine des interfaces par cellule basée sur l'information des fractions volumiques et les centroïdes de chaque matériau. La première étape d'initialisation consiste à calculer la fraction volumique et le centroïdes de l'intersection de la surface initiale avec chaque cellule. Ce calcul nécessite l'utilisation de méthodes numériques d'intégration qui sont couteuses et peu précises. Dans le cas particulier de l'initialisation d'un ellipsoïde dans un maillage cartésien, on propose une nouvelle approche complètement analytique qui permet de faire le calcul des moments de manière exacte. Il est basé sur le calcul de l'intégrale de fonctions caractéristiques sur un cuboïde. Les résultats numériques montrent que l'approche proposée est bien plus rapide (plusieurs ordres de grandeurs) et plus précise que les approches de la littérature.

Solving stochastic inverse problems for CFD using data-consistent inversion and an adaptive stochastic collocation method

• Anca Belme (Institut Jean le Rond d'Alembert - Sorbonne Université)

Room: amphithéâtre Hermite Abstract : The talk focuses on the coupling of two stochastic methods to solve uncertain (compressible) CFD problems in which singularities—such as shocks—may propagate into the parameter space. Specifically, it addresses an inverse stochastic problem whose goal is to update the prior distribution of uncertain parameters so that, after forward propagation, the resulting posterior distribution matches a given distribution of observed data. This data-consistent framework requires a forward uncertainty-propagation model (or surrogate), which is constructed here using an adaptive simplex stochastic collocation method. Additional samples are selected adaptively to minimize the stochastic error, ensuring that the surrogate remains sufficiently accurate to capture singularities that may travel from the physical domain into the space of uncertain parameters.

A hybrid kinetic-fluid relaxation scheme for particle-fluid flows

• Codor KHODR (CEA - Bruyères)

Room: amphithéâtre Hermite We present a novel numerical method for the treatment of particle-fluid flows, in which a dispersed phase, composed of mesoscopic particles, interacts with a continuous fluid phase. It is well known that the kinetic description of such flows is very accurate in the dilute regime but suffers from robustness issues when the volumic fraction of particles increases. It is therefore common, in practical simulations, to perform a relaxation towards an eulerian two-phase mixture (typically a 4 or 5 equation model) when dense, highly collisional regimes are reached. In this presentation, we derive such a model by doing a modal decomposition of the density function (PDF) of the particles into a “kinetic” and an “eulerian” mode, according to some physically sound criterion, and taking the hydrodynamic limit of the latter. This gives rise to a coupled Vlasov-Euler system which contains a non-conservative mixing operator, representing the flux of particles going from the kinetic to the fluid sub-space of the phase space. This approach is related to so-called “hybrid” or “micro-macro” methods, which were initially developed in the context of rarefied gas dynamics and plasma transport. Numerically, this involves coupling a Lagrangian PIC solver with a finite-volume Euler solver through a relaxation scheme which aims at integrating the mixing source terms and enforcing thermodynamic equilibrium. The method is analyzed and illustrated on a test case : the interaction of a shock with a dense cloud of water particles, with addition of a simple break-up kernel as a driver of mixing.