Séminaire Equations aux dérivées partielles
organisé par l'équipe Modélisation et contrôle
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Amina Mecherbet
Autour de l'équation de Transport-Stokes
10 janvier 2023 - 14:00Salle 301
L'équation de Transport-Stokes modélise la sédimentation d'une suspension de particules à faible fraction volumique dans un fluide visqueux. Le système est un couplage entre une équation de Stokes pour le fluide et une équation de transport pour la fonction de densité qui désigne la probabilité de présence des particules dans le fluide.
Dans cet exposé je rappellerai dans un premier temps l'origine de la dérivation d'un tel modèle ainsi que les résultats d'existence et d'unicité connus pour des données initiales de type $L^1\cap L^\infty$.
Je présenterai ensuite des résultats récents obtenus en collaboration avec Franck Sueur concernant certaines propriétés des solutions : existence et unicité pour des données initiales de type $L^1 \cap L^p$, $p\geq3$, analyticité des trajectoires et contrôlabilité du système.
Enfin si le temps le permet, j'évoquerai certaines questions ouvertes liées à la modélisation de la sédimentation d'une gouttelette. -
Emmanuel Franck
Soutenance HDR
17 janvier 2023 - 14:00Salle de conférences IRMA
Soutenance de l'HDR d'Emmanuel Franck, à 14h en salle de conférences de l'IRMA. Sujet de l'HDR : Numerical methods for conservation laws. Application to gas dynamics and plasma physics -
Eloi Martinet
REPORTÉ (trains supprimés) -- Maximisation des valeurs propres du Laplacien avec condition de Neumann
24 janvier 2023 - 14:00Salle de conférences IRMA
On s'intéresse au problème d'optimisation de formes consistant à maximiser les valeurs propres du Laplacien avec conditions de Neumann homogènes. Ces valeurs propres interviennent notamment dans des problèmes acoustiques ou thermiques et sont en particulier liées à la "hot spot conjecture". Contrairement aux valeurs propres de Dirichlet, celles associées au problème de Neumann sont de nature plutôt instables, ce qui rend le problème d'optimisation difficile. On verra comment certaines explorations numériques du problème pour des domaines du plan et de la sphère ont permis de mettre en évidence certaines propriétés des optima. -
Stefania Fresca
Deep learning-based reduced order models for parametrized PDEs
31 janvier 2023 - 14:00Salle de conférences IRMA
The solution of differential problems by means of full order models (FOMs), such as, e.g., the finite element method, entails prohibitive computational costs when it comes to real-time simulations and multi-query routines. The purpose of reduced order modeling is to replace FOMs with reduced order models (ROMs) characterized by much lower complexity but still able to express the physical features of the system under investigation. Conventional ROMs anchored to the assumption of modal linear superimposition, such as proper orthogonal decomposition (POD), may reveal inefficient when dealing with nonlinear time-dependent parametrized PDEs, especially for problems featuring coherent structures propagating over time. To overcome these difficulties, we propose an alternative approach based on deep learning (DL) algorithms, where tools such as convolutional neural networks (CNNs) are used to build an efficient nonlinear surrogate. In the resulting DL-ROM, both the nonlinear trial manifold and the nonlinear reduced dynamics are learned in a non-intrusive way by relying on DL models trained on a set of FOM snapshots, obtained for different parameter values [Fresca et al. (2021a), Fresca et al. (2022)]. Accuracy and efficiency of the DL-ROM technique are assessed in several applications, ranging from cardiac electrophysiology [Fresca et al. (2021b)] to fluid dynamics [Fresca et al. (2021c)], showing that new queries to the DL-ROM can be computed in real-time. Finally, with the aim of moving towards a rigorous justification on the mathematical foundations of DL-ROMs, error bounds are derived for the approximation of nonlinear operators by means of CNNs. The resulting error estimates provide a clear interpretation on the role played by the hyperparameters of dense and convolutional layers. Indeed, by exploiting some recent advances in Approximation Theory, and unvealing the intimate relation between CNNs and the discrete Fourier transform, we are able to characterize the complexity of the neural network in terms of depth, kernel size, stride, and number of input-output channels [Franco et al. (2023)]. References S. Fresca, A. Manzoni, L. Dede’ 2021a A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs. Journal of Scientific Computing, 87(2):1-36. S. Fresca, A. Manzoni 2022 POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition. Computer Methods in Applied Mechanics and Engineering, 388, 114181. S. Fresca, A. Manzoni L. Dede’, A. Quarteroni 2021b POD-enhanced deep learning-based reduced order models for the real-time simulation of cardiac electrophysiology in the left atrium. Frontiers in Physiology, 12, 1431. S. Fresca, A. Manzoni 2021c Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models. Fluids, 6(7), 259. N. R. Franco, S. Fresca, A. Manzoni, P. Zunino 2023 Approximation bounds for convolutional neural networks in operator learning. Neural Networks, Accepted. -
Andrea Thomann
REPORTÉ (grève) -- Semi-implicit schemes for all Mach number flows
7 février 2023 - 14:00Salle de séminaires IRMA
When considering multi-physics applications described by hyperbolic models, flow regimes, in comparison to single phase flows described by the Euler equations, are not characterized by one Mach number only. Examples are two fluid flows, where each phase is characterized by its own Mach number depending on the sound speed of the respective medium, or the simulation of elastic materials where, in addition to the standard acoustic Mach number, a shear Mach number depending on the shear modulus, describing the elastic shear stiffness of the material, can be defined.
The characteristic speeds of these models scale with the inverse Mach number inflicting a very restrictive CFL condition on the time step for standard explicit schemes.
Consequently, to avoid vanishing time steps, for near incompressible flows especially, implicit or implicit-explicit time integrators are necessary.
Moreover, the monitoring of sound waves is usually less in the focus of a numerical simulation.
Following the slower material waves and contact waves yields a less restrictive, Mach number independent CFL condition, which is advantageous when these slow dynamics are observed over a long time.
In this talk we address implicit explicit time integration approaches for hyperbolic models involving the above mentioned applications as well as issues and difficulties arising in the construction of the corresponding finite volume scheme. -
Guillaume Ferriere
Théorie de Cauchy et ondes progressives pour l'équation de Gross-Pitaevskii logarithmique
28 février 2023 - 14:00Salle de séminaires IRMA
On s'intéresse dans cet exposé à l'équation de Gross-Pitaevskii logarithmique (logGP), qui n'est autre que l'équation de Schrödinger non-linéaire logarithmique (logNLS) dans le contexte de solutions dont le module tend vers 1 à l'infini. La première partie concerne le problème de Cauchy, pour lequel les techniques classiques pour Gross-Pitaevskii avec non-linéarité polynomiale mais également celles utilisées pour logNLS se sont révélées infructueuses. Pour obtenir une bonne théorie de Cauchy, notre preuve de l'existence d'une solution adapte la méthode par compacité utilisée par Ginibre et Velo pour NLS. L'unicité découle du caractère lipschitzien du flot dans L^2 comme pour logNLS. Dans un deuxième temps, on s'intéresse aux ondes progressives, et en particulier au cas 1d, pour lequel plusieurs conclusions similaires au cas avec non-linéarité polynomiale découlent : au-delà d'une certaine vitesse critique explicite, aucune onde progressive n'existe; en deçà, les ondes progressives non-constantes sont uniques à invariants près. Ce travail a été réalisé en collaboration avec R. Carles. -
ANNULÉ (grève)
7 mars 2023 - 14:00Salle de conférences IRMA
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Nelly Boulos Al Makary
Analysis of the Shallow Water equations with two velocities
14 mars 2023 - 14:00Salle de conférences IRMA
In this work, we are interested in the analysis of the "Shallow water model with two velocities". First, we study the steady state solutions using the Bernouilli's principle for $C^1$ regular solutions and the Rankine-Hugoniot relations through discontinuities. Then, we present the types of solutions, their existence and their uniqueness depending on the boundary conditions. Second, we propose several finite volume approximate Riemann solvers for the resolution of the homogeneous Shallow water model with two velocities. The construction of the schemes is based on a recent analysis of the Riemann problem. We present several test cases to illustrate the behavior and the properties of the schemes. Afterwards, we extend these schemes for the model with topography and we propose a suitable numerical approximation of the source term. We prove that the proposed schemes are well-balanced and ensure the positivity of the water heights. Finally, we study the numerical stability of the stationary solutions. -
Frédéric Valet
Collision of two solitary waves for the Zakharov-Kuznetsov equation
21 mars 2023 - 14:00Salle de conférences IRMA
The Zakharov-Kuznetsov (ZK) equation in dimension 2 is a generalization in plasma physics of the one- dimensional Korteweg de Vries equation (KdV). Both equations admit solitary waves, that are solutions moving in one direction at a constant velocity, vanishing at infinity in space. When two solitary waves collide, two phenomena can occur: either the structure of two solitary waves is conserved without any loss of energy and change of sizes (elastic collision), or the structure is lost or modified (inelastic collision). As a completely integrable equation, KdV only admits elastic collisions. The goal of this talk is to explain the collision phenomenon for two solitary waves having almost the same size for ZK, and to describe the inelasticity of the collision. The talk is based on current works with Didier Pilod. -
Olivier Hurisse
A Random-Choice Scheme for Scalar Advection
28 mars 2023 - 14:00Salle de conférences IRMA
This talk is dedicated to a numerical method based on a random choice as proposed in Glimm's scheme. It is applied to the problem of advection of a scalar quantity. The numerical scheme proposed here relies on a fractional step approach for which: the first step is performed using any classical finite-volume scheme, and the second step is a cell-wise update. This second step is a projection based on a random choice. The resulting scheme possesses a very low level of numerical diffusion. In order to assess the capabilities of this approach, several test cases have been investigated including convergence studies with respect to the mesh-size. The algorithm performs very well on one-dimensional and multi-dimensional problems. This algorithm is very easy to implement even for multi-processor computations. -
Eloi Martinet
Maximisation des valeurs propres du Laplacien avec condition de Neumann
4 avril 2023 - 14:00Salle de conférences IRMA
On s'intéresse au problème d'optimisation de formes consistant à maximiser les valeurs propres du Laplacien avec conditions de Neumann homogènes. Ces valeurs propres interviennent notamment dans des problèmes acoustiques ou thermiques et sont en particulier liées à la "hot spot conjecture". Contrairement aux valeurs propres de Dirichlet, celles associées au problème de Neumann sont de nature plutôt instables, ce qui rend le problème d'optimisation difficile. On verra comment certaines explorations numériques du problème pour des domaines du plan et de la sphère ont permis de mettre en évidence certaines propriétés des optima. -
Angèle Niclas
Defect reconstruction in waveguides using resonant frequencies
11 avril 2023 - 14:00Salle de conférences IRMA
This talk aims at introducing a new multi-frequency method to reconstruct width defects in waveguides. Different inverse methods already exist. However, those methods are not using some frequencies, called resonant frequencies, where propagation equations
are known to be ill-conditioned. Since waves seem very sensible to defects at these particular frequencies, we exploit them instead. After studying the forward problem at these resonant frequencies, we approach the wavefield and focus on the inverse problem. Given partial wavefield measurements, we reconstruct slowly varying width defects in a stable and precise way and provide numerical validations and comparisons with existing methods. -
Léo Meyer
Modeling the size distribution of adipose cells using a Lifshitz-Slyozov model
9 mai 2023 - 14:00Salle de conférences IRMA
Adipose cells or adipocytes are the specialized cells composing the adipose tissue in a variety of species.Their role is the storage of energy in the form of a lipid droplet inside their membrane. Based on the amount of lipid they contain, one can consider the distribution of adipocyte per amount of lipid and observe a peculiar feature : the resulting distribution is bimodal, thus having two local maxima. The aim of this talk is to introduce a model built from the work in Soula & al. (2013) that is able to reproduce this bimodal feature using a Lifshitz-Slyozov model. Additionally we present some result on this model and its relation to the Becker-Döring model. We can show that under some assumptions the later converges to the former and by looking at higher order term we can build an extended diffusive Lifshitz-Slyozov model which better describes the dynamics of adipose cells. I will also present some probabilistic insight into this convergence and some numerical simulations. -
Wassim Tenachi
Recovering physical laws from data using deep reinforcement learning
16 mai 2023 - 14:00Salle de séminaires IRMA
Symbolic Regression is the study of algorithms that automate the search for analytic expressions that fit data. I will introduce the state-of-the-art techniques of the field and give the basic principles of symbolic computational maths.
I will then present our work which was motivated by the fact that although recent advances in deep learning have generated renewed interest in symbolic regression, efforts have not been focused on physics, where we have important additional constraints due to the units associated with our data. I will present Φ-SO, our Physical Symbolic Optimization framework for recovering analytical symbolic expressions from physics data using deep reinforcement learning techniques by learning units constraints (https://arxiv.org/abs/2303.03192). -
Karim Ramdani
Homogénisation pour les problèmes indéfinis (horaire inhabituel)
23 mai 2023 - 13:30Salle de conférences IRMA
On s’intéresse à un problème scalaire d’homogénéisation périodique faisant intervenir deux matériaux isotropes de conductivités de signes opposés : un matériau classique et un métamatériau négatif. En raison du changement de signe des coefficients apparaissant dans les équations, il n’est pas facile d'obtenir des estimations d'énergie uniformes pour pouvoir appliquer les techniques d'homogénéisation usuelles. En utilisant la méthode de T-coercivité, on prouve le caractère bien posé du problème microscopique de départ et du problème homogénéisé, ainsi qu’un résultat de convergence. Ces résultats sont obtenus sous réserve que le contraste (négatif) entre les deux matériaux soit assez grand ou assez petit en module. -
Andrea Thomann
Semi-implicit schemes for all Mach number flows
30 mai 2023 - 14:00Salle de séminaires IRMA
When considering multi-physics applications described by hyperbolic models, flow regimes, in comparison to single phase flows described by the Euler equations, are not characterized by one Mach number only. Examples are two fluid flows, where each phase is characterized by its own Mach number depending on the sound speed of the respective medium, or the simulation of elastic materials where, in addition to the standard acoustic Mach number, a shear Mach number depending on the shear modulus, describing the elastic shear stiffness of the material, can be defined.
The characteristic speeds of these models scale with the inverse Mach number inflicting a very restrictive CFL condition on the time step for standard explicit schemes.
Consequently, to avoid vanishing time steps, for near incompressible flows especially, implicit or implicit-explicit time integrators are necessary.
Moreover, the monitoring of sound waves is usually less in the focus of a numerical simulation.
Following the slower material waves and contact waves yields a less restrictive, Mach number independent CFL condition, which is advantageous when these slow dynamics are observed over a long time.
In this talk we address implicit explicit time integration approaches for hyperbolic models involving the above mentioned applications as well as issues and difficulties arising in the construction of the corresponding finite volume scheme. -
Gero Schnücke
Moving mesh nodal discontinuous Galerkin methods to solve hyperbolic conservation laws
19 septembre 2023 - 14:00Salle de conférences IRMA
Real world applications, e.g. the simulation of turbulent flows around airfoils, require adaptive discretizations to reduce the computational costs and degrees of freedoms. The r-adaptive method involves the re-distribution of the mesh nodes in regions of rapid variation of the solution. In comparison with h-adaptive discretizations, where the mesh is refined and coarsened by changing the number of elements in the tessellation, the r-adaptive method has some advantages, e.g. no hanging nodes appear and the number of elements does not change. On the other hand a r-adaptive method can be only used when the effect of mesh movement is appropriately accounted for the discretization.
Discontinuous Galerkin (DG) methods offer benefits for the discretization of hyperbolic conservation laws on a complex mesh geometry, since no inter-element continuity is required. Furthermore, it is known that on a static mesh DG methods have some useful theoretical properties, e.g. these methods satisfy a cell entropy inequality.
In this talk, the focus is on the construction of moving mesh nodal DG methods that satisfy a discrete entropy inequality. Thereby, a proper methodology to compute the grid point distribution to move the mesh will be not discussed. Numerical experiments as well as results from the simulation of turbulent flows around airfoils will be presented to validate the capabilities of these methods. -
Thomas Chambrion
Averaging techniques for the control of conservatives bilinear quantum systems with mixed spectrum
26 septembre 2023 - 14:00Salle 301
The time evolution of well isolated quantum systems can be modeled by the bilinear Schrödinger equation, i.e., a standard linear Schrödinger equation, formally x’(t)=A x(t), where the state x(t) of the system at time t is a point in some L^2 space endowed with its Hilbert structure and A is a skew-adjoint linear operator. When submitted to a sufficiently weak external excitation, the dynamics of the system can be formally written x’(t)=A x(t) + u(t) B x(t). That is, the original system is perturbed by a bilinear term u(t) B x(t), where u is the real valued control and B is a fixed (usually unbounded) skew-symmetric operator.
When the unperturbed Schrödinger operator A has a pure point spectrum (that is, the ambient Hilbert space admits a basis made of eigenvectors of the Schrödinger operator) and under reasonable regularity assumptions, this bilinear system is well posed. Moreover, a classical and efficient control strategy to steer the system from one eigenstate of A to (a small neighborhood of) another is to use a periodic control law whose frequency is proportional to the difference of the corresponding eigenvalues.
In this talk, we will expose how this result can be generalized in the case where the eigenvectors of the Schrödinger operator do not span a basis of the ambient space anymore. The proof amounts to an “averaging version” of the celebrated RAGE theorem. This is an ongoing work in collaboration with Nabile Boussaïd and Marco Caponigro. -
Demi-Journée De L'équipe
MOCO + TONUS
10 octobre 2023 - 14:00Salle de conférences IRMA
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Hung Truong
Aerodynamics of insect flight and mathematical modeling of wing flexibility
17 octobre 2023 - 14:00Salle de conférences IRMA
The remarkable flight capabilities of flapping insects are attributed to their wings, often approximated as flat, rigid plates. However, real wings consist of delicate structures, comprising veins and membranes that can undergo substantial deformation. In this presentation, we offer comprehensive numerical simulations of these deformable wings, focusing on two models: a bumblebee (Bombus ignitus) wing and a blowfly (Calliphora vicina) wing. We employ a mass-spring system that utilizes a functional approach to model the distinct mechanical behaviors of the veins and membranes of the wings. Subsequently, we conduct numerical simulations of tethered flapping insects with flexible wings using a fluid-structure interaction solver. This solver couples the mass-spring model for the flexible wing with a pseudo-spectral code that solves the incompressible Navier-Stokes equations. We apply the no-slip boundary condition through the volume penalization method, describing the time-dependent complex geometry with a mask function. This approach enables us to solve the governing fluid equations on a regular Cartesian grid. Our implementation, designed for massively parallel computers, empowers us to conduct high-resolution computations with up to 500 million grid points. The findings from this study provide insights into the role of wing flexibility in flapping flight. We observed that wing flexibility made only a minor contribution to lift or thrust enhancement. However, a significant reduction in required power suggests that wing flexibility plays a crucial role in conserving the energetic cost of flight. -
Davide Ferrari
An extension and numerical solution of a multi-phase hyperbolic model of continuum mechanics in the Baer-Nunziato type form
14 novembre 2023 - 14:00Salle de conférences IRMA
We present an extension and numerical solution of a multi-phase first order hyperbolic Unified Model of Continuum Mechanics in the Baer-Nunziato type form. It is a hyperbolic formulation of multi-phase flows, by which compressible Newtonian and non-Newtonian, inviscid and viscous fluids as well as elasto-plastic solids can be described.
Past and current research on multi-phase flow modelling mostly focuses on two-phase mathematical models. One of the most relevant, is the one originally proposed by Baer and Nunziato [1]. However, it is known that the model is not closed, i.e. the definition of these interphase terms is not unique and the generalisation of the model with more than two phases is not
clear. For this reason, in this work we intend to illustrate again how a closed multiphase model of the Baer-Nunziato type can be derived from the original theory of the SHTC systems. The SHTC theory of mixtures was first proposed by Romenski in [11, 12] for the case of two fluids and it was generalized to the case of arbitrary number of constituents in [10].
Furthermore, the Eulerian hyperelasticity equations of Godunov and Romenski are used to introduce viscous and elastic forces into this Baer-Nunziato type multi-phase hyperbolic model derived from the SHTC theory. This formulation of hyperelasticity in Eulerian coordinates, rather than the Lagrangian framework more commonly adopted in solid mechanics, is based on
the work of Godunov and Romenski [3, 4, 6, 5, 8], and in [9], Peshkov and Romenski presented the key insight that the Godunov-Romenski model can be applied not only to elasto-plastic solids, but also to fluid flows.
Hence, once the GPR theory is also introduced, we have an hyperbolic formulation of multiphase flows, by which compressible Newtonian and non-Newtonian, inviscid and viscous fluids as well as elasto-plastic solids can be described. The resulting system is large and includes highly nonlinear stiff algebraic source terms as well as non-conservative products. Consequently, the numerical solution of a multi-phase system in the multi-dimensional case, even if on a Cartesian grid, is a great challenge. For this purpose, we propose to employ a robust second-order explicit MUSCL-Hancock method on Cartesian meshes and a path-conservative technique of Castro and Pares for the treatment of non-conservative [7] products, in the context of the diffuse interface approach. Furthermore, the scheme employs a semi-analytical time integration method for the nonlinear stiff source governing the deformation relaxation, which is a rather challenging task, especially in the context of multi-phase flows. This temporal integration approach, which involves a polar decomposition of the stretching and rotation components of the distortion field, has been extended to the complete equations of the Unified Model of Continuum Mechanics in the fluid regime in [2] by Chiocchetti and Dumbser.
References
1. M.R. Baer and J.W. Nunziato. A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials. International Journal of Multiphase Flow, 6:861–889, 1986.
2. S. Chiocchetti and M. Dumbser. An exactly curl-free staggered semi-implicit finite volume scheme for a first order hyperbolic model of viscous two-phase flows with surface tension. Journal of Scientific Computing, 94:24, 2023.
3. S.K. Godunov. Elements of mechanics of continuous media. 1978.
4. S.K. Godunov, T.Y. Mikhaîlova, and E.I. Romenskî. Systems of thermodynamically coordinated laws of conservation invariant under rotations. Siberian Mathematical Journal, 37(4):690–705, 1996.
5. S.K. Godunov and E.I. Romenski. Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates. Journal of Applied Mechanics and Technical Physics, 13:868–885, 1972.
6. S.K. Godunov and E.I. Romenski. Elements of Continuum Mechanics and Conservation Laws. 2003.
7. Carlos Parés. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM Journal on Numerical Analysis, 44(1):300–321, 2006.
8. I. Peshkov, M. Pavelka, E.I. Romenski, and M. Grmela. Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations. Continuum Mechanics and Thermodynamics, 30(6):1343–1378, 2018.
9. I. Peshkov and E.I. Romenski. A hyperbolic model for viscous Newtonian flows. Continuum Mechanics and Thermodynamics, 28:85–104, 2016.
10. Evgeniy Romenski, Alexander A. Belozerov, and Ilya M. Peshkov. Conservative formulation for compressible multiphase flows. Quarterly of Applied Mathematics, 74(1):113–136, dec 2016.
11. E I Romensky. Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics. Mathematical and computer modelling, 28(10):115–130, 1998.
12. Evgeniy I Romensky. Thermodynamics and Hyperbolic Systems of Balance Laws in Continuum Mechanics. In E. F. Toro, editor, Godunov Methods, pages 745–761. Springer US, New York, NY, 2001. -
Roland Badeau
Statistical Wave Field Theory
21 novembre 2023 - 14:00Salle de conférences IRMA
The Statistical Wave Field Theory provides the general equations which govern the statistics of a reverberant acoustic field, expressed as a function of the geometric and physical parameters of a room. In order to introduce this theory, I will make a parallel with two well-known physical theories:
- Statistical physics establishes the macroscopic thermodynamic properties of gases from the laws of quantum mechanics governing microscopic particles. In the same spirit, the statistical wave field theory allows us to determine the macroscopic properties of a wave field in an enclosure (in terms of power distribution and statistical dependencies, through space, time and frequencies), from local physical laws: the wave equation in 3 dimensions and its boundary conditions (Neumann and Robin).
- The theory of relativity is twofold: in its special version, space-time is described as a flat 4-dimensional space; in its general version, which is a relativistic theory of gravitation, space-time is described as a curved space (using Riemannian geometry). The statistical wave field theory will also be presented in two parts. In its special version, we will consider rigid walls (Neumann's boundary condition) and we will show that the statistical properties are described in a flat, Euclidean space; in its general version, we will consider non-rigid walls (Robin's boundary condition) and we will show that the statistical properties are described in a curved space (the wave vector space).
In acoustics, the statistical wave field theory allows us to retrieve all the well-known properties of reverberation:
- time-frequency distribution (Polack formula);
- spatial correlation over frequency in the case of a diffuse wave field (Cook formula);
- modal density over frequency (Balian and Bloch formula);
- reverberation time in the case of a diffuse wave field (Eyring equation).
The statistical wave field theory might be a prominent tool in room acoustics, in particular because it should lead to dramatic computational savings. It should also be useful in most audio signal processing applications (including, of course, artificial reverberation and dereverberation), especially those involving spatial data (analysis/synthesis of sound scenes, source separation and localization, spatialization, etc.). Finally, since this theory is entirely based on the wave equation, it could also find applications in a variety of fields, including electromagnetism, optics and nuclear physics. -
Alessia Del Grosso
From supersonic to low Mach flows using multi-point numerical methods
28 novembre 2023 - 14:00Salle de conférences IRMA
An entropy stable, positivity preserving Godunov-type scheme for multidimensional hyperbolic systems of conservation laws on unstructured grids was presented by Gallice et al. in [1]. A specific feature of their Riemann solver is coupling all cells in the vicinity of the current one thanks to a nodal parameter: the velocity of the nodes. Consequently, this Riemann solver is no longer 1D across one edge. Contrarily, it encounters genuine multidimensional effects. In this presentation, we extend their work to handle source terms, with a specific application to the shallow water system. The scheme we obtain is well balanced in 1D and 2D. We show that the numerical scheme appears to be insensitive to the numerical instability known as Carbuncle in supersonic flows. We also investigated the reasons behind the good behaviour of this numerical scheme with respect to this instability. To conclude the presentation, we discuss possible research paths. In particular, we are investigating (with promising results) whether the knowledge of multidimensional effects can improve the numerical results for low-Mach flows. References [1] G. Gallice, A. Chan, R. Loubère, P.-H. Maire. Entropy Stable and Positivity Preserving Godunov-Type Schemes for Multidimensional Hyperbolic Systems on Unstructured Grid. Journal of Computational Physics, Volume 468, 2022, 111493, ISSN 0021-9991, https://doi.org/10.1016/j.jcp.2022.111493. -
Miranda Boutilier
Multi-Domain Solutions of PDEs Posed on Perforated Domains
5 décembre 2023 - 14:00Salle de conférences IRMA
This talk addresses linear and nonlinear elliptic problems on non-periodic, perforated domains. With the domains representing realistic urban geometries, our goal is to model floods in urban areas. For the linear model, we introduce a novel coarse space that is spanned by locally discrete harmonic basis functions that are piecewise polynomial along subdomain boundaries. We combine the coarse approximation with local subdomain solves in a two-level domain decomposition method. For the nonlinear Diffusive Wave model, we present nonlinear preconditioning techniques that allow us to significantly reduce iteration counts when compared to Newton's method.